MODELING OF DAMAGE PROPAGATION IN COHESIVE-FRICTIONAL MATERIALS

Transcription

1 MODELING OF DAMAGE PROPAGATION IN COHESIVE-FRICTIONAL MATERIALS

2 MODELING OF DAMAGE PROPAGATION IN COHESIVE-FRICTIONAL MATERIALS By EHSAN HAGHIGHAT, B.Eng., M.Sc. A Thesis Submitted to the School o Graduate Studies in Partial Fulilment o the Requirements or the Degree Doctor o Philosophy in Civil Engineering McMaster University Copyright by Ehsan Haghighat, November 2014

3 DOCTOR OF PHILOSOPHY (2014) (Civil Engineering) McMaster University Hamilton, Ontario TITLE: Modeling o Damage Propagation in Cohesive Frictional Materials AUTHOR: Ehsan Haghighat, M.Sc. (Shari University o Technology, Iran) SUPERVISOR: Proessor S. Pietruszczak NUMBER OF PAGES: viii, 109 ii

4 Abstract The primary ocus in this research is on proposing a methodology or modeling o discrete crack propagation in geomaterials such as soil, rock, and concrete. Structures made o such materials may undergo damage due to several reasons. Here, mechanical loading and chemo-mechanical interactions that result in degradation o strength parameters are considered as the sources o damage initiation. Both tensile and compressive cracks are investigated. For analysis o crack propagation, two dierent methodologies are employed; the Constitutive Law with Embedded Discontinuity (CLED) and the Extended Finite Element Method (XFEM). The CLED approach is enhanced here to describe the discrete nature o crack propagation. This is done by coupling the CLED with explicit modeling o crack path using the Level-Set method. The XFEM is used as a veriication tool to check the results rom CLED analysis. An algorithm is proposed or crack initiation and propagation that results in stable and a mesh-independent solution. The CLED approach is urther improved by developing the return-mapping and closest-point projection algorithms. Extensive numerical investigations are conducted that include mode I cracking in a three point bending test, mode I cracking in notched cantilever beam, mixed cracking mode in a plate subjected to shear and tension, and a mixed mode cracking in a notched beam under our point loading. For rictional interaces, the shear band ormation in a sample subjected to bi-axial compression and the shear band ormation in a geo-slope are studied. iii

5 The thesis also addresses the topic o the response o unsaturated cohesive soils undergoing an iniltration process. The problem is approached within the ramework o Chemo-Plasticity. It is assumed that the complex chemo-mechanical interactions are the controlling actors or degradation o strength parameters during this process. A return mapping integration scheme is developed and the approach is employed to investigate the stability o a geoslope subjected to a heavy rainall. Analysis o shear band ormation is urther investigated in the context o sedimentary rocks. The microstructure tensor approach is used to describe the inherent anisotropy in this class o materials. The orientation o the shear band is deined by invoking the Critical Plane approach and the closest-point projection algorithm is developed or numerical integration o the governing constitutive relations. The model is used along with CLED or analysis o the mechanical response o Tournemire argillite. It is shown that the riction between loading platens and sample can play an important role in the process o shear band ormation and the associated assessment o the ultimate load. A mesh-sensitivity analysis employing the CLED ramework is also conducted here. The research clearly demonstrates that the discrete representation o crack path propagation is essential or an accurate analysis o ailure in various engineering structures. It is shown that i the classical smeared Constitutive Law with Embedded Discontinuity is enhanced to simulate the discrete nature o the damage process, it can yield very accurate results that are virtually identical to those obtained rom discrete approaches such as XFEM. iv

6 Acknowledgements I would like to express my special appreciation and thanks to my advisor Proessor Stan Pietruszczak, you have been a tremendous mentor or me. I would like to thank you or encouraging my research and your countless support in these years. Your advice on both research as well as on my carrier has been priceless. I would also like to thank my committee members, Proessor Ken S. Sivakumaran and Proessor Peijun Guo, or their guidance and support at all stages o this research. A special thanks to my amily or all their love and encouragement. Words cannot express how grateul I am to my parents or all o the sacriices you have made on my behal and or all o the support I have got in all my pursuits. And most o all, I would like express appreciation to my beloved wie Aida who was always my support at all stages o this research. Ehsan Haghighat McMaster University November 2014 v

7 List o Publications This dissertation consists o ollowing papers: Paper I Pietruszczak, S., & Haghighat, E. (2013). Assessment o slope stability in cohesive soils due to a rainall. International Journal or Numerical and Analytical Methods in Geomechanics, 37(18), Paper II Haghighat, E., & Pietruszczak, S. Discrete modeling o cohesive crack propagation via an enhanced continuum approach. Submitted (May 2014) to International Journal or Numerical Methods in Engineering. Paper III Haghighat, E., & Pietruszczak, S. On modeling o discrete propagation o localized damage in cohesive-rictional materials. Submitted (Aug 2014) to International Journal or Numerical and Analytical Methods in Geomechanics. Paper IV Pietruszczak, S., & Haghighat, E. On the modeling o shear band ormation in anisotropic rocks. Submitted (Nov 2014) to International Journal o Solids and Structures. Paper V Pietruszczak, S, & Haghighat, E. Modeling o delayed ailure o embankments due to water iniltration. Accepted in Journal o Architecture Civil Engineering Environment. vi

8 Co-Authorship: This dissertation has been prepared in accordance with the regulations or a sandwich thesis ormat and the papers have been co-authored with my supervisor. Chapter 2: Assessment o slope stability in cohesive soils due to a rainall by: S. Pietruszczak and E. Haghighat The chemo-plasticity ramework has been proposed by Dr. S. Pietruszczak. The displacement ormulation and the numerical integration scheme have been developed and implemented in the FEM code by E. Haghighat. The simulations have been conducted by E. Haghighat in consultation with Dr. S. Pietruszczak. Chapter 2 was prepared by E. Haghighat and then revised/inalized by Dr. S. Pietruszczak. Chapter 3: Discrete modeling o cohesive crack propagation via an enhanced continuum approach by: E. Haghighat and S. Pietruszczak The development o the discrete representation o CLED and its implementation was done by E. Haghighat in consultation with Dr. S. Pietruszczak. The XFEM code was developed by E. Haghighat. The numerical simulations have been conducted by E. Haghighat in consultation with Dr. S. Pietruszczak. Chapter 3 was written by E. Haghighat and revised by Dr. S. Pietruszczak. Chapter 4: On modeling o discrete propagation o localized damage in cohesiverictional materials by: E. Haghighat and S. Pietruszczak The development o an enhanced numerical procedure or simulating the shear band localization in rictional materials was done by E. Haghighat in collaboration with Dr. S. Pietruszczak. The numerical simulations were conducted by E. Haghighat in consultation with Dr. S. Pietruszczak. Chapter 4 was prepared by E. Haghighat and revised by Dr. S. Pietruszczak. Chapter 5: On the modeling o shear band ormation in anisotropic rocks by: S. Pietruszczak and E. Haghighat The coupling o the micro-structure tensor approach with shear band ormation has been proposed by Dr. Pietruszczak. The numerical model or simulating the localized ailure has been developed and programed by E. Haghighat in consultation with Dr. S. Pietruszczak. The numerical simulations have been conducted by E. Haghighat in consultation with Dr. S. Pietruszczak. Chapter 5 was written by E. Haghighat and revised by Dr. S. Pietruszczak. Chapter 6: Modeling o delayed ailure o embankments due to water iniltration by: S. Pietruszczak and E. Haghighat This is an extension o the work reported in Chapter 2. The constitutive model has been proposed by Dr. S. Pietruszczak, while the numerical integration scheme incorporating the eect o localization was developed and implemented in FEM code by E. Haghighat. The simulations have been conducted by E. Haghighat in consultation with Dr. S. Pietruszczak. Chapter 6 was written by E. Haghighat and subsequently revised by Dr. S. Pietruszczak. vii

9 List o Contents Abstract... iii Acknowledgements... v List o Publications... vi Co-Authorship... vii Chapter 1 Thesis summary Goals and motivations Background Methodologies and developed tools Contributions Concluding remarks Suggestions or uture work Reerences Chapter 2 Assessment o slope stability in cohesive soils due to a rainall Chapter 3 Discrete modeling o cohesive crack propagation via an enhanced continuum approach Chapter 4 On modeling o discrete propagation o localized damage in cohesiverictional materials Chapter 5 On the modeling o shear band ormation in anisotropic rocks Chapter 6 Modeling o delayed ailure o embankments due to water iniltration viii

10 Ph.D. Thesis E. Haghighat McMaster University Civil Engineering Chapter 1- Thesis summary In this chapter, a summary o the research conducted in this study is provided. The thesis is ramed around the topic o modeling o damage propagation in geomaterials such as soils, rocks and concrete. Two approaches are addressed here. The main ocus is on the Constitutive Law with Embedded Discontinuity (CLED) approach, while the Extended Finite Element Method (XFEM) is used as a veriication tool. Contributions made in this research are provided in the orm o journal papers as separate chapters. In the overview presented below, a brie summary o each paper is given. The chapter starts by pointing out the motivation and objectives behind this thesis. Subsequently, a short background on the numerical modeling o damage propagation in engineering structures is provided. The methodologies and tools that have been developed and applied in this dissertation are then reviewed, ollowed by a summary o contributions made in each paper. This chapter closes with conclusions and provides some guidelines or uture studies. 1.1 Goals and motivations During the last ew decades, an intensive amount o research has been conducted in the area o modeling o the onset and propagation o damage. Over the years a number o dierent methodologies have been developed which include Node Separation (Ngo & Scordelis, 1967; Nilson, 1968), Smeared Cracking (Bažant & Cedolin, 1979; Nayak & Zienkiewicz, 1972; Pietruszczak & Mroz, 1981; Rashid, 1968), Boundary Integral (Blandord & Ingraea, 1981), Element-Free (Belytschko et al., 1995; Liu et al., 1997), 1

11 Ph.D. Thesis E. Haghighat McMaster University Civil Engineering and Extended Finite Element (Belytschko & Black, 1999; Moës et al., 1999) approaches. Among those, the Extended Finite Element Method that was introduced in 1999 appears to be the most commonly used approach or analysis o crack propagation. The primary reason is that the analysis is mesh independent and there is no need or adaptive mesh reinement, which results in a aster solution. Although XFEM is similar to standard FEM, there are certain disadvantages associated with its implementation, so that most sotware companies have still not incorporated it into their packages. The main diiculty is dealing with additional degrees o reedom and the associated enrichment unctions. In addition, a special integration scheme must be incorporated and also a special treatment o data structures is required. Finally, the numerical problems arise due to stress singularities in locations where the interace is close to a node. The main motivation in this study is to develop a methodology that does not have the issues associated with XFEM, yet is still able to properly model the damage propagation process or a broad range o cohesive-rictional materials. The methodology pursued here incorporates an enhanced orm o the Constitutive Law with Embedded Discontinuity approach that is capable o modeling the discrete nature o crack propagation very eiciently within the standard FEM mesh. A comprehensive investigation is conducted to veriy the accuracy o the proposed approach in relation to XFEM and its applicability to various class o problems. Two main sources o damage in structures are investigated here, i.e. damage due to mechanical loading and damage due to degradation o material properties triggered by chemo-mechanical interactions. 2

12 Ph.D. Thesis E. Haghighat McMaster University Civil Engineering The starting point o this research is the incorporation o the eect o chemical interaction between water and clay particles in cohesive rictional soils using the chemo-plasticity ramework. A scalar parameter is deined that monitors the degradation o material properties due to an increase o saturation associated with a heavy rainall. The research is then continued into modeling o localized ailure associated with CLED and XFEM analysis o cracking under various loading scenarios. 1.2 Background Modeling o damage in geomaterials such as soil, rock, and concrete is one o the most challenging problems, as it involves very complex computations due to the highly nonlinear response o these materials. Analysis o damage propagation, which is usually associated with localized deormation, is very important speciically or the assessment o the loss o stability. The latter is one o the main sources o loss o lie in such natural disasters as earthquakes and landslides. In this thesis, the problem o damage initiation and propagation due to mechanical loading and chemo-mechanical interactions is addressed. The main ocus is on the development o models which are not only capable o accurately describing these phenomena but can also be easily incorporated into any commercial package. The research reported here was initiated with the ormulation o a chemo-plasticity ramework or the modeling o loss o cohesion in clayey soils ater an intense rainall. It is well known that a period o heavy rainall can trigger a loss o stability o cohesive/cementitious slopes, where the water iniltration leads to a chemical interaction resulting in degradation o the mechanical properties o the material. This is addressed 3

13 Ph.D. Thesis E. Haghighat McMaster University Civil Engineering classically using the notion o suction pressure as a state parameter (Alonso et al., 1990). However, the suction depends on microstructure o saturation and at very low degrees o saturation its measurement is not possible or clayey soils due to complex chemical interactions between particles o clay and the water. Recognizing this limitation, the approach implemented here is based on a phenomenological ramework o chemoplasticity (c. Hueckel, 1997; Pietruszczak et al., 2006). A scalar variable is deined that monitors the evolution o strength parameters. The proposed methodology is believed to be a pragmatic alternative to both the micro-mechanical approach and the classical notions o unsaturated soil mechanics. Within the proposed ramework, the injection o water is said to trigger a volume change (swelling/collapse) that is coupled with reduction in suction pressures that, in turn, results in degradation o strength and deormation properties. It needs to be emphasized that this approach is used only in the range o very low degrees o saturation when the water phase is discontinuous, i.e. water is present only near the interparticle contacts. The study o stability o clayey soils is ollowed by the introduction o an enhanced orm o the constitutive law with embedded discontinuity capable o modeling damage initiation and propagation in a discrete manner. This is in contrast to the original studies (Pietruszczak, 1995; Pietruszczak & Mroz, 1981) which were ocused on smeared modeling o shear band localization in geomaterials. As mentioned earlier, modeling o damage initiation and propagation has been one o the most intensely researched topics over the last ew decades. Existing analytical solutions are restricted to an elastic material and involve simple geometries and boundary conditions. 4

14 Ph.D. Thesis E. Haghighat McMaster University Civil Engineering Thereore, they are not directly relevant to practical engineering problems. The latter require, in general, a numerical simulation that typically involves the use o the Finite Element Method (FEM). The early methodologies or capturing the progressive damage were based on tracing the crack propagation on the element boundaries (Schellekens & de Borst, 1993). In addition, some smeared techniques have also been proposed that incorporated plasticity-based strain-sotening relations (Lubliner et al., 1989; Parks, 1977; Rots & Blaauwendraad, 1989). Later, in order to avoid remeshing in the context o FEM, the mesh-ree Galerkin method was introduced (Belytschko et al., 1996; Fleming et al., 1997). More recently, ater introduction o the partition o unity approach (Melenk & Babuška, 1996), the Extended Finite Element Method (XFEM) was developed (Belytschko & Black, 1999; Dolbow et al., 2000; Moës et al., 1999; Moës et al., 2003), which enables the incorporation o discontinuous displacement ields as well as tip enrichments into the approximation space. The XFEM has shown good predictive abilities in modeling discrete cracking problems; however, it is still not a practical approach in engineering applications although a great amount o research has been conducted on that. The primary diiculty arises rom its implementation. Although the approach is based on FEM, special elements and integration schemes along with special algorithms or dealing with the increased number o degrees o reedom o the system are required. The solution shows artiicial singularities or the cases where an interace passes through the proximity o an FEM node, which is unavoidable in general engineering applications. In contrast to this methodology, the smeared approaches have been widely used in engineering applications due to their simplicity in 5

15 Ph.D. Thesis E. Haghighat McMaster University Civil Engineering implementation. The main problem with the early ormulations was the sensitivity o the solution to discretization. The issue has been addressed by incorporating a length scale into the ormulation, resulting in a mesh-independent ramework (Pietruszczak & Mroz, 1981). This approach was later revised (Pietruszczak, 1999) and applied to a broad range o practical engineering problems (e.g., Pietruszczak & Gdela, 2010; Shieh-Beygi & Pietruszczak, 2008; Xu & Pietruszczak, 1997). The main diiculty, nonetheless, with smeared approaches is the lack o stability in numerical simulations that in turn causes convergence issues. It is believed here that a discrete representation o crack propagation can resolve the convergence issue and provide a stable algorithm capable o advancing the analysis beyond the peak load to model the post ailure response. As most geomaterials show anisotropic behaviour that is mainly related to their microstructure, the next part o this research is ocused on the modeling o shear band ormation in sedimentary rocks. As an example, Tournemire argillite has closely spaced bedding planes and exhibits a strong anisotropic response, so that its strength as well as deormation properties are directionally dependent. A comprehensive review on this topic, examining dierent approaches, is provided in the articles by Duveau and Henry (1998) and Kwasniewski (1993). One o the irst attempts to describe the conditions at ailure in anisotropic rocks was the work reported by Pariseau (1968), which was an extension o Hill s criterion (Hill, 1967). This was ollowed by more complex tensorial representations (Amadei, 1983; Boehler & Sawczuk, 1977; Nova, 1980). A simple and pragmatic approach, which incorporates a scalar anisotropy parameter that is a unction o a mixed invariant o the stress and the structure orientation tensor, was next developed by Pietruszczak (2002) 6

16 Ph.D. Thesis E. Haghighat McMaster University Civil Engineering and Pietruszczak and Mroz (2000). This approach was later applied to the modeling o sedimentary rocks (Lade, 2007; Lydzba et al., 2003). Here, the microstructure tensor approach is combined with the CLED or modeling o shear band ormation under axial compression. The eect o boundary conditions on crack ormation and peak load is also investigated. 1.3 Methodologies and developed tools As discussed earlier, two approaches are investigated here, standard FEM with CLED and XFEM. For this purpose, a two-dimensional nonlinear inite element program is developed. The program is designed in a robust way so that it can be easily implemented in ABAQUS as a user element library. The computer program can also be run separately or conducting quasi-static analysis. Thus, the problem can be completely deined in the ABAQUS environment, and an ABAQUS inp input ile can be used to perorm the analysis. 1.4 Contributions In this section a summary o the papers that comprise this dissertation is presented. Paper I: Assessment o the slope stability in cohesive soils due to a rainall (Pietruszczak, S., & Haghighat, E., Assessment o slope stability in cohesive soils due to a rainall. International Journal or Numerical and Analytical Methods in Geomechanics, 37(18), ) The loss o stability o geoslopes ater a heavy rainall is still a source o major natural disasters. An example here is the recent slope ailure in Oso, Washington, that resulted in 43 deaths and a signiicant damage to the surrounding area. In this paper, a methodology is 7

17 Ph.D. Thesis E. Haghighat McMaster University Civil Engineering proposed or the assessment o stability o natural/engineered slopes in clayey soils subjected to water iniltration. In natural deposits o ine-grained soils, the presence o water in the vicinity o minerals results in an interparticle bonding. This eect cannot be easily quantiied as it involves complex chemical interactions at the micromechanical level. Here, the ramework o chemo-plasticity is employed to address the evolution o material properties. The degradation o strength, including the apparent cohesion resulting rom initial suction at the irreducible luid saturation, is taken into account by incorporating a scalar parameter that monitors the progress o chemical interaction. It should be stresses again that the proposed chemo-plasticity ramework addresses the mechanical response o cohesive soil only at very low degrees o saturation when water does not exist in a ree state. At the stage when the water phase becomes continuous, the soil properties are commonly described by considering suction pressure as a state variable. In general, however, the behaviour strongly depends on the microstructure o saturation. Thereore, it seems more appropriate to employ some averaging procedures whereby the compressibility o the mixture is expressed as a unction o properties o constituents (ree water and air) and the microstructure o saturation itsel. The paper provides irst the governing constitutive relation, ollowed by the development o a return mapping integration scheme. Subsequently, the coupled transient ormulation is presented. The ramework is applied to examine the stability o a slope subjected to a prolonged period o intensive rainall. It is shown that the ormulation can adequately describe the water iniltration problems, including the assessment o the stability o geo-structures. 8

18 Ph.D. Thesis E. Haghighat McMaster University Civil Engineering Paper II: Modeling o cohesive crack propagation: a constitutive law with embedded discontinuity vs XFEM (Submitted to International Journal or Numerical Methods in Engineering, May 2014) The research reported here is ocused on the problem o damage propagation in brittle materials. The approach incorporates a constitutive law with embedded discontinuity (CLED). As mentioned earlier, this methodology has been previously used within the context o smeared modeling o localization problems. The primary novelty here is an extension o this ramework to model the discrete nature o racture propagation process. This has been achieved by coupling the original approach with the Level-set method to capture the path o the crack propagation and to enable an accurate assessment o the characteristic length parameter. Thus, within the proposed ramework the discontinuity is deined at the element level rather than at an integration point, which is in contrast to the classical smeared approaches. The paper provides a new analytical representation o the governing equations. In addition, a new implicit integration scheme is developed to impose the condition o continuity o traction along the interace. The eiciency and the accuracy o the proposed approach are veriied using the ramework o XFEM. The numerical simulations include a simple tension test, a three-point bending test, and a mixed mode cracking test in which a specimen is subjected to both shear and tension. It is shown that the approach can provide very accurate results or modeling discrete cracking with a very simple implementation through a user material subroutine, which can be done within almost all FEM packages. It is also concluded that the new approach is computationally more eicient as there is no need to incorporate additional degrees o reedom associated with discontinuous motion. 9

19 Ph.D. Thesis E. Haghighat McMaster University Civil Engineering Paper III: On modeling o discrete propagation o localized damage in cohesiverictional materials (Submitted to International Journal or Numerical and Analytical Methods in Geomechanics, August 2014) Here, the study initiated in the previous paper is extended to model the localized deormation associated with shear band ormation in rictional media. Again, the research is ocused on discrete tracing o the path o crack/shear band propagation. The CLED approach is extended here to deal with elasto-plastic materials. Some new numerical simulations are conducted and the results are compared again with XFEM to show urther applications o the approach. The simulations include a mode I cohesive crack propagation in a cantilever beam, a mixed mode cracking problem in a our-point beam test, a problem o shear band ormation in a biaxial compression test conducted on dense sand, and the evolution o localized damage in a cohesive slope. It is demonstrated that the CLED approach can be applied to a broad range o problems in the same way as XFEM, without the need or changing the main subroutines o the program as required in XFEM implementation. Paper IV: Modeling o deormation and localized ailure in anisotropic rocks (Submitted to International Journal o Solids and Structures, November 2014) The introduced methodology or modeling o damage propagation, i.e. CLED, is urther investigated in the context o materials exhibiting a strong inherent anisotropy. The main ocus here is on shear band ormation and propagation in sedimentary rocks. In order to consider the material's anisotropy, the microstructure tensor approach is employed and a simple criterion is deined or the inception o shear band. The critical plane approach is 10

20 Ph.D. Thesis E. Haghighat McMaster University Civil Engineering then applied in order to ind the orientation o the crack. The closest-point projection algorithm is developed or both the anisotropic constitutive model and the constitutive law with embedded discontinuity or accurate integration o the constitutive relations. The problem o shear band ormation in biaxial plane strain compression tests is studied or samples at dierent orientations o the bedding planes relative to the loading direction. It is demonstrated that the model can accurately reproduce a mesh- independent response. The eect o riction between loading plates and sample is also studied here. It is shown that it plays an important role in the damage evolution process and may aect the peak load signiicantly. Paper V: Modeling o delayed ailure o embankments due to water iniltration (Journal o Architecture Civil Engineering Environment, in print) The problem that was initiated in Paper I is urther investigated taking into account the localized ailure mechanism associated with the shear band ormation triggered by water iniltration. The ramework incorporating CLED is extended here to deal with a coupled hydro-mechanical loading conditions. Note that in this article some passages rom paper I, in relation to the numerical example provided, have been quoted verbatim. This, however, is consistent with publisher policy on reusing author s own article. 1.5 Concluding remarks In paper I, a methodology or modeling a coupled hydro-mechanical response o clayey soils has been presented. The ollowing main conclusions emerge rom this work 11

21 Ph.D. Thesis E. Haghighat McMaster University Civil Engineering Within the proposed ramework, as long as the water phase is discontinuous, the material is treated as a single phase medium undergoing evolution o mechanical properties as a result o chemo-mechanical interactions. At this level o saturation, the approach does not require a speciication o suction-saturation relationship as the suction pressure is said to be the product o interaction o a complex system o mineralogical and chemical actors. The developed return mapping combined with the subincrementation approach signiicantly improves the convergence o the global solution or unsaturated low in porous media. The analysis o a geoslope subjected to an intense rainall revealed that the water iniltration may trigger a loss o stability resulting rom degradation o strength properties. It is also concluded that or an accurate assessment o stability, a methodology that is capable o modeling the localized ailure associated with a shear band ormation is required, which is the main study in ollowing papers. The research reported in paper II that dealt with the discrete modeling o crack propagation using the constitutive law with embedded discontinuity and the extended inite element method, led to the ollowing major conclusions The work clearly demonstrated that the constitutive law with embedded discontinuity can be successully employed or modeling o discrete damage propagation. This has been accomplished by coupling the approach with the level-set method or tracing the crack path. It was demonstrated that a discrete representation o crack propagation process results in a stable, mesh-independent methodology that reproduces almost identical results to those obtained rom XFEM. The developed 12

22 Ph.D. Thesis E. Haghighat McMaster University Civil Engineering implicit integration scheme extends the range o applicability o this approach to any interacial description, including traction ree and cohesive models. The other signiicant conclusion is that the CLED methodology is computationally more eicient than the XFEM, since there are no additional degrees o reedom/enriched interpolations involved in the calculations. The CLED ramework can be easily incorporated into any FEM package through the constitutive model subroutine. This is in contrast to XFEM, which requires development o special elements and algorithms or dealing with additional DOFs. Finally, a simple procedure was developed or tracing the crack initiation and propagation that is based on the averaging o the values o the ailure unction and the crack orientation in the domain adjacent to the crack tip. The result is a smooth cracking pattern that is meshinsensitive. The work reported in paper II was urther extended in paper III to deal with more complex problems that involved rictional interaces leading to localization into a shear band. The inal remarks emerging rom this study are as ollows It was shown that the approach incorporating CLED can be applied to problems in which the localization is associated with the onset and discrete propagation o a shear band. In contrast to XFEM, where artiicial singularities (when interace passes through the vicinity o a node) may cause diiculties in the numerical analysis, the CLED approach proved very eicient in the context o elasto-plastic simulations. The return mapping scheme that was developed or both the intact material and the one 13

23 Ph.D. Thesis E. Haghighat McMaster University Civil Engineering with embedded discontinuity showed a stable response and proved to be capable o accurately modeling the post-ailure behaviour o the structure. In paper IV, the problem o description o the deormation process in argillaceous rocks that display a strong inherent anisotropy was investigated. The scope o the work and the primary conclusions are summarized below The mechanical characteristics o Tournemire shale were examined and the eect o boundary condition on the shear band ormation was investigated or a series o plane strain biaxial compression tests. It was shown that in the case o a rictionless interace between loading platens and the sample, the deormation ield remains homogeneous and the ailure mode is diused. With a presence o riction, however, the stress state is signiicantly perturbed which results in ormation o a shear band/macrocrack. In this case, the ultimate strength o the sample is noticeably less than the one attained under rictionless conditions. A series o simulations or samples with dierent orientations o bedding planes was also conducted. It was shown that in samples with horizontal and vertical bedding planes, the ailure mode is diused or both rictionless and ully constrained cases; however the peak strength is still noticeably dierent or both these cases. The results o simulations clearly demonstrated that the kinematic constraints can play an important role in the process o evolution o damage and may signiicantly aect the strength characteristics that are commonly perceived as a material property. This is particularly the case or inclined samples. The mesh-dependency o the solution was also examined here. It was concluded that by invoking the constitutive law with embedded discontinuity, 14

24 Ph.D. Thesis E. Haghighat McMaster University Civil Engineering which incorporates a characteristic dimension, the solution is virtually mesh independent. In paper V, which was an extension to the study conducted in paper I, the problem o shear band propagation in materials undergoing a chemo-mechanical degradation was investigated and ollowing conclusion was arrived at Comparison o the results with those presented in paper I shows that incorporation o the localized deormation mode signiicantly aects the displacement ield as well as the stress distribution and thereby impacts the assessment o the saety actor o the slope. The solution incorporating the localized ailure mode is, in general, more accurate and consistent with the ield observations. 1.6 Suggestions or uture work In the next step, the research should ocus on the application o CLED approach to three dimensional problems. In this case, XFEM simulations are expensive and computationally ineicient. Further detailed analysis on the mesh-sensitivity and the rate o convergence o the proposed approach should be conducted. The accuracy o the CLED approach or can be urther investigated within the context o more accurate meshree methods and the recently developed isogeometric approach. Further applications o the proposed methodology to a broad class o problems involving hydro-thermo-mechanical interaction could be undertaken. The simplicity 15

25 Ph.D. Thesis E. Haghighat McMaster University Civil Engineering o this approach, while retaining the accuracy, can be an asset in terms o analysis o these complex coupled problems. The random-based simulations o cracking and shear banding can also be eectively investigated using this approach, due to its simplicity o implementation. 1.7 Reerences Alonso, E. E., Gens, A., & Josa, A. (1990). A constitutive model or partially saturated soils. Géotechnique, 40(3), Amadei, B. (1983). Rock Anisotropy and the Theory o Stress Measurements. Springer- Verlag. Bažant, Z. P., & Cedolin, L. (1979). Blunt crack band propagation in inite element analysis, 105(EM2), Belytschko, T., & Black, T. (1999). Elastic crack growth in inite elements with minimal remeshing. International Journal or Numerical Methods in Engineering, 45, Belytschko, T., Krongauz, Y., Organ, D., Fleming, M., & Krysl, P. (1996). Meshless methods: an overview and recent developments. Computer Methods in Applied Mechanics and Engineering, 139(1), Belytschko, T., Lu, Y. Y., & Gu, L. (1995). Crack propagation by element-ree Galerkin methods. Engineering Fracture Mechanics, 51(2), Blandord, G. E., & Ingraea, A. R. (1981). Two dimensional stress intensity actor computations using the boundary element method. International Journal or Numerical Methods in Engineering, 17, Boehler, J. P., & Sawczuk, A. (1977). On yielding o oriented solids. Acta Mechanica, 27(1-4), Dolbow, J., Moës, N., & Belytschko, T. (2000). Discontinuous enrichment in inite elements with a partition o unity method. Finite Elements in Analysis and Design, 36(3), Duveau, G., & Henry, J. P. (1998). Assessment o some ailure criteria or strongly anisotropic geomaterials. Mechanics o Cohesive Frictional Materials, 3(1),

26 Ph.D. Thesis E. Haghighat McMaster University Civil Engineering Fleming, M., Chu, Y. A., Moran, B., Belytschko, T., Lu, Y. Y., & Gu, L. (1997). Enriched element-ree Galerkin methods or crack tip ields. International Journal or Numerical Methods in Engineering, 40(8), Hill, R. (1967). The essential structure o constitutive laws or metal composites and polycrystals. Journal o the Mechanics and Physics o Solids, 15(2), Hueckel, T. (1997). Chemo-plasticity o clays subjected to stress and low o a single contaminant. International Journal or Numerical and Analytical Methods in Geomechancis, 21(1), Kwasniewski, M. A. (1993). Mechanical behavior o anisotropic rocks. Comprehensive rock engineering. Lade, P. V. (2007). Modeling ailure in cross-anisotropic rictional materials. International Journal o Solids and Structures, 44, Liu, W. K., Li, S., & Belytschko, T. (1997). Moving least-square reproducing kernel methods (I) methodology and convergence. Computer Methods in Applied Mechanics and Engineering, 143, Lubliner, J., Oliver, J., Oller, S., & Onate, E. (1989). A plastic-damage model or concrete. International Journal o Solids and Structures, 25(3), Lydzba, D., Pietruszczak, S., & Shao, J. F. (2003). On anisotropy o stratiied rocks: homogenization and abric tensor approach. Computers and Geotechnics, 30, Melenk, J. M., & Babuška, I. (1996). The partition o unity inite element method: basic theory and applications. Computer Methods in Applied Mechanics and Engineering, 139(1), Moës, N., Cloirec, M., Cartraud, P., & Remacle, J.-F. (2003). A computational approach to handle complex microstructure geometries. Computer Methods in Applied Mechanics and Engineering, 192(28), Moës, N., Dolbow, J., & Belytschko, T. (1999). A inite element method or crack growth without remeshing. International Journal or Numerical Methods in Engineering, 46(1), Nayak, G. C., & Zienkiewicz, O. C. (1972). Elasto plastic stress analysis. A generalization or various contitutive relations including strain sotening. International Journal or Numerical Methods in Engineering, 5(1), Ngo, D., & Scordelis, A. C. (1967). Finite element analysis o reinorced concrete beams. ACI Journal Proceedings, 64(3), Nilson, A. H. (1968). Nonlinear analysis o reinorced concrete by the inite element method. ACI Journal Proceedings, 65(9),

27 Ph.D. Thesis E. Haghighat McMaster University Civil Engineering Nova, R. (1980). The ailure o transversely isotropic rocks in triaxial compression. International Journal o Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 17(6), Pariseau, W. G. (1968). Plasticity Theory For Anisotropic Rocks And Soil. Parks, D. M. (1977). The virtual crack extension method or nonlinear material behavior. Computer Methods in Applied Mechanics and Engineering, 12(3), Pietruszczak, S. (1995). Undrained response o granular soil involving localized deormation. Journal o Engineering Mechanics, 121(12), Pietruszczak, S. (1999). On homogeneous and localized deormation in water-iniltrated soils. International Journal o Damage Mechanics, 8, Pietruszczak, S. (2002). Modelling o inherent anisotropy in sedimentary rocks. International Journal o Solids and Structures, 39(3), Pietruszczak, S., & Gdela, K. (2010). Inelastic Analysis o Fracture Propagation in Distal Radius. Journal o Applied Mechanics, 77(1), Pietruszczak, S., & Mroz, Z. (1981). Finite element analysis o deormation o strainsotening materials. International Journal or Numerical Methods in Engineering, 17(3), Pietruszczak, S., & Mroz, Z. (2000). Formulation o anisotropic ailure criteria incorporating a microstructure tensor. Computers and Geotechnics, 26(2), Pietruszczak, S., Lydzba, D., & Shao, J. F. (2006). Modelling o deormation response and chemo mechanical coupling in chalk. International Journal or Numerical and Analytical Methods in Geomechanics, 30(10), Rashid, Y. R. (1968). Ultimate strength analysis o prestressed concrete pressure vessels. Nuclear Engineering and Design, 7(4), Rots, J. G., & Blaauwendraad, J. (1989). Crack models or concrete, discrete or smeared? Fixed, multi-directional or rotating? Delt University o Technology,. Schellekens, J., & de Borst, R. (1993). On the numerical integration o interace elements. International Journal or Numerical Methods in Engineering, 36(1), Shieh-Beygi, B., & Pietruszczak, S. (2008). Numerical analysis o structural masonry: mesoscale approach. Computers & Structures, 86(21-22), Xu, G., & Pietruszczak, S. (1997). Numerical analysis o concrete racture based on a homogenization technique. Computers & Structures, 63(3),

28 ASSESSMENT OF THE SLOPE STABILITY IN COHESIVE SOILS DUE TO A RAINFALL S. Pietruszczak *, and E. Haghighat Department o Civil Engineering, McMaster University, Canada ABSTRACT: The primary ocus in this work is on proposing a methodology or the assessment o stability o natural/engineered slopes in clayey soils subjected to water iniltration. In natural deposits o ine grained soils, the presence o water in the vicinity o minerals results in an interparticle bonding. This eect cannot be easily quantiied as it involves complex chemical interactions at the micromechanical level. Here, the evolution o strength properties, including the apparent cohesion resulting rom initial suction at the irreducible luid saturation, is described by employing the ramework o chemo-plasticity. The paper provides irst the ormulation o the problem; this involves speciication o the constitutive relation, development o an implicit return mapping scheme as well as the outline o a coupled transient ormulation. The ramework is then applied to examine the stability o a slope subjected to a prolonged period o intensive rainall. It is shown that the water iniltration may trigger the loss o stability resulting rom the degradation o material properties. KEYWORDS: chemo-plasticity, rainall iniltration, slope stability, unsaturated low 1. INTRODUCTION It is well known that a period o heavy rainall can trigger a loss o stability o slopes. This is particularly the case or slopes constructed in cohesive soils, such as clay or a cemented soil, where the water iniltration leads to a chemical interaction resulting in degradation o mechanical properties o the material. The problem o stability o natural and engineered slopes has been a subject o research or a number o decades. In particular, the notion o inluence o suction pressure on the stability has become an important issue. In recent years, several case histories have been documented (e.g., in Hong Kong, Korea, Malaysia, Singapore) whereby the loss o stability was related to the loss o suction triggered by the local weather conditions [1-3]. The primary diiculty in modeling the loss o stability due to a heavy rainall lies in assessing the in-situ conditions and in describing the coupling between the time-dependent process o water iniltration and the evolution o stress ield. The problem is typically analyzed by integrated sotware in which the transient seepage analysis is coupled with traditional limit * Correspondence to: S. Pietruszczak, Civil Engineering Department, McMaster University, Hamilton, On, Canada. pietrusz@mcmaster.ca 19

29 equilibrium slope stability analysis [4-6]. Alternatively, the rameworks or unsaturated soil are implemented in which the suction pressure is considered as a state parameter and an optimization technique is used to search or a critical slip surace (e.g.,[4]). In general, the conventional methods or assessing the stability o unsaturated soils, based on limit equilibrium approach, signiicantly underestimate the saety actors. Thereore, more accurate techniques are required. Speciication o properties o clays is diicult as it involves a very complex system o mineralogical and chemical actors. There are signiicant variations in composition, size and relative orientation o the mineral phase, as well as composition and the amount o aqueous phase. Thereore, a micro-mechanically based description o physical properties o clays represents an overwhelming task. In clays, the bond strength increases rapidly with decreasing water content. It is rather apparent that ree water in clays has low compressibility and virtually no viscosity. The water in the vicinity o minerals, however, has quite dierent properties which cannot, in act, be quantiied due to complex chemical interactions. Thereore, the measurement and/or control o suction pressures are diicult. Recognizing the above limitations, a dierent methodology is proposed here as an alternative to a micro-mechanical approach. At the range o irreducible saturation (c. [7]), when the water phase is discontinuous, the behavior o the material is described based on a phenomenological ramework o chemo-plasticity (c. [8-10]). Within this ramework, an increase in water content due to wetting is said to trigger a reduction in the interparticle bonding and the corresponding degradation o strength and deormation properties at the macroscale. At the stage when the water phase becomes continuous, the behaviour can then be deined in mechanical terms alone; or example, by employing an averaging procedure in which the compressibility o the mixture is expressed as a unction o properties o constituents (ree water and air) and the microstructure o saturation ([11,12]). Note such an approach incorporates the average pore size as an independent characteristic dimension. Alternatively, the problem may be phrased using the classical notions o unsaturated soil mechanics [13-15]. In the next section, the ormulation o the problem is discussed including the development o implicit (backward Euler) integration scheme. Later, the governing equations describing the transient hydro-mechanical coupling are reviewed and the ramework is applied to examine the stability o a slope in cohesive soil, subjected to a period o intense rainall. In solving the problem, the evolution o the phreatic surace is monitored and is coupled with mechanical analysis incorporating the chemical interaction. 2. FORMULATION OF THE PROBLEM 2.1 Constitutive relation The general approach or describing the evolution o properties o clays in the presence o the interparticle bonding is based on the ramework o chemo-plasticity, similar to that o re. [10]. Within this approach, the progress in chemo-mechanical interaction is monitored by a scalar parameterζ. Here, this parameter may be interpreted as the change in the initial suction pressure 0 u s, at the irreducible wetting luid saturation, in REV; i.e., The evolution law can be taken in a simple linear orm ζ ( u u ) / u, so that ζ 0, s s s 20

30 ζ B1 ζ ; dt gdt (1) t where g 0,1 depends on the chemical composition o the clay minerals and water, and B is a material constant. In the elastic range, the constitutive relation takes the orm e 1 e : 1 ε σ ε : σ 1 : σ 1 : σ ζ 1 ζ : σ where σ is the eective stress and 1 is the elastic compliance operator. Note that the dierential orm o eq. (2) may be expressed as e σ : ε 1 ζ : ζ : σ Invoking now the additivity postulate, total strain rate can be written as (2) (3) e p ε ε ε (4) where is the volumetric strain rate due to wetting. The latter may be taken as ζ I, where is the maximum expansion/contraction in the stress-ree state. The plastic strain rates deined ollowing standard plasticity ormalism, i.e. where,, p ε are p σ,, ; ε ψ κ ζ 0 λ (5) σ p σ κ ζ is the yield unction, ( ε ) is the hardening parameter and ( σ, ζ ) is the plastic potential unction. Using the decomposition (4), the constitutive relation deining the stress rate may be expressed as p 1 ζ ζ σ : ε : ε : : σ (6) 2.2 Implicit integration-scheme The constitutive relation deined by eq. (6) is subjected to the ollowing constraints σ, κ, ζ 0 λ 0 During an active loading process, there is 0 latter represents the consistency condition and takes the orm λ 0 and σ, κ, ζ 0, so that σ,, (7) κ ζ 0. The : σ κ ζ 0 (8) σ κ ζ Following now the general return-mapping scheme [16,17], the stress and plastic strain at time t t can be expressed as 21

31 ε ε ε p p p tδt t tδt p ε Δλ ψ σ, ζ σ σ σ t tδt σ tδt tδt tδt t tδt Δt Δt p 1 Δ Δ Δ Δζ : ζ : ε : ε : σ tδt tδt tδt tδt tδt tδt tδt Dropping the subscript t t and solving the second equation or σt Δ t, one obtains ε p ε Δλ ψ p t 1 σ : σ : Δε Δ : Δε t σ p (9) (10) where ζ ζ. Note that the primary unknowns here 1 1 tt ( : ζ ) tt ( : ζ ) are σ σ tt and t t, while the parameter is a scalar valued unction o plastic deormation, so that it is a dependent variable. In order to solve the above set o equations, the general Newton-Raphson procedure is implemented. Following this scheme, the residuals can be deined as, 1 r σ : σ : Δε Δ : Δε p k k t k 1 σ : σ : Δε Δ Δλ : ψ (11) k t k σ k σ, κ, Δλ k k k k Expanding these residuals, using Taylor expansion, yields r r : δ σ r δλ 0 k σ k k Δλ k k : δ σ δλ 0 k σ k k Δλ k k where 1 σrk k : : σσ and k Solving now the above set o equation, one has and k r 1 : : σ. : δλ δ σ r r 1 k k k Δλ k k k (12) (13) 1 k σ k : : rk δλk (14) 1 σ k : : : σψ k Δλ k which ully deines the stress and plastic strain corrections during the Newton-Raphson iterations. In order to speciy the tangential stiness operator, which is required or the global Newton- Raphson solution algorithm, the stress increment can be expressed as 1 : 1 : : 1 : 1 Δσ Δε Δ : : σ ψ Δλ (15) 1 1 Deining now an operator : :, and using the consistency condition, one obtains 22

32 so that Ph.D. Thesis E. Haghighat McMaster University Civil Engineering Δ : Δσ : Δλ σ Δλ ε ψ : : Δ Δ : : Δλ 0 σ σ σ Thus, the stress increment may be written as where σ Δλ : : ψ σ : : Δε Δ T σ Δλ : ε T, i.e. the tangential stiness operator, is deined as T Δλ (16) (17) Δσ Δ Δ (18) ( : σψ ) ( σ : ) : : ψ σ In the numerical examples provided later, the above backward Euler scheme is combined with subincrementation. This is primarily due to the act that the chemical degradation is very ast and small time increments are required in order to obtain an accurate solution or the local stress trajectories. 3. APPLICATION OF CHEMO-PLASTICITY FRAMEWORK TO MODELING OF SOIL INFILTRATION 3.1 Description o unsaturated low in porous media In order to trace the evolution o phreatic surace during the rainall iniltration, the problem is deined by invoking a coupled ormulation or low through unsaturated porous media[18]. Within this ramework, the porous material is considered as a mixture o solid grains and voids; the latter illed with water and/or air. Neglecting the inertia orces, the momentum balance equation or the solid-luid mixture takes the orm subject to σ Δλ (19) σˆ ˆ ρg 0 (20) v v on Γ σˆ n t on Γ Here, ˆσ is the total stress, g is the acceleration due to gravity, while v and t represent the velocity and traction prescribed on the boundaries d and t, respectively. Furthermore, ˆ is the density o the mixture; the latter, neglecting the density o air, may be deined as ˆ ( M s M w) / V ( 1 n) s Snw, where n and S are the porosity and the degree o saturation, while the indexes s, w reer to solids and water, respectively. d t (21) 23

33 For unsaturated soil, the stress transmitted by the skeleton, i.e. the eective stress σ, is typically deined using Bishop s relation σ σˆ χ u 1 χ u I σˆ χ u I (22) w a w where I is the identity tensor, uw, u a is water and air pressures, respectively, and is the Bishop s parameter. Apparently, i the air phase is continuous, the excess o air pressure is usually neglected, so that ua 0 in eq. (22). Note again that at the range o irreducible saturation, when the aqueous phase is present only in the vicinity o minerals, the material is conceptually considered as a continuum with an apparent cohesion. At this stage, an increase in water content due to wetting activates the chemomechanical interaction and leads to a microstructure o saturation in which the water phase becomes continuous. For the latter scenario, the simulations presented here were completed by identiying the parameter χ with the degree o saturation S, which is consistent with a similar assumption adopted by other investigators (e.g., [19,20]). It is noted, however, that the decomposition (22) is semi-empirical in nature and the problem may, in act, be phrased by assuming χ 1 (i.e. Terzaghi s principle) and considering speciic eatures o the geometry o microstructure o saturation. As shown in res. [11,12], when the water phase is continuous, the evolution o suction pressure as well as the compressibility o the immiscible air-water mixture may be deined as an explicit unction o the surace area o grains (or the air-water menisci) and the properties o constituents (i.e., ree water/air). Consider now the mass balance equation or the luid phase, which may be expressed, ater normalizing with respect to, in the ollowing orm [18] 0 w 0 w 1 d ρ dt 0 (23) ρ Sn ρ SnI : ε ρ Snv w w w w Assuming that solid grains are undeormable while the water is linearly compressible, the ollowing relations may be established or a representative dierential volume dv dvs dvv dvs dvw dva ρ dv dv ΔdV 1 ρ ρ ρ 0 s s s s 0 s dvs dvs dv dv ΔdV u 0 w w w w w 1 0 w dvw dvw Kw dv dv dv dv 0 0 s s s 1 0 n JsJ 1 n 0 0 dv dvs dv dv (24) where J dv dv 0 / det F and F is the deormation gradient. Substituting the above relations into eq. (23) and taking time derivatives (note that J J I : ε ), the ollowing expression is obtained 24

34 Introducing now the deinitions 0 ρw 1 n Sn ρw ds ρ w S n I : ε n u w Snvw 0 (25) ρw J Kw ρw duw ρw the mass balance relation can be expressed as 0 ρ w 1 n α n 0 ρw J 1 Sn ρ ds n Q K du w 0 w ρw w and is subjected to the ollowing boundary conditions (26) 1 ρ w SαI : ε u w Snv w 0 (27) 0 Q ρw Snv w u u on Γ w w w n q on Γ The luid low in porous media is governed by Darcy s law, which takes the orm q (28) Snv kˆ (29) w In the equation above, v w is the relative velocity o water, is the piezometric head and k ˆ k s k is the permeability o the skeleton; with k representing the permeability under ully saturated condition and ks ks ( S). Note that, i the soil is isotropic with respect to permeability, then kˆ k ˆ sk I k I. The piezometric head can be expressed as x g / g uw / g w, where the irst term deines the elevation head and g is the magnitude o the gravitational acceleration. Given the above deinitions, Darcy s law may be written in an alternative orm, as Snv kˆ ρ g w w w w u ρ g (30) Now, using stress decomposition (22), the weak orm o eq. (20) can be expressed as δ ε : σ dv δ ε : χ u I dv δ v t ds δ v ˆ ρ g dv (31) Ω Ω w Γt Ω s where ε v. Also, using the Darcy s law (30), the mass-balance may be expressed in a weak orm as 1 kˆ δ u SαI : ε dv δ u u dv δ u u dv Ω w Ω w w Ω w 0 w Q ρwg ρ kˆ δ u q dv δ u ρ g dv Γq w w 0 Ω w 0 w ρw ρwg (32) 25

35 Here, the velocity v and pressure u w are both smooth and continuous unctions that satisy the boundary conditions, i.e. v, v v 0, v v on Γ 0 u, u u, u u on Γ w w w w w w where and are the set o kinematically admissible velocity and pressure spaces. Note that u and uw are also required to have similar properties as those in eq. (33), while both must vanish on the respective boundaries, i.e δ v, δ v δ v C, δ v 0 on Γ δ u, δ u δ u C, δ u 0 on Γ w w w w w 0 Finally, note that under the restrictions o small deormation theory, i.e. J 1, n n, and 0 w w, the material parameters deined in eq. (26) become identical to those deined in re.[18], i.e., 3.2 Finite element discretization 1 ds Sn α 1 ; n Q du K In order to derive the set o discretized FE equations, a proper approximation or both velocity and pressure ields is required. Assuming the ollowing approximations s, t, t d t, t d t u x, t N xu t u x, t B xu t v x u x N x u u x B x u w w w w w w the space-discretized orm o equations (31) and (32) can be derived as, where, T 1 B σ dv Qu 0 d w 2 Qu Su Hu 0 w w w w d d (33) (34) (35) (36) 26

36 Q B mn Ω Ω T 1 S Nw Nw dω Ω Q ˆ T k H Bw Bw dω Ω g Γt Γq T d T w 0 w dω 1 T T N t ds N ˆ g dω d w Q N SmB d Ω dω 2 kˆ N q dω B g dω d T w T w 0 w 0 w w wg and m is the matrix orm o the identity tensor I. The set o equations (36) must be discretized in time as well. Using a conventional Euler-backward integration scheme, the ollowing relations are obtained 1 1 T t t r B σ dω Q u tt d tt w tt Ω tt r Q u S t H u t 2 (2) tt tt tt tt tt tt w tt The above equations are nonlinear and a general Newton-Raphson scheme is required. Using an iterative algorithm, leads to where, and 1 tt Ktt Qt t u r tt t 2 Q S H u r tt tt tt tt w i tt i tt tt tt u u u i1 i i tt tt tt uw uw uw i1 i i Ω (37) (38) (39) (40) T K B d d TBd Ω (41) where T is the tangential stiness operator which has been deined in the previous section. 4. NUMERICAL SIMULATIONS 4.1 Deviatoric hardening model The simulations presented here employ the ramework o deviatoric hardening (c. [21]). σ, κ, ζ is deined as Within this approach, the loading surace 27

37 3 g c cot 0 (42) m Here, m σ : I / 3, J 2 1/ 2 s : s, and deviator and J : interval / 6 / 6, while ( ) sin ( 3 3 J / 2 ) / 3 ; where s is the stress s s s. The parameter represents Lode s angle and is deined within the g satisies g / 6 1 and g / 6 K, where K is a constant. In equation (42), is the riction angle and c is the cohesion. The speciic mathematical expression or g ( ) implemented here is that proposed by Willam-Warnke [22]. The hardening parameter ( ) is deined using the ollowing hyperbolic orm A where A is a material constant and deines the value o at ailure, i.e. or. Assuming that the condition at ailure are consistent with Mohr-Coulomb criterion, we have 6sin / ( 3 sin ). Furthermore, the low rule is assumed to be non-associated and the potential unction is taken in the orm where, const. is a material constant. c 0 m (43) m c cot 3 c ( m c cot) ln 0 (44) In order to incorporate the deviatoric-hardening model in the chemo-plasticity ramework, the strength parameters and c, as well as the Young s modulus E, are assumed to undergo a progressive degradation in the course o chemical interaction. The evolution laws are taken in a simple linear orm 0 1 G1 0 c c 1 G2 0 E E 1 G3 where G s are material constants and the kinetics o the interaction process, viz. evolution o ζ, is governed by eq.(1). Note that, given the last equation in (45), the elastic operator satisies so that G ζ / G ζ 3 : 3 (45) (46) 1 G3 : ζ g( ζ ) ; g ( ζ ) (47) 1 G ζ Hence, the ourth-order tensor deined in the previous sections takes the orm, : ζ 1 g( ) ζ ζ ζ 1 ζ g( ζ ) (48) 28

38 In what ollows, some numerical examples are provided. The irst one, which deals with the point integration algorithm, is ocused on examining the eect o water injection in a sample subjected to a sustained deviatoric load o a prescribed intensity. The second example deals with an initial boundary-value problem, which involves assessment o slope stability under conditions o an intense rainall. 4.2 Example # 1 The irst example involves sample o a cohesive soil that is tested in triaxial compression under some initial conining pressure. The sample, in its natural state o compaction, is subjected to an axial load o a prescribed magnitude, which is subsequently kept constant while the water is said to be injected. The latter results in time-dependent degradation o mechanical properties, which stems rom chemical interaction taking place in the neighborhood o interparticle contacts. The mechanical properties o the material, in its initial state, are taken as K 100 MPa; G 45 MPa; c 50 kpa; 1.4; 1.2; A ; Note that c represents here the apparent cohesion, which results rom interparticle bonding and it s subsequently lost upon wetting. The degradation parameters, related to chemical interaction, are assumed as G 20%; G 100%; G 50%; B 0.92; The constant B is chosen in such a way that at t 5sec, the parameter, eq. (1), reaches the value o 0.99, i.e. 99% o the reaction is said to be completed. The simulations correspond to initial conining pressure o p 500 kpa. The water injection is assumed to be instantaneous, so that the problem is solved by employing the point integration scheme, as discussed earlier. The key results are presented in Figs Fig.1 shows the time-dependent history o axial strain corresponding to dierent deviatoric stress intensities q 3 1, ranging rom 800kPa kPa. Note that or the irst stage o loading, i.e. application o deviatoric stress q, the response is time-independent as it is governed by standard plasticity ramework. Fig.2 shows the time history o /. The value o the parameter is evaluated here rom eq. (42), so that 3 g c cot while at ailure, there is. Thus, the ratio / 0,1 is indicative o the extent o damage within the material. It is evident rom Figs.1-2 that i the wetting commences at low values o deviatoric stress intensities, i.e. / 0.7, the stationary conditions are reached. At higher intensities, the degradation o strength properties upon wetting triggers a spontaneous ailure o the sample. m c (49) 29

39 Figure 1- Displacement history versus time Figure 2- Value o / versus time 30

40 4.3 Example # 2 The numerical analysis presented here, involves a slope in a cohesive soil (clay) subjected to a period o an intense rainall. The slope examined in this study has dimensions typical o engineered slopes in Singapore; it is also representative o shallow slopes in the province o Manitoba (Canada) that underwent a translational ailure in the late 1990 s. A major rainall event o a prescribed intensity is considered. Note that the actual amount o rainall that can iniltrate the ground at a given time ranges rom zero to iniltration capacity, which depends on moisture content and porosity o the speciic soil. Apparently, i the precipitation rate exceeds the iniltration rate, the runo will usually occur. In the simulations presented here, no antecedent rainall is applied prior to the major event. The numerical analysis incorporates the transient coupled ormulation or unsaturated low, as described in Section 3. As indicated earlier, the primary purpose o employing this ramework lies in its ability o tracing the evolution o the phreatic surace. In the simulations, the progress o the wetting ront is monitored and the ramework o chemo-plasticity (Section 2.1) is used to model the evolution o mechanical characteristics o clay. The overall stability o the slope is assessed by examining the time history o the parameter /, eq. (49), which deines the extent o damage. The simulations were carried out assuming the ollowing material parameters E MPa c kpa A ; 0.35; 0.98; c 0.77; 20 ; ; The kinetics o the chemical interaction and the degradation constants were selected as G 10%; G 75%; G 10%; B 460.5; while the parameters governing the luid low were taken as k I m / day ; k 1.0; e 1.0; Note that while the choice o these parameters is somewhat arbitrary, their values are typical or lightly-overconsolidated clays. It needs to be emphasized that the example given here serves primarily as an illustration o the proposed methodology, so that the quantitative aspects are rather secondary. The loading process incorporated two stages. The irst one involved the solution due to own weight o the material, while the second one was the simulation o the iniltration process and its coupling with the mechanical response. Fig.3 shows the geometry o the problem and the boundary conditions corresponding to the irst stage. The total height o the slope was taken as H 10 m. The gravity load was applied incrementally in ive layers, in order to relect the construction sequence. The key results are shown in Figs.4-5. Fig.4 shows the distribution o /, while Fig.5 gives the corresponding distribution o equivalent plastic strains at the end o construction stage. Here, the maximum value o / is 0.8 which clearly indicates that the slope is stable. s 31

41 Fig.6 presents the boundary conditions or the second stage o the analysis, i.e. iniltration process. In this phase, the slope is said to be exposed to a heavy rainall (i.e., precipitation in excess o 0.75 cm per hour). Along the ground surace, the water pressure is assumed to increase linearly rom an initial value o 5kPa, which corresponds to S 5%, to zero in a period o 4hr and then is maintained constant. Such boundary conditions are analogous to those in re.[20] and imply that the horizontal suraces can absorb water at the rate which depends on the permeability, while the water cannot congregate along the slope. In addition, a negative pressure 0 uw 5kPa is applied at the bottom. This ensures that water can low out o the domain which, in turn, implies that the groundwater level (which was initially ar below the ground surace) cannot be aected by this rainall. The iniltration analysis was perormed or a period o 30 days. Fig.7 shows the distribution o the degree o saturation at the end o rainall, while Fig.8 presents the corresponding contours o /. It is evident now that, in a large area around the slope, there is / 1. The latter is indicative o a ailure o the slope in this region. Figure 3- Geometry and boundary conditions or dry analysis under its own weight Figure 4- Value o / at the end o dry analysis Figure 5- Equivalent plastic strain at the end o dry analysis 32

42 Figure 6- Geometry and boundary conditions in iniltration analysis ollowed by dry analysis Figure 7- Saturation at the end o rainall ( ) Figure 8- Value o / at the end o rainall ( ) Figure 9- Equivalent plastic strain at the end o rainall ( ) 33

43 5. FINAL REMARKS In this paper, a methodology or modeling a coupled hydro-mechanical response o clayey soils has been presented. Within this ramework, as long as the water phase is discontinuous, the material is treated as a single phase medium undergoing evolution o mechanical properties as a result o chemo-mechanical interactions. At this level o saturation, the approach does not require a speciication o suction-saturation relationship as the suction pressure is said to be the product o interaction o a complex system o mineralogical and chemical actors. A return-mapping algorithm has been developed or the proposed chemo-plasticity ramework. In analyzing the problem o slope iniltration, the backward Euler scheme has been combined with subincrementation. This was primarily due to the act that the chemical degradation is very ast and small increments are required in order to obtain an accurate solution or the local stress trajectories. This algorithm signiicantly improves the convergence o the global solution or unsaturated low in porous media. The governing equations or the luid low in unsaturated media have been derived here using a continuum ramework that incorporates the notion o a deormable control volume. The inal set o equations or the discretized system is the same as that obtained using a classical approach, viz. re. [18]. The primary application given here involved a coupled hydro-mechanical analysis o a slope exposed to an intense rainall. It was demonstrated that the water iniltration may trigger a loss o stability resulting rom degradation o strength properties. The assessment o stability itsel was based on examining the distribution o the ratio / that is indicative o the extent o damage. For quantitative purposes, such an assessment is rather restrictive and a more accurate representation is required. Thereore, the uture work will ocus on development and implementation o numerical procedures that account or modeling o localized ailure. These will involve the use o Extended Finite Element Method [23] and/or the procedures incorporating the volume averaging in the neighborhood o the shear zone [24]. 34

44 REFFRENCES [1] Blatz JA, Ferreira NJ, Graham J. Eects o near-surace environmental conditions on instability o an unsaturated soil slope. Canadian Geotechnical Journal 2004;41: [2] Griiths DV, Mines US, Lu N. Unsaturated slope stability analysis with steady iniltration or evaporation using elasto-plastic inite elements. International Journal For Numerical And Analytical Methods In Geomechanics 2005;29: [3] Chen Q, Zhang L. Three-Dimensional analysis o water iniltration into the Gouhou rockill dam using saturated unsaturated seepage theory. Canadian Geotechnical Journal 2006;43: [4] Schmertmann JH. Estimating slope stability reduction due to rain iniltration mounding. Journal O Geotechnical And Geoenvironmental Engineering 2006;132(9):1219. [5] Tsaparas I, Rahardjo H, Toll D, Leong EC. Controlling parameters or rainall-induced landslides. Computers And Geotechnics 2002;29(1):1-27. [6] Cho SE, Lee SR. Instability o unsaturated soil slopes due to iniltration. Computers And Geotechnics 2001;28(3): [7] Nitao J, Bear J. Potentials and their role in transport in porous media. Water Resources Research 1996;32(2): [8] Pietruszczak S. On the mechanical behaviour o concrete subjected to alkali-aggregate reaction. Computers & Structures 1996;58(6): [9] Hueckel T. Chemo-Plasticity o clays subjected to stress and low o a single contaminant. International Journal For Numerical And Analytical Methods In Geomechanics 1997;21(1): [10] Pietruszczak S, Lydzba D, Shao JF. Modelling o deormation response and chemomechanical coupling in chalk. International Journal For Numerical And Analytical Methods In Geomechanics 2006;30(10): [11] Pietruszczak S, Pande G. On the mechanical response o partially saturated soils at low and high degrees o saturation. In Numerical Models In Geomechanics, Pietruszczak S And Pande G (eds) 1995: [12] Pietruszczak S, Pande G. Constitutive relations or partially saturated soils containing gas inclusions. Journal O Geotechnical Engineering 1996;122: [13] Fredlund DG, Rahardjo H. Soil Mechanics or Unsaturated Soils. Wiley-Interscience, [14] Nuth M, Laloui L. Eective stress concept in unsaturated soils: Clariication and validation o a uniied ramework. International Journal For Numerical And Analytical Methods In Geomechanics 2008;32: [15] Alonso E, Gens A, Josa A. A constitutive model or partially saturated soils. Géotechnique 1990;40: [16] Crisield MA. Advanced Topics, Volume 2, Non-Linear Finite Element Analysis o Solids and Structures. Wiley,

45 [17] Simo J. Computational inelasticity. Springer Verlag, [18] Zienkiewicz O, Chan A, Pastor M, Schreler B. Computational geomechanics [19] Sanavia L. Numerical modelling o a slope stability test by means o porous media mechanics. Engineering Computations 2009; 26(3): [20] Borja RI, White JA. Continuum deormation and stability analyses o a steep hillside slope under rainall iniltration. Acta Geotechnica 2010;5(1):1-14. [21] Pietruszczak S. Fundamentals o Plasticity in Geomechanics. CRC Press, [22] Willam K, Warnke E. Constitutive model or the triaxial behaviour o concrete, IABSE Seminar on Concrete Structures Subjected to Triaxial Stresses III-1, 184, 1974 n.d. [23] Belytschko T, Black T. Elastic crack growth in inite elements with minimal remeshing. International Journal For Numerical Methods in Engineering 1999;45(5): [24] Pietruszczak S. On Homogeneous and Localized Deormation in Water-Iniltrated Soils. International Journal O Damage Mechanics 1999;8(3):

46 Discrete modeling o cohesive crack propagation via an enhanced continuum approach E. Haghighat and S. Pietruszczak * Department o Civil Engineering, McMaster University, Hamilton, Ontario, Canada SUMMARY In this paper, a methodology or modeling o discrete crack propagation in brittle materials is outlined. Within the proposed approach, the discontinuity is explicitly embedded in the constitutive relation that governs the discontinuous motion within a representative elementary volume. This methodology, which was originally developed or smeared modeling o localized deormation, is enhanced here to deal with the discrete crack propagation problems. In particular, the approach is coupled with the level-set method in order to explicitly trace the crack path trajectory. In addition, a new implicit scheme is developed or accurately imposing the continuity condition along the interace. Some numerical examples are provided incorporating the proposed approach. The simulations include a simple tension test, a three-point bending test as well as a mixed mode cracking test in which a specimen is subjected to both shear and tension. It is demonstrated that the ramework incorporating an enhanced constitutive relation with embedded discontinuity gives results that are very close to those obtained using Extended FEM, while the ormer requires signiicantly less computational eort. A comparison with a standard smeared approach is also provided in order to highlight the nature o the contribution. KEYWORDS: discrete crack propagation, embedded discontinuity, extended FEM, constitutive modeling 1. INTRODUCTION This study ocuses on the development and implementation o a methodology or describing the discrete crack propagation within the context o a constitutive law with embedded discontinuity (CLED) that has been previously used or smeared modeling o localized deormation [1,2]. Here, a rigorous analytical derivation o the governing equations is provided irst, which is based on averaging the discontinuous motion in the neighborhood o the macrocrack. The approach is then coupled with the Level-set method or capturing the trajectory o crack propagation and to enable an accurate assessment o the characteristic length parameter that appears in the ormulation. An implicit integration scheme is developed or imposing the continuity condition along the interace and a simple, yet eicient algorithm is introduced or inding the direction o crack propagation. The ramework is applied to numerical analysis o problems involving cohesive crack propagation and the results are compared with those obtained rom Extended Finite Element simulations. A comparison o the proposed methodology with a standard smeared approach is also provided in order to highlight the nature o the contribution. Modeling o damage initiation and propagation has been one the most intensely researched topics over the last ew decades. The existing analytical solutions are restricted to an elastic material and involve simple * Correspondence to: S. Pietruszczak, Department o Civil Engineering, McMaster University, Hamilton, Ontario pietrusz@mcmaster.ca 37

47 geometries and boundary conditions. Thereore, they are not directly relevant to practical engineering problems. The latter require, in general, a numerical simulation that typically involves the use o the Finite Element Method (FEM). The early methodologies or capturing the progressive damage were based on tracing the crack propagation on the element boundaries [3]. In addition, some smeared techniques have also been proposed that incorporated plasticity-based strain-sotening relations [4-6]. Later, in order to avoid remeshing in the context o FEM, mesh-ree Galerkin method was introduced [7,8]. More recently, ater introduction o the partition o unity approach [9], the Extended Finite Element Method (XFEM) has been developed [9], which enables incorporation o discontinuous displacement ields as well as the tip enrichments into the approximation space. XFEM was initially ormulated or mesh-independent modeling o tensile crack propagation [10,11]. The irst extension o the original ramework incorporated the crack tip enrichment and the contact condition [12,13]. Later, the ramework was combined with the Level-set method or tracking the propagating crack in the solution domain [14]. Modeling o cohesive crack within XFEM ramework was irst introduced in re.[15,16]. The XFEM was also used in the context o dynamic problems [17,18], localization problems [19,20], and modeling o multiphase media [21,22]. The primary diiculty in implementing XFEM is the need to deal with additional degrees o reedom. In particular, a special treatment or activating enriched DOFs is required that generally increases the computational eort compared with the standard FEM. The main advantage o this method is the ability to incorporate any asymptotic unction in the discretized model. Along with the development o the X-FEM procedures, a level-set method was introduced [14,23] or tracing the crack propagation path. The latter representation is particularly suitable or problems involving rictional interaces (e.g., a shear band localization), however serious numerical complexities may arise in problems involving multiple-cracking/branching, which require the use o multiple level-sets. In order to overcome these problems, an eicient approximation was introduced in re. [24], whereby instead o dealing with a continuous level-set, the crack growth was represented discretely by activation o crack suraces at individual particles (or nodes), so that no explicit representation o the crack's topology was required. A similar approximation has also been employed within the mesh-ree approach [25,26], where the crack has been represented as a set o adjacent particles (nodes). The above methodologies proved to be quite robust or static/dynamic problems involving multiple cohesive cracks; their eiciency, however, has not yet been veriied or problems involving shear band localization. In parallel with advances in inite element techniques, methodologies have also been developed to describe discontinuous/localized deormation by enhancing the standard phenomenological constitutive laws. The irst such approach [1] advocated the use o volume averaging to estimate the properties o an initially homogeneous medium intercepted by a shear band/interace. The proposed constitutive relation incorporated the properties o constituents (viz. intact material and interace) as well as a characteristic dimension associated with the structural arrangement. This approach was later revised [27] and applied to a broad range o practical engineering problems (e.g., [28-30]). In addition, other continuum rameworks have been developed employing various regularizations techniques that included the use o micro-polar continua [31,32] non-local theories [33], as well as gradient-dependent ormulations [34-36]. It needs to be mentioned that the latter approach was recently employed to eliminate stress/strain singularities at the crack tips ([37,38]). These results are o signiicance in the context o using tip enrichments within the XFEM. In general, all methodologies mentioned above are primarily intended or smeared modeling o evolution o damage in cohesive-rictional materials. In this paper, the problem o cohesive crack propagation in a class o brittle materials, like concrete, is investigated. The main ocus here is on the extension o the methodology employing a constitutive law with embedded discontinuity (CLED) to model the discrete nature o the process. This is achieved by enhancing this approach, which was ormerly used in a smeared sense, with the level-set approximation. The results presented here are compared with those obtained using XFEM approach. In addition, a comparison with the standard smeared approach is also provided. In section 2.1, the analytical representation o a discontinuous motion is reviewed, along with the space discretization employed in XFEM. The approach incorporating a constitutive model with embedded discontinuity is presented in section 2.2., ollowed by 38

48 the ormulation o an implicit integration scheme or imposing the continuity condition along the interace. In section 3, the constitutive model or the interacial material is described and the strategy or modeling the crack propagation is outlined. In the subsequent section, the ield equations o the problem are briely reviewed, or completeness. In section 5, a number o numerical examples are provided that involve simulations o a simple extension test, a three-point bending test as well as a mixed mode cracking test in which a specimen is subjected to both shear and tension. It is demonstrated that both rameworks, i.e. XFEM and FEM incorporating an enhanced constitutive law with embedded discontinuity (CLED), yield virtually identical response, thus giving advantage to the latter one as it does not require the incorporation o any additional degrees o reedom. The conclusions emerging rom this study are presented in section 6. All simulations conducted here are based on FEM/CLED and XFEM programs that are developed by authors. 2. DESCRIPTION OF A DISCONTINUOUS MOTION A discontinuous motion v( x, t) in the domain that contains a discontinuity surace d can be deined as [39] v( x, t) vˆ ( x, t) v ( x, t) (1) where, vˆ ( x, t) and v ( x, t) are continuous unctions in the solution domain and ( ) is the Heaviside step unction that can be expressed in its symmetric orm as d 1 0 ( ) 2 ( ) d 1 (2) 1 0 Here, ( x ) is the signed distance rom the discontinuity interace d, and ( ) is the Dirac delta unction which is deined as being singular at 0 and equal to zero elsewhere. Denoting a jump o a unction on the discontinuity interace by aces, i.e. g, can be deined as ˆ ˆ where g v v v v v v d, the rate o separation between the opposite crack (3) d d where, based on the representation (2), the jump in the step unction is 2. Considering that ( ) and ( ) ( ) d, the velocity gradient o the discontinuous motion (1) can be expressed as s s s s v vˆ v ( v n) (4) d d where n is the normal to the interace, and the superscript s reers to the symmetric part o the gradient operator Space discretization or XFEM strategy Within the XFEM strategy, a discontinuous ield can be incorporated into the approximation space by introducing enrichment unctions and additional degrees o reedoms. Thus, the discontinuous motion (1) can be approximated by h h h v ( x, t) vˆ ( x, t) v ( x, t) Nˆ ( x) d ˆ ( t) N ( x) d ( t) N( x) d ( t) (5) d d 39

49 where, ˆN and N are standard inite element shape unctions that may have dierent orders o approximation [40] and ˆd and d are standard and enriched degrees o reedom, respectively. In order to achieve a better representation o the enriched approximation and to avoid the use o blending elements, the shited orm o approximation (5) can be employed [40], viz. h v ( x, t) Nˆ ( x) dˆ ( t) N ( x) Ψ( x) d ( t) N( x) d ( t) (6) Here, Ψ( x ) is the shited orm o the step unction deined as ( ) ( 1 ) ( ) ( 2) 0 ( ) Ψ x 0 0 ( ) ( n ) and n is the number o interpolation unctions. Note that since Ψ x) ( the identity matrix,the crack opening g h ( x, t) can be expressed as (7) ( ) I I, where I is h h g ( x, t) v ( x, t) Nˆ ( x) dˆ ( t) N ( x) Ψ( x) d ( t) N ( x) d ( t) 2.2. The approach incorporating a constitutive law with embedded discontinuity (CLED) Averaging the velocity gradient (4) over a Representative Elementary Volume (REV), which includes the discontinuity interace, one obtains 1 s 1 s s s dv ˆ dv ( ) v v v dv d v dv d v v v v v v n 1 s s s ˆ dv dv ( ) da d v v v v v v n a where, v is the volume o REV and a is the area o the discontinuity inside the REV. Assuming that the variations o the integrands in (9) are small, one has v v a s v s vˆ s v ( v n) s (10) v v Deining now a / v and ( v v ) / ( v), and using g v one obtains s ε v ε ε ε where s ε ( vˆ g ) (11) s ε ( g n) In eq. (11), ε represents the deormation in the intact material while ε is the strain rate due to discontinuous motion along the interace averaged over REV. In general, ε may include both elastic and plastic deormations in the intact material. Note that the representation (11) may be simpliied by assuming that the discontinuity divides the element into two approximately equal volumes, in which case approaches zero, i.e. 0. (8) (9) 40

50 Using now the additivity postulate and ollowing the standard plasticity procedure, the stress rate in the intact material σ can be deined as, Ω σ ε ( ε ε Ω ) (12) where is the ourth order stiness operator. The traction vector across the interace must remain continuous, i.e. nσω nσγ. Thus, imposing this constraint and writing the constitutive relation or the interacial material in the rate orm as σ n t K g, one obtains, Γ (13) * T n : ε K g where K R K R In equation (13), K is the tangential stiness operator or the interacial material in the global coordinate * system while K deines the same operator related to the local coordinate system along the interace. Using eq. (13), together with eq.(11) and eq. (12), one obtains ε : : ε where χ n K χ 1 n n n Now, substituting eq. (14) into the constitutive relation (12) yields σ T : ε Ω where T : : (15) Note that using matrix notation, relations (11), (14) and (15) can be expressed as s where, ε ε ε Lv vˆ g ε ng ε ε ε where ε L 1 nk n σ ε where L and T (14) where (16) T K K n n T T n x n y n 0 0 z n2 n1 0 y x 0 n n z 0 x 0 n n 0 3 n ; L 3 2 z y (17) The above deined relations are identical to those given in res.[27], while the starting point in this representation is the analytical expression or the discontinuous motion (1). Examining the ormulation o the problem, it can be seen that the primary dierence between the XFEM and the CLED approach is the way in which the discontinuity is perceived. In XFEM, the discontinuity is deined using an enriched approximation space that includes the discontinuous motion, while in 41

51 FEM/CLED, it is deined as an internal variable embedded within a plasticity based approach through a volume averaging. Thus, no enrichments and/or additional DOFs are required, so that it can be easily implemented as a user-deined material subroutine within any commercial FEM package Implicit integration o the constitutive relation In order to calculate the increment g, the continuity condition n σ t must be imposed. For this purpose, an implicit scheme is developed here. Based on eq.(12), the state o stress at t t can be expressed as t σ σ : ε : ε where ε g n (18) Note that i the intact material is assumed to be elastic, which is the case here, the primary unknown in the expression (18) is g. This implies that or a prescribed motion along the interace, the stress state is known. Thus, the discontinuous motion along the interace can be deined by imposing the continuity condition. The latter can be expressed in the residual orm as Using a standard Newton-Raphson method, one can write where Thus, the stress correction can be expressed as while the key variables can be updated as k k k r n σω t I (19) k k 1 k1 k r k k1 r k r r δ g 0 δ g r (20) g g k k k r σω t I n χ nn K g g g 1 δ σ χ : δ g n : δ ε k k k Ω Δg Δg δ g k 1 k k 1 Δε Δε δ ε k 1 k k 1 Δσ Δσ δ σ k 1 k k 1 Ω Ω Ω Finally, using the stress increment σ : ( ε ε ) and imposing the continuity condition in its incremental orm, i.e. nσ I t, the tangential stiness operator can be deined as k I s (21) (22) (23) Δσ : Δ ε wher e : : (24) T which completes the implicit integration scheme or the considered constitutive relation. T 42

52 3. DESCRIPTION OF INTERFACIAL MATERIAL AND THE CRACK PROPAGATION PROCESS For the crack initiation, the simple maximum tensile stress criterion is used and the subsequent behavior is described using the ramework o strain-sotening. The propagating crack is traced by the Level-set method or both XFEM and FEM/CLED techniques Constitutive model or the cohesive zone In order to describe the mechanical characteristics o the interacial material, a simple damage model, similar to that proposed in re.[15], is employed. Within this ramework, an exponential relation or the evolution o cohesive orces acting on the crack aces is assumed, i.e. Ft gn c t ( gn) Ft (25) ( gn c ) G Fe t gn c where, c is the critical separation or imposing the contact condition in a penalty approach, F t is the initial tensile strength o the material, and ( g ) g. The ailure criterion is written as t n is the tensile strength at separation n ( t, g ) t ( g ) (26) n n n t n During an active loading process, there is ( tn, gn) 0, so that the normal traction t n can be expressed as F t tn Ft exp ( gn c ) G The shear traction t t can be deined in terms o discontinuity in the tangential component o displacement g t (27) t d K g K g (28) t t t t t t Ft where d t / Ft is the damage parameter and K t is the shear stiness o the interacial material. Reerring the problem to the local coordinate system along the crack, the incremental orm o eqs. (27) and (28) can be expressed as Ft G Ft Ft g n t t k g g g 2 Ft n 11 n G exp ( ) n c n t k g k g exp ( g ) K g d K g, i 1,2 ti i1 n i2 ti G G n c t ti ti where the indexes 1, 2 deine the in-plane orientations. Thus, the constitutive relation or the interacial material can be expressed in the rate orm, (29) 43

53 where * K is the tangential stiness operator. k * * * * t K g where K k21 k22 0 (30) k31 0 k The strategy or tracing the crack propagation within XFEM and FEM/CLED As shown in several previous studies [14,23], the geometry o crack propagation can be traced by the levelset method that is introduced within XFEM. Based on this approach, a moving/propagating interace Γ ( ) can be deined as the zero level set o a unction ( x, t), i.e., Γd ( t) { x ( x, t) 0}. The unction ( x) itsel can be expressed as the signed distance unction ( x) sign nγ ( x xγ ) min x x (31) Γ where n Γ is a normal to the direction o propagation, x is an arbitrary point in Ω, and x Γ is the point located on the interace at the minimum distance rom x. For two dimensional XFEM simulations, as perormed in this study, the interace is deined as a polygon o line segments passing through elements in which a crack has ormed. Hence, or a node that is common or two adjacent enriched elements, the level set unction can be deined as the minimum distance rom this node to the respective line segments associated with these elements. For each element, the values o the level-set unction at Gauss points can be determined rom nodal values using FEM interpolation unctions, i.e. Ni. i In the case o FEM/CLED, i.e. the ramework incorporating a constitutive law with embedded discontinuity, the same approach is used or tracing the propagating crack. Thus, by analogy to XFEM, the crack is introduced at the element level and the characteristic dimension χ is evaluated based on the volume o the element and the geometry o the propagating crack. Consequently, at all Gauss points associated with this element the constitutive relation employing the volume averaging is used. Given the similarity o this procedure to the general methodology or tracing the interace adopted in XFEM, the results obtained rom these two approaches are very similar. Also, the strain-sotening response is numerically very stable. d t (i) Determination o the direction o propagation The problems involving the crack propagation in brittle materials are typically dealt with using the ramework o racture mechanics. The approach involves evaluation o an integral, known as J-integral, which is a measure o the rate o racture energy. For a linearly elastic material, the J-integral is in act the strain energy release rate and it can be related to the stress intensity actors [41] which, in turn, can be used or speciying the direction o crack propagation. In this case, the assessment o the crack orientation involves the stress state in the neighborhood o the crack tip and, as a result, an accurate/smooth crack path can be captured. However, the main limitation o this approach is that the relation between the direction o propagation/j-integral and the stress intensity actors is valid only or an elastic material subjected to tension. Thus, or inelastic materials and/or in the compressive stress regime, this approach is not applicable. A more general strategy, which is employed here, is to deine the onset o cracking as well as the direction o propagation based on a ailure criterion that is embedded in the constitutive relation. For ailure in the tensile regime, or example, the orientation o crack is commonly assumed to be orthogonal to the 44

54 direction o the major principal stress and the onset o cracking is deined based on the critical value o this stress component. This critical value is deined as 1 Ft, so that or 1 the interace (crack) is assumed to be ormed. The main problem associated within this strategy is that the stress state at the crack tip is not, in general, as accurate as required. Although this may not aect the global accuracy o the solution, it can be important in terms o predicting the crack paths propagation. A simple enhancement that can make this criterion more accurate is to adopt a procedure that is conceptually similar to that employed in assessment o the J-integral. Thus, the crack orientation can be established at each integration point in a region around the tip element and then averaged to assess the actual direction o propagation. This procedure can be used or modeling o damage in both tensile and compressive regime; it can also be used or evaluating the onset o cracking. As shown later in the numerical examples given, such a strategy results in a smooth pattern o the crack propagation, similar to the one obtained based on the J-Integral calculations. Fig. 1 illustrates the algorithm employed or these simulations. Here, the candidate elements are the same as those embedded within the contour o the J-integral, except the ones in which the crack has already ormed. At the same time, the candidate integration points are the ones or which 1 F t 1. The computational procedure is briely summarized in Table 1. Standard Elements Enriched Elements Tip Element Candidate Elements Candidate Integration Points Regular Integration Points Fig.1 Strategy or tracing the crack propagation used within XFEM and FEM/CLED Table 1. The low chart or analysis and identiication o cracked elements 1. Apply the load increment 2. Form the stiness and internal load operators and solve the nonlinear system For intact elements, the constitutive relation describing the homogeneous deormation mode is used at each Gauss point. For cracked elements, the XFEM or CLED approach is employed 3. Check the ailure criterion and orm the new cracked elements a. Find the average value o 1 Ft or tip and standard elements b. Deine the new elements that must be enriched c. For the crack, update the level-set or deine new level-set 45

55 (ii) On computational eort associated with both methodologies In standard FEM packages, all data structures (i.e. number o DOFs, nodal connectivity, etc.) are allocated prior to the analysis. However, or the crack propagation problems remeshing is required and, as a result, the size o the data structures is progressively changing. In addition, the transer o data is required orm the old mesh to the new one, which signiicantly impacts the eiciency o this approach. For the XFEM simulations, the mesh remains ixed and the data transer is not required; however, the total number o DOFs is changing as the crack propagates. Thus, within this approach the size o the problem is progressively increasing. Perhaps the simplest strategy to resolve this issue is to introduce the required enrichments at all nodes or just within an estimated domain where the crack may potentially propagate. Such a scheme will ensure that the size o data structures remains constant during the crack propagation. At the same time, however, the total number o equations and the size o required data structures are likely to be signiicantly larger than those or the same mesh without enrichment. This will increase the computational eort signiicantly as compared to standard FEM. Another problem that will impact the computational cost o the XFEM, is the modiied integration scheme associated with the enriched elements and one time data transer that is required at the transition stage. The best scheme that results in an accurate integration within these elements is the triangulation technique that divides the element into two dierent regions, one with 0 and the other with 0. Thus, the shape unctions are continuous in both these domains and the standard Gauss quadrature can be used or the integration process. The modiied scheme will increase the computational eort, as more integration points are involved; however, in most cases the number o enriched elements is much less than total number o elements, so that the impact is not overly signiicant. In the XFEM code developed or the purpose o this study, the ollowing strategies are used. First, additional enriched DOFs are assigned a priori to nodes that are located in the region where the crack is likely to orm. Second, all these additional DOFs are treated as boundary conditions at the stage when they are not yet activated. This is done by setting all terms associated with these DOFs to zero in the stiness matrix and in the orce vector. Third, the routine that is used or solving the global system o equations is modiied so that the zero rows/columns can be removed rom the global system. These remedies help to improve the numerical eiciency; however the computational eort is still signiicantly higher than that associated with the standard FEM. It should be stressed that or the FEM incorporating the constitutive model with embedded discontinuity, none o above mentioned diiculties arise. The data structures and the solution strategy are the same as those or standard FEM and a continuous body. The discontinuity is taken into account at the level o constitutive relation. There are no additional DOFs involved and no special integration scheme is required. Thus, the main advantage o FEM/CLED over the XFEM is that the ormer maintains the eiciency o the standard FEM, while, the initiation/propagation routine is the same or both methodologies. As demonstrated through the simulations provided in section 5, the results based on both these rameworks are very close. 4. THE GOVERNING EQUATIONS OF THE PROBLEM Consider a body that includes a discontinuity surace d. The body is subjected to the traction t on t, velocity v on v, and the gravity orce b on, as shown in Fig.2. Along the discontinuity surace d, the interacial orces are present within the part I. 46

56 Fig. 2 A schematic representation o the boundary value problem The equations o equilibrium can be written as and are subjected to the ollowing boundary conditions where n is a unit vector normal to n σ t n σ t σ b 0 (32) v v I n σ 0 on on on on with t, I, d. v t In order to ormulate the weak orm o eq.(32) satisying the boundary conditions (33), the integral o the inner product o the test unction w and the equilibrium unction (32) must vanish over the solution domain Ω, i.e. I d (33) w σ ρb dω 0 (34) Ω Based on the minimum continuity condition or the trial and test unctions v and w, and also the kinematic boundary conditions, both these unctions must lie in and spaces, known as kinematically admissible trial and test spaces, i.e. v v v v v 0,, on w w w w 0 0,, on v Expanding eq.(34) using the discontinuous divergence theorem (see eq.(39) in Appendix) and imposing the boundary conditions (33), one obtains h h h h a) w : σ ( v ) d w t d w b d t h h h h h b) w : σ ( v ) d w ti d w t d w b d I t Here, the irst equation (36a) is the weak orm that is used in FEM/CLED approach while the second one, eq. (36b), applies to the case that the domain is discontinuous, i.e. or XFEM. The discretized orm and an v (35) (36) 47

57 implicit solution method or equations (36) are discussed in the Appendix. This completes the ormulation o the problem that is employed in the numerical simulations provided in ollowing section. 5. NUMERICAL SIMULATIONS In this section, some numerical examples are provided that are solved using the two dierent methodologies, i.e. XFEM and FEM incorporating a constitutive law with embedded discontinuity. As explained earlier, in both cases, the Level-set method is used or tracing the crack propagation. The irst problem involves a direct tension test. The second simulation presented here is the three-point bending test; the problem has the geometry analogous to that employed in re. [16]. The third problem is based on an experimental study perormed by Nooru-Mohamed, as described in re. [42], and involves a sample subjected to a combination o shear and tension. All simulations are carried out under two-dimensional plane-strain conditions Simple tension test The irst simple illustrative example involves a direct tension test conducted on a plate with dimensions o (in mm), discretized into 9 quadrilateral elements ( 3 3 ). The plate is ixed in the horizontal direction along the let edge and in the vertical direction rom the let-bottom corner. A horizontal displacement is applied along the right boundary, see Fig.3. Mechanical properties o the material are assumed as ollows, 3 E 2910 MPa; v 0. 25; F 3. 0MPa; G 0. 1 N/mm; δ mm t c where, E is the elasticity modulus, v is the Poisson s ratio, F t is the tensile strength, G is the racture energy appearing in the cohesive law, eq. (25), and δ c is the critical displacement at the onset o cracking. Note that or the given boundary conditions, the stress state is uniorm, while the crack is assumed to orm in the middle o the sample, in the direction normal to the imposed displacement. 1.6E+04 X-FEM FEM/CLED - + Force (N) d Fig. 3 Direct tension test Fig. 4 Load-displacement response o the structure The load-delection response using both XFEM and FEM/CLED techniques is plotted in Fig.4. It is evident rom this igure that the result corresponding to both methodologies are the same Three point bending problem 0.0E Displacement (mm) The second example given here involves a simply supported concrete beam subjected to an increasing vertical displacement applied in the middle o the span, Fig.5. The geometry is taken rom the re.[16]. The 48

58 beam has dimensions l 600 mm and b t 150 mm, where t is the out o plane thickness, and the material properties are as ollows E v F G 3-4 = MPa; =0.1; t =3.19 MPa; =0.05 N/mm; δ c 1 10 mm l b Fig. 5 Three point bending test on a concrete beam ( l 4b 4t 600 mm ) The key results o the analysis are presented in Figs.6-8. Figs.6 shows the damage pattern. The ailure process involves development o tensile cracks near the middle o the span and subsequent propagation o a dominant vertical crack in the center o the beam. These results are now compared in Fig.7 with the solution using the original smeared cracking approach, i.e. CLED without the enhancement or discrete representation o crack path. The load-displacement (δ) response is shown in Fig.8. The behaviour becomes unstable ater reaching the peak (see the igure on the let-hand side). It is evident that the solution based on the enhanced FEM/CLED is virtually identical to that obtained using XFEM methodology, while there is a markable dierence in the results o CLED approach corresponding to discrete and smeared cracking. Fig.6 Top: Cracking pattern dipicted on 11 countor or XFEM or FEM/CLED analysis; Bottom: Cracking pattern and deormation vector ploted on the deormed shape (Scale=100) 49

59 Fig.7 The crack pattern obtained using the classical smeared FEM/CLED approach Fig. 8 Reaction orce vs a) vertical displacement and b) crack mouth opening displacement 5.3. Nooru-Mohamed mixed mode cracking test The third problem that is studied here is based on the experimental test perormed by Nooru-Mohamed [42] and involves a mixed mode cracking. The geometry o the problem is shown in Fig.9. The specimen has the dimensions o l b 200 mm, c 25 mm, and out o plane thickness t 50 mm. Material properties, are as reported in [42], i.e. E 2910 MPa; MPa; 0.05 N/ mm; 1 10 mm ; Ft G c The loading process consists o two dierent stages. First, a horizontal displacement o 0.005mm is applied along the vertical aces under y 0. At this stage, reerred to as the shearing stage, no cracks are ormed. Then, a vertical displacement y is imposed along the horizontal aces, while the x remains constant. This stage results in onset and propagation o tensile cracks. The response o the structure is shown Figs Fig.10 shows the cracking pattern superimposed on the contours o horizontal and vertical displacements, while Fig.11 gives a similar representation in terms o displacement vectors. In this case, two macrocracks orm, at the notches, and propagate towards the center o the specimen. The racture x 50

60 pattern is consistent with the experimental evidence [42] and it s identical or both XFEM and FEM/CLED approaches. Figs present the evolution o the components o the reaction orce against imposed boundary displacements as well as the crack tip opening. Again, as the vertical displacement is imposed, the response becomes unstable. The global characteristics are very similar or both methodologies, i.e. XFEM and FEM/CLED. This is particularly evident both prior to as well as at the early stages o the onset o global instability, which is o primary interest or practical engineering purposes. At very advanced stages o deormation, the enhanced volume averaging method predicts less crack opening than the XFEM. y x c b x l y Fig. 9 Mixed mode cracking test ( l b 200 & c 25 mm ) Fig.10 Cracking pattern superimposed on the contours o horizontal (u) and vertical (v) displacements (mm) 51

61 Fig.11 Let: Cracking pattern and the displacement vector plotted on the deormed shape (scale actor=100); Right: Experimental results [42] 6.0E E+04 X-FEM FEM/CLED X-FEM FEM/CLED RX (N) RY (N) 0.0E DY (mm) 0.0E DY (mm) Fig.12 Reaction components (RX and RY) vs. vertical displacement (DY) 6.0E E+04 X-FEM FEM/CLED X-FEM FEM/CLED RX (N) RY (N) 0.0E CMOD (mm) 0.0E CMOD (mm) Fig. 13 Reaction orce vs. CMOD (crack mouth opening displacement) 52

62 6. CONCLUDING REMARKS In this work, the problem o cohesive crack propagation has been addressed using two conceptually dierent methodologies. The main approach involved incorporation o a constitutive law with embedded discontinuity. Such a methodology is conceptually similar to that introduced earlier or smeared modeling o strain localization. Here, an enhancement to this approach has been proposed that allows or modeling o the discrete nature o crack propagation. A smeared representation o damage results, in general, in a less accurate assessment o the ultimate load o the structure. Furthermore, the crack is said to orm at each Gauss point and, as a result, it may propagate within the equilibrium iterations. The latter leads to numerical instabilities developing near the peak load, which present diiculties in advancing the analysis past this stage. Nevertheless, the simplicity o this ramework makes it still very attractive in terms o application to practical engineering problems. The main ocus in this study is on incorporation o discrete representation o crack within the CLED methodology. For this purpose, the level-set method has been implemented to trace the topology o damage. The latter is deined at the level o an element, which improves the numerical stability and allows advancing the analysis into the range o the globally unstable (sotening) response without major convergence issues. The original CLED approach has been also enhanced by developing an implicit scheme or imposing the continuity condition along the interace. The second methodology employed here, which has been primarily used as a veriication tool or the CLED simulations, was the Extended Finite Element method. Within this ramework, the approximation space is enhanced by incorporating a discontinuous displacement ield. The methodology is attractive, but computationally more costly compared to standard FEM. The primary advantage o the XFEM is its ability to incorporate any type o enrichment into the discretized system based on the analytical solution o the problem, while the major gain rom CLED is the computational eiciency and simplicity o implementation, which are both important in the context o practical engineering applications. In order to properly compare these two approaches, a series o numerical simulations have been conducted. Those included a direct tension test, a three point bending test and a numerical analysis o the Nooru-Mohamed mixed mode cracking test. It was demonstrated that the discrete representation o CLED yields very similar results to the ones obtained rom XFEM in terms o the evolution o damage propagation and the assessment o the ultimate load. At the same time, as mentioned earlier, the approach based on standard FEM incorporating a constitutive model with embedded discontinuity (FEM/CLED) is numerically more eicient than XFEM approach as it does not require any additional DOF. REFERENCES 1. Pietruszczak S, Mroz Z. Finite element analysis o deormation o strain sotening materials. Int J Numer Meth Engng. 1981;17(3): Pietruszczak S. Undrained response o granular soil involving localized deormation. Journal o Engineering Mechanics. American Society o Civil Engineers; 1995;121(12): Schellekens J, de Borst R. On the numerical integration o interace elements. Int J Numer Meth Engng. 1993;36(1): Rots JG, Blaauwendraad J. Crack models or concrete, discrete or smeared? Fixed, multi-directional or rotating? Delt University o Technology; Lubliner J, Oliver J, Oller S, Oñate E. A plastic-damage model or concrete. International Journal o Solids and Structures. 1989;25(3):

63 6. Parks DM. The virtual crack extension method or nonlinear material behavior. Comput Methods Appl Mech Engrg. 1977;12(3): Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P. Meshless methods: an overview and recent developments. Comput Methods Appl Mech Engrg. 1996;139(1): Fleming M, Chu YA, Moran B, Belytschko T, Lu YY, Gu L. Enriched element-ree Galerkin methods or crack tip ields. Int J Numer Meth Engng. 1997;40(8): Melenk JM, Babuška I. The partition o unity inite element method: basic theory and applications. Comput Methods Appl Mech Engrg. 1996;139(1): Belytschko T, Black T. Elastic crack growth in inite elements with minimal remeshing. Int J Numer Meth Engng. 1999;45: Moës N, Dolbow J, Belytschko T. A inite element method or crack growth without remeshing. Int J Numer Meth Engng. 1999;46(1): Dolbow J, Moës N, Belytschko T. Discontinuous enrichment in inite elements with a partition o unity method. Finite Elements in Analysis and Design. 2000;36(3): Dolbow J, Moës N, Belytschko T. An extended inite element method or modeling crack growth with rictional contact. Comput Methods Appl Mech Engrg. 2001;190(51): Stolarska M, Chopp DL, Moës N, Belytschko T. Modelling crack growth by level sets in the extended inite element method. Int J Numer Meth Engng. 2001;51(8): Wells GN, Sluys LJ. A new method or modelling cohesive cracks using inite elements. Int J Numer Meth Engng. 2001;50(12): Moës N, Belytschko T. Extended inite element method or cohesive crack growth. Engineering Fracture Mechanics. 2002;69(7): Belytschko T, Chen H, Xu J, Zi G. Dynamic crack propagation based on loss o hyperbolicity and a new discontinuous enrichment. Int J Numer Meth Engng. 2003;58(12): Belytschko T, Chen H. Singular enrichment inite element method or elastodynamic crack propagation. Int J Comput Methods. 2004;01(01): Samaniego E, Belytschko T. Continuum-discontinuum modelling o shear bands. Int J Numer Meth Engng. 2005;62(13): Liang J, Huang R, Prévost J-H, Suo Z. Evolving crack patterns in thin ilms with the extended inite element method. International Journal o Solids and Structures. 2003;40(10): Réthoré J, Borst R de, Abellan M-A. A two-scale approach or luid low in ractured porous media. Int J Numer Meth Engng. 2006;71(7): Khoei AR, Haghighat E. Extended inite element modeling o deormable porous media with arbitrary interaces. 2011;35:

64 23. Sukumar N, Chopp DL, Moës N, Belytschko T. Modeling holes and inclusions by level sets in the extended inite-element method. Comput Methods Appl Mech Engrg. 2001;190(46): Remmers J, de Borst R, Needleman A. A cohesive segments method or the simulation o crack growth. Computational Mechanics. 2003;31: Rabczuk T, Belytschko T. Cracking particles: a simpliied meshree method or arbitrary evolving cracks. Int J Numer Meth Engng Rabczuk T, Belytschko T. A three-dimensional large deormation meshree method or arbitrary evolving cracks. Comput Methods Appl Mech Engrg. 2007;196(29-30): Pietruszczak S. On homogeneous and localized deormation in water-iniltrated soils. International Journal o Damage Mechanics. 1999;8(3): Xu G, Pietruszczak S. Numerical analysis o concrete racture based on a homogenization technique. Computers and Structures. 1997;63(3): Shieh-Beygi B, Pietruszczak S. Numerical analysis o structural masonry: mesoscale approach. Computers and Structures. 2008;86(21-22): Pietruszczak S, Gdela K. Inelastic Analysis o Fracture Propagation in Distal Radius. Journal o applied mechanics. 2010;77(1): Mühlhaus HB, Vardoulakis I. The thickness o shear bands in granular materials. Géotechnique. 1987;37(3): Borst R de. Simulation o strain localization: a reappraisal o the cosserat continuum. Engineering Computations. 1991;8(4): Pijaudier-Cabot G, Bažant ZP. Nonlocal damage theory. 1987;113(10): Aiantis EC. On the microstructural origin o certain inelastic models. Journal o engineering materials and technology. American Society o Mechanical Engineers; 1984;106(4): Mühlhaus HB, Aiantis EC. A variational principle or gradient plasticity. International Journal o Solids and Structures. 1991;28(7): de Borst R, Mühlhaus HB. Gradient dependent plasticity: Formulation and algorithmic aspects. Int J Numer Meth Engng. 1992;35(3): Askes H, Aiantis EC. Gradient elasticity in statics and dynamics: An overview o ormulations, length scale identiication procedures, inite element implementations and new results. International Journal o Solids and Structures. 2011;48(13): Konstantopoulos I, Aiantis E. Gradient elasticity applied to a crack. J Mech Behav Mater. 2013;22: Wells GN, Sluys LJ. Three-dimensional embedded discontinuity model or brittle racture. International Journal o Solids and Structures; 2000;38(5):

65 40. Bordas S, Legay A. X-FEM Mini-Course. Ecole Polytehnique Federale de Lausanne; pp Gdoutos EE. Fracture Mechanics. Springer. Springer; Nooru-Mohamed MB. Mixed-mode Fracture o Concrete. Delt University o Technology. Delt University o Technology;

66 (i) Discontinuous divergence theorem Consider a body that includes a discontinuity APPENDIX d, as shown in the igure below. - d n t + Fig. 14 Region that includes the discontinuity The integral over the domain can now be decomposed into two separate integrals, one over other over, i.e. d and the d d d (37) Based on the divergence theorem, the integrals over the domains and can be converted to the surace integrals over and. Thus, the discontinuous orm o the divergence theorem can be expressed as. where on d d d d d n d n d n n d Thus, using the above discontinuous divergence theorem and the chain rule, the irst term in eq. (34) can be expressed as wdiv σ d div d : d wσ w σ d Substituting (39) into eq. (34), one obtains d w σn d w σn d w : σ d d : d w σ d w σn d w σn d w b d (40) d (ii). Space discretization and the solution methodology For both the XFEM and FEM/CLED, the approximation unctions can be introduced based on classical FEM approach as (38) (39) 57

67 h v ( x, t) N( x) d ( t) h v ( x, t) B( x) d ( t) h w ( x, t) N( x) w( t) h w ( x, t) B( x) w( t) (41) where, N is the set o all shape unctions, i.e. standard FEM interpolations as well as enrichments, and d, w are the sets o nodal degrees reedom associated with these shape unctions. Substituting the approximation (41) into weak orm (36), dropping w, the residual orce vector R can be written as T B T σ d T T coh d d d N t N t N b R U W (42) coh This relation is, in general, nonlinear due to nonlinearity in the material response. Thus, in order to ind the solution, Newton-Raphson procedure is typically used. Following a standard Newton-Raphson scheme, the residual (42) at time step t t at k th iteration can be expressed as Using the Taylor expansion, one can write k k k tt tt tt t R U W (43) k k 1 k 1 k R t t k k Δ 1 1 Rt Δt k tδt t Δt δ d δ d t Δt R R 0 R (44) Δd Δd where the global tangential stiness matrix is deined as R Δd and the relation or updating nodal unknowns are k t Δ t T k T k B B d Ω N K N d Ω Γ coh Δd Δd δ d k1 k k1 tδt tδt d d Δd tδt t tδt Note that the integration scheme or the constitutive relation and the interacial orces is discussed in the main body o the paper. Γ (45) (46) 58

68 On modeling o discrete propagation o localized damage in cohesive-rictional materials E. Haghighat a and S. Pietruszczak *, Department o Civil Engineering, McMaster University, Hamilton, Ontario, Canada SUMMARY In this paper, the problem o propagation o localized deormation associated with ormation o macrocracks/shear bands is studied in both tensile and compressive regimes. The main ocus here is on enhancement o the constitutive law with embedded discontinuity to provide a discrete representation o the localization phenomenon. This has been accomplished by revising the ormulation and coupling it with the level set method or tracing the propagation path. Extensive numerical studies are conducted involving various racture modes, ranging rom brittle to rictional, and the results are compared with the experimental data as well as those obtained using XFEM methodology. KEY WORDS: Embedded discontinuity model, XFEM, Constitutive modeling, Level set method, Localized deormation 1. INTRODUCTION The research reported here is ocused primarily on the problem o damage propagation in rictional materials. The approach incorporates a constitutive law with embedded discontinuity (CLED). This methodology, developed in res. [1,2], has been previously used within the context o smeared modeling o localization problems. The primary novelty here is an extension o this ramework to describe the discrete nature o the rictional damage process associated with elastoplastic deormation. This has been achieved by revising the original approach and coupling it with the level-set representation to capture the path o the macrocrack/shear band propagation. Within the proposed ramework the discontinuity is deined at the element level rather than at an integration point, which is in contrast to the classical smeared approach. Thus, in the presence o a discontinuous motion, the volume o a inite element itsel is perceived as a representative elementary volume (REV). In what ollows, an analytical derivation o the governing equations is provided irst based on averaging the motion characteristics in the vicinity o discontinuity. An implicit return mapping algorithm is then developed or integration o the governing elasto-plastic constitutive relation. A simple numerical strategy is also introduced or speciying the direction o propagating macrocrack based on orientation-averaging in the domain adjacent to the tip. The ramework is subsequently applied to numerical analysis o problems involving a cohesive/mixed * Correspondence to: S. Pietruszczak, Department o Civil Engineering, McMaster University, Hamilton, Ontario, L8S 4L7, Canada Address: pietrusz@mcmaster.ca 59

69 mode racture as well as strain localization associated with ormation o a shear band, and the results are compared with those obtained rom XFEM simulations. Since the rise o computational era in the 1960 s, an intensive amount o research has been conducted on modeling o the evolution o damage. The ocus was initially on brittle materials and one o the irst attempts, within the inite element ramework, was based on separating the element edges during the crack ormation process ([3,4]). In order to improve the accuracy, this approach was later combined with adaptive re-meshing ([5-7]); however, it proved to be computationally ineicient. The other methodology that was pursued was the smeared cracking approach. This approach was initiated in late 1960 s in a work by Rashid ([8]). Because o its simplicity and suitability in FEM ormulation, it has been widely used in 1970 s and 1980 s or modeling o damage ([9-12]). The approach is well suited or simulating the evolution o cracking in brittle materials, like concrete, at early stages o the loading process, as the damage is then associated with ormation o multiple micro-cracks within the considered domain. However, the marcocracking and its discrete nature cannot be properly modeled within this approach. Furthermore, the ormulation suers rom mesh-size dependency. An alternative approach to address the issues related to discrete crack propagation involves the boundary integral ormulation [13]. The ramework is consistent with that o the linear racture mechanics; it can provide very accurate results in relation to the analytical solutions, however, its extension to deal with material nonlinearity is complex. The next approach that was developed was the one presented in re. [1]. The methodology was based on a smeared representation, incorporating volume averaging, and the ocus was on resolving the issue related to the mesh-size dependency. This was accomplished by incorporating a characteristic dimension that was explicitly related to geometry o the discretized domain. An attempt conceptually similar to this work was later presented in re. [14] ollowing a dierent mathematical ormat. Another ways to incorporate the strong discontinuities within the FEM approximation space was the use o elements with embedded discontinuities [15] and/or regularized discontinuous inite elements [16]. In addition, element ree Galerkin methods have also been employed or simulating the damage evolution as they show higher accuracy compared to standard FEM interpolations ([17]). Introduction o reproducing kernel and partition o unity concepts [18-20] has opened a new door in numerical modeling o crack propagation and led to the introduction o the Extended Finite Element Method or XFEM [21-23]. This approach is well suited or describing the discrete nature o the crack propagation process and an extensive research has been conducted on this topic over the last ew decades. The approach has been coupled with the level-set method or tracing the propagating crack [24,25] and later used in modeling the discontinuities in a wide range o applications including dynamic problems [26], hydro-mechanical problems [27-29], thermoelasticity [30-32], contact problems [33,34], and shear band ormation [35,36]. The approach has also been used or discontinuous modeling within the newly developed isogeometric method [37]. In this article, the problem o damage propagation associated with shear band localization is investigated. As mentioned earlier, the main ocus here is on employing an enhanced approach incorporating a constitutive law with embedded discontinuity (CLED). The enhancement deals with proposing a strategy to capture the path o damage propagation in a discrete way and coupling o this methodology with the level-set representation. The results are compared with experimental evidence and/or those obtained using XFEM approach. In section 2.1, the analytical representation o a discontinuous motion along with its implementation in the XFEM is reviewed. Section 2.2 ocuses on the ormulation o a constitutive law incorporating a strong discontinuity, which is then ollowed, in section 3, by the ormulation o a new return mapping scheme. The enhanced crack 60

70 propagation strategy that is used within this ramework is outlined in section 4. In section 5, the results o extensive numerical studies are discussed. First, some illustrative examples dealing with cohesive crack propagation are provided, which include a double cantilever problem investigated in [38] and the simulation o an mixed-mode cracking test perormed in [39]. Later, the problems involving a shear band localization are examined including the simulation o a plane strain biaxial compression test [40], as well an assessment o a slope stability. It is demonstrated that both rameworks, i.e. FEM incorporating the enhanced constitutive law with embedded discontinuity (CLED) and XFEM, yield virtually identical response, thus giving advantage to the ormer one as it does not require the incorporation o any additional degrees o reedom. The conclusions emerging rom this study are presented in section 6. All simulations conducted here are based on FEM/CLED and XFEM programs that were developed by the authors. 2. DISCONTINUOUS MOTION: XFEM VS. CONSTITUTIVE LAW WITH EMBEDDED DISCONTINUITY Consider a body, as shown in Fig. 1, that includes a discontinuous surace d. The discontinuity can be described as d x x ( x ) 0 where ( ) is the signed distance unction that can be expressed as ( x ) sign n ( x x ) min x x, while x x x x is the coordinate indicator. The normal vector n. The latter is directed rom i, i, i to and d can be deined as n to the interace i and thus, the tangential vector expressed as a result o counterclockwise 90 rotation o n, as shown in Fig. 1. i m may be i d m n Fig. 1. A body with a discontinuity Within this body, a discontinuous motion vi ( x, t) can be described as a sum o two continuous unctions vˆ i ( x, t) and v i ( x, t) combined with a discontinuous step unction, i.e. d d v ( x, t) vˆ ( x, t) v ( x, t) (1) i i d i Here ( ) is the Heaviside unction that can be expressed in its symmetric orm as d 61

71 1 0 ( ) 2 ( ) d 1 (2) 1 0 where ( ) is the Dirac delta unction that is deined as singular at 0 and zero elsewhere. Denoting a jump o a unction at the point x x located on discontinuity surace d as, the discontinuous motion g can be deined as i g v v (3) i i i where, is the jump o Heaviside unction at x x and can be evaluated as. Based on the representation (2), it can be shown that 2. Considering that, i( ), i and ( ( x )) ( ( x )), the velocity gradient o the discontinuous motion (1) can be expressed as d v x vˆ x v x t n (4) i, j(, t) i,j (, t) i, j (, t) gi( ) j where n i is the normal to the interace, as deined earlier., i d d 2.1. Incorporation o discontinuous motion into the FE approach: XFEM Based on the partition o unity property o FEM interpolations, the enriched shape unctions can be directly incorporated into the approximation space through the generalized FEM or Extended FEM. Thus, the discontinuous motion (1) can be approximated by h h h v ( x, t) vˆ ( x, t) v ( x, t) i i d i I Iˆ ˆ ( ) ˆ N x d ( t ) ( ( x )) N ( x ) d ( t ) I Ii I I I Ii (5) where, N ˆ I and N I are standard inite element shape unctions, Î and I are sets o standard and enriched nodes, and ˆ d ( t) and d ( t) are standard and enriched degrees o reedom associated Ii Ii with node I and direction i, respectively. In order to achieve a better representation o the enriched approximation and to avoid the use o blending elements, the shited orm o the I I enrichment unction can be used, i.e. ( x ) ( ( x )) ( ( x )), as introduced in [41]. h h Note that the crack opening g i ( x, t) vi ( x, t) can be expressed as (, ) ˆ ( ) ˆ g x t N x d ( t) ( x ) N ( x ) d ( t) N ( x ) d ( t) (6) h I i I Ii I Ii I Ii I Iˆ I I I I where is the jump in shited enrichment unction that is. 62

72 2.2. Constitutive law with embedded discontinuity (CLED) In order to incorporate a discontinuous motion into a constitutive model, one can invoke the additivity postulate, similar to that employed in plasticity. Thus, the total strain rate ij can be decomposed into a continuous part ij and an additional part, ij, that is due to discontinuous motion along the interace, i.e. (7) ij ij ij The continuous strain rate itsel can be decomposed into an elastic and a plastic part, viz. e p. In order to deine a proper measure or the discontinuous strain rate, the ij ij ij representation (4) can be employed which provides an analytical representation o a motion that includes a discontinuity. As can be seen rom this equation, there are two parts associated with the motion; one is a continuous part that is deined through v ˆ i, j vi, j vi, j and the other one is associated with the discontinuous motion along the interace, i.e. (g n ). It is clear that the d i j continuous part o the motion will produce a strain rate in the continuous part o body, known as the intact material, while the discontinuous part will generate the strain rate due to deormation within the interacial material. It is evident rom the nature o the Dirac delta unction that this component acts only along the interace and it can be distributed over the small enough REV through averaging procedure. Thus, taking the volume average o this term over the REV that includes the discontinuity, the discontinuous strain ij can be deined as d 1 s 1 s ij (g n ) d (g ) v d i j v a i j a v v n d (8) where, v is the volume o REV and a is the area o the discontinuity inside the REV. Considering the last integral to represent an average value o the dyadic product g n over the dierential area a and deining a / v, one can approximate eq.(8) as a (g ) s ij i n (9) j v In conclusion, the strain decomposition including discontinuous motion can now be expressed as e p (g n ) s (10) ij ij ij ij ij i j Note that the representation (10) is, in act, identical to the strain decomposition introduced in re. [2]. In eq. (10), one can interpret ij as the deormation in the intact material and as the ij discontinuous motion averaged over a REV. It should be noted that in contrast to XFEM, where additional DOFs are introduced as external variables, the current representation employs unknown rates o velocity discontinuities g i, which can be deined through a plasticity based approach by imposing the continuity condition along the interace. i j 63

73 The strain localization is typically associated with an elastic response o the intact material. Thus, in order to ormulate the problem, we can invoke the elastic constitutive operator, so that e e e D ( ) D D ( g n ) (11) ij ijkl kl kl ijkl kl ijkl k l Now, the interacial constitutive model relates the rate o traction, which is a unction o the discontinuous motion, to the velocity discontinuity g i. Thus, t i K ij g j where K ij is the tangential stiness operator or the interace material. By imposing the continuity condition ni ij t j K jig i along the interace, it can be shown that E D ; E n ( K n D n ) n (12) e e 1 ij ijpq pqkl kl ijpq i jp r rjps s q Thereore, the constitutive relation or the case o embedded discontinuity can be written as e e e D ; D D D E D (13) ij ijkl kl ijkl ijkl ijpq pqrs rskl It should be pointed out that the discontinuous motion involving the presence o localization is deined here via eq.(12). By combining eqs.(12) and (9), the velocity discontinuity can be expressed as an explicit unction o a given macroscopic strain rate. Unlike in the original smeared representation, the discontinuity is deined here at element level, not at individual Gauss points, and it s traced by the level-set method; thus, the location and orientation o the crack is known exactly, as in XFEM. In this way, the value o the characteristic dimension a / v can be accurately assessed. It is noted that i the intact material undergoes plastic deormation, the stress rate σ ij should be deied as p D ( ) (14) ij ijkl kl kl In this case, the yield unction and the non-associated low rule can be expressed as p ( ij, ) 0; ( ij ) const; ij (15) ij where is a hardening parameter. For an active loading process, the consistency condition can be written as ij 0 (16) ij Substituting now the stress rate deined in equation (14) one obtains, ater some algebraic manipulations 1 H D ijkl kl; H D ijkl (17) ij ij kl 64

74 Thus, substituting relations (17) back into (14), the tangential stiness operator can be deined as 1 ij D ijklkl ; Dijkl D ijkl D ijkl D rskl (18) H In conclusion, or the elements that experience the localized deormation in the orm o a macrocrack/shear band, the trial stress rate can be ound using equation (13), i.e. assuming linear response o the intact material. I the trial stress does not violate the loading condition, no correction is required. I not, i.e. in the case o an active plastic process, the stress increment and the tangential operator must be updated based on relations (18). kl rs 3. IMPLICIT INTEGRATION SCHEME FOR CLED For the completeness o the presentation, the numerical integration scheme is briely discussed here. In case o XFEM analysis, the nonlinearity o the interacial material is handled within the Newton-Raphson solver itsel, as the discontinuous motion g i is an external variable. At the same time, the nonlinearity o the intact material can be dealt with using a standard implicit integration scheme. The implementation o CLED requires, however, a development o an appropriate a return mapping algorithm which is presented below. The procedure invokes the speciication o trial stress that is assessed using the elastic stiness operator. Thus, or an element with embedded discontinuity, the trial stress can be expressed as D (19) trial t ij ij ijkl kl where D ijkl is deined in eq. (13). Once the trial stress is determined, the value o the yield unction trial trial t must be evaluated. In the case when (, ) 0 or (, ) 0 ( ) 0, the material remains in the elastic range, so that the trial trial t ij ij ij representation (13) results in correct response. For the case o an active loading process, i.e. trial trial t (, ) 0, according to the return mapping scheme [42], the residuals at iteration ij can be written in terms o stress and plastic strain increment at t r t p, ij ij ij Dijkl ( kl kl ) (, ) ij t as Using Newton-Raphson algorithm, residuals (20) can be expressed as ij (20) r r r 0 1 ij ij ij ij mn mn r 1 mn mn 0 (21) 65

75 Solving now equation (21) or, ), yields ( ij Q r 1 ijkl kl ij ; ij Dijkl 1 Q kl ijkl D klmn ij mn (22) where Q. Note that i increments are small enough, 0, one 2 ijkl ij kl ( ij kl ) Q obtains ijkl ij kl. Thus, stress updates can be expressed as 1 ij ij ij 1 (23) This correction process must be continued until the convergence is achieved, i.e. 1 1 ( ij, ) 0. Note that at the end o each iteration, g,, D must be updated as the discontinuous motion g i is a unction o the stress state. i ij ijkl 4. CRACK PROPAGATION STRATEGY: INITIATION AND TRACING THE DIRECTION The onset o cracking is typically assessed based on a ailure criterion speciic to the material. Thus, i the trajectory describing the averaged stress state, within a inite element, approaches the ailure surace, one may assume that the deormation localizes into a macrocrack. For the simulations conducted here, the maximum tensile strength criterion (or the cohesive crack propagation) and the Mohr-Coulomb criterion (or the description o rictional shear band ormation) were employed. As proposed in a number o previous studies, e.g. [24], the level set method can be used or tracing the propagating crack. Based on this methodology, a moving/propagating interace Γ ( ) can be deined as the zero level set o a unction ( x α, t), i.e., Γd ( t) { x α ( x α, t) 0 }. A unction that can represent such a property is the signed distance unction as proposed in [43]. For the numerical simulations conducted here, the level-set method has been coupled with both XFEM and FEM/CLED. In most o the published works, the J-integral method is used to evaluate the stress intensity actors and to deine the direction o the crack propagation. However, the main drawback associated with this method is that it is limited to elastic-brittle materials that undergo a tensile damage. A simple and eective way o deining the onset o racture and the direction o the propagating crack is to invoke a speciic orm o the ailure criterion. For ailure in tension, or example, the maximum tensile stress criterion may be employed which stipulates that the direction o crack is perpendicular to that o major principal stress. In case o rictional materials, commonly described by Mohr-Coulomb criterion, the crack is also said to orm at a prescribed orientation o 45 / 2 relative to the direction o minor principal stress. Note that or an elasto-plastic strainhardening material, the direction o localization is oten deined as a biurcation problem [44]. It d t 66

76 should be pointed out that the integral approaches, such as the J-integral method, have an implicit averaging nature; i.e., they provide average measures deined over a domain o interest. Using a similar concept o average property, a simple algorithm is implemented here or assessing the direction o propagation o the discontinuity surace. In a typical scenario, the stress state at most integration points in the vicinity o the crack tip is close to the ailure envelope. Thereore, the scheme used here or assessing the direction o propagation is based on checking the ailure criterion at the integration points adjacent to the tip element. The average direction o propagation is then established based on the orientations associated with the neighboring integration points. This algorithm is illustrated schematically in Fig. 2. Note that by implementing this methodology, a stable crack pattern propagation is achieved or both XFEM and CLED, which is not the case without incorporation o the averaging scheme. Standard Elements Enriched Elements Tip Element Candidate Elements Candidate Integration Points Regular Integration Points Fig. 2. Crack propagation methodology used or both XFEM and FEM/CLED 5. NUMERICAL SIMULATIONS In this section, dierent boundary value problems are examined involving both FEM with a discontinuity embedded within the constitutive relation and XFEM. Note that XFEM is used here as a veriication tool or assessing the accuracy o the results based on FEM/CLED ormulation. The primary ocus is on examining the damage propagation in rictional materials; however, or a more complete representation, some illustrative examples involving cohesive crack propagation are also provided. The key results are discussed in two separate sections; each section starting with a brie review o the interacial model that is employed or the associated numerical study. For the case o crack propagation, the irst example is based on the work reported in [38], while the second one incorporates the results o an experimental study conducted in [39]. For the strain localization involving a shear band ormation, a plane strain biaxial test reported in re. [40] is simulated irst. This study also involves an examination o the issue o sensitivity o the solution to mesh size/alignment. Later, numerical simulations involving an assessment o stability o a slope excavated in a material with an apparent cohesion are provided. All simulations reported here involve two-dimensional conigurations. 67

77 5.1. Cohesive crack propagation: interace model and the results o numerical simulations For this study, a cohesive crack model with an exponential decay has been employed (ater re. [45]). Within this ramework, the tensile strength o the material is deined as Ft t ( gn) Ft exp( gn) (24) G where, F t is the initial strength at the onset o racture, G is the racture energy release rate, and t( gn) is the tensile strength at separation g n. The ailure unction is deined as ( t, g ) t ( g ) (25) n n n t n For an active loading process, setting ( tn, gn) 0 will clearly result in tn t ( gn). For the simulations conducted here, it was assumed that tangential component o cohesive orce is negligible, i.e. tm 0. Thereore, the constitutive relation in the local coordinate system, along the crack, can be expressed as 2 Ft Ft * * * * G exp( ) 1 G g n i j ti Kij g j ; Kij (26) 0 otherwise where * K ij is the tangential operator Double cantilever beam test The irst example involves the simulation o mode 1 crack propagation in a double cantilever beam. The geometry and the boundary conditions are shown in Fig. 3 and were chosen based on the inormation provided in re. [38]. The beam has the dimensions o mm with a thickness o 1000mm ; the notch is mm. Due to the symmetry, only hal o the domain was discretized. Following the original reerence, a total number o 1005 structured quadrilateral elements were employed. The beam was analyzed under plane stress condition. 68

78 Fig. 3. Geomerty and boundary conditions or the double cantilever beam The load consisted o applying the vertical displacement at point A o the beam and the material properties were taken as E 36.5GPa, 3MPa, v 0.18, G 3 N/m (27) t Note that the value o G, assumed ater re. [38], is quite low which is indicative a relatively high strain sotening rate. The re. [38] provides a detailed discussion on the inluence o G on the solution, in particular on the issue o convergence. The resultant orce vs displacement characteristics are depicted in Fig. 4 or FEM/CLED as well as XFEM simulations. It is noted that or both solutions the descending branch exhibits some oscillations, which are purely numerical. The problem may be rectiied by incorporating more reined implicit/explicit integration schemes (c. [38]). The cracking pattern superimposed on the vertical displacement ield is shown in Fig. 5. It is evident here that virtually identical response is obtained rom both methodologies, in terms o the cracking pattern as well as the mechanical characteristics. 3.0E+04 XFEM FEM/CLED Force (N).0E E+00 Displacement (mm) 1.0E-04 Fig. 4. Force displacement response o the structure Fig. 5. Cracking pattern orm both FEM/CLED and XFEM simulations superimposed on vertical displacement contours 69

79 Mixed mode cracking o concrete The example given here involves the simulation o experimental tests conducted by Galvez and his co-workers [39] at Delt University. The geometry o the problem is shown in Fig. 6. The problem involves our-point bending o a notched concrete beam under the action o two independent actuators. Fig. 6. Mixed mode cracking problem; geometry and boundary conditions In the original experimental studies, the specimens were tested at dierent values o D and K. The simulations conducted here correspond to D 300mm and involve two limiting cases o the value o K, viz. K=0 (no constraint) and K (point A ixed). The problem was again considered as plane-stress and was analyzed as displacement-controlled (viz. ). The material properties were taken as ollows, E 38GPa, v 0.2, 3MPa, G 70 N/m (28) while the thickness was assumed to be equal to 50mm. The cracking pattern, or both cases considered in the analysis, is shown in Fig. 7 together with the relevant experimental data. Fig. 8, shows the scaled images o the deormed shape or FEM/CLED as well as XFEM simulations. The load vs. crack mouth opening displacement (CMOD) and the load-vertical displacement characteristics are plotted in Fig. 9 and Fig. 10 respectively. It is quite evident the predictions using both methodologies are virtually identical and are airly consistent with the experimental data reported in re.[34]. t 70

80 Y X Fig. 7. Let: Cracking pattern rom the FEM/CLED and XFEM analysis. Right: Experimental results [39] Type I Type II Fig. 8. Deormed shape or both FEM/CLED and XFEM simulations; let: K=0 and right: K Fig. 9. Load vs CMOD and load vs vertical displacement response (K=0) Fig. 10. Load vs CMOD and load vs vertical displacement response ( K ) 71

81 5.2. Shear band localization: interace model and the results o numerical simulations For the case involving localized deormation associated with the inception o a shear band, a rictional contact model is incorporated. The yield and the plastic potential suraces are deined as ( t, t ) t ( )( t c) 0; ( t ) t const; g (29) n m m n m m m where ( tn, t m ) are the normal and tangential components o the traction vector, g m is the tangential displacement along the interace, and is the strain-sotening parameter that is identiied with g m. The sotening unction is assumed in an exponential orm ( )exp( ) (30) r 0 where, 0 is the value o riction coeicient at ailure, r is the residual value and the parameter controls the rate o rate o sotening. The constitutive relation can be ormulated by invoking the standard plasticity ormalism Modeling o localized deormation in a biaxial test on dense sand The numerical analysis carried out here involves the simulation o a biaxial (plane strain) test conducted on a dense Ottawa sand at the coninement o 100kPa [40]. The sample had the dimensions o mm and the deormation was recorded by digital monitoring o nodal displacements o the grid that was imprinted on the membrane surace (see Fig. 11). The simulations were carried out assuming that the material remains elastic prior to the onset o localization. The latter was deined as in the classical Mohr-Coulomb criterion. In the sotening regime, the response was said to be associated with localized deormation mode, whereby the behaviour o the shear band material was deined through the interacial constitutive model deined above. The key material properties, as reported in re.[40], were as ollows r E 23 MPa, v 0.3, 48.2 where is the riction angle. For the interace, three dierent values o the residual riction coeicient were selected, viz. r 0 (perectly-plastic response), r 0.80 (i.e., 20% degradation) and r The boundary conditions involved no riction at the end platens while the localization was triggered by introducing an inhomogeneity in the center o the specimen (20 % increase in the value o E). For the localized deormation mode, it was assumed that the shear band within an element orms at 57 0 with respect to the horizontal, which was the actual value measured in the experiment. Note that this value is lower than the usual estimate based on Mohr-Coulomb criterion, i.e. 45 / 2, which is likely due to the act that the material displayed some inherent anisotropy. It is noted again that the orientation o the localization plane can, in general, be determined through an independent criterion, such as that associated with the biurcation properties o the constitutive relation (c.[44,46]). The main results o simulations are presented in Fig. 11. The igures on the let show the deormation mode, both the predicted and experimentally observed, while the igure on the right 72

82 gives the corresponding material characteristics. The results o simulations are, in general, airly consistent with the experimental data. 9 7 Principle Stress Ratio 5 3 CLED-PP CLED-20% CLED-70% XFEM-PP XFEM-20% XFEM-70% EXP. (a) Axial Strain (%) (b) Fig. 11. Shear band ormation in biaxial plane strain test: a) cracking pattern and post localization deormation mode; b) Load-axial displacement curve Fig. 12. Post ailure response o the sample; let: FEM/CLED and right: XFEM analysis It is noted that the analysis presented above employs a mesh that is oriented along the shear band in order to avoid potential numerical issues. In general, however, the results o simulations based on FEM/CLED are virtually independent o discretization, as the mathematical ramework incorporates a characteristic dimension. In order to demonstrate this, a sensitivity analysis has been conducted employing, in addition to previous mesh, two structured grids o dierent size. The results, which are shown in Fig.13, clearly show that the predicted localization patterns as well as the global mechanical characteristics are almost the same or all three types o FE meshes. 73

83 9 Principle Stress Ratio Mesh 1 Mesh 2 Mesh Axial Strain (%) Fig. 13. Mesh sensitivity analysis o shear band ormation in biaxial-compression test Modeling o shear band initiation and propagation in a cohesive slope subjected to oundation loading The last example given here involves an assessment o stability o a slope in cohesive soil (overconsolidated silty clay) subjected to a oundation loading. The geometry adopted is shown Fig. 13. The soil was modeled using a plasticity based deviatoric hardening model [47,48], while the oundation was assumed to be rigid-elastic. The material properties or the soil were taken as E 100 MPa; 0.35; 0.98; 0.77; C 40 kpa; A c 5 where and c are slope o Mohr-Coulomb ailure line and the dilation lines in p q space, respectively, C is the cohesion, and A is the hardening parameter. For the interacial properties, a cohesionless Mohr-Coulomb criterion was assumed, as described earlier. Here, the riction coeicient is measured as /, where and are normal and tangential values o traction acting along the interace at the onset o localization. A degradation o 30% over a sliding o 8 0.1mm, along with kn km 10 N m, was assumed or the interacial model. The boundary conditions are shown in Fig. 13. The irst step involved the analysis due to own weight o soil. Later, the oundation load was imposed through 100 increments. The criterion or the onset o localization was set as 0.9. The direction o the shear band was established as 45 / 2, which corresponds to the orientation that maximizes the Coulomb ailure unction. Fig. 15 shows the distribution o the ailure ratio / at three dierent stages o the loading process. The shear band is initiated at the let corner o the oundation. Subsequently, the main localized zone orms at the right hand side and propagates until a complete ailure o the slope. The load displacement response and the cracking pattern at the end o simulation are plotted in Fig. 15 and Fig. 16. Once again, it is evident that the value o the critical load triggering the loss o stability is virtually the same or both methodologies, i.e. FEM/CLED and XFEM. 74

84 Fig. 13. Geometry o the slope and the boundary conditions; H=10m Fig. 14. Failure ratio at dierent stages o shear band propagation; top: FEM/CLED, bottom: XFEM simulations Fig. 15. Force displacement response rom both FEM/CLED and XFEM analysis Fig. 16. Shear band pattern 75

85 6. CONCLUDING REMARKS The present study was ocused on simulation o cohesive crack propagation as well as modeling o the onset and propagation o localized deormation associated with a shear band ormation. The primary methodology employed here involved a constitutive relation with embedded discontinuity. A new ormulation was presented based on the description o discontinuous motion and an implicit integration scheme was derived incorporating nonlinear properties o the intact material. The approach was coupled with the level-set method or explicit modelling o the discrete crack propagation process. Four dierent sets o numerical simulations were conducted. The irst example involved cohesive mode 1 crack propagation. Next, a mixed mode cracking was examined by simulating the experimental study o Galvez et al. [34]. Subsequently, the localized deormation associated with ormation o a shear band was investigated. The analysis involved the simulation o a plane strain test on a granular material as well as the problem involving the assessment o slope stability. For all studied problems, a close correlation between the results rom FEM/CLED and XFEM was achieved in terms o both the ailure mechanism, i.e. the cracking/shear band pattern, as well as the global load-displacement response; in particular, the assessment o ultimate load. REFERENCES 1. Pietruszczak S, Mroz Z. Finite element analysis o deormation o strain sotening materials. Int J Numer Meth Engng 1981;17(3): Pietruszczak S. On homogeneous and localized deormation in water-iniltrated soils. International Journal o Damage Mechanics 1999;8(3): Ngo D, Scordelis AC. Finite element analysis o reinorced concrete beams. ACI Journal Proceedings 1967;64(3): Nilson AH. Nonlinear analysis o reinorced concrete by the inite element method. ACI Journal Proceedings 1968;65(9): Ingraea AR. Nodal grating or crack propagation studies. Int J Numer Meth Engng 1977;11(7): Saouma VE, Ingraea AR, Gergely P, White RN. Interactive inite element analysis o reinorced concrete: A racture mechanics approach. Report 81 5, Department o Structural Engineering, Cornell University Shephard MS, Yehia NA, Burd GS, Weidner TJ. Computational strategies or nonlinear and racture mechanics problem. Computers & Structures 1985;20(1-3): Rashid YR. Ultimate strength analysis o prestressed concrete pressure vessels. Nuclear Engineering and Design 1968;7(4): Nayak GC, Zienkiewicz OC. Elasto plastic stress analysis. A generalization or various contitutive relations including strain sotening. Int J Numer Meth Engng 1972;5(1): Bažant ZP, Cedolin L. Blunt crack band propagation in inite element analysis. Journal o the Engineering Mechanics Division 1979;105(EM2):

86 11. Cedolin L, Bažant ZP. Eect o inite element choice in blunt crack band analysis. Comput Methods Appl Mech Engrg 1980;24: Bazant ZP, Cedolin L. Finite element modeling o crack band propagation. Journal o Structural Engineering 1983;109: Blandord GE, Ingraea AR. Two-dimensional stress intensity actor computations using the boundary element method. Int J Numer Anal Meth Geomech 1981;17: Oliver J. Modelling strain discontinuities in solid mechanics via strain sotening constitutive equations. Part 1: undamentals. Int J Numer Meth Engng 1996 Nov 15;39(21): Belytschko T, Fish J, Engelmann BE. A inite element with embedded localization zones. Computer Methods in Applied Mechanics and Engineering Elsevier; 1988;70(1): Simo JC, Oliver J, Armero F. An analysis o strong discontinuities induced by strain-sotening in rate-independent inelastic solids. Computational Mechanics 1993;12(5): Belytschko T, Lu YY, Gu L. Element ree Galerkin methods. Int J Numer Meth Engng 1994;37(2): Liu WK, Jun S, Zhang YF. Reproducing kernel particle methods. Int J Numer Meth Fluids 1995;20(8-9): Liu WK, Li S, Belytschko T. Moving least-square reproducing kernel methods (I) methodology and convergence. Comput Methods Appl Mech Engrg 1997;143: Melenk JM, Babuška I. The partition o unity inite element method: basic theory and applications. Computer Methods in Applied Mechanics and Engineering 1996;139(1): Belytschko T, Black T. Elastic crack growth in inite elements with minimal remeshing. Int J Numer Meth Engng 1999;45: Moës N, Dolbow J, Belytschko T. A inite element method or crack growth without remeshing. Int J Numer Meth Engng 1999 Sep 10;46(1): Sukumar N, Moës N, Moran B, Belytschko T. Extended inite element method or three dimensional crack modelling. Int J Numer Meth Engng 2000;48(11): Sukumar N, Chopp DL, Moës N, Belytschko T. Modeling holes and inclusions by level sets in the extended inite-element method. Computer Methods in Applied Mechanics and Engineering 2001;190(46): Stolarska M, Chopp DL, Moës N, Belytschko T. Modelling crack growth by level sets in the extended inite element method. Int J Numer Meth Engng 2001;51(8): Belytschko T, Chen H, Xu J, Zi G. Dynamic crack propagation based on loss o hyperbolicity and a new discontinuous enrichment. Int J Numer Meth Engng 2003;58(12): Réthoré J, Borst R de, Abellan M-A. A two-scale approach or luid low in ractured porous media. Int J Numer Meth Engng 2006 Dec 12;71(7): Khoei AR, Haghighat E. Extended inite element modeling o deormable porous media with arbitrary interaces. Applied Mathematical Modelling 2011;35:

87 29. Khoei AR, Moallemi S, Haghighat E. Thermo-hydro-mechanical modeling o impermeable discontinuity in saturated porous media with X-FEM technique. Engineering Fracture Mechanics 2012; 96: Areias PMA, Belytschko T. Two-scale method or shear bands: thermal eects and variable bandwidth. Int J Numer Meth Engng 2007;72(6): Zamani A, Eslami MR. Implementation o the extended inite element method or dynamic thermoelastic racture initiation. International Journal o Solids and Structures 2010;47(10): Khoei AR, Moallemi S, Haghighat E. Thermo-hydro-mechanical modeling o impermeable discontinuity in saturated porous media with X-FEM technique. Engineering Fracture Mechanics 2012;(96): Dolbow J, Moës N, Belytschko T. An extended inite element method or modeling crack growth with rictional contact. Computer Methods in Applied Mechanics and Engineering 2001;190(51): Liu F, Borja RI. A contact algorithm or rictional crack propagation with the extended inite element method. Int J Numer Meth Engng 2008;76(10): Borja RI. A inite element model or strain localization analysis o strongly discontinuous ields based on standard Galerkin approximation. Comput Methods Appl Mech Engrg 2000;190(11-12): Sanborn SE, Prévost JH. Frictional slip plane growth by localization detection and the extended inite element method (XFEM). Int. J. Numer. Anal. Meth. Geomech. 2011; 35: Benson DJ, Bazilevs Y, De Luycker E. A generalized inite element ormulation or arbitrary basis unctions: rom isogeometric analysis to XFEM. Int. J. Numer. Meth. Engng 2010; 83: Oliver J, Huespe AE, Blanco S, Linero DL. Stability and robustness issues in numerical modeling o material ailure with the strong discontinuity approach. Computer Methods in Applied Mechanics and Engineering 2006;195(52): Gálvez JC, Elices M, Guinea GV, Planas J. Mixed mode racture o concrete under proportional and nonproportional loading. Int J Fract. 1998;94(3): Alshibli KA, Sture S. Shear band ormation in plane strain experiments o sand. Journal o geotechnical and geoenvironmental engineering. American Society o Civil Engineers 2000;126(6): Zi G, Song J-H, Budyn E, Lee S-H, Belytschko T. A method or growing multiple cracks without remeshing and its application to atigue crack growth. Modelling Simul Mater Sci Eng 2004;12(5): Simo J. Computational inelasticity. Springer Verlag; Chessa J, Belytschko T. An enriched inite element method and level sets or axisymmetric two-phase low with surace tension. Int J Numer Meth Engng. 2003;58(13):

88 44. Rudnicki JW, Rice JR. Conditions or the localization o deormation in pressure-sensitive dilatant materials. Journal o the Mechanics and Physics o Solids 1975;23(6): Wells GN, Sluys LJ. Three-dimensional embedded discontinuity model or brittle racture. International Journal o Solids and Structures 2000;38(5): Borja RI, Song X, Rechenmacher AL, Abedi S. Shear band in sand with spatially varying density. Journal o the Mechanics and Physics o Solids. 2013;61: Pietruszczak S. Fundamentals o plasticity in geomechanics. Balkema: CRC Press; Pietruszczak S, Haghighat E. Assessment o slope stability in cohesive soils due to a rainall. Int J Numer Anal Meth Geomech. 2013;37:

89 Modeling o deormation and localized ailure in anisotropic rocks S. Pietruszczak 1 and E. Haghighat Department o Civil Engineering, McMaster University, Hamilton, ON, Canada ABSTRACT This paper deals with description o the deormation process in argillaceous rocks that display a strong inherent anisotropy. Both, the homogeneous and the localized deormation modes are considered. The eects o anisotropy are incorporated by invoking the microstructure tensor approach. The strain localization is assumed to be associated with ormation o a macrocrack the orientation o which is deined using the critical plane approach. The propagation o damage is traced within the context o a boundary value problem by employing a constitutive law with embedded discontinuity. The crack path is monitored in a discrete manner by using the level-set method. The closest-point projection algorithm is developed or the integration o the constitutive relations at both stages o the anisotropic deormation process, i.e. the homogenous mode as well as that involving an embedded discontinuity. The problem o macrocrack ormation in a biaxial plane strain compression test is studied. It is demonstrated that riction between loading platens can play an important role in the process o evolution o damage and may signiicantly aect the compressive strength. Keywords: Constitutive modeling, anisotropy, localization, discrete crack propagation, embedded discontinuity 1. Introduction Many geomaterials display a structural anisotropy which is closely related to their microstructure. The primary ocus here is on the argillaceous rocks, which are sedimentary rocks ormed rom clay deposits. These rocks, which include shales and argillites, are characterized by the presence o closely spaced bedding planes and exhibit a strong directional dependence o strength as well as deormation properties. The understanding o the mechanical behaviour o argillaceous rocks is o a signiicant importance due to their widespread applications in many types o geotechnical projects, including petroleum extraction, carbon dioxide sequestration as well as a deep geological disposal o radioactive waste. The primary concern in this case is the onset and propagation o damage due to excavation, transport o pore luids and/or the elevated temperature. The description o the mechanical behaviour requires, irst o all, the speciication o conditions at ailure under an arbitrary stress state. In addition, a general ramework must be provided or the evaluation o the deormation ield, which may include discontinuities such as macrocracks. Over the last ew decades, an extensive research eort has been devoted to modeling o the mechanical behaviour o anisotropic rocks. A comprehensive review on this topic, examining dierent approaches, is provided in the articles by Duveau and Henry (1998) and Kwasniewski (1993). One 1 Corresponding author. Tel: +1 (905) ; Ext: address: pietrusz@mcmaster.ca 80

90 o the irst attempts to describe the conditions at ailure in anisotropic rocks was the work reported by Pariseau (1968), which was an extension o Hill s criterion (Hill, 1950). This was ollowed by more complex tensorial representations (Amadei, 1983; Boehler and Sawczuk, 1977; Nova, 1980). The application o the latter criteria to practical problems is generally diicult due to a large number o independent material unctions/parameters that appear in the ormulation. A simple and pragmatic approach, which incorporates a scalar anisotropy parameter that is a unction o a mixed invariant o the stress and the structure orientation tensor, has been developed by Pietruszczak and Mroz (2001). This approach was later applied to modeling o sedimentary rocks (Lade, 2007; Pietruszczak et al., 2002). Numerical approaches dealing with crack propagation in continuous isotropic media have been investigated or several decades. Many attempts have been made to accurately model both the smeared nature o the onset o cracking at initial stages o loading as well as the discrete nature o macrocrack propagation. The research started in early 1960 s when the node separation technique was introduced (Ngo and Scordelis, 1967). This was ollowed by the adaptive remeshing (Ingraea, 1977) as well as introduction o the smeared damage models (Bažant and Cedolin, 1979; Nayak and Zienkiewicz, 1972; Rashid, 1968). The issue o mesh-size dependency was irst addressed in the early 1980 s within the context o a constitutive law with embedded discontinuity (Pietruszczak and Mroz, 1981; see also Pietruszczak, 1999). Later, several attempts were made to incorporate the discontinuous motion into the interpolation unctions (Belytschko et al., 1994; Fish and Belytschko, 1988; Simo et al., 1993). The concept o reproducing kernels (Liu et al., 1995) and partition o unity (Melenk and Babuška, 1996) resulted in development o a methodology or modeling crack propagation through the Extended Finite Element Method X-FEM (Belytschko and Black, 1999; Moës et al., 1999; Sukumar et al., 2000). This approach was then combined with the level-set method or tracing o the paths crack propagation (Chessa and Belytschko, 2003) and was applied to a broad class o problems providing stable, mesh independent results. The only concern in relation to this approach is the necessity o dealing with additional enrichment unctions or the cracked elements. The latter requires updating mesh data structures throughout the analysis as well as the use o a special integration technique. In a recent study, the idea o tracing the propagating crack in a discrete way was combined with the embedded discontinuity approach (Haghighat and Pietruszczak, 2013) and applied to modeling o crack/shear band inception and propagation in brittle materials. The approach shows an accurate mesh-independent response without requiring any special techniques in terms o implementation. This paper is an extension o the work reported earlier (Pietruszczak et. al., 2002) and deals with description o the deormation process in argillaceous rocks. Both, the homogeneous and the localized modes are considered here while the eects o anisotropy are incorporated by invoking the microstructure tensor approach. The strain localization is assumed to be associated with ormation o a macrocrack and a simple methodology is proposed or identiying the orientation o the crack based on the critical plane approach. In section two, the plasticity ramework incorporating an anisotropic deviatoric hardening model is presented, ollowed by the ormulation o a constitutive model with embedded discontinuity. The section is concluded by introducing a simple criterion or initiation o crack and speciication o its orientation. In section three, the closest-point projection algorithm is developed or the integration o the constitutive relations at both stages o the anisotropic deormation process. Section our ocuses on numerical modeling o shear band ormation in Tournemire argillite. The crack path is monitored in a discrete manner by using the level-set method. The eects o boundary conditions, orientation o bedding planes, and mesh-sensitivity o the approach are studied. It is demonstrated that riction between loading 81

91 platens can play an important role in the process o evolution o damage and may signiicantly aect the strength characteristics that are commonly perceived as a material property. 2. Formulation o the problem The deormation process in argillaceous rocks incorporates two main stages; the irst one is associated with a homogeneous deormation and the second one involves a ailure mode associated with inception o a macrocrack. Thus, in order to describe the mechanical behaviour, one has to ormulate a constitutive relation dealing with an inherently anisotropic response prior to ailure, an appropriate ailure criterion, and a ramework describing the post-ailure response involving the localized deormation. These three key points are addressed below. First, an anisotropic deviatoric hardening model is presented describing the response prior to the onset o ailure. Then, a ailure criterion is introduced which incorporates the critical plane approach or identiying the direction o the macrocrack. Finally, a constitutive law with embedded discontinuity is introduced that is capable o describing the post localization response within a REV. In the last case, the dominant actor controlling the mechanical characteristics is the interacial response Constitutive relation describing the homogeneous deormation mode The elasto-plastic ramework employed in this study incorporates the microstructure tensor approach (Pietruszczak and Mroz, 2001). In order to deine the anisotropy parameter(s), the ormulation employs a generalized loading vector that is deined as L e t t e σ) e σ) (1) ( ) 2 ( ) ( ) ( ) ( ) L ; L ( ( ( where e ), 1, 2,3, are the base vectors, which speciy the preerred material axes. Thus, the components o L represent the traction moduli on planes normal to the principal material axes. Introduce now a microstructure tensor a, which is a measure o material abric. While dierent descriptors may be employed to quantiy the abric, the eigenvectors o this operator are said to be ( ) collinear with e. The projection o the microstructure tensor on the direction L becomes 1/2 a ; L / ( L L) (2) Here, the scalar variable, reerred to as anisotropy parameter, speciies the eect o load orientation relative to material axes and can be deined as the ratio o joint invariant o stress and microstructure tensor tr( σ a σ) to the stress invariant tr( σ σ ). It is a homogeneous unction o stress o the degree zero, so that the stress magnitude does not aect its value. Note that Eq.(2) can be expressed as (1 A) (1 ); A (3) 0 0 where A dev( a) / 0 is a symmetric traceless tensor with 0 tr( a ) / 3. The representation (3)can be generalized by employing a polynomial expansion which incorporates higher order terms in dyadic products, i.e. (1 a a a ) (4)

92 where a, a, a,... are the expansion coeicients. Using the notion o this scalar anisotropy parameter, as deined viz. eq.(4), any isotropic ailure criterion can be extended to the case o anisotropy by assuming where I1, I2, I 3 are the basic stress invariants. F( σ ) F( I, I, I, ) (5) The parameter is typically identiied with a relevant strength descriptor, whose value is then assumed to depend on the orientation o the sample relative to the direction o loading. In this work a simple linear orm o eq.(5) has been adopted, which corresponds to the well-known Mohr- Coulomb criterion, i.e. F( σ ) F(,,, ) 3 ( ) g( )( C) 0 (6) m m Here, tr( σ ) / 3, tr( ss ) 2, where s is the deviatoric part o stress tensor; while, m 1 3 which is deined as sin ( 27 / 4 J3 ) / 3 with J3 tr( ss s ) / 3, denotes the Lode s angle. Moreover, 3 sin 6sin g( ) ; ; C c cot 2 3 cos 2sin sin 3 sin where, c are the angle o riction and cohesion, respectively. Note that the strength descriptors, in this case and C, are assumed to be orientation-dependent and have the representation analogous to that o eq.(4). However, in the context o the ailure criterion (6), the parameter C is associated with a hydrostatic stress state. The latter is, in act, invariant with respect to orientation o the sample. Thus, the eects o anisotropy can be primarily attributed to variation in the strength parameter. The general plasticity ormulation can be derived by assuming the yield/loading surace in the orm consistent with representation (6), i.e. where 2 p p 3 g( ) m C 0; (, ) ( ) ; d e : e (8) A 3 p e is the deviatoric part o the plastic strain rate and A and are material parameters. According to the hardening rule, or there is, where 1. The parameter is introduced here in order to deine the transition to localized deormation, which is assumed to occur at. Note that the latter equality implies that F, so that the conditions at ailure are consistent with Mohr-Coulomb criterion (6). The plastic potential can be chosen as m C 3 c g( ) m Cln 0 (9) where, c is the dilatancy coeicient deined as c with assumed as a material constant. 0 m 1/2 (7) 83

93 Now, the constitutive relation can be expressed in the general orm e p p σ : ε : ( ε ε ), ε σ (10) where the yield and potential unctions are deined by eqs. (8) and (9). Here, is the anisotropic elastic operator. For an elastic loading trajectory, 0, which implies ε p 0. For an active (plastic) loading process, ( σ, ) 0 and 0. Imposing the consistency condition, the plastic multiplier can be expressed as where e : : : 0 σ ε σ σ : : σ σ m σ σ σ m σ σ m σ σ σ m Substituting eq.(11) back into eq.(10), the tangential elasto-plastic operator σ T : ε, T σ σ (11) (12) T can be deined as ( : σ ) ( σ : ) (13) : : The integration scheme or the above constitutive relation is presented in section 3. σ σ 2.2. Description o localized deormation: a constitutive law with embedded discontinuity (CLED) A discontinuous motion within a body can be expressed in the orm v( x, t) vˆ ( x, t) ( ) v ( x, t) (14) where ˆv and v are two continuous unctions and ( ) is the Heaviside step unction. The latter depends on ( x, t), which is the level-set unction that represents the geometry o the crack. The symmetric part o the gradient operator o (14) can be expressed as s v( x, t) s vˆ ( x, t) ( ) s v ( x, t) ( ) ( n v ) s (15) s Note that ( ) ( ), where is the Dirac delta unction deined as being singular at 0 and zero elsewhere. The gradient o a level-set unction represents the normal to the surace, i.e. n. The level-set that is employed here is the signed distance unction, i.e. ( x) sign{ n ( x - x)} min x - x, as introduced in Sukumar et al. (2001). As mentioned in the introduction, dierent methodologies can be used to incorporate the discontinuous motion into the solution. In the approach pursued here the discontinuity is embedded in the constitutive law by averaging the localized deormation over the REV. 84

94 Examining eq. (15), it can be noted that the irst two terms describe the motion in the intact part o the REV while the third term that involves the Dirac unction is associated with the localized deormation along the crack. This term can be averaged over the dierential volume v 1 v v s 1 s ( )( n g ) d ( ) d v n g (16) a Here, a represents the surace area o the crack within the REV, while g is the discontinuous motion along the interace, i.e. g v v. Ignoring the variation o this discontinuous motion within a small-enough REV, one may express eq. (16) as 1 s s a ( ) ( ) d ( ) with v n g n g (17) v v Thus, the total strain rate can be divided into two elementary parts. The irst one, denoted as ˆε, is associated with the intact part o the REV, while and the other one, reerred to as ε, represents the discontinuous motion along the crack averaged over the REV, i.e. ε εˆ ε where ε ( n g ) s (18) Adopting the additivity o strain rates, the stress ield within the REV can be deined as σ : εˆ : ( ε ε ) (19) In order to determine the velocity discontinuity g that is embedded in the constitutive relation (19), the continuity o the rate o traction along the interace is imposed, i.e. t n σ. Thus, n σ n ε n ( n g ) t K g 1 ε : : ε ; n ( K n n) n where K is the tangential operator which deines the interacial properties. Thus, the constitutive relation within the region experiencing the discontinuous motion may be inally expressed as where is the tangential operator. (20) σ : ε where : : (21) 2.3. Speciication o the orientation o macrocrack at its inception The issue o the onset o localization may be perceived as a biurcation problem (Rudnicki and Rice, 1975). Such an approach, although mathematically rigorous, strongly relies on the constitutive description o homogeneous deormation, including the type o hardening and/or low rule, and the predictions are not always consistent with the experimental evidence. A simpler approach is to assume that the inception o localized damage occurs when the ailure unction reaches a critical value. For the analysis conducted in this work, the Mohr-Coulomb criterion (6) is checked at every Gauss point within the volume o a inite element that is considered as a REV. I the average value o F approaches zero, the element is said to undergo a discontinuous motion; the level-set is updated and the next load increment is applied. In order to carry out the integration o the constitutive law (21), the orientation o the macrocrack must be deined. This is accomplished by invoking the critical plane approach as explained below. 85

95 Within the context o the critical plane ramework (Pietruszczak and Mroz, 2001), the ailure unction is deined as a unction o the components o traction vector t nσ acting on a plane with unit normal n. For reproducing the bias in the special distribution o strength parameters, say h( n ), a scalar descriptor is deined such that F F [ t, h ( n )]; h h ˆ( 1 b b b ), nωn (22) where, b1, b2, b 3 are constants and Ω is a symmetric traceless tensor whose eigenvectors coincide with the principle material axis. Thus, the Coulomb ailure unction can be expressed here as F ( c) (23) where, are tangential and normal components o traction t, and the riction coeicient as well as cohesion c are both assumed to be orientation dependent, i.e. ( n) and c c( n ), and their distribution is deined viz. representation (22). The problem o the speciication o the direction o macrocrack can then be deined as a constrained optimization problem or ailure unction (22), i.e. max F max F[ t, h( n)] F 0 nn 1 (24) n n The above equation may be solved by any known optimization technique, such as Lagrange multipliers, and the result would deine the direction o the localization plane at any integration point. It should be noted that the conditions at ailure as stipulated by eqs. (23) and (24) are consistent with the Mohr-Coulomb representation (6) provided the strength parameters h( n ), eq. (22), are properly identiied. This issue is addressed urther in section 4, which deals with the numerical implementation. In general, the solution will provide two conjugate orientations. Here, the orientation that maximizes the product ε : ( n m ), where m is the direction o discontinuous motion g, is selected to deine the prevailing orientation o the localization plane, as suggested by Rabczuk and Belytschko (2007). 3. The numerical integration schemes As indicated in the previous section, the deormation process involves two stages; one deals with a homogeneous deormation that is governed by the constitutive law (13) and the other one is associated with localized deormation, viz. eq. (21). In what ollows, the implicit integration schemes or both these cases are derived The integration scheme or elasto-plastic model governing the homogeneous deormation mode In this case, in order to simpliy the algebra, it is convenient to reer the problem to the coordinate system associated with the principal material axes. According to eq.(10), the constitutive relation takes the orm 86

96 p p σ σt : ( ε ε ), ε σ (25) where is the elastic stiness operator, is the potential unction deined in eq.(9) and is the plastic multiplier. Here, the variables without a subscript reer to time t t. Using the closest point projection approach (Simo, 1998), the plastic strain and the yield unction residuals at time t t can be expressed as R ε ε ( k ) p( k ) p ( k ) ( k ) n σ ( σ, ) ( k ) ( k ) ( k ) Linearizing the above equations, one can write R R : σ R 0 ( k ) ( k ) ( k ) ( k ) ( k ) σ : σ 0 ( k ) ( k ) ( k ) ( k ) ( k ) σ ( k ) 2 ( k ) with the derivatives expressed as σr σ and R σ, where is the 1 compliance operator. Denoting σ R, the plastic multiplier, stress and plastic strain corrections can be determined as ( k ) : : R ( k ) ( k ) ( k ) ( k ) σ ( k ) ( k ) ( k ) ( k ) : : σ ( k ) ( k ) ( k ) ( k ) ( k ) σ p ( k ) ( k ) (26) (27) σ :[ R ] (28) ε σ : σ and the updated variables can be speciied as ( k1) ( k ) ( k ) σ σ σ ( k1) ( k ) ( k ) ε ε ε p ( k1) p ( k ) p ( k ) Note that the anisotropy parameter is embedded in the gradient operators, indicated in eq.(12) 3.2. The integration scheme or constitutive model with embedded discontinuity Considering eq.(19), one can express the stress state in the intact material at t (29) σ σ as t as s σ σ : ( ε ε ) where ε ( n g) (30) t The interacial constitutive model can be deined as p p t t K ( g g ), g ψ (31) t In the examples provided in section 4, the yield and plastic potential unctions or the interace material are taken in a simple linear orm t 87

97 ( t, g) ( g) 0, ψ ψ( t) g) r 0 r ( )exp( (32) where g is the tangential component o plastic part o velocity discontinuity and is the riction coeicient. The latter is assumed to be monotonically decreasing according to an exponential relation (32), in which 0 r, are the initial and residual values o and the rate o degradation is governed by the parameter. Imposing the consistency condition =0 along the interace, one obtains ep ep ( K tψ) ( t K) t K g where K K (33) K ψ In order to ormulate the return mapping scheme, the ollowing residuals are introduced Linearizing these residuals R ε ε ( n g ) R ( k ) ( k ) ( k ) s 1 t n σ t ( k ) ( k ) ( k ) 2 R R : σ R g 0 ( k ) ( k ) ( k ) ( k ) ( k ) 1 σ 1 g 1 R R : σ R g 0 ( k ) ( k ) ( k ) ( k ) ( k ) 2 σ 2 g 2 t t g (34) (35) and substituting or the derivatives, one has R : σ ( n g ) 0 ( k ) ( k ) ( k ) s 1 R nσ K g ( k ) ( k ) ( k ) 2 0 Solving now eqs.(36) or stress and strain corrections, one obtains with g [ ( n n) K ] ( R n : R ) ( k ) ( k ) 1 ( k ) ( k ) ( k ) 2 1 σ : ( R ( n g ) ε ( k ) ( k ) ( k ) s 1 : g ( k ) ( k ) g g g ( k 1) ( k ) ( k ) σ σ σ ( k 1) ( k ) ( k ) ε ε ε ( k 1) ( k ) ( k ) Note that the tangential operator is deined here viz eq.(21). ) (36) (37) (38) 4. Numerical study: bi-axial plane strain tests on Tournemire argillite The numerical study presented here deals with the description o localized damage in Tournemire argillite rock subjected to plane strain biaxial compression. The ocus is on examining 88

98 the inluence o boundary conditions, as well as the orientation o the bedding planes, on the ailure mechanism and the resulting assessment o compressive strength. The study makes use o the results o triaxial tests that have been reported by Abdi and Evgin (2013). Those results were employed to identiy the material parameters/unctions that enter the ormulation, as discussed in section 2. The details on the identiication procedure, which is based on a comprehensive examination o a series o tests conducted at dierent orientation o the sample relative to the direction o loading and dierent conining pressures, are provided in the orthcoming proceedings o the ISRM Congress (Haghighat and Pietruszczak, 2015). Based on that study, the ollowing material parameters were identiied or the anisotropic deviatoric hardening model describing the homogeneous deormation mode, viz. section 2.1 E 12.5GPa, E 21GPa, G 4.57 GPa, v 0.16, v 0.08 n t nt nt ˆ , A , a , 0.99, 1.25, C 10.61MPa, t 1 A where n,t deine the normal and tangential direction, respectively, in the coordinate system attached to a bedding plane. As an illustration, the spatial variation o the strength parameter, eq.(6), is depicted in Fig.1 below. tt (39) Fig. 1- Variation o strength parameter vs let) microstructure parameter and right) loading angle Given the distribution o as a unction o the loading angle, the corresponding values o the riction angle and the cohesion c can be calculated rom and 1 sin ( 3 ( 6+ )) c C (3 si n ) / (2 3 cos), respectively. The latter values are required or identiication o the critical plane ramework, which is employed to deine the orientation o the localization plane. The best-it approximation, which retains the second-order terms, i.e. 2 2 ˆ(1 n n A ( n n ) ), c cˆ(1 n n A ( n n ) ) and tan, results in the ij i j 1 ij i j ollowing set o coeicients ij i j 1 ij i j ˆ , cˆ , , A (40) t 1 89

99 The corresponding spatial variation o strength parameters is plotted in Fig.2. It should be noted that since C is deined as orientation independent, the distribution o both descriptors displays a similar bias. Friction angle Cohesion Bedding plane orientation, β Bedding plane orientation, Fig. 2- Distribution o riction angle and cohesion c with respect to the orientation o the bedding planes 0 For the interacial material, once the localization occurs the parameter, eq.(32), is determined based on the ratio o the components o the traction vector, i.e /. The simulations presented here were conducted assuming the ollowing set o elastic moduli and the degradation coeicients, eq.(32) K K 110 MPa/mm, 0.7, 5.45mm (41) 5 r 0 1 n t These values were selected on a rather intuitive basis as no explicit experimental evidence is available. The set up or the biaxial test and the corresponding boundary conditions are shown in Fig.3. The sample has in plane dimensions o mm and out o plane thickness o 50 mm. The load was applied incrementally in two stages. The irst one involved subjecting the sample to a conining pressure P 0. Ater this, the displacement ield was set to zero and the vertical displacement was applied. 0 Fig. 3- Biaxial test coniguration and boundary conditions 90

100 In order to simulate the actual test conditions, a simple elastic spring model was used to incorporate the eect o riction between the loading platens and the sample. Five dierent stiness 4 10 coeicients were employed ranging rom K 110 to 1 10 N/mm which are representative o rictionless and ully constrained (sticking riction) conditions, respectively. Fig.4 shows the evolution o the interacial displacement in an inclined sample, 45, or dierent values o coeicients K. Fig. 4- Average horizontal sliding between plate and sample or the case o 45 Fig. 5 shows the distribution o the damage ratio /, which is deined according to eqs.(8) and (6). Note that 0 1, with 1signiying the onset o macrocrack ormation. The / / distribution corresponds to the instant just beore the onset o localization and reers to a sample tested at 45. The results shown in Fig.5 correspond again to dierent rictional constraints at the end platens (viz. the coeicients K ). It is evident rom this igure that or low values o K 1, i.e. K 1 10 N/mm, the stress and deormation ields remain homogeneous. On the other hand, 3 or K 1 10 N/mm, i.e. when the riction is more signiicant, the deormation is localized into a shear band/macrocrack that orms at the corners and/or at the center o the sample. Fig. 5- Damage ratio / or dierent values o riction at the end platens; 45 91

101 Fig. 6 shows the corresponding load-displacement characteristics. As can be seen here, the value o the coeicient K signiicantly aects the ultimate (peak) load. This implies that the riction between loading platens and the sample plays an important role in the assessment o both the load capacity and the actual ailure mode o the sample. Fig. 6- Load displacement curves or dierent values o riction at the end platens; 45 Fig. 7 shows the evolution o the damage ratio / just beore the onset o localization or tests conducted at dierent orientations o the bedding planes. The results correspond to 5 K 1 10 N/mm. It is evident here that in case o horizontal and vertical bedding planes, the ailure mode is largely diused while or the inclined samples, there is an indication o a shear band/macrocrack ormation. The actual ailure mechanism, as obtained rom numerical simulations, is depicted in Fig. 8 or various bedding planes orientations. 92

102 Fig. 7- Damage ratio / at the crack initiation or dierent orientations o bedding planes ; 5 K 1 10 Fig. 8-The predicted ailure pattern in samples tested at dierent orientation o bedding planes The last study conducted here deals with the issue o the mesh dependency o the solution based on the approach ollowed in this work. The key results are shown in Figs. 9 and 10. The simulations involved three dierent discretizations. The basic mesh, i.e. mesh 1, is identical to the one employed in all simulations presented earlier. Mesh 2 is a reined structured mesh while mesh 3 is an unstructured mesh that is oriented along the direction o the shear band corresponding to the localization pattern predicted rom the previous meshes. The load- displacement cures are provided in Fig. 9, whereas Fig. 10 shows the deormed shape with the scale actor o 10. It is quite evident here that the solution is mesh-independent, i.e. the ultimate load as well as deormation pattern are virtually the same. 93

103 Fig. 9- Load displacement response or various FE meshes; the case o 45 with 5 K 1 10 Fig. 10- Deormed mesh plots or various discretizations; scale actor o Concluding remarks In this study, the problem o description o deormation and progressive damage in anisotropic rock ormations was examined. The primary ocus was on modeling o the discrete propagation pattern associated with strain localization. The deormation prior to ailure was described using an elasto-plastic ormulation incorporating the microstructure tensor approach. The onset o ailure was deined by employing a simple stress criterion ormulated within the context o a deviatoric 94

104 hardening model. The orientation o the localization plane was then established based on the critical plane approach by converting the strength parameters used in the elasto-plastic model. The constitutive law with embedded discontinuity was used to model the post ailure response associated with localized ailure mode. The discrete propagation o damage was monitored trough the level-set method. The closest-point projection integration scheme was derived or both the anisotropic deviatoric hardening model and the constitutive law with embedded discontinuity. The mechanical characteristics o Tournemire shale were examined and the eect o boundary condition on the shear band ormation was investigated or a series o biaxial plane strain compression tests. It was shown that in the case o a rictionless interace between loading platens and the sample, the deormation ield remains homogeneous and the ailure mode is diused. With a presence o riction, however, the stress state is signiicantly perturbed which results in ormation o a shear band/macrocrack. In this case, the ultimate strength o the sample is noticeably less than the one attained under rictionless conditions. A series o simulations or samples with dierent orientations o bedding planes was also conducted. It was shown that in samples with horizontal and vertical bedding planes, the ailure mode is diused or both rictionless and ully constrained cases; however the peak strength is still noticeably dierent or both these cases. In summary, the results o simulations clearly demonstrated that riction between loading platens can play an important role in the process o evolution o damage and may signiicantly aect the strength characteristics that are commonly perceived as a material property. This is particularly the case or inclined samples. Finally, the mesh-dependency o the solution was examined using three dierent discretizations. Two structured meshes o dierent size and one mesh oriented along the direction o shear band were employed. It was concluded that by invoking the constitutive law with embedded discontinuity, which incorporates a characteristic dimension, the solution is virtually mesh independent. Reerences Abdi, H., Evgin, E., Laboratory characterization, modeling, and numerical simulation o an excavation damaged zone around deep geologic repositories in sedimentary rocks. CNSC Technical Report RSP-0287, Ottawa. Amadei, B., Rock anisotropy and the theory o stress measurements. Springer-Verlag, Berlin. Bažant, Z.P., Cedolin, L., Blunt crack band propagation in inite element analysis. Int. J. Eng. Mech. ASCE. 105, Belytschko, T., Black, T., Elastic crack growth in inite elements with minimal remeshing. Int. J. Num. Meth. Eng. 45, Belytschko, T., Gu, L., Lu, Y.Y., Fracture and crack growth by element ree Galerkin methods. Modelling Simul. Mater. Sci. Eng. 2,

105 Boehler, J.P., Sawczuk, A., On yielding o oriented solids. Acta Mechanica 27, Chessa, J., Belytschko, T., An enriched inite element method and level sets or axisymmetric two phase low with surace tension. Int. J. Num. Meth. Eng. 58, Duveau, G., Henry, J.P., Assessment o some ailure criteria or strongly anisotropic geomaterials. Mechanics o Cohesive rictional Materials 3, Fish, J., Belytschko, T., Elements with embedded localization zones or large deormation problems. Computers & Structures 30, Haghighat, E., Pietruszczak, S., On the description o racture propagation in brittle materials, in: Computational Geomechanics. Pietruszczak & Pande (Ed), ICCE Publ., pp Haghighat, E., Pietruszczak, S., Numerical investigation o the mechanical behaviour o Tournemire argillite, in:. Proceedings ISRM. doi: /bf Hill, R. The mathematical theory o plasticity, 1950, 328. Oxord University Press. Ingraea, A.R., Nodal grating or crack propagation studies. Int. J. Num. Meth. Eng. 11, Kwasniewski, M.A., Mechanical behavior o anisotropic rocks. Comprehensive rock engineering. Pergamon Presss, Oxord, pp Lade, P.V., Modeling ailure in cross-anisotropic rictional materials. Int. J. Solids and Structures 44, Liu, W.K., Jun, S., Zhang, Y.F., Reproducing kernel particle methods. Int. J. Num. Meth. Fluids 20, Melenk, J.M., Babuška, I., The partition o unity inite element method: basic theory and applications. Comp. Meth. Appl. Mech. and Eng. 139, Moës, N., Dolbow, J., Belytschko, T., A inite element method or crack growth without remeshing. Int. J. Num. Meth. Eng. 46, Nayak, G.C., Zienkiewicz, O.C., Elasto plastic stress analysis. A generalization or various contitutive relations including strain sotening. Int. J. Num. Meth. Eng. 5, Ngo, D., Scordelis, A.C., Finite element analysis o reinorced concrete beams. ACI Structural Journal Proceedings 64, Nova, R., The ailure o transversely isotropic rocks in triaxial compression. Int. J. Rock Mech. Mining Sci. & Geomech. Abstr. 17, Pariseau, W.G., Plasticity theory or anisotropic rocks and soil. Proceedings 10 th Inten. Symp. Rock Mech., Amer. Rock. Mech. Assoc. Pietruszczak, S., On homogeneous and localized deormation in water-iniltrated soils. Int. J. Damage Mech. 8, Pietruszczak, S., Lydzba, D., Shao, J.F., Modelling o inherent anisotropy in sedimentary rocks. Int. J. Solids and Structures 39, Pietruszczak, S., Mroz, Z., Finite element analysis o deormation o strain sotening materials. Int. J. Num. Meth. Eng. 17,

106 Pietruszczak, S., Mroz, Z., On ailure criteria or anisotropic cohesive rictional materials. Int. J. Num. Anal. Meth. Geomech. 25, Rabczuk, T., Belytschko, T., A three-dimensional large deormation meshree method or arbitrary evolving cracks. Comp. Meth. Appl. Mech. Eng. 196, Rashid, Y.R., Ultimate strength analysis o prestressed concrete pressure vessels. Nucl. Eng.& Design 7, Rudnicki, J.W., Rice, J.R., Conditions or the localization o deormation in pressuresensitive dilatant materials. J. Mech. Phys. Solids 23, Simo, J., Computational inelasticity. Springer-Verlag, Berlin. Simo, J.C., Oliver, J., Armero, F., An analysis o strong discontinuities induced by strainsotening in rate-independent inelastic solids. Comp. Mech. 12, Sukumar, N., Chopp, D.L., Moës, N., Belytschko, T., Modeling holes and inclusions by level sets in the extended inite-element method. Comp. Meth. Appl. Mech. Eng. 190, Sukumar, N., Moës, N., Moran, B., Belytschko, T., Extended inite element method or three dimensional crack modelling. Int. J. Num. Meth. Eng. 48,

107 MODELING OF DELAYED FAILURE OF EMBANKMENTS DUE TO WATER INFILTRATION S. PIETRUSZCZAK a & E. HAGHIGHAT b a Pro.; Department o Civil Engineering, McMaster University, Hamilton, Ont. L8P3H3, Canada address: pietrusz@mcmaster.ca b PhD Student; Department o Civil Engineering, McMaster University, Hamilton, Ont. L8P3H3, Canada Abstract The primary ocus here is on modeling o racture propagation in soils with apparent cohesion subjected to a period o intense rainall. In this case, a micromechanically-based description represents an overwhelming task due to a very complex system o mineralogical and chemical actors. This is particularly evident at the range o irreducible saturation. Recognizing this limitation, the approach ollowed here is based on the ramework o chemo-plasticity. The ormulation incorporates an assumption that the injection o water triggers a volume change (swelling/collapse) that is coupled with a reduction in suction pressures which, in turn, results in degradation o the strength and deormation properties. The modeling o localized ailure mode is based on a constitutive law ormulated through volume averaging in the neighborhood o the embedded discontinuity. The latter is enhanced by employing the level set method. The governing equations are applied to examine the stability o a slope in cohesive soils, subjected to a period o intense rainall. Keywords: Chemo-plasticity; Slope-stability; Shear band modeling. 1. INTRODUCTION Increased precipitation oten leads to a loss o stability o geotechnical structures. Examples include here the natural slopes and embankments constructed in cohesive soils. In recent years, several case histories have been documented, both in Canada as well as in other parts o the world (e.g., in China, Korea, Malaysia, South America), whereby the ailure o engineered and/or natural slopes was related to the loss o apparent cohesion triggered by the local weather conditions [1-3]. The primary diiculty in modeling the loss o stability due to a heavy rainall lies in assessing the in-situ conditions and in describing the coupling between the time-dependent process o water iniltration and the evolution o the stress/pore pressure ield. The problem is typically analyzed by integrated sotware in which the transient seepage is coupled with traditional limit equilibrium slope stability analysis [4-6]. Alternatively, the rameworks or unsaturated soil are implemented in which the suction pressure is considered as a state parameter and an optimization technique is used to search or a critical slip surace (e.g., [4]). In general, the conventional methods or assessing the stability o unsaturated soils, based on the limit equilibrium approach, signiicantly underestimate the saety actors. Thereore, more accurate techniques are required. The speciication o properties o unsaturated clayey soils is diicult. This is particularly the case when dealing with low degrees o saturation, i.e. within the range o irreducible saturation, as the latter involves a very complex system o mineralogical and chemical actors. In clays, the 98

108 bond strength increases rapidly with decreasing water content. The water in the vicinity o minerals, however, has quite dierent properties which cannot, in act, be quantiied due to complex chemical interactions. Thereore, the assessment o suction pressures and their evolution is diicult, which is the main reason why the developments in the area o mechanics o unsaturated soils have not ound their utility in a parallel development o design methodologies. Recognizing the above limitations, a dierent approach is pursued here. In particular, at the range o irreducible saturation (c.[7]), when the water phase is discontinuous, the behavior o the material is described based on a phenomenological ramework o chemo-plasticity (c. [8-10]). Within this ramework, an increase in water content due to wetting is said to trigger a reduction in the interparticle bonding and the corresponding degradation o strength and deormation properties at the macroscale. At the stage when the water phase becomes continuous, the behaviour can then be deined in mechanical terms alone; or example, by employing an averaging procedure in which the compressibility o the pore space is expressed as a unction o properties o constituents (ree water and air) and the microstructure o saturation ([11,12]). The research presented here is an extension o the work recently reported by [13], and it is ocused on the development o a general methodology that includes modeling o the onset and propagation o ailure in geotechnical structures subjected to water iniltration. In the next section, a brie overview is given pertaining to the evolution o microstructure o saturation during the iniltration process. In the subsequent section, the ormulation o the problem is discussed, viz. chemo-plasticity, including the notion o modeling o localized deormation. Two numerical examples are given. The irst one, aimed at illustrating the proposed methodology, deals with simulation o a biaxial test on dense sand. The second one involves a transient hydromechanical analysis investigating the stability o a slope in cohesive soils subjected to a period o intense rainall. In solving the problem, the evolution o the phreatic surace is monitored and coupled with mechanical analysis incorporating the propagation o localized damage triggered by the chemical interaction. 2. ON MICROSTRUCTURE OF SATURATION IN GRANULAR SOILS During the iniltration process the microstructure o saturation undergoes a progressive evolution, which should be accounted or in the course o speciication o hydraulic/mechanical properties o the material. In general, reerring to Fig.1, our dierent types o microstructure can be distinguished in partially saturated soils [14]: (A) At very low degrees o saturation, the gas phase is continuous while the liquid phase is discontinuous (i.e., the liquid phase is present only within the interparticle contact areas) (B) At higher degrees o saturation, both the gas and the liquid phase remain continuous (C) As the degree o saturation is urther increased, the gas phase becomes discontinuous (e.g., bubbles embedded in the liquid phase) (D) Large bubbles may be entrapped in saturated matrix ( gassy soil). The common types are (A) to (C) and during the iniltration process the microstructure will abruptly change rom one type to another. The last structure (D) is ormed when the gas (produced by decomposition o organic matter) pushes against the soil skeleton creating gas voids o a size that is much larger than the average particle size. It should be noted that or soil types B, C and D the liquid (water) phase is continuous and has known mechanical properties. In this case, or each speciic geometric arrangement o the 99

109 microstructure o saturation, the mechanical properties can be deined in terms o properties o the skeleton, air and the ree water. In this case, the ormulation incorporates the degree o saturation as well as the average pore size which is considered to be an independent characteristic dimension (c.. [11,12]). Alternatively, the problem has also been phrased using the notions o unsaturated soil mechanics [15] whereby the suction pressure is introduced as an independent state variable. Note, however, that the latter approach does not make any explicit reerence to the type o microstructure. At the range o irreducible saturation, which is o main ocus here, the water phase is discontinuous and the soil microstructure is o the type A. In this case, the speciication o properties, particularly in clayey soils, is diicult as the problem involves complex physicalchemical interactions and the properties o water in the vicinity o minerals cannot be easily quantiied. Thus, the control/measurement o suction pressures is rather problematic and the problem cannot be approached as purely mechanical one. In view o these diiculties, the approach adopted here or the type A soil is based on the phenomenological ramework o chemo-plasticity which is preerred to the classical notions o unsaturated soil mechanics. Figure 1. Microstructure o partially saturated soils: Soil types A through D 3. FORMULATION OF THE PROBLEM 3.1. Chemo-plasticity ramework The general approach or modeling the evolution o properties o clays in the presence o the interparticle bonding is based on the ramework o chemo-plasticity. The particular ormulation outlined here is analogous to that described in the recent article by [13]. Within this approach, the progress in chemo-mechanical interaction is monitored by a scalar parameter, which may be interpreted as the change in the initial suction pressure 0 0 saturation, in REV; i.e. ( us us ) / us, so that 0,1. The evolution law can be taken in a simple linear orm B 1 ; dt gdt t 0 u s, at the irreducible wetting luid where g 0,1 depends on the chemical composition o the clay minerals and water, and B is a material constant. In the elastic range, the constitutive relation takes the orm C (2) e e ij ijkl kl ij (1) 100

110 Here, σ ij is the eective stress, e C ijkl is the elastic compliance operator and the last term represents the volumetric strain due to wetting, with being the maximum expansion/contraction in the stress-ree state. Note that the dierential orm o eq. (2) may be expressed as e e e ij ijkl kl ijkl kl ij C C (3) In order to speciy the plastic strain rates, the unctional orm o the yield criterion 0 is assumed to be aected by the chemical interaction, i.e. p p ij,, 0; ( ij ); ij (4) where ( p ) is the hardening parameter and (, ) is the plastic potential unction. ij Employing now the consistency condition, the plastic multiplier λ can be deined as 1 0 ij ; H p H ij ij ij Thus, invoking the additivity postulate and using the equations (3)-(5), the constitutive relation may be expressed as e 1 e 1 ij Cijkl kl b ij ; bij Cijkl kl ij ; Cijkl Cijkl (6) H H ij ij ij ij kl Note that the inverse orm, deining the stress rates or given strain rates, can be expressed as 1 ; D b D C (7) ij ijkl kl kl ijkl ijkl (5) 3.2. Description o localized deormation In this work, the propagation o localized ailure is modeled by employing the volume averaging to estimate the properties o an initially homogeneous medium intercepted by a shear band/interace [16,17]. The constitutive relation incorporates the properties o constituents (i.e., intact material and interace) as well as a characteristic dimension associated with the structural arrangement. This approach is later enhanced by incorporating the level set method, similar to that used in Extended Finite Element Method [18,19], in order to capture a discrete nature o the shear band propagation process. A discontinuous motion within a representative volume V which contains a discontinuity surace Γ, can be deined as v ( x, t) v ˆ ( x, t) v ( x, t) (8) i i i i i i where, v ( x, t ) and ˆv ( x, t ) are continuous unctions and i i i i Denoting the velocity discontinuities across the interace as g i vi velocity gradient v s i, j can be expressed is the Heaviside step unction., a symmetric part o the 101

111 where, within a representative volume viz. which implies s v vˆ v (g n ) (9) s s s i, j i, j i, j is the Dirac delta unction. The procedure or assessing the equivalent properties i j V intercepted by a shear band is based on averaging scheme, s s s v, vˆ i j i, j vi, j 1 1 s dv dv (g i n j ) dv V V V V V (10) s v vˆ k v ( g n ) (11) s s s i, j i, j i, j i j Here, v i, j s and g i are volume averages o the respective variables deined in eq.(10), A / V and k ( V V ) / V, while A is the surace area o the interace/shear band within the representative volume. Note that the decomposition (11) may be simpliied by assuming that the discontinuity divides the representative element into two approximately equal volumes, in which case there is k 0. Identiying now the symmetric parts o the displacement gradients with the corresponding strain rates, one can write In eq.(12), s ; vˆ k v ; ( g n ) (12) s s ij ij ij ij i, j i, j i, j i j ij deines the strain rate in the intact material, while ij is the strain rate due to discontinuous deormation along the interace averaged over the representative volume. In general, ij may include both elastic and plastic components. Within the context o the chemo-plasticity ramework, as discussed in the previous section, the average stress rates in the intact material can now be deined as The stress rate D l kl ; ij ( g i j ) s ij Dijkl kl bkl ij ij ijkl kl k b n (13) σ ij is subjected to the continuity condition that requires n t K g (14) ij j i ij j where t i is the traction along the interace and Kij deines the stiness properties o the interacial material. Note that the latter can be described using the plasticity ormalism where, p i p p t i Kij g j ; ( ti, ); gi ; ( gi ) (15) t g is the plastic part o the velocity discontinuity,, are the yield and plastic potential unctions, respectively, and κ is the sotening parameter. Combing representations (13) and (14) leads, ater some algebraic transormations, to the localization rule g n E D b ; E K D n n (16) 1 i p ij jpkl kl kl ij ij iklj k l i 102

112 which deines the local velocity discontinuities in terms o average macroscopic strain rates. Note that or a standard rate-independent plasticity there is 0 and the representation given in [17] is recovered. The strategy or monitoring the propagation o shear band within the context o inite element (FE) analysis is similar to that explained in the companion paper that deals with modeling o racture process in brittle materials [13]. The interace is traced using the level set method, so that it is represented as a polygon o line segments passing through elements in which the shear band develops. The characteristic dimension is then evaluated based on the geometry o the element and that o the propagating localization zone. 4. APPLICATION OF CHEMO-PLASTICITY FRAMEWORK TO MODELING OF SOIL INFILTRATION In order to trace the evolution o phreatic surace during the rainall iniltration, the problem is deined by invoking a coupled ormulation or low through unsaturated porous media. Within this ramework, the porous material is considered as a mixture o solid grains and voids; the latter illed with water and/or air. For the mathematical details pertaining to FE ormulation o the initial boundary-value problem the reader is reerred to the original article [13]. The numerical simulations presented here are based on the classical plasticity approach incorporating the notion o deviatoric hardening [20]. Within this approach, the loading surace,, is deined as ij, 1/2 3 h m c cot 0; A (17) 1 3 Here, m ii / 3 sijsij and sin ( 3 3 J3 / 2 ) / 3 ; where s ij is the stress deviator and J s ij s jk s ki. The parameter represents Lode s angle and the unction h( ) implemented here is that proposed by [21]. Furthermore, in eq.(17), is the riction angle, c is the cohesion and the hardening eects are attributed to accumulated plastic distortions, i.e. p p e 1/2 ij eij where e ij is the strain deviator. Note that in the hardening unction given above, A is a material constant and deines the value o at ailure, i.e. or. Assuming that the condition at ailure are consistent with Mohr-Coulomb criterion, we have 6sin / ( 3sin ). Furthermore, the low rule is assumed to be non-associated and the plastic potential unction is taken in the orm m ccot 3 c( m c cot)ln 0 (18) where, c const. is a material constant. In order to incorporate the deviatoric-hardening model within the chemo-plasticity ramework, the strength parameters and c, as well as the Young s modulus E, are assumed to undergo a progressive degradation in the course o chemical interaction. The evolution laws are taken in a simple linear orm 0 m 103

113 1 ; 1 ; G1 c c G2 E E G3 (19) where G s are material constants and the kinetics o the interaction process, viz. evolution o, is governed by eq.(1). In the localized regime, the interacial constitutive relation is derived by invoking the classical Coulomb criterion and attributing the strain sotening eects to irreversible sliding along the interace. Thus, t m ( ) t n c 0; t m const. (20) i i i i i i where n i is a unit vector normal to the shear band/interace, m i is an arbitrary vector normal to n, c is the cohesion and deines the rictional properties. The latter are assumed to degrade i p as a unction o discontinuity in tangential component o velocity g m, i.e. ( ) exp (21) 0 r where is a constant and r deines the residual value o. In what ollows, two numerical examples are given. The irst one is aimed at illustrating the proposed methodology and involves a numerical simulation o the onset and propagation o a shear band in a sample o dense sand subjected to axial compression under plane strain conditions. The second example deals with a coupled hydro-mechanical analysis, which involves assessment o slope stability under conditions o an intense rainall Modeling o localized deormation in a biaxial test on dense sand The numerical analysis carried out here involves the simulation o a biaxial (plane strain) test conducted on a dense Ottawa sand at the coninement o 100kPa [22]. The sample had the dimensions o mm and the deormation was recorded by digital monitoring o nodal displacements o the grid that was imprinted on the membrane surace (see Fig.2). The simulations were carried out assuming that the material remains elastic prior to the onset o localization. The latter was deined as 0.99 in the Mohr-Coulomb criterion (16). In the sotening regime, the response was said to be associated with localized deormation mode, viz. eq.(15), whereby the behaviour o the shear band material was deined through eqs.(19) and (20). The key material properties, as reported by [22], were as ollows E 23 MPa, v 0.3, 48.2 while or the interace, the ollowing material constants were employed k k 1000 N/ mm; 0.6 ; 0.2 mm N T r where kn, k T are the elastic moduli. Note that since prior to the onset o localization the material is said to be elastic, the value o E represents the secant Young s modulus. For the localized deormation mode, the simulations were completed assuming that the shear band orientation was 57 0 with respect to the horizontal, which was the actual value measured in the experiment. In general, however, this value should be determined through an independent criterion, such as that associated with the biurcation properties o the constitutive relation (c. [23]). r 0 1 i i 104

114 The boundary conditions involved no riction at the end platens while the localization was triggered by introducing an inhomogeneity in the center o the specimen (25 % increase in the value o E).The main results o simulations are presented in Fig.2. The igures on the let show the deormation mode, both the predicted and experimentally observed, while the igure on the right gives the corresponding material characteristics. The results o simulations are, in general, airly consistent with the experimental data. (a) (b) Figure 2. Shear band ormation in biaxial plane strain test: a) cracking pattern and post localization deormation mode; b) load-delection curve 4.2. Modeling o shear band initiation and propagation in clayey slopes subjected to a heavy precipitation The analysis presented here is an extension o the recent work reported by [13]. The study involves a slope in a cohesive soil (silty clay) subjected to a period o an intense rainall. The slope has the geometry typical o engineered slopes in Singapore; it is also representative o shallow slopes in the province o Manitoba (Canada) that underwent a translational ailure in the late 1990 s. In order to trace the evolution o the phreatic surace, a transient coupled analysis incorporating unsaturated low was conducted. In the simulations, the history o iniltration was monitored and the ramework o chemo-plasticity (Section 2.1) was used to model the degradation o mechanical properties o clay. The overall stability o the slope was assessed by examining the time history o the onset and propagation o localized damage. The simulations were carried out assuming the same material parameters as in the original reerence, i.e. 5 E 100 MPa; 0.35; 0.98; 0.77; c 20 kpa; A c The constants governing the kinetics o the chemical interaction and the rate o degradation were taken as 1 G 0.10; G 0.75; G 0.10; B 460.0sec while the parameters deining the shear band properties were selected as k k MN/m; 0.6 ; m N T r 5 The soil permeability was assumed as 1 10 m/sec. The transition to localized deormation was deined again in terms o the critical ratio o / as 0.95, while the local orientation

115 0 o the shear band was assumed to be at 45 / 2 with respect to the direction o the minor principal stress. The loading process incorporated two stages. The irst one involved the solution due to own weight o the material, while the second one dealt with the simulation o the iniltration process and its coupling with the mechanical response. The total height o the slope was taken as H 10 m. The gravity load was applied incrementally in ive layers, in order to relect the construction sequence. By the end o irst stage, the maximum value o / was in the range o 0.8, indicating that no localized deormation developed (see [13]). Fig.6 presents the boundary conditions or the second stage o the analysis, i.e. the iniltration process. In this phase, the slope was said to be exposed to a heavy rainall (i.e., precipitation in excess o 0.75 cm per hour). The boundary conditions or this stage o analysis are shown in Fig.3. Along the ground surace, the water pressure was assumed to increase linearly rom an initial value o 5 kpa, which corresponds to S 5%, to zero in a period o 4hr and then was maintained constant. Such boundary conditions are analogous to those assumed in the article by [24] and imply that the horizontal suraces can absorb water at the rate which depends on the permeability, while the water cannot congregate along the slope. The iniltration analysis was perormed or a period o 30 days. Fig.4 shows the distribution o the degree o saturation at the end o rainall, while Fig.5 presents the corresponding contours o / and those o accumulated plastic distortions, respectively. It is evident here that, in the area around the toe o the slope, there is 1. Finally, Fig.6 shows the predicted shear band ormation.the latter is indicative o a ailure mechanism orming in this region. / Figure 3. Geometry and boundary conditions or the iniltration analysis Figure 4. Saturation at the end o rainall (30 days) 106

116 (a) Figure 5. a) Value o / at the end o rainall; b) Equivalent plastic strain at the end o rainall (30 days) (b) Figure 6. Shear band pattern and deormed mesh (scale actor=200) 5. FINAL REMARKS In this paper, the problem o shear band propagation in cohesive-rictional materials has been investigated. The primary ocus was on modeling o racture process in soils with apparent cohesion subjected to a period o intense rainall. The approach was based on employing (at very low degrees o saturation) the ramework o chemo-plasticity, whereby the injection o water was assumed to trigger a reduction in initial suction pressures which, in turn, resulted in degradation o the strength and deormation properties. The modeling o racture propagation at the macroscale incorporated a constitutive law ormulated through volume averaging in the neighborhood o the embedded discontinuity. This methodology was enhanced by coupling with the level set method that has been previously used or the same class o problems within the context o the Extended Finite Element approach. A new analytical ormulation has been presented or the decomposition o strain rates in the presence o a discontinuous motion. Besides the coupled hydro-mechanical analysis dealing with environmental loads, another illustrative example has been provided that ocused on the evolution o localized damage triggered by mechanical load. In particular, the propagation o racture was examined within the context o a sample o dense sand subjected to biaxial compression under initial hydrostatic pressure. The results o simulations clearly demonstrate the ability o the volume averaging approach to describe the process o onset and propagation o localized deormation in geomaterials. This paper was presented at the 3 rd Intern. Con. on Computational Geomechanics (ComGeo III), Krakow, August