Multilaminate and Microplane Models: Same Principles and Different Solutions for Constitutive Behaviour of Geomaterials

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1 The 12 th International Conference of International Association for Computer Methods and Advances in Geomechanics (IACMAG) 1-6 October, 2008 Goa, India Multilaminate and Microplane Models: Same Principles and Different Solutions for Constitutive Behaviour of Geomaterials F. Sánchez, P.C. Prat Dept. of Geotechnical Engineering and Geosciences, Technical University of Catalonia, Spain V. Galavi, H.F. Schweiger Computational Geotechnics Group, Inst. Soil Mech. & Foundation Engng., Graz Univ. of Technology, Austria Keywords: multilaminate models, roplane models, angular discretization of space ABSTRACT: In the last decades an important number of constitutive models based on the concept of angular discretization of space have been developed. Two of the most important families of this kind of models are referred to as roplane (MPM) and multilaminate (MLM) theories. Both frameworks represent the behaviour of the material by considering the response on several so-called integration planes or roplanes, where the yield surface and plastic potential are defined individually. Thus an initially isotropic material becomes anisotropic after loading, capturing plastic flow induced anisotropy intrinsically. Also, both frameworks allow considering inherent anisotropy as a particular case in a straightforward manner without mathematical complexity. In this paper the basics of each formulation will be explained and their similarities and differences, as well as their corresponding advantages, disadvantages, capabilities and limitations will be discussed. Some numerical simulation of simple laboratory tests will be performed to compare and illustrate the response of both formulations. 1 Introduction The original idea on which both models are based is due to Taylor (1938) who proposed that the stress-strain relation could be specified in an independent way on different planes of the material assuming that the stress vector acting on a certain plane is the resultant of the overall (macroscopic) stress tensor (static constraint), or that the strain vector is the resultant of the overall strain tensor (kinematic constraint). The first works supported in this approximation only used the static compatibility and the response of the different direction planes was related to the macroscopic level through simple superposition. Initially the formulation with static constraint was developed for polycrystalline metals by Batdorf & Budianski (1949) and Sanders (1955) and it is known as the slip theory of plasticity. This theory was seen as the best available description for the plastic behaviour of metals. Later, the model with static constraint was applied to soils and rocks under the name of multilaminate model (Zienkiewicz & Pande, 1977) and also to creep of clays (Bažant & Kim, 1986; Bažant & Prat, 1987). In soil mechanics the multilaminate concept was introduced for clay by Pande & Sharma (1983), with which they formulated a critical state model on 13 independently acting planes to account for the rotation of principal stress axes and anisotropy due to the change of stresses. Thereon many constitutive studies and mathematical proposals have been made using the multilaminate framework such as: comparisons with results from standard invariant formulations of the critical state model (Sharma, 1980 and Varadarajan et al.1990); incorporation of shear and volumetric hardening (Krajewski, 1986); a model including deviatoric hardening with a non-associated flow to model strain localisation in dense sands (Karstunen, 1999); etc. The development by Wiltafsky (2003), namely the Multilaminate Model for Clay (MMC) forms the basis for the model presented in this paper. Bažant & Oh (1983) introduced the term roplane arguing that the expression slip theory had plasticity connotations ( the model is not restricted to plastic slip ) and did not fit into the description of damage (i.e. rocracking) in quasi-brittle materials. The prefix ro was referred to the fact that the behaviour could be characterised in the different direction planes located within the rostructure (contact planes between concrete aggregates). Earlier roplane models developed at Northwestern University where originally focused to the appropriate modelling of strain softening of concrete. Many subsequent roplane models have been developed by the group of Bažant. Those have been practically applied and experimentally calibrated and verified, especially for concrete. Bažant (2000) has labelled these models as M1 through M4 (model M5 appeared in 2005). 911

2 Since the early 2000 s, roplane models follow two different directions according to the scopes of research and the type of materials to be modelled: the group of models developed by Bažant and co-workers, mainly focused on the behaviour of concrete, but also to other materials, and the models produced at the Technical University of Catalonia (Spain), that focus into the constitutive behaviour of geomaterials. The later are based on the introduction of the elastoplasticity theory within the roplane framework and forms the basis of the formulations presented here. 2 Fundamentals of the multilaminate framework Generally, the multilaminate framework establishes a simplified relationship between the ro-scale mechanical behaviour of a material and the macro-scale response. The ro-level stress (normal and tangential stresses acting on a certain plane) are obtained by means of the projection of the macroscopic stress tensor over the plane (static constraint). Yield functions are specified independently on the sampling planes, thus, during deviatoric loading, some yield surfaces are activated (those with the most favourable orientation for plastic slip) while others stay untouched. The yield function for the i th sampling plane can be written as a function of the ro level stress σ i and a set of strength parameters χ i. fi = f i( σ i, χ i) (1) According to the theory of plasticity, the plastic strain increments are proportional to the derivative of the plastic potential function (or yield function in the case of associated flow) with respect to the stresses. The increment of plastic strain dε p i of the i th sampling plane is given by: p εi = λ i( gi / σ i) (2) in which λ i is the increment of the ro level plastic multiplier and g i is the ro level plastic potential function. According to the additivity postulate of the theory of plasticity, the overall deformation of a soil mass is the summation of elastic and plastic deformations. Therefore, the total strain increment ε consists of the elastic strain increment ε e and the plastic strain increment ε p. Only plastic strains ε p i are calculated on sampling planes (ro-level), elastic strains ε e are calculated on global (macro-level). The increment of ro level plastic strain ε p i is only calculated for the activated sampling planes, i.e. sampling planes in which the yield function is zero (f i =0). To obtain the global plastic strain increment ε p, the contributions from all sampling planes have to be taken into account by transformation of the ro level plastic strain increment ε p i and the numerical integration over the surface of the unit sphere: np p p gi gi ε = T i ε i ds = λ T i i ds = λi i wi T (3) σ S S i i= 1 σ i S denotes the surface of the unit sphere and T i is the transformation matrix of the sampling plane i. 3 Fundamentals of the roplane framework In the roplane framework one can also establish a simplified relationship between the ro-scale mechanical behaviour and the overall response. In this case the ro-level volumetric and deviatoric stresses (σ V and σ D ) are obtained by means of their relations (elastic or elastoplastic) to the previously projected macroscopic strain tensor over the plane (utilising the kinematic constraint one obtains ε V and ε D ). Yield functions are also specified independently on the sampling planes. During deviatoric loading, some yield surfaces are activated while others stay initially untouched. However, since equilibrium of the system is achieved at the macro-level only (not in roplanes), if loading continues, stresses on roplanes continue augmenting so that successive yield surfaces are activated in different plasticity events, until a saturated, fully plastic state is achieved, i.e., all yield surfaces in all roplanes are active. The yield function for the i th roplane is also written as a function of the ro level stress σ i and a set of strength parameters χ i (eq. 1). To obtain the global stress increment σ, the contributions from all roplanes are taken into account by the numerical integration over the surface of the unit sphere: np ( k + 1) 3 σ = i σids i σiwi 2π T = T (4) S i= 1 where S denotes the surface of the unit sphere and T i is the transformation matrix of the roplanes. 4 Assumptions for comparison To compare both formulations at a very fundamental level, simplifying assumptions have to be made, i.e.: for the states of stress, only the total stress tensor will be considered; for the elastic response, only the case of linear elasticity will be assumed; finally, these comparison would be valid for any simple yield criterion. In a later section, more complex kinds of behaviour will be studied utilising higher order constitutive relations. 912

3 Figure 1. Flow diagram for a step-wise computation. 4.1 Basic parts of a step-wise calculation with the multilaminate framework Starting with an initial state of stress and deformation (step k), an increment of strain is prescribed for the step k+1; in the proposed multilaminate model an implicit return stress algorithm is utilised to calculate plastic strain increments; a trial macroscopic state of stress is calculated due to purely linear elastic behaviour; the macroscopic trial stress tensor is transformed to each plane to obtain the normal and tangential components of the ro level stress vector (σ n, τ); the trial stress is transferred back to the yield surface by means of an iterative subalgorithm which utilises plastic strain components; these values and the actual strength parameters are used to evaluate the yield condition f i ; the roscopic plastic strain increment is only calculated for activated sampling planes (planes in which f i 0). After doing the above process on all sampling planes, the macroscopic plastic 913

4 strain is calculated by means of a numerical integration over the surface of a unit sphere and a new trial stress state is obtained; the above procedure is performed until f i <0 for all sampling planes; after convergence the latest trial stress state is the macroscopic stress state (Figure 1). 4.2 Basic parts of a step-wise calculation with the roplane framework Again, from an initial state of stress and deformation (step k), prescribing an increment of strain (step k+1), in the roplane framework a loop over roplanes is started; in each plane the macroscopic strain increment tensor is projected via the projection tensors V, D (namely the volumetric and deviatoric components) and the roplane incremental strains are obtained ( ε V, ε D ); the trial state of ro stresses (σ V, σ D ) is calculated with the elastic roplane relation D e and the previous step stress components; with these values and the actual strength parameters, the yield condition f i is evaluated; if the trial state of ro stress is inside the elastic domain, calculations continue for the next plane, whereas if the yield condition is violated plasticity is solved with a stress point algorithm; once stresses converge, the components of this vector go inside the integration formula to get the corresponding macro-level stress increment components; finally the macroscopic stress increments are added to the previous step macroscopic stress components to get the new state of stress (Figure 1). 5 Constitutive relations 5.1 Multilaminate model The multilaminate formulation used in this paper is based on the developments of Wiltafsky (2003) who derived a model for the application to normally consolidated (or slightly overconsolidated) clays, Scharinger (2007) who introduced small strain stiffness behaviour and Galavi (2007) who enhanced the model to cover inherent anisotropy, destructuration and strain softening. The yield criterion of the Multilaminate Model for Clay (MMC) is defined independently for every sampling plane and in the case of isotropy has exactly the same shape in all planes. All mathematical equations presented are defined at the level of a representative sampling plane. It should be noted that the continuum mechanics sign convention is used, i.e. tensile stresses and strains are positive Yield function, hardening law and plastic flow The yield function of the MMC consists of three independent functions (f d, f v and f t ) of effective stress components, namely the deviatoric, volumetric and tension parts of the envelope curve, respectively (Figure 2). The deviatoric yield function f d (Figure 2) is an extended Mohr-Coulomb criterion, which introduces a mobilized friction angle ϕ mob. p ε ( ) tan ϕ = tanϕ + tanϕ tanϕ mob i mod i p εγ, d where ϕ i, ϕ peak and ϕ mod are initial, ultimate and modified effective friction angles respectively, A mat is a parameter that governs the rate of deviatoric hardening and R f = tan ϕ peak /tan ϕ mod is a failure ratio. Mobilization of friction angle is controlled by plastic shear strains on the deviatoric part of yield function ε p γ,d. γ, d + Amat (5) Figure 2. Yield curve and failure criterion on a sampling plane. In the volumetric part of yield function f v (Figure 2), σ' nc is the effective preconsolidation stress state resulting from the loading history on the respective sampling plane (Equation. 6) and M α governs shape of f v. p Κ εnv, nc σ nc, i e σ = (6) The hardening parameter Κ is defined as Κ = ( 1 + e)/( λ κ ). λ and κ denote the compression and swelling 914

5 index obtained from the ln p e diagram, respectively. The third part of the yield surface f t is a tension cut-off criterion (Figure 2). In the proposed model, a non-associated flow rule is assumed for f d and an associated flow rule for f v and f t. The plastic potential function of the deviatoric yield surface is defined by: gd = τ + σ n tanψmob (7) where p sinϕ sin sin sin sin mob ϕ cv ϕ ϕ mob peak ψ sinψmob = with sinϕ cv = (8) sinϕmob sinϕcv sinϕpeak 1 1 sinϕ peak sinψ This formulation corresponds to the theory proposed by Rowe (1962) and modified by Søreide et al. (2002). 5.2 Microplane model The roplane formulation used in this paper is based on the developments of Sánchez and Prat (2004, 2005) and Sánchez & González (2005). As proposed in those papers, in the case of roplanes the only way to solve the elastoplastic return at the level of planes in a consistent way is to define the strain projections as Volumetric and Deviatoric (V-D formulation) instead of using the normal-tangential projections (N-T formulation) of traditional roplane models. The main reason is that the V-D formulation allows defining thermodynaally consistent elasticity moduli for the elastic stress-strain relation over the planes (matrix D e ). On the other hand, the strength definitions, i.e. the yield or failure criterions are still defined in the normal-tangential stress space although solved elastoplastically at the volumetric-deviatoric stress space. The later is fundamental for having a dependency of strength on the Lode s angle at the macroscopic level, which is common for constitutive models used in geotechnics Definitions of strain stress and elasticity relations at the level of roplanes. The expressions to obtain the roplane strain quantities from the projection of their macroscopic counterparts are: δij ε εij ε ε r ijdrij ε 1 1 V = ε: V = ; εd = ε: D D = = ij ( δrinj + δrjni ) δijnr (9) Then, defining the consistent roplane stressesσ V andσ D r as work conjugates of strains ε V and ε D, r and according to the fundamental assumption of Carol et al. (2001) that the macroscopic free energy may be written as the integral of the free energy at the roplane level, one gets the moduli: E E EV = K ED G ν = 3 ; = + ν = 2 (10) where K and G are the macroscopic bulk and shear modulus respectively. For linear elasticity one can establish the simplest explicit constitutive relation, which assumes that each stress component depends exclusively on its corresponding strain conjugate, i.e. σ = ε E ; σ = ε E V V V Dr Dr D Yield function As in the case of the multilaminate formulation, this roplane model includes a three surface yield envelope defined at the σ N -τ stress space (Figure 3). The deviatoric yield function f d is a parabolic expression that accounts for the critical state defined by the point ( σcrit, τcrit ) and is defined in terms of the Mohr-Coulomb line ( c, ϕ ) at the level of roplanes. The other parameter involved in the equation for f MC d is expressed as: ref τ crit c A = 2 (11) Note that the cohesive parameter c is not the same as the Mohr-Coulomb parameter c MC.The tensile cut-off (f t ) and the yield cap (f v ) surfaces are defined in terms of the parameters of f d in such a way that during hardening-softening all three surfaces evolve together. σ crit 915

6 Figure 3. Yield curve and failure criterion on a roplane. For the three surfaces an associated flow is assumed so the plastic potentials are the same as the yield surfaces. Note that, although the yield criterion is defined in the σ N -τ stress space, plasticity is solved at the volumetricdeviatoric space, which implies that the yield functions have to be projected from the N-T to the V-D space, having different shapes for differently oriented roplanes (Sánchez & González, 2005). For the cases presented in this paper no hardening or softening rules will be considered. 6 Comparison of stress strain behaviour In this section a simple triaxial test for a hypothetical geotechnical material will be performed with both formulations, using a confinement pressure of σ conf = 50 kpa. Since the present multilaminate model is more suitable for soils and the roplane model is developed for rock, an intermediate material was defined, a sample with characteristics between a hard soil and a soft rock. In both cases a plane configuration of 33 planes according to Bažant & Oh (1986) was utilised. The parameters of the multilaminate model are: initial cohesion c ini = 33 kpa; peak cohesion c peak = 50 kpa; initial friction angle ϕ i = 19º; peak friction angle ϕ peak = 27.5º; ψ = 9 ; p = 0; λ = and κ = ; Poisson ratio ν = 0.20, deviatoric hardening parameter A mat = The parameters of the roplane model are: roplane Mohr-Coulomb cohesion c = 33 kpa; roplane Mohr-Coulomb friction MC angle φ = 19º; volumetric consolidation stress X MC V = 510 kpa; Young s modulus E = 40,000 kpa; Poisson ratio ν = Note that roplane parameters act like initial parameters (those that define the onset of elastoplastic behaviour). These parameters coincide with the ones used as initial parameters in the multilaminate model and they produce a peak response similar to the multilaminate model, according to what is explained in Sánchez & Prat (2008). Figure 4a shows the results of the stress strain behaviour, while in Figure 4b the dilatant response of the models is plotted. Note that in the case of the roplane formulation hardening and dilatancy are generated naturally by the model (without the introduction of a flow rule), so the multilaminate response had to be adjusted through the calibration of parameters A mat and exponent p from the modified stress-dilatancy theory (Eq. 8). 7 Comparison of anisotropic response It has been previously mentioned that inherent anisotropy can be introduced without mathematical complexity as a particular characteristic of the material. For the present comparison anisotropy will be considered in terms of strength by introducing different values for the peak cohesion and friction angle on the planes. The mathematical formulation corresponds to that one proposed by Pietruszczak & Mroz (2000) who consider that in rostructure models the strength of soil at any direction can be modelled by a so called rostructure tensor. The rostructure tensor A can be considered as a measure of material fabric. It can describe the spatial distribution of material parameters, for example spatial distribution of voids or cracks or arrangement of inter-granular contacts (Galavi, 2007). This approach has been used by Pietruszczak & Pande (2001) and with some changes by Galavi (2007) for studying the capabilities of the multilaminate model. The anisotropy function based on the rostructure tensor A is expressed, for the case of cross anisotropy, by the following equation: Ar αu α 1 ( uv ) 2 = (12) Ar

7 where Ar = αv / αh is called anisotropy ratio (Eq. 13) and α v and α h are the values of any parameter associated with the vertical and horizontal directions respectively. α u will be the value for a certain plane, while α 0 is the mean value of this parameter α 0 = (α v +2α h )/3 and u v is the vertical component of unit vector u at the direction of the shear stress vector on the respective plane. This function is applied to the friction angle and the cohesion of both models for this comparison. The trace of the failure surface in the deviatoric plane for both formulations is depicted in Figure 5 for three values of A r, namely A r = 0.5, 1.0 (isotropic behaviour) and 2.0. Note that the response of both models is very similar. Therefore one can conclude that for this simple concept the two formulations are basically equal. However, in the case of the roplane model, these results correspond with the onset of the plastic behaviour; while in the multilaminate model they correspond with the failure (hardened) state, considering that the roplane parameters are the same as the peak parameters of the multilaminate model. If the roplane model is driven to peak strength, the result will show a very different shaped failure surface, basically much more rounded. Figure 4. Results from the simulation of a drained triaxial test. Figure 5. Failure surfaces in the deviatoric plane for different values of A r and for both models. 8 Discussion on similarities and differences In this paper the basic principles of the roplane (MPM) and multilaminate (MLM) models were presented. Both derive from the concept of angular discretization of space into planes of various orientations and use the projec- 917

8 tion of macroscopic tensorial quantities on the planes. They take the advantage of solving the elastoplastic constitutive relations on each plane, and after similar integration procedures recovering the macroscopic constitutive stress-strain response. This allows modelling advanced features of the material behaviour. The main difference is that MLM use the projection of macroscopic stresses to calculate plastic strains on the planes, and recover the macroscopic stress strain relation from the integration of such plastic strains, whereas MPM use the projection of macroscopic strains to calculate the roplane stresses, and recover the macroscopic stress strain relation from the integration of the roplane stresses. The main implication of this difference is that in MPM successive events of plasticity occur until all the roplanes are under plastic flow, whereas in the MLM, near failure, plasticity concentrates on one or a few planes while the rest of the planes remain elastic. Thus, in the MLM, one needs to define a hardening rule and a plastic potential function with dilatancy control, while in MPM hardening and dilatancy happen in a natural fashion. However, because of that, with MPM the shape of the yield surfaces has to be controlled as a function of the orientation with respect to the macroscopic principal stresses making these models more complicated. 9 References Batdorf, S. & Budiansky, B. (1949). A mathematical theory of plasticity based on the concept of slip. Technical Note 1871, National Advisory Committee for Aeronautics. Bažant, Z.P. & Oh, B.H. (1983). Microplane model for fracture analysis of concrete structures. Proc. Symp. on the Interaction of Non-Nuclear Munitions with Structures. U.S. Air Force Academy, Colorado Springs, Bažant, Z.P. & Oh, B.H. (1986). Efficient Numerical Integration on the Surface of a Sphere. Zeitschrift für angewandte Mathematik und Mechanik (ZAMM), 66: Bažant, Z.P. & Kim, J.-K. (1986). Creep of anisotropic clay: roplane model. ASCE J. Geotech Engrg. 112(4): Bažant, Z.P. & Prat, P.C. (1987). Creep of anisotropic clay: new roplane model. ASCE J. Engrg. Mech. 103(7): Bažant, Z.P., Caner, F.C., Adley, M.D. & Akers, S.A. (2000). Microplane model M4 for concrete. I: formulation with work conjugate deviatoric stress. J. Eng. Mech. ASCE 126(9): Carol, I., Jirásek, M. & Bažant, Z.P. (2001). A thermodynaally consistent approach to roplane theory: Part I. Free energy and consistent roplane stresses. Int. J. Solids and Structures. 38: Galavi, V. (2007). A multilaminate model for structured clay incorporating inherent anisotropy and strain softening. PhD thesis, TU Graz, Austria, Hest 32. Karstunen, M. (1999). Numerical modelling of strain localization in dense sands. Acta Polytechnica Scandinavia, The Finnish Academy of Technology, Espoo, Finland, No Krajewski, W. (1986). Mathematisch-numerische und experimentelle Untersuchung zur Bestimmung der Tragfähigkeit von in Sand gegründeten, vertikal belasteten Pfählen. Veröff. des Institutes für Grundbau, Bodenmechanik, Felsmechanik und Verkehrswasserbau der RWTH Aachen, Germany, Vol. 13. Pande, G.N. & Sharma, K.G. (1983). Multilaminate model of clays a numerical evaluation of the influence of rotation of principal stress axes. International Journal of Numerical and Analytical Methods in Geomechanics, 7(4): Pietruszczak, S. & Mroz, Z. (2000). Formulation of anisotropic failure criteria incorporating a rostructure tensor. Computers and Geotechnics, 26: Pietruszczak, S. & Pande, G.N. (2001). Description of soil anisotropy based on multilaminate framework. International Journal of Numerical and Analytical Methods in Geomechanics, 25: Rowe, P.W. (1962). The stress-dilatancy relation for static equilibrium of an assembly of particles in contact. Proc. Royal Society of London, Mathematical and Physical Sciences, Series A, 269: Sánchez, F. & González, N.A. (2005). Elastoplasticity within the framework of roplane models. Part II: Applicable models for their use in geotechnical analyses. In IACMAG-11. Torino, Italy. Balkema, 1: Sánchez, F. & Prat, P.C. (2004). Constitutive roplane model for geotechnical materials. Cyclic Behaviour of Soils and Liquefaction Phenomena. Triantafyllidis, Th. (editor), Balkema, Netherlands, Sánchez, F. & Prat, P.C. (2005). Elastoplasticity within the framework of roplane models. Part I: Basic concepts. In IACMAG-11. Torino, Italy. Balkema, 1: Sánchez, F. & Prat, P.C. (2008). A multiple plane plasticity model for rock materials. Part I: definitions of strength. Proceedings IACMAG-12, Goa, India. Sanders, J.L. (1955). Plastic stress-strain relations based on linear loading functions. Proc. 2 nd United States National Congress on Applied Mechanics, ASME, Scharinger, F. (2007). A multilaminate model for soil incorporating small strain stiffness. PhD thesis, TU Graz, Austria, Heft 31. Sharma, K.G. (1980). Numerical models of soils under monotonic and cyclic loading. PhD thesis, U. of Wales, Swansea, UK. Søreide, O.K., Nordal, S. & Bonnier, P.G An implicit friction hardening model for soil materials. Proc. 5 th Europ. Conf. on Numerical Methods in Geotechnical Engng (NUMGE), Mestat (ed.), Paris, France. Presses de l ENPC/LCPC, Taylor, G.I. (1938). Plastic strain in metals. Journal of the Institute of Metals, Vol. 62, Reprinted in: The Scientific Papers of G.I. Taylor 1, 1958, Cambridge University Press, Cambridge (UK). 918

9 Varadarajan, A.; Sharma, K.G. & Pande, G.N. (1990). Prediction of anisotropic behaviour of clays by the multilaminate model. Proc. 2nd European Conference on Numerical Methods in Geomechanics, Santander, Spain. Wiltafsky Ch. (2003). A multilaminate model for normally consolidated clay. PhD thesis, TU Graz, Austria, Heft 18. Zienkiewicz, O.C.& Pande, G.N. (1977). Time-dependent multilaminate model of rocks A numerical study of deformation and failure of rock masses. International Journal for Numerical and Analytical Methods in Geomechanics, 1(3):