AN ORTHOTROPIC CONTINUUM MODEL FOR THE ANALYSIS OF MASONRY STRUCTURES

Transcription

1 Delft University of Technology Faculty of Civil Engineering AN ORTHOTROPIC CONTINUUM MODEL FOR THE ANALYSIS OF MASONRY STRUCTURES Author : P. B. LOURENÇO Date : June 1995 TU-DELFT report no TNO-BOUW report no. 95-NM-R712 TNO Building and Construction Research Computational Mechanics

2 Summary A continuum model for the analysis of masonry structures subjected to in-plane loading is proposed. The model combines anisotropic elastic behaviour with anisotropic plastic behaviour. The proposed composite yield surface includes a Hill type failure criterion and a Rankine type failure criterion. Different uniaxial strengths and post-peak behaviour are predicted by the model along the material axes both in tension and compression. The formulation of the elastoplastic algorithm is made in modern plasticity concepts, including implicit Euler backward return mapping schemes and consistent tangent operators for all regimes of the model. The problem of localization is tackled, in an engineering way, by the introduction of an equivalent length h related to the element size. The performance of the implementation and behaviour of the model is assessed by means of single element tests. A comparison between numerical results and experimental results available in the literature shows good agreement both for ductile and brittle failure modes, providing that the size of the structure is large enough to permit a macro-modelling strategy. Acknowledgements The author wishes to express his gratitude to Dr. ir. J.G. Rots and Dr. ir. P. Feenstra for their support during the course of this work. The financial support by the Netherlands Technology Foundation (STW) under grant DCT is gratefully acknowledged. The calculations have been carried out with the Finite Element Package DIANA of TNO Building and Construction Research on a Silicon Graphics Indigo R4 workstation of the Delft University of Technology.

3 TNO-95-NM-R INTRODUCTION An accurate analysis of masonry structures in a macro-modelling (or composite) perspective requires a material description for all stress states. The difþculties are, however, quite strong. This is due, not only, to the fact that almost no comprehensive experimental results (including pre- and post-peak behaviour) are available, but also to intrinsic difþculties in the formulation of orthotropic inelastic behaviour. It is noted that a representation of an orthotropic yield surface in terms of principal stresses or stress invariants is not possible. For plane stress situations, which is the case of the present report, a graphical representation in terms of the full stress vector ( x, y and τ xy ) is necessary. The material axes are assumed to be deþned by the bed joints direction (x direction) and the head joints direction (y direction). In some of the pictures shown below, the yield surface and the experimental results are plotted in terms of principal stresses and a angle θ. The angle θ measures the rotation between the principal stress axes and the material axes. Clearly, different principal stress diagrams are found according to different values of θ. Tw o different strategies for the macro-modelling of masonry can be used, namely: Extend the conventional formulation of isotropic quasi brittle materials in order to describe orthotropic behaviour. Current approaches consider different inelastic criteria for compression and tension. A possible extension of conventional models is to use a Hill type yield criterion for compression and a Rankine type à yield criterion for tension (see Fig. 1). This approach will be discussed in this report; Describe the material behaviour with a single yield criterion. The Hoffman yield criterion is quite ßexible and attractive to use, see Schellekens and de Borst (199) and Scarpas and Blaauwendraad (1993), but yields a non-acceptable Þt of the masonry experimental values (see Fig. 2). A least squares Þt of the experimental results from Page (1981) with a Hoffman type yield criterion turns out to show no tensile strength in uniaxial behaviour and a manual Þt through the different uniaxial strengths plus the compressive failure obtained upon loading with 1 = 2 and θ = giv es a very poor representation of the diagrams for the other θ values. In fact a single surface Þt of the experimental values would lead to an extremely complex yield surface with a mixed hardening/softening rule in order to describe properly the inelastic behaviour. The author believes that this strategy is practically non-feasible. ( ) The word type is used here because the original authors, see Hill (1948) and Hoffman (1967), assumed a 3-dimensional formulation. The inßuence of the out-of-plane direction is generally unknown and will be not considered in the present report. The proposed yield surface for compression should in fact be considered as a particular case of the complete quadratic formulation from Tsai and Wu (1971). (à) The word type is used here because the Rankine yield criterion represents the material strength along the maximum principal stress. For an anisotropic material such deþnition is clearly not possible. The proposed yield surface for tension represents only a Þt of the experimental results.

4 TNO-95-NM-R θ θ θ = b) θ = 22.5 c) θ = 45. Fig. 1 - Comparison between a Hill type + Rankine type composite yield surface and experimental results from Page (1981) Through axes + biaxial compression fit Best fit with least squares Through axes + biaxial compression fit Best fit with least squares 2 1 θ 2 1 Through axes + biaxial compression fit Best fit with least squares 2 1 θ 2 1 θ = b) θ = 22.5 c) θ = 45. Fig. 2 - Comparison between a Hoffman type single yield surface and experimental results from Page (1981) The composite yield surface to be presented features anisotropic behaviour in tension and compression as well as non-isotropic softening. The formulation of the model is given in modern plasticity concepts including fully implicit Euler backward return mapping, a local Newton-Raphson method to solve the return mapping, proper handling of the corners and tangent operators consistent with the integration of the update equations for all modes of the model, including the apex and the corner regimes. The application of the model is limited, at present, to a plane stress conþguration. The model is implemented in the DIANA Þnite element package. For the purpose of compatibility with the current code and for simpler future extension of the model to a 3-dimensional stress-state, the formulation will be given in 4 stress components (i.e. plane strain). The expansion/compression mechanism from 3 stress components to 4 stress components is described in de Borst (1991) and will not be reviewed here. Only a few authors tried to develop speciþc macro-models for the analysis of masonry structures, in which anisotropic elasticity is combined with anisotropic inelastic behaviour. To the knowledge of the author of this report only Dhanasaker et al. (1985,1986) and Seim (1994) dealt with the implementation of a speciþc numerical model for masonry. Both of these authors fail to include rationally softening in the model: brittle softening was included for tension, which leads to mesh sensitive results and numerical instabilities, and compressive softening was either absent or brittle. Moreover, the numerical analyses of masonry walls carried out by these authors included interface elements in the boundaries. This yields a weak assessment of the material macro-model because the interface elements were responsible for most of the inelastic behaviour. Finally, the complex yield

5 TNO-95-NM-R surfaces suggested by the above authors almost preclude the use of modern plasticity concepts. The model proposed by Dhanasaker et al. (1985,1986) for solid units masonry consists of three elliptical cones, see Fig. 3. The model is based on the experimental Þndings of Page (1981) and Þts the data extremely well, see Fig. 4. However, it is difþcult to handle this model in modern plasticity concepts. Not only the composite yield surface contains several corners and apexes but also the inclusion of ÒrealisticÓ inelastic behaviour is practically impossible - how to control the expansion/shrinkage of the yield surface? y x y xy x Fig. 3 - Yield surface for solid units masonry proposed by Dhanasekar (1986) with iso-shear stress lines Contour spacing:.1 f mx θ θ 2 1 θ = b) θ = 22.5 c) θ = 45. Fig. 4 - Comparison between DhanasekarÕs (1986) masonry yield surface and experimental results from Page (1981) The model proposed by Seim (1994) is based on the failure surface proposed by Ganz (1989) for hollow masonry units, see Fig. 5. The failure surface was derived from the theorems of Limit Analysis, assuming rigid-perfectly plastic behaviour for the masonry components. The assumptions for the material behaviour of the components include a Mohr-Coulomb yield surface for the units and a Coulomb friction law for the joints. The head joints do not feature any strength. This yield surface is even more difþcult to handle numerically in a consistent way, specially when softening is included in the model with different softening parameters for all regions. Seim (1994) assumed ideally plastic behaviour in compression and brittle tension failure.

6 TNO-95-NM-R y x y xy x Fig. 5 - Yield surface for hollow units masonry proposed by Ganz (1989) with iso-shear stress lines Contour spacing:.1 f mx Finally, it should be realised that a masonry macro-model always includes some degree of approximation. The basic features of a two-material composite cannot be reproduced but only smeared out in the continuum. The Þeld of applications of these models are indeed large structures where the state of stress and strain across a macro-length can be assumed uniform. It is noted that, due to the difþculties of carrying out experiments in large structures, the examples used in the present report for the assessment of the model performance (extracted from available literature) are, in general, ÒsmallÓ for a macro-modelling strategy. The model proposed in the present report seems however capable of reproducing the global behaviour of the analysed structures. Satisfactory predictions of collapse loads are also found provided that the structures do not show a highly localized failure mode. In such cases the interaction between units and mortar can be of capital importance and a micro-model, in which both masonry components are modelled separately, should be used instead.

7 TNO-95-NM-R TENSION - A RANKINE TYPE ANISOTROPIC YIELD SURFACE In this section a possible extension of the standard Rankine yield criterion to an orthotropic formulation is given. It is clear that the yield surface obtained cannot be derived from the material strength in the maximum principal stress direction. The proposed yield surface has to be regarded as pure curve Þtting from existing experimental results. The yield surface is coined as Rankine type as the derivation is based on the original Rankine yield surface. The difþculties of formulating the Rankine yield criterion in the principal stress state are addressed, for example, in Feenstra (1993). Consider a plane-stress situation in which the major principal stress 1 and the minor principal stress 2 are deþned by means of a MohrÕs circle as 1, 2 = x + y 2 2 ± x y 2 + τ 2 xy. (1) The hardening behaviour is assumed to be described by two internal variables κ 1 and κ 2 which govern the corresponding principal stresses. The yield functions are then given by the principal stress j and an equivalent stress j as a function of an internal variable κ j according to f I = x + y 2 f II = x + y x y x y 2 + τ 2 xy I (κ I ) + τ 2 xy II (κ II ) The yield function is depicted in Fig. 6 in the principal stress space. The problem which occurs with the two yield functions is that the transformation between the stress space and the principal stress space has to be deþned uniquely for two yield functions with different hardening models. 2. (2) 1 Fig. 6 - Rankine yield surface in the principal stress space In Feenstra (1993), the formulation is given by a single function which is governed by the Þrst principal stress and one equivalent stress which describes the softening behaviour of the material. The assumption of isotropic softening is not completely valid for a material such as concrete or masonry which can be loaded to the tensile strength even if in the perpendicular direction the strength has

8 TNO-95-NM-R712 been reduced due to softening of the material. This problem is partially solved in Feenstra (1993) by using kinematic softening such that the yield surface is shifted in the direction of the Þrst principal stress. It is noted that the above formulation of kinematic softening is also not quite realistic: let us assume that the material is loaded initially along a certain direction until softening is completed. If now the material is loaded in a direction orthogonal to the crack previously open, ideal plastic behaviour is found. This is due to the fact that all the fracture energy has been consumed during the opening of the Þrst crack. An elegant solution is found if two independent softening parameters control the shifting of the yield surface. Such a formulation for the Rankine yield surface is given in Louren o et al. (1995) and reproduces exactly the material feature in tension just described. It is shown by Louren o et al. (1995) that the response of the model seems to lie between the Þxed and rotating crack models and, therefore, comprises the beneþt of a model with memory and a ßexible shear response. Unfortunately, for certain values of the trial stress the return mapping becomes illposed and an almost singular Jacobian is found close to the solution. This precludes the use of such a yield surface in large scale computations due to the lack of robustness and need to consider extremely small steps. Here a different approach is used aiming at an orthotropic Rankine type yield surface controlled by only one scalar that measures the amount of softening simultaneously in the two material axes but, still, two corresponding different fracture energies are considered. This approach is less attractive from a physical point of view but leads to a more robust algorithm and should be preferred in practice. The Rankine yield surface reads, cf. eqs (1,2), f 1 = x + y x y 2 This expression can however be rewritten as f 1 = ( x t (κ t )) + ( y t (κ t )) 2 + τ 2 xy t (κ t ). (3) 2 ( + x t (κ t )) ( y t (κ t )) 2 + τ 2 xy, (4) where coupling exists between the stress components and the equivalent stress. Setting forth a Rankine type yield surface for an orthotropic material, with different tensile strengths along the x, y directions, see Þg. 7, is now straightforward if eq. (4) is modiþed to f 1 = ( x t1 (κ t )) + ( y t2 (κ t )) 2 2 ( + x t1 (κ t )) ( y t2 (κ t )) 2 + ατ 2 xy. (5) Note that a parameter α is introduced to calibrate the shear strength. The parameter α reads α = f tx f ty τ 2 u, (6) where f tx, f ty and τ u are, respectively, the uniaxial tensile strengths in the x and y directions and the pure shear strength. Note that the material axes are now Þxed and it shall be assumed that all stresses and strains for the elastoplastic algorithm are given in the material reference axes.

9 TNO-95-NM-R τ xy τ u x f tx f ty y Fig. 7 - Orthotropic Rankine type yield surface (plotted for τ xy ) Eq. (5) can be recast in a matrix form as f 1 = ( 1 / 2 ξ T P t ξ ) 1 / / 2 π T ξ, (7) where the projection matrix P t reads 1/ 2 1 / 2 P t = 1 / 2 1/ 2, (8) 2α the projection vector π reads π = { 1, 1,, } T, (9) the reduced stress vector ξ reads ξ = η (1) and the back stress vector η reads η = { t1 (κ t ), t2 (κ t ),, } T. (11) Exponential tensile softening is considered for both equivalent stress-equivalent strain diagrams, with different fracture energies (G fx and G fy ) for the yield values, and reads t1 = f tx exp h f tx κ G t fx and t2 = f ty exp h f ty G fy κ t. (12) Here, the scalar κ t controls the amount of softening and the equivalent length h, see Baºant and Oh (1983), is assumed to be related to the area of an element A e by, see Feenstra (1993), h = α h A e = α h n ξ Σ n η ξ = 1 η = 1 1 /2 Σ det(j) w ξ w η, (13) in which w ξ and w η are the weight factors of the Gaussian integration rule as it is tacitly assumed

10 TNO-95-NM-R712 that the elements are always integrated numerically. The local, isoparametric coordinates of the integration points are given by ξ and η. The factor α h is a modiþcation factor which is equal to one for quadratic elements and equal to 2 for linear elements, see Rots(1988). With this approach the results which are obtained in the analyses are reasonably objective with regard to mesh reþnement. It is however possible that the equivalent length of an element results in a snap-back at the constitutive model if the element size is large. Then, the concept of fracture energy which has been assumed is no longer satisþed. In such a case, the strength limit has to be reduced in order to obtain an objective fracture energy by a sudden stress drop, resulting at a certain stage in brittle failure, see Rots (1988). The condition of maximum equivalent length is given by h G f E ft 2, (14) where E is the YoungÕs modulus in the respective material axis. If the condition is violated, for any of the material axes, the tensile strength in the respective axis is reduced according to f t = 1 G f E /2. (15) h It is noted that eq. (15) yields a reduction on the material strength without any physical ground. The idea is solely to obtain an energy release independent of the mesh size but this objective should be accomplished by means of a mesh reþnement and not with a strength reduction. Finally, for an orthotropic material with different yield values along the material axes, it would seem only natural to assume two different equivalent lengths along the material axes (this is of course irrelevant in the special case of a mesh with square elements). However, the equivalent length depends on so many factors that more complex assumptions are disregarded. The ßow rule is written in a standard fashion (non-associated softening) as úεúε p = ú g 1 λ t, (16) where the plastic potential g reads g 1 = ( 1 / 2 ξ T P g ξ ) 1 / / 2 π T ξ (17) and the projection matrix P g is given by 1/ 2 1 / 2 P g = 1 / 2 1/ 2. (18) 2α g The parameter α g is taken equal to the unit value (Rankine plastic ßow), unless otherwise stated. The inelastic behaviour is described by a modiþed strain softening hypothesis given by úκ t = úε p α = úε p x + úε p y / 2 (úε x p úε y) p (úγ xy) p 2. (19) α g It is noted that the above expression represents a ÒmodiÞedÓ maximum principal plastic strain, in which the shear component has been averaged by the inverse of α g. The expression for the softening scalar rate úκ t can be recast in a matrix form and reads

11 TNO-95-NM-R where úκ t = úε p α = 1 /2 1 / 2 (úεúε p ) T Q úεúε p + 1 / 2 π T úεúε p, (2) 1/ 2 1 / 2 Q = 1 / 2 1/ 2 1 2α g. (21) 2.1 Return mapping algorithm - Tension regime The integration of the constitutive equations given above is a problem of evolution that can be regarded as follows. At a stage n the total strain and plastic strain Þelds as well as the hardening parameter (or equivalent plastic strain) are known: {ε n, ε p n, κ t, n } given data. (22) Note that the elastic strain and stress Þelds are regarded as dependent variables which can be always be obtained from the basic variables through the relations ε e n = ε n ε p n and n = D ε e n. (23) Therefore, the stress Þeld at a stage n+1is computed once the strain Þeld is known. The problem is strain driven in the sense that the total strain ε n+1 is trivially updated according to the exact formula ε n+1 = ε n + ε n+1. (24) It remains to update the plastic strains and the hardening parameter. These quantities are determined by integration of the ßow rule and hardening law over the step n n+1. In the frame of a fully implicit Euler backward integration algorithm this problem is transformed into a constrained optimization problem governed by discrete Kuhn-Tucker conditions as shown by Simo et al. (1988). It has been shown in different studies, e.g. Ortiz and Popov (1985) and Simo and Taylor (1986), that the implicit Euler backward algorithm is unconditionally stable and accurate. This algorithm results in the following discrete set of equations: ε n+1 = ε n + ε n+1 n+1 = trial λ t, n+1 D g 1 ε p n+1 = ε n p g 1 + λ t, n+1 n+1 n+1, (25) κ t, n+1 = κ t, n + κ t, n+1 in which κ t, n+1 results from the integration of the rate equation, eq. (19), and the elastic predictor step returns the value of the elastic trial stress trial as trial = n + D ε n+1. (26) The above equations must be satisþed and simultaneously the yield criterion must be fulþlled

12 TNO-95-NM-R712 f 1 = ( 1 / 2 ξ T n+1p t ξ n+1 ) 1 / / 2 π T ξ n+1 =. (27) It is noted that the update of the softening scalar κ t, n+1 reduces to the particularly simple expression κ t, n+1 = λ t, n+1. (28) The above equations can be reduced to the following set of Þve equations containing 5 unknowns ( n+1 and κ t, n+1 = λ t, n+1 ) D 1 ( n+1 trial g ) 1 + λ t, n+1 = n+1 f 1 = ( 1 / 2 ξ n+1p T t ξ n+1 ) 1 / / 2 π T ξ n+1 = Due to the coupling of the n+1 and κ t, n+1 values it is not possible to obtain an explicit one variable non-linear equation. The system of non-linear equations is therefore solved with a regular Newton- Raphson method. The Jacobian necessary for this procedure reads (note that the subscript n+1 is dropped in the derivatives and matrices for convenience). (29) D 1 2 g 1 + λ t, n+1 2 J = T f 1 + g 1 + λ 2 g 1 t, n+1 κ t f 1 κ t, (3) where f 1 = P t ξ n+1 2( 1 / 2 ξ T n+1p t ξ n+1 ) 1 / / 2 π ; g 1 = P g ξ n+1 2( 1 / 2 ξ T n+1p g ξ n+1 ) 1 / / 2 π 2 g 1 2 = P g P g ξ n+1 ξ n+1p T g 2( 1 / 2 ξ n+1p T g ξ n+1 ) 1 / 2 4( 1 / 2 ξ n+1p T. g ξ n+1 ) 3 / 2 g 1 κ t = g 1 T η ; κ t 2 g 1 = 2 g 1 κ t 2 η κ t ; η = t1, t2,, T κ t κ t κ t (31) The numerical algorithm explained above is howev er not stable through all the stress domain. In the apex of the yield surface the gradient of the plastic potential, cf. eq. (31.1), is not deþned. It is further noted that the proposed plastic potential can be written in a quadratic form as g 1 = 1 / 2 ξ T P g ξ + 1 / 4 ξ T π ξ T π. (32) However, this formulation does not overcome the problem of a non-deþned gradient in the apex. The new expression for the gradient reads g 1 = P gξ + 1 / 2 π ξ T π, (33) which degenerates to a point in the apex ( g 1 = ) due to the singularity of yield surface. For the apex regime, the stress update, cf. eq. (25.2), in case of plane stress conþguration, is independent of the trial stress. It is simply a return mapping to the apex and reads

13 TNO-95-NM-R n+1 = η n+1. (34) However, the present yield surface is implemented in the expansion/compression concept for plane strain (four stress components), see de Borst (1991). This concept yields several advantages for multi-purpose Þnite elements packages but for the present yield surface no advantage is found. On the contrary, eq. (34) cannot be used because the third normal stress component is not zero during the global/local iteration procedure. In this case ( n+1 ) z must be calculated so that ( ε p n+1 ) z =. This can be easily done from and results in ( ε p n+1 ) z = D 1 ( trial n+1 ) z = (35) ( n+1 ) z = ( trial ) z + d 1 31 d 1 33 trial n+1 + d 1 32 x d 1 33 trial n+1. (36) y Here the values d 1 ij are terms from the compliance matrix D 1. The others stress components are given from eq. (34) and read ( n+1 ) x = t1 (κ t ), ( n+1 ) y = t2 (κ t ) and (τ n+1 ) xy =. (37) The above expression can be advantageously recast in a matrix format and, after some manipulations, the stress update reads n+1 = A 1 η n+1 + A 2 trial, (38) where the auxiliary matrices A 1 and A 2 are given by A 1 = 1 d 1 31 d d 1 32 d and A 2 = d 1 31 d 1 33 d 1 32 d (39) The stress update given is sufþcient to fulþll f n+1 =. It remains to update the softening scalar according to eq. (2). For this purpose a non-linear equation in one variable can be written as F = F( κ t, n+1 ) = κ t, n+1 ε p α,n+1 = 1 /2 1 / 2 ( ε p n+1 )T Q ε p n / 2 π T ε p n+1 =, (4) where the increment of the plastic strain vector ε p n+1 can be calculated from ε p n+1 = D 1 ( trial n+1 ) (41) and the update of the stress vector n+1 is given by eq. (38). The secant method is used to solve this non-linear equation instead of the regular Newton method. This has proven robust and fast, see Louren o (1994) and the Appendix A. 2.2 Consistent tangent operator - Tension regime In order to obtain quadratic convergence when making use of a Newton-Raphson iterative solving procedure at the structural level, a tangent operator consistent with the integration algorithm must be used, see Simo and Taylor (1985). For the standard part of the yield criterion, differentiation of

14 TNO-95-NM-R712 the update equations and the consistency condition (d f 1, n+1 = ) results in J d n+1 = dε n+1. (42) dλ t, n+1 Then, the consistent tangent operator is given by D ep = = J (43) ε n+1 in which J 1 4x4 is the top-left 4 4 submatrix of the inverse of J. The consistent tangent operator can also be written in other fashion by means of a condensation of the matrix J. Let us deþne the modi- Þed compliance matrix H t and the modiþed ßow direction vector γ t as H t = D 1 2 g 1 + λ t, n+1 2 and γ t = g 1 + λ 2 g 1 t, n+1. (44) κ t Condensation of the Jacobian J and the Sherman-Morrison formula yield, after algebraic manipulation, D ep = H 1 f 1 = H 1 t γ t t ε T n+1 f 1 T H 1 t H 1 t γ t f 1 κ t. (45) The consistent tangent for the apex regime is obtained from differentiation of eqs. (38,4-41). This results in the following system with I T κ t ε p D 1 p n+1 + η A 1 κ t 1 d n+1 = dκ t, n+1 A 2 D dε n+1 T κ t ε p dε n+1, (46) κ t ε p = Q ε 2( 1 / 2 ( ε p n+1 )T Q ε p + / 1 n+1 )1 2 / 2 π. (47) Let us call the matrix above A. Then, the consistent tangent operator is given by D ep = A 1 4 4A 2 D + A T κ t ε p, (48) where the A and A are submatrices of the inverse of A. The consistent tangent operator can also be written in other fashion by means of a condensation of the matrix A and reads D ep = 1 η κ = I + A t 1 ε κ n+1 t ε p D 1 η κ t A 1 κ t ε p T T + A 2 D. (49) An investigation on the performance of the numerical implementation is given in Appendix A.

15 TNO-95-NM-R Features of the model - Tension regime One plane stress element with unit dimensions is loaded under different conditions in order to discuss the behaviour of the model. Before considering orthotropic material behaviour, a Þrst example with isotropic material behaviour is presented. This is extremely relevant for a comparison with different models and an assessment of the assumptions made in the previous section Isotropic material behaviour. Tension-shear model problem The elementary problem proposed by Willam et al. (1987) introduces biaxial tension and shear loading in the element. This causes a continuous rotation of the principal strain axes after cracking, as it is typical of crack propagation in smeared Þnite element analysis. The element is subjected to tensile straining in the x-direction accompanied by lateral Poisson contraction in the y-direction to simulate uniaxial loading. Immediately after the tensile strength has been reached, the element is loaded in combined biaxial tension and shear strain, see Fig. 8. The ratio between the different strain components is given by ε x : ε y : γ xy =. 5 :. 75 : 1. The material properties are given in Table 1. Table 1 - Material properties (isotropic - α = 1.) Material properties E 1 N/mm 2 ν.2 f t 1. N/mm 2 G f.15 N.mm/mm 2 The behaviour of different crack models for this problem can be found in Rots (1988). A comparison with different smeared ÒcrackingÓ formulations (total, tangential and Rankine plasticity) can be found in Feenstra (1993). The analyses from Rots (1988) of this problem with the multi-directional crack model showed that the shear response becomes softer with decreasing threshold angle, resulting in the limiting case of the rotating crack model with zero threshold angle as the most ßexible response. The analyses from Feenstra (1993) showed that the Rankine model with kinematic hardening is in very good agreement with the rotating crack model. A comparison between the proposed model and the commonly used smeared crack models is relevant to assess the adequacy of the model to describe cracking behaviour. The results for the different stress and strain components are depicted in Fig. 9-Fig. 11. ε y ε y =-νε x γ xy y x ε x γ xy ε x a) Tension up to cracking b) Biaxial tension with shear after cracking Fig. 8 - Tension-shear model problem

16 TNO-95-NM-R712 Initial shear modulus Proposed model xy[mpa] τ.1 Fixed crack model ( =.5) β. Rotating crack model γxy[1 ] Fig. 9 - Comparison of ÒcrackingÓ models.τ xy - γ xy response 1. x [MPa] Fixed crack model Rotating crack model Proposed model εx [1 ] Fig. 1 - Comparison of ÒcrackingÓ models. x - ε x response 1. Fixed crack model y [MPa] Rotating crack model Proposed model εy [1 ] Fig Comparison of ÒcrackingÓ models. y - ε y response The main conclusions from the above results are: Ñ The shear stress-strain behaviour gives a good impression about the model because a ßexible response is obtained. This is clearly in opposition with the Þxed crack model with constant shear retention factor but is close to the rotating crack model;

17 TNO-95-NM-R Ñ The normal stress-strain response in the x-direction shows an implicit coupling between normal stress and shear stress. This is also in opposition with the Þxed crack model, where no coupling is found, but also characterizes the rotating crack model, though to a less extent; Ñ The normal stress-strain response in the y-direction shows the implicit coupling between normal stresses. This is also in opposition with the Þxed crack model, where no coupling is found, but also characterizes the rotating crack model, though to a less extent. The larger amount of coupling found in the proposed model is due to the isotropic softening Orthotropic material behaviour The orthotropic behaviour of the model is now discussed in a single element test under pure uniaxial tension. The material properties given in Table 2 are assumed, in which the y-direction is penalized by a factor 2. Two different fracture energies are considered for the y-direction: G fx /2 and 5 G fx (almost ideally plastic behaviour). Table 2 - Material properties (orthotropic - α = 1.) Material properties E x 1 N/mm 2 E y 5 N/mm 2 ν xy.2 G xy 3 N/mm 2 f tx 1. N/mm 2 f ty.5 N/mm 2 G fx.2 N.mm/mm 2 G fy Case 1 Case 2.1 N.mm/mm 2.1 N.mm/mm 2 The values chosen for the material properties conþrm the fact that completely different behaviour along the two material axes can be reproduced, see Fig. 12. In the Þrst example isotropic softening is considered. This means that the ratio of the material strength along the material axes is constant, see Fig. 13a, during any load history. It is important that this deþnition is not confounded with the deþnition of isotropy used in damaged models. Isotropic softening is related to the current yield strength values and, not necessarily, to all the components the current stress vector. When all the fracture energy is exhausted a no-tension material is recovered, see Fig. 14a. In the second example ideally plastic behaviour in the y-direction is considered. This means that the ratio of the material strength along the material axes ( f tx / f ty ) tends to zero, see Fig. 13b. The yield surface is only allowed to shrink along the x-axis, see Fig. 14b.

18 TNO-95-NM-R712 [MPa] x-direction y-direction [MPa] x-direction y-direction ε [1 ] ε [1 ] a) Case 1 (isotropic softening) b) Case 2 (ideally plastic behaviour in y-direction) Fig Stress-strain response in uniaxial tension along the two material axes / f ty f tx / f ty f tx κ [1 ] κ [1 ] a) Case 1 (isotropic softening) b) Case 2 (ideally plastic behaviour in y-direction) Fig Equivalent plastic strain vs. strength ratio along the material directions (tension regime) y Initial yield surface y Initial yield surface x x Residual yield surface Residual yield surface a) Case 1 (isotropic softening) b) Case 2 (ideally plastic behaviour in y-direction) Fig Trace of the yield surface in the plane τ xy = (tension regime)

19 TNO-95-NM-R COMPRESSION - A HILL TYPE ANISOTROPIC YIELD SURFACE In this section, a possible Þt of the experimental results in the compression regime is given, see also Fig. 1. The Hill type yield criterion here introduced is capable of reproducing different behaviour along two orthogonal material axes. The yield surface is coined Hill type because the formulation is limited to plane stress material properties. The properties in the out-of-plane direction are usually unknown and are not included in the model, in opposition to the original formulation from Hill (1948). The proposed yield surface should, in fact, be considered a particular form of the complete quadratic formulation from Tsai and Wu (1971). τ xy τ u x f mx f my y Fig The Hill type yield surface (plotted for τ xy ) The simplest yield surface that features different compressive strength along the material axes is a rotated centered ellipsoid in the full plane stress vector ( x, y and τ xy ), see Fig. 15. The expression for such a quadric is f 2 = A 2 x + B x y + C 2 y + Dτ 2 xy 1 =, (5) where A, B, C and D are four material parameters such that B 2 4AC <, in order to ensure convexity. For the numerical implementation the yield surface will be recast in a square root matrix form and the variables will be rewritten in a more amenable way. Thus, the proposed yield surface is given by f 2 = ( 1 / 2 T P c ) 1 / 2 c (κ c ), (51) where the projection matrix P c reads P c = 2 c2(κ c ) c1 (κ c ) β β 2 c1(κ c ) c2 (κ c ) the yield value c is given by 2γ, (52)

20 TNO-95-NM-R712 c = c1 c2 (53) and the single scalar κ c controls the amount of hardening and softening. The current yield stress values along the materials axes ( c1 (κ c )and c2 (κ c )) follow the inelastic law giv en below as a function of the material strength along the material axes (respectively f mx and f my as shown in Fig. 15). It is noted that the β and γ values introduced in eq. (52) are additional material parameters that determine the shape of the yield surface. The parameter β controls the coupling between the normal stress values and must be obtained from one additional experimental test, e.g. biaxial compression with a unit ratio between principal stresses. If this test is used to obtain the parameter β, the collapse load under biaxial compression ( x = y = f 45 and τ xy = ) leads to β = 1 f f 2 mx 1 f 2 my f mx f my. (54) The parameter γ, which controls the coupling between the normal stress values and the shear strength, can be obtained from γ = f mx f my τ 2 u, (55) where τ u is the material pure shear strength. Parabolic hardening followed by parabolic/exponential softening is considered for both equivalent stress-equivalent strain diagrams, with different compressive fracture energy (G fcx and G fcy ) along the material axes. The problem of mesh objectivity of the analyses of strain softening materials is a well debated issue, at least for tensile behaviour. Due to localization of deformation in a single element or row of elements the stress-strain diagram must be adjusted according to a characteristic length h to provide an objective energy dissipation. Here, the same expression for h is used as for the tension regime even if it is recognized that tensile fracture is a surface driven process and compressive failure is a volume driven process. The inelastic law shown in Fig. 16 features hardening, softening and a residual plateau of ideally plastic behaviour. The compressive fracture energy G fc (shaded area in Fig. 16) is deþned only as the non-local contribution of the stress-strain diagram. The basis for the present deþnition is only numerical, so that objective analyses with regard to mesh reþnement are obtained, see also Section 3.4. Howev er some experimental evidence exists on a local and non-local component for the total compressive fracture energy, see Vonk (1992). The peak strength value is assumed to be reached simultaneously on both materials axes, i.e. isotropic hardening, followed by anisotropic softening as determined by the different fracture energies. A residual strength value is considered to avoid a cumbersome code (precluding the case when the compressive mode falls completely inside the tension mode) and a more robust code (precluding degeneration of the yield surface to a point).

21 TNO-95-NM-R c p m i r I (κ c ) II (κ c ) G fc III (κ c ) κ p κ m κ c I (κ c ) = i + ( p i) 2κ c κ p κ 2 c κ 2 p II (κ c ) = p + ( m p ) 2 κ c κ p κ m κ p III (κ c ) = r + ( m r )exp m κ c κ m m r with m = 2 m p κ m κ p Fig Hardening/softening law for compression For practical reasons, it is assumed that all the stress values for the inelastic law are determined from the peak value p = f m as following: i = 1 / 3 f m, m = 1 / 2 f m and r = 1 / 1 f m. The equivalent plastic strain corresponding to the peak compressive strength, κ p, is assumed to be an additional material parameter. In the case that no experimental results are available, this material parameter can be calculated assuming a total peak stress equal to Then, in order to obtain a mesh independent energy dissipation the parameter κ m is given by κ m = G fc h f m + κ p. (56) To avoid a possible snap-back at constitutive lev el, the condition κ m f m E + κ p (57) must be fulþlled. Otherwise, the strength limit, in order to obtain an objective fracture energy, is reduced to f m = G fc E /2. (58) h The ßow rule is written in a standard fashion (associated softening) as úεúε p = ú f 2 λ c. (59) The inelastic behaviour is described by a work hardening hypothesis given by úκ c = 1 T úεúε p = ú λ c. (6) c 3.1 Return mapping algorithm - Compression regime The return mapping algorithm in the frame of a implicit Euler backward integration scheme is given in Section 2.1 and, for the present yield surface, results in the following set of Þve equations containing 5 unknowns ( n+1 and κ c, n+1 = λ c, n+1 ), cf. eqs. (29),

22 TNO-95-NM-R712 D 1 ( n+1 trial f ) 2 + λ c, n+1 = n+1 f 2 = ( 1 / 2 n+1p T c, n+1 n+1 ) 1 / 2 c, n+1 = This set of equations can be reduced to one non-linear equation, namely f 2 ( λ c, n+1 ) =, if the stress update is manipulated to obtain. n+1 = I + λ 1 c, n+1 DP c, n+1 n+1 trial. (62) 2 c, n+1 This approach will, however, not be followed here so that a constant framework is obtained for the several modes of the composite yield surface. Alternatively, the system of 5 non-linear equations is solved with a regular Newton-Raphson method. The Jacobian necessary for this procedure reads (note that the subscript n+1 is dropped in the derivatives and matrices for convenience) (61) D 1 2 f 2 + λ c, n+1 2 J = T f 2 + f 2 + λ 2 f 2 c, n+1 κ c f 2 κ c, (63) where f 2 = P c n+1 2( 1 / 2 T n+1 P c n+1 ) 1 / 2 ; 2 f 2 2 = P c 2( 1 / 2 T n+1 P c n+1 ) 1 / 2 P c n+1 T n+1p c 4( 1 / 2 T n+1 P c n+1 ) 3 / 2 f 2 κ c = n+1 T P c n+1 κ c 4( 1 / 2 n+1 T P c n+1 ) 1 / 2 c1 κ c c2, n+1 + c2 κ c c1, n+1 2 c, n+1 2 f 2 κ c = P c n+1 ( T P c n+1 n+1 )P c n+1 κ c κ 2( 1 / 2 n+1 T P c c n+1 ) 1 / 2 8( 1 / 2 n+1 T P c n+1 ) 1 / 2. (64) c2 P c c1, n+1 c1 κ = diag 2 c κ c c1, 2 n+1 c2, n+1 κ c,2 3.2 Consistent tangent operator - Compression regime c1 κ c c2, n+1 c2 2 c2, n+1 c1, n+1 κ c,, Differentiation of the update equations and the consistency condition (d f 2, n+1 = ) results, after algebraic manipulation, in

23 TNO-95-NM-R [MPa] D ep = d H 1 f 2 = H 1 c γ c c dε T n+1 f 2 T H 1 c H 1 c γ c f 2 κ c, (65) where the modiþed compliance matrix H c and the modiþed ßow direction vector γ c read H c = D 1 2 f 2 + λ c, n+1 2 and γ c = f 2 + λ 2 f 2 c, n+1. (66) κ c An investigation on the performance of the numerical implementation is given in Appendix A. 3.3 Features of the model - Compression regime One plane stress element with unit dimensions is loaded under different conditions in order to discuss the behaviour of the model. The material properties given in Table 3 are assumed, in which the material strength and YoungÕs modulus in the y-direction are penalized by a factor 2. Three different fracture energies are considered for the y-direction:. 3G fcx,g fcx / 2 (isotropic softening) and 5 G fcx (almost ideally plastic behaviour). Table 3 - Material properties (β = -1., γ = 3. and κ p =.5) Material properties E x 1 N/mm 2 E y 5 N/mm 2 ν xy.2 G xy 3 N/mm 2 f mx 1. N/mm 2 f ty 5. N/mm 2 G fcx.5 N.mm/mm 2 G fcy Case 1 Case 2 Case 3.15 N.mm/mm 2.25 N.mm/mm N.mm/mm 2 The orthotropic behaviour of the model is now discussed in a single element test under pure uniaxial compression. The uniaxial stress-strain responses for the different cases considered are illustrated in Fig. 17. As shown in this picture, the model is capable of reproducing different behaviour along the two material axes. x-direction y-direction [MPa] x-direction y-direction ε [1 ] ε [1 ] a) Case 1 b) Case 2 (isotropic softening) Fig Stress-strain response in uniaxial compression along the two material axes (cont.)

24 TNO-95-NM-R [MPa] x-direction y-direction ε [1 ] c) Case 3 (ideally plastic behaviour in y-direction) Fig Stress-strain response in uniaxial compression along the two material axes (contd.) Further insight on the behaviour of the model can be obtained from Fig. 18 and Fig. 19. Fig. 18 gives the ratio of the material strength along the material axes and Fig. 19 shows the trace of the yield surface in the plane τ xy =. In the Þrst case, the material strength in y-direction degrades faster than the material strength in x-direction. In the second case, degradation of the material strength in both directions occurs with the same rate and isotropic softening is obtained. Finally, in the third case, degradation only occurs in the material strength in x-direction. Note that, in the case of isotropic softening, the post-peak stress-strain diagrams under uniaxial loading conditions along the two material axes, see Fig. 17b, is not scaled by a factor 2. This is solely due to the deþnition of the softening scalar and the fact that the yield value c is not equal to the uniaxial strength along each material axis. This also means that the deþnition of the Òcompressive fracture energyó can be argued because a perfect equivalence to the stress-strain diagram is not obtained. This limitation of the model can be solved by e.g. if a unit norm is used for the plastic ßow vector and a strain softening hypothesis is used for the softening scalar. The additional difþculty introduced in the formulation ( úκ c ú λ c ) is not particularly difþcult to solve but, with the notorious lack of experimental results on the material behaviour, the initial assumptions are kept in order to simplify the implementation / f my f mx / f my f mx κ c [1 ] κ c [1 ] a) Case 1 b) Case 2 (isotropic softening) Fig Equivalent plastic strain vs. strength ratio along the material directions (compression regime) (cont.)

25 TNO-95-NM-R / f my f mx κ c [1 ] c) Case 3 (ideally plastic behaviour in y-direction) Fig Equivalent plastic strain vs. strength ratio along the material directions (compression regime) (contd.) Residual yield surface y Residual yield surface y Peak yield surface Peak yield surface x x a) Case 1 b) Case 2 (isotropic softening) Residual yield surface y Peak yield surface x c) Case 3 (ideally plastic behaviour in y-direction) Fig Trace of the yield surface in the plane τ xy = (compression regime) 3.4 About the definition of a mesh independent energy release For strain-softening materials, the need to introduce an equivalent length h in the stress-strain diagram to obtain analyses which are objective with respect to mesh reþnement is a well debated issue

26 TNO-95-NM-R712 since the original work of Baºant and Oh (1983). As stated in Section 2., in the present work h is assumed to be related to the area of an element, cf. eq. (13). However, this approach is generally used in engineering practice only for modelling tensile behaviour with linear elastic pre-peak behaviour followed by inelastic softening until total degradation of strength. The constitutive relation shown in Fig. 16 features pre-peak hardening and a residual plateau. Clearly, the hardening branch of the constitutive relation is stable and should not be adjusted as a function h but also the residual plateau is constant and independent of the h value. To demonstrate the veracity of the deþnition of a mesh independent release of energy upon mesh reþnement an example of a simple bar loaded in uniaxial is given. The problem is similar to the well-known problem of a simple bar loaded in tension proposed by CrisÞeld (1982). Consider the bar shown in Fig. 2 which is divided in n elements with n = 1, 2 and 4 elements. The length of the bar is 5 mm and the tranversal section of the bar has unit dimensions (1. mm 2 1. mm 2 ). The compressive fracture energies are assumed to equal G fcx = 1. N/mm and G fcy = 5. N/mm. For the rest of the material properties the values used in the previous section are assumed. One element is slightly imperfect (1%) to trigger the localization: f cx = 9. N/mm 2, f cy = 4. 5 N/mm 2,G fcx = 9. N/mm and G fcy = 4. 5 N/mm. The other material parameters remain the same. Imperfect element Fig. 2 - Simple bar with imperfect element loaded in compression The load-displacement response of the bar is depicted in Fig. 21a for the energy-based regularization method (note that ÒdisplacementÓ is understood as the relative displacement between the ends of the bar). It can be observed that the response is totally independent from the number of elements. The response of the bar with a constitutive model which has not been modiþed by the size of the Þnite element mesh, see Fig. 21b, shows a dramatic mesh-dependent behaviour in the post-peak response. The brittleness of the response increases with an increasing number of elements Load [MN] n = 1, 2 and 4 Load [MN] n = 2 n = n = Displacement [1 mm] Displacement [1 mm] a) Energy-based regularization b) No regularization Fig Load-displacement diagram for simple bar with imperfect element

27 TNO-95-NM-R A COMPOSITE YIELD SURFACE FOR MASONRY The two yield surfaces detailed in the previous section are now combined in a composite yield surface as illustrated in Fig. 22. y x y xy x Fig Proposed composite yield surface with iso-shear stress lines f tx = 1., f ty =. 5, f mx = 1., f my = 5. [N/mm 2 ] α = 1., β = 1., γ = 3. Contour spacing:.1 f mx A full description of the compressive and tensile parts of the composite yield surface are given in the previous sections and will not be repeated here. Only the aspects relative to the corner regime will be addressed in the present section. One of the most important issues of multi-surface plasticity is to deþne the number of active yield surfaces. Simo et al. (1988) have proposed an algorithm in which the assumption is made that the number of active yield surfaces in the Þnal stress state is less than or equal to the number of active yield surfaces in the trial stress state. This implies that it is not possible that a yield surface, which is inactive in the trial state, becomes active during the return mapping. This is not valid for the proposed yield surface. Due to the small number of yield surfaces (two), a trial and error iterative procedure is used, in which the return mapping process is restarted if a non-admissible solution is found, see Louren o (1994) for a complete description. From the experience of the author, this leads to a robust and always convergent algorithm. The disadvantage is that the return mapping algorithm might have to be restarted before the correct solution is obtained. 4.1 Return mapping algorithm - Corner regime The return mapping algorithm in the frame of a implicit Euler backward integration scheme is given in Section 2. for single surface plasticity. For multisurface plasticity, the most important assumption is KoiterÕs (1953) generalisation of the plastic strain rate as úεúε p = ú g 1 λ t + ú f 2 λ c. (67) Note that no coupling is assumed between the compressive and tensile regimes. Upon algebraic manipulation, the return mapping algorithm for the corner regime results in the following set of six equations containing 6 unknowns ( n+1, κ t, n+1 = λ t, n+1 and κ c, n+1 = λ c, n+1 ), cf. eqs. (29,61),

28 TNO-95-NM-R712 D 1 ( n+1 trial g ) 1 f 2 + λ t, n+1 + λ c, n+1 = n+1 n+1 f 1 = ( 1 / 2 ξ n+1p T t ξ n+1 ) 1 / / 2 π T ξ n+1 =. f 2 = ( 1 / 2 n+1p T c, n+1 n+1 ) 1 / 2 c, n+1 = This system of non-linear equations is solved with a regular Newton-Raphson method and the Jacobian necessary for this procedure reads (note that the subscript n+1 is dropped in the derivatives and matrices for convenience) (68) D 1 2 g 1 + λ t, n λ 2 f 2 c, n+1 2 T J = f 1 T f g 1 + λ 2 g 1 t, n+1 κ t f 1 κ t + + f 2 + λ 2 f 2 c, n+1 κ c f 2 κ c. (69) 4.2 Consistent tangent operator - Corner regime Differentiation of the update equations and the consistency conditions (d f 1, n+1 = and (d f 2, n+1 = ) results in J d n+1 dε n+1 dλ t, n+1 = dλ c, n+1. (7) Then, the consistent tangent operator is given by D ep = = (J 1 ) 4 4 ε n+1, (71) in which (J 1 ) 4x4 is the top-left 4 4 submatrix of the inverse of J. The consistent tangent operator can also be written in other fashion by means of a condensation of the matrix J. The derivation of an expression equivalent to eqs. (45,65) is given in Louren o (1994). An investigation on the performance of the numerical implementation is given in Appendix A.