and 2 nd International ILEM Conference on Strain Hardening Cementitious Composites 12-14 December 2011, io de Janeiro, Brazil FACTUE OCESS ZONE EXTENT AND ENEGY DISSIATION IN SILICATE COMOSITES WITH DIFFEENT COHESIVE BEHAVIOU Václav Veselý (1), Tomáš ail (1), etr Frantík (1) Stanislav Seitl (1) Institute of Structural Mechanics, Faculty of Civil Engineering, Brno University of Technology, Czech epublic (2) Institute of hysics of Materials, Academy of Sciences of the Czech epublic, Czech epublic (2) Abstract This paper deals with estimation of the size and shape of the zone of material failure evolving around the crack tip during fracture of silicate-based composites. A technique developed for the task by the authors is briefly sketched and then verified via numerical simulations utilizing several modelling approaches. The study is performed on numerically simulated responces of three-point bending tests on notched beams of two considerably different sizes made of silicate-based materials with three different compositions resulting in significantly dissimilar cohesive behaviour (from quasi-brittle to quasi-ductile/strain hardening). Numerical simulations by means of physical discretization of continuum (a spring network/lattice-particle type model), fictitious crack model (specialized FEM code), and crack band model (commercial FEM software) are compared with results of the developing technique for estimation of the extent of the fracture process zone (FZ). The conducted analysis provides information which the authors intend to utilize within the specification of the energy dissipated during fracture by the volume of the material undergoing failure, which shall provide a true fracture parameter/s of the material. 1. INTODUCTION, MOTIVATION The scope of the present paper is the tensile failure (fracture) of silicate-based quasi-brittle materials, particularly the description of aspects of the phenomenon related with the volume in which the material fails, and the amount of energy which dissipates within it. The research is motivated by a need for an appropriate characterization of the above-mentioned materials from the perspective of the tensile failure by means of a unique set of parameters which do not depend on the size and shape of the test specimen(s) and laboratory set-up/conditions, see e.g. Bažant and lanas (1998) as a representative reference. Such parameters shall serve as relevant inputs into computational tools for simulations of the structural behaviour and characterize the material s ability to resist the failure propagation. 259
of 2 nd International ILEM Conference on Strain Hardening Cementitious Composites 12-14 December 2011, io de Janeiro, Brazil ecently, the authors have been developing such a technique (termed the efraro UeUconstruction of UFraUcture UroUcess technique) which shall in particular be applied on the determination of the fracture properties of cementitious quasi-brittle materials from experimental tests on laboratory specimens (Veselý and Frantík 2010). In order to satisfy the conditions of the independence of the determined fracture parameters from the specimen s size, geometry and free boundaries, the possibilities of specification of the energy dissipated within the FZ by the volume of the FZ rather than by the cracked ligament area, as is usual in the ILEM work-of-fracture method (ILEM 1985), have been investigated and proposed for the developing procedure. An amalgamation of classical nonlinear fracture models for concrete, multi-parameter fracture mechanics and plasticity theory is employed for a reconstruction of the FZ volume by which the amount of the energy consumed during the fracture process should be specified. The technique has been already partially validated (Veselý and Frantík 2010, Frantík et al. 2011) using published results of experimental measurements (acoustic emission scanning). However, only a limited number of works on this topic is available, typically with slightly different focus or containing data that are not directly usable for its sound validation. Moreover, they are often concerning only with normal concrete or mortar, i.e. quasi-brittle material with characteristic length in the order of magnitude of units of cm/dm, not fibre- or otherwise reinforced, i.e. quasi-brittle/quasi-ductile material of characteristic length of units of meters. Therefore, numerical simulations of fracture tests have been carried out to gain better evidence and understanding/verifying of fundamental issues for the developing efraro method. Computational tools based on physical discretization of continuum (a model from the class of lattice-particle models) and finite element method with implementation of a cohesive crack model (both fictitious crack model and crack band model) have been employed. The results of such numerical simulations are considered to answer questions regarding the evolution of the FZ, the amount of energy dissipated within it, etc. 2. THEOETICAL BACKGOUND, USED METHODS 2.1 Estimation of fracture process zone extent The procedure of FZ extent estimation is based on three fundamental approaches to model material failure. It exploits the theory of linear elastic fracture mechanics (LEFM), classical nonlinear fracture models for concrete, i.e. the effective crack model (ECM, Nallathambi and Karihaloo 1986) and the fictitious crack model (FCM, Hillerborg et al. 1976), and plasticity theory. For the general description of the stress field in cracked bodies, more terms of the Williams power series (Williams 1957) are taken into account. It is, therefore, referred to as multi-parameter LEFM in this context. A detailed description of the method of FZ extent estimation is given in Veselý and Frantík (2010); here we only sketch out the fundamental ideas behind the procedure. The method is utilized within the processing of fracture test records; typically, load displacement ( d) diagrams. For individual stages of the fracture process, the length of the equivalent elastic crack is estimated by means of the EFM (Nallathambi and Karihaloo 1986). Then, the stress state in the body with the effective crack is approximated through the Williams power series (Williams 1957). Subsequently, the extent of the zone where the untilnow elastic material starts to fail is determined by comparing the tensile strength ft the material to a relevant characteristic of the stress state around the crack tip (some sort of 260
(i.e. 2 nd International ILEM Conference on Strain Hardening Cementitious Composites 12-14 December 2011, io de Janeiro, Brazil equivalent stress, σeq, for cementitious composites e.g. the ankine or Drucker-rager failure criterion can be employed). This zone referred to as ΩZ (index Z stands for lastic Zone). Then, in agreement with the cohesive crack approach, the FZ (marked as ΩFZ) is supposed to extend from the ΩZ around the current crack tip up to an ΩZ corresponding to the prior crack tip (i.e. currently a point at the crack faces), where the value of crack opening displacement w reaches its critical value wc the value of cohesive stress σ(w) drops to zero here). Thus, the region ΩFZ results in a union of ΩZ within the field of activity of the cohesive stress. In order to project the shape of the cohesive stress function σ(w) into the shape of the ΩFZ, the individual plastic zones ΩZ,i creating the entire FZ are scaled by a factor of σ(wi)/ft in this method. The envelope of the ΩFZ zones for each stage of the fracture represents a region in which some sort of damage has taken place during the fracture process throughout the entire specimen ligament, referred to as the cumulative damage zone (CDZ), ΩCDZ. The inspiration for the formulation of the procedure was taken from the field of metallic materials. Irwin and Dugdale were the first who proposed models for estimation of plastic zone ahead of the crack tip. For determination the size and shape of plastic zone at the crack tip two yield criteria are commonly used for metallic material: i) Tresca the material yields when the maximum shear stress exceeds the yield strength in shear, and ii) von Mises material yields when the octahedral shear stress exceeds the maximum shear yield strength, see e.g. Anderson (1995). Consideration of T-stress corresponding to the second term of the Williams series within the calculations improves, in general, the accuracy of such estimation. 2.2 Verification technique physical discretization of continuum modelling hysical discretization is based on the idea that a phenomenon can be studied on an alternative structure which can be much more easily modelled as a numerical representation by the computer than the original structure. This procedure is well known in the theory of nonlinear dynamical systems, where it leads to extremely simple models. In this case, the modelled specimen is substituted by a set of mass points mutually connected by simple translational springs. They load the neighbouring mass points by forces whose value depends on the elongation of the springs. The springs behaviour can be generally defined as fully nonlinear; in this case a multi-linear force elongation function was used (fig. 1a). The model is then defined as a nonlinear dynamical system described by equations of motion including damping. The technique for the continuum discretization within this model is based on a procedure developed for the rigid body spring networks (Bolander et al. 1999). This means that the mass of the mass points and the parameters of the translational springs are computed based on Delaunay triangulation and Voronoi tessellation. Every mass point has its own Voronoi cell, which specifies its mass according to cell area, material density and thickness. Similarly, every translational spring has an edge of a Voronoi cell which specifies its cross-sectional area. This area is then used for calculation of the spring parameters. More details about the used computational and discretization procedures can be found in Frantík et al. (2011) 2.3 Verification technique cohesive zone FE modelling The adapted cohesive zone based model is applied as computationally efficient approach 261
2 nd International ILEM Conference on Strain Hardening Cementitious Composites 12-14 December 2011, io de Janeiro, Brazil for elementary numerical investigation of a fracture process with quasi-brittle response. The following are general simplifying assumptions regarding the model: i) only a symmetrical half of the specimen is considered, discretized by linear elastic finite elements in plane stress state; ii) the cohesive crack is prescribed explicitly on the symmetry axis of the specimen in front of the initial crack (the notch of zero width). The initiation and propagation of the pre-defined cohesive crack is controlled, in accordance with the FCM, by the cohesive stress vs. crack opening curve (i.e. cohesive function). The half specimen segment starts to separate from the symmetry axis while the tensile strength is reached due to increasing load so that cohesive crack begins to form ahead of the notch. After its formation, the cohesive crack is being opened and transfers the cohesive stress according to the prescribed traction separation law. The equilibrium between the crack opening and the transferred cohesive stress is solved iteratively for each loading step as a set of nonlinear algebraic equations based on the finite element method. However, there are some limitations of the adapted model causing some inconsistency with the postulates of FCM. The assumption that value of maximum principal stress does not exceed a tensile strength in regions outside the main crack path is not guaranteed. This discrepancy is distinguished especially in the case of the cohesive function including hardening (see section Discussion). a) b) c) in [mm] W B S L a 0 α 0 = a 0 /W SIZE I 160 80 800 960 16 0.1 SIZE II 640 80 3200 3840 64 0.1 Fig. 1: a) Scheme of the force elongation function used within the FyDiK simulation technique, b) TB test geometry with indication of the middle part where results on damage zone extents are displayed, c) specimen dimensions 3. NUMEICAL STUDY The investigation of issues related to the damaged material extent have been conducted on records of virtual tests on notched beams of two sizes subjected to three-point bending. Three material sets differing in the cohesive behaviour were included in the study, which is a reason why two considerably different sizes of specimens were considered the shape of cohesive law influences the structural response of small and large beams in different way. The virtual experiment was carried out using commercial software (Červenka et al. 2005). 3.1 Models The scheme of the test geometry is shown in fig. 1b; the dimensions of the considered beams are tabulated in fig. 1c. The mid-span relevant parts of the geometrical models corresponding to the used computational tools are depicted in fig. 2. The mesh density for 262
2nd International ILEM Conference on Strain Hardening Cementitious Composites 12-14 December 2011, io de Janeiro, Brazil both the small and large beam within the either simulation technique was used constant: the element size of 8 mm in the case of FE models, the average mass points distance of cca 5 mm in the case of FyDiK models, and the element size of 9 mm for models. 2D W = 160 mm W = 640 mm FyDiK W = 160 mm W = 160 mm W = 640 mm W = 640 mm Fig. 2: Meshes of mid-span parts of the FE model, FyDiK lattice-particle model, and FE model (symmetrical half) for the small and large TB specimen, respectively 1 load (kn) BILINEA 6.0 4.0 0 5 0.10 0.15 0.20 0.25 crack opening, w (mm) 0.30 8.0 load (kn) 4.0 5 0.10 0.15 0.20 0.25 crack opening, w (mm) 8.0 B 1 D 4.0 5 0.10 0.15 0.20 0.25 crack opening, w (mm) 0.30 C D 0.2 0.4 0.6 0.8 0.5 1.0 1.5 45.0 1 3 5.0 15.0 0.5 1.0 1.5 3 TILINEA#2 FyDiK A 2 C 2.5 0.30 6.0 0 FyDiK B 5.0 15.0 TILINEA#1 6.0 0 SECIMEN W = 640 mm 3 A 7.5 load (kn) coh. stress, σ (Ma) coh. stress, σ (Ma) coh. stress, σ (Ma) SECIMEN W = 160 mm 8.0 1.0 3.0 8 2 6 4 1 2 0.5 1.0 1.5 4.0 6.0 Fig. 3: Cohesive functions considered in the simulations (left); comparisons of the d diagrams simulated by different used models for the corresponding cohesive functions and specimen sizes, respectively (middle and right) 3.2 Materials Three materials differing only in cohesive behaviour, see fig. 3 left, were considered: i) a common quasi-brittle material with bilinear cohesive function, ii) a material of the sort of fibre reinforced composites with trilinear cohesive function, and iii) a strain hardening fibrereinforced material with trilinear cohesive function. Their other mechanical/physical properties are considered in a way corresponding to the material definition for the software. In this software, a set of mechanical, fracture and physical parameters was generated for a concrete of cubic compressive strength f cub = 85 Ma. This material set was then held as the reference set. The basic parameters of this set are as follows: specific gravity ρ = 2400 kg/m 3, modulus of elasticity E = 44.5 Ga, oisson s ratio ν = 0.2, tensile strength f t = 5.17 Ma, and compressive strength f c = 85.0 Ma. The input parameters were considered the same as in the reference set; the above-defined cohesive functions were employed for the tensile failure behaviour. 263
2 nd International ILEM Conference on Strain Hardening Cementitious Composites 12-14 December 2011, io de Janeiro, Brazil FyDiK A simple multi-linear force elongation function with linear descending branch in tension has been utilized (fig. 1a). This function is set to correspond to the initial descending branch of the cohesive functions used for the simulations. Therefore, the results of the FyDiK simulation can be partially compared only with the simulations with the bilinear cohesive function. Any (reasonable) non-linear dependence can be generally used. However, such models are not implemented in FyDiK code yet. The numerical model was created from a homogeneous elastic isotropic material with parameters corresponding to the reference material set (used for simulations). 3.3 esults The d diagrams from the simulations are regarded as records of experimental test in this study. As such, they are used as inputs into the efraro technique for the estimation of the FZ and CDZ extents. Other inputs into the procedure are: 4 terms of the Williams series, ankine failure criterion, and the corresponding cohesive function. The d diagrams from the virtual tests are plotted in fig. 3 in the middle and right column for the small and large specimen, respectively. The FZ extents (including the indication of the cohesive stress distribution, see the scale in Ma) in 4 stages of the fracture process (marked by letters A, B, C and D in the d curves in the top row in fig. 3) in the specimens made of material with bilinear cohesive function are shown in fig. 4. The extents of the CDZ for both small and large beams and all considered cohesive functions are depicted in fig. 5. Fig. 5 contains also the results of FyDiK simulations the representations of the CDZ, i.e. the cumulative number of failure events (including the indication of the intensity of the energy dissipation, see the scale in mj). Fig. 4, on the other hand, contains also the FyDiK interpretations of the FZ, i.e. the union of the failure events within the cohesive function impact. The failure events with low energy release (less than 2 mj) are filtered. The d diagrams from the FyDiK and the, simulations are compared to the ones in fig. 3. In fig. 5, the CDZ estimated by the efraro technique can be compared also to the interpretations of the same phenomena obtained using the software (displayed as areas with cracks) and the code (displayed as area, where principal stress exceeds 75% of tensile strength). 4. DISCUSSION AND CONCLUSIONS The paper presents an analysis aimed at the verification of the (semi-)analytical technique for estimation of the extent (size and shape) of the FZ and the CDZ in quasi-brittle silicatebased composites during tensile failure. The method was applied on virtually tested materials exhibiting also rather ductile, even strain hardening, cohesive behaviour. The FZ extents estimated using the efraro and FyDiK approach are in reasonable agreement, both in the characteristics of the size and shape and the parameter describing the failure (the intensity of cohesive forces and energy dissipation, respectively). The abovementioned feature of the FZ shape reconstruction, i.e. the scaling of the FZ width by the relative cohesive stress value, works well here. The extent of the overall failure zone estimated using the efraro technique agrees with the FE and lattice-particle model simulations quite well, however only for the bilinear cohesive function. In the cases of the trilinear ones the width of the damage zone is significantly overestimated, which is caused by the scaling technique. So, this method is inappropriate in these cases. The area of the damaged material 264
2 nd International ILEM Conference on Strain Hardening Cementitious Composites 12-14 December 2011, io de Janeiro, Brazil resulting from the ATENA simulation is considerably narrower and of different shape than the ones estimated by the efraro method and the FyDiK and simulations. The used FyDiK model, although somewhat simplified in aspects regarding the cohesive function, providedd a fairly good approximation of both the current and cumulative failure volume, including their energy consumption. esults of such simulations can serve as a significant source of new knowledge which can be obtained regarding the energy dissipation mechanisms in the FZ in the case of quasi-brittle materials. For its application to fibre- reinforced/strain hardening composites other cohesive laws have to be implemented. Fig. 4: FZ extents estimated by the efraro method and the FyDiK model with indication of the cohesive stress intensity and the energy dissipation intensity, respectively, for 4 stages of the fracture process and two specimen sizes (the middle part of the beam is displayed) Nevertheless, it should be noted that the conclusions coming out from the comparisons are achieved via different quantities and should be regarded as provisional. Subsequent analyses of the results are planned. This research shall include studies regarding the cohesive stress distribution over the FZ, with which the energy dissipation is connected. 265
2nd Internatioonal ILEM M Conferencce on Strain Hardening Cementitiou C us Compositees 12-144 Decemberr 2011, io de d Janeiro, Brazil B Fig. 5: 5 CDZ exteents estimatted by the A simulaation, the efraro meethod, the FyDiK F model, and the B s simulation f three coh for hesive functtions and tw wo specimen n sizes OWLEDG GEMENTS ACKNO This ouutcome has been achieeved with the t financiaal support from f the Grrant Agency of the Academ my of Sciennces of the C (KJB 200410901), the Miniistry of Eduucation, Yo outh and Sports (ME ( 10030)) and the Czzech Scientiific Foundation (GA C 104/11//0833). ENCES EFE [1] Andderson, T.L L. 'Fracturee Mechaniccs Fundameentals and Applicationns', 2nd Ed dn (CC reess, 1995). [2] Bažžant, Z.., lanas, J. 'Frracture and size effect in i concrete and other quuasi-brittle materials' m (C C ress Bocca aton, 1998). [3] Bollander, J.E., Yoshitake, K., K Thomuree, J. 'Stress analysis a usinng elasticallyy uniform rig gid-bodyspriing networkss', J. Struct. Mech. M Earthqquake Engng g., 633(1999)) 125-132. [4] Čerrvenka, V. ett al. 'ATENA A rogram Doocumentation n' (Cervenkaa Consulting,, rague, 200 05). [5] Franntík,. 'FyD DiK applicatioon', http://ww ww.kitnarf.cz/fydik, 20077-2011. 266
Int. 2 nd International ILEM Conference on Strain Hardening Cementitious Composites 12-14 December 2011, io de Janeiro, Brazil [6] Frantík,., Veselý, V., Keršner, Z. 'Efficient lattice modelling of fracture process zone extent in nd cementitious composites', in. Iványi, B.H.V. Topping (eds.). roc. of the 2 Conf. on arallel, Distributed, Grid and Cloud Computing (Civil-Comp ress, Stirlingshire, 2011). [7] Hillerborg, A., Modéer, M., etersson,.e. 'Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements', Cem. Concr. es., 6 (1976) 773-782. [8] Nallathambi,., Karihaloo, B.L. 'Determination of specimen-size independent fracture toughness of plain concrete', Mag. Concr. es., 38 (1986) 67-76. [9] ILEM Committee FMT 50. 'Determination of the fracture energy of mortar and concrete by means of three-point bend test on notched beams', Mater. Struct., 18 (1985) 285-290. [10] Veselý, V., Frantík,. 'econstruction of a fracture process zone during tensile failure of quasi-brittle materials', Appl. Comp. Mech., 4 (2010) 237-250. [11] Williams, M.L. 'On the stress distribution at the base of stationary crack', ASME J. Appl. Mech., 24 (1957) 109-114. 267