INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng. 42, 703?727 (1998) AN ANISOTROPIC ELASTOPLASTIC-DAMAGE MODEL FOR PLAIN CONCRETE G. MESCHKE ? , R. LACKNER AND H. A. MANG Institute for Strength of Materials, Vienna University of Technology, A-1040, Karlsplatz 13, Vienna, Austria This paper is dedicated to Prof. F. Ziegler on the occasion of his 60th birthday ABSTRACT A material model for plain concrete formulated within the framework of multisurface elastoplasticity-damage theory is proposed in this paper. Anisotropic stiness degradation as well as inelastic deformations are taken into account. The applicability of the model encompasses cracking as well as the non-linear response of concrete in compression. The eect of dierent softening laws on the stress?strain relationship and on the dissipation is investigated in the context of a 1D model problem. The integration of the evolution laws is based on the standard return map scheme. Further computational issues include the stability of the local iteration procedure and the treatment of the apex region of the damage surface. The model is employed for re-analyses of a cylinder splitting test and of a notched concrete beam. Results from the composite elastoplastic-damage model are compared with test results and results from other material models for concrete, respectively. ? 1998 John Wiley & Sons, Ltd. KEY WORDS: damage; plasticity; concrete; cracking; Rankine criterion; nite element analysis 1. INTRODUCTORY REMARKS Brittle materials such as geomaterials and concrete exhibit distributed as well as localized degradation of the mechanical properties with increasing loading. The phenomenological response of plain concrete subjected to predominantly tensile stresses is characterized by a more or less linear ascending branch of the stress?displacement curve followed by a progressively decreasing residual strength resulting in the formation of macrodefects in the form of discrete cracks. When unloaded in the post-peak regime, non-recoverable deformations as well as a degradation of the stiness of the unloading branch is observed. From a microstructural point of view, the progressive degradation of the elastic moduli, commonly referred to as damage, is the result of growth and coalescence of existing microcracks and microvoids along the interfaces of the cement paste and the aggregates. This deterioration process prevents a complete closure of microcracks in unloading processes. As a consequence, permanent strains develop. On the phenomenological level, this eect is often modelled by means of classical plasticity theory. Depending on the level of hydrostatic ? Correspondence to: G. Meschke, Institute for Strength of Materials, Vienna University of Technology, Karlsplatz 13=202, A-1040 Wien, Austria. E-mail: [email protected] CCC 0029?5981/98/040703?25$17.50 ? 1998 John Wiley & Sons, Ltd. Received 19 February 1997 Revised 13 November 1997 704 G. MESCHKE, R. LACKNER AND H. A. MANG stress, a more or less gradual transition from highly localized fracture under tension to distributed microstructural deterioration leading to a more or less ductile material behaviour when subjected to triaxial compressive stresses is observed.1 Continuum damage mechanics is concerned with the modelling of microstructural degradation processes on a phenomenological level. The bulk of the existing continuum damage approaches is concerned with isotropic damage evolution, leading to a degradation of Young?s modulus as a function of a scalar damage parameter by exploiting the notion of eective stress, see e.g. References 2?5 for a review on this subject. Stiness and strength degradation as well as permanent deformations in concrete has been considered in References 6? 9. In contrast to the assumption of isotropy, the formation of cracks in concrete induces a directional bias to the material properties. Several formulations have been proposed to extend the concept of eective stresses to anisotropic damage models (see e.g. References 10 ?12 and 8). In formulations based on the microplane concept,13; 14 anisotropic damage was attributed to the reduction of the stress-carrying area fraction associated with the respective microcrack orientation. Recently, Simo et al.15 and Govindjee et al.16 have proposed an anisotropic damage model, using the principle of maximum dissipation for dening the evolution of the compliance tensor. This postulate is also taken as the starting point for the present material model. In contrast to Reference 16, however, it is used within the framework of multisurface damage-elastoplasticity allowing for stiness degradation as well as for the modelling of inelastic deformations. Cracking as well as the non-linear response of concrete in compression is taken into account. Cracks are represented within the framework of the smeared crack concept.17 Conceptually, the proposed model is similar to elastoplastic multisurface models recently developed by Feenstra and DeBorst18 for plain concrete and by Meschke19 for shotcrete. These models employ the maximum tensile stress criterion (Rankine criterion) to determine the tensile strength of concrete and a suitable yield function to describe mixed tensile-compressive and multiaxial compressive states of stress. The attractiveness of this class of models stems from their relative simple structure and their computational eectiveness. The proposed model makes an attempt to extend the range of applicability of this type of multisurface elastoplastic models to anisotropic damage mechanics without loss of computational eciency. It will be shown that the algorithmic structure remains essentially unchanged. It should be noted, however, that a completely dierent physical mechanism is represented by the proposed model. The aforementioned elastoplastic models suggested in References 18 and 19 are included as special cases. As far as the numerical modelling of cracks in plain concrete is concerned, classical local models lead to mesh-dependent results. To circumvent this problem, several methods such as non-local or gradient-dependent formulations, viscoplastic regularization, the use of the Cosserat theory and the ?fracture energy approach? characterized by a softening modulus adjusted to the element size and to the specic fracture energy (see Referene 20 for a comparison of approaches) may be used. We remark that this issue is not addressed in this paper. For the numerical simulations contained in this paper, however, the fracture energy approach 21 is employed. The remainder of the paper is organized as follows: Section 2 contains the theoretical formulation of the proposed plasticity-damage theory. From the denition of the Helmholtz energy function, the evolution laws for the compliance matrix and for the plastic strains are derived. The algorithmic formulation is addressed in Section 3. In Section 4, a one-dimensional model problem is used to investigate the eect of dierent softening laws on the stress?strain behaviour and on the dissipation. ? 1998 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 42, 703?727 (1998) AN ANISOTROPIC ELASTOPLASTIC-DAMAGE MODEL 705 Section 5 contains the ingredients of a 2D multisurface plastic-damage model for plain concrete. For the modelling of mode-I cracking, the Rankine damage surface is employed. The non-linear ductile material behaviour of concrete in compression is taken into account by means of a hardening=softening Drucker?Prager elastoplastic model. This section also contains selected algorithmic aspects of the model. Results from numerical analyses of a notched concrete beam and of a split cylinder test obtained from the proposed composite plastic-damage model are contained in Section 6. The analysis results are compared with respective test results and with results from other material models for plain concrete. 2. GOVERNING EQUATIONS OF PLASTICITY-DAMAGE MECHANICS In this section, the theoretical foundation of an anisotropic elastoplastic-damage model for plain concrete which incorporates both damage degradation and growth of inelastic strains is given. It is based on a continuum formulation of damage degradation recently proposed in References 15 and 16. The tensor of linearized strains is decomposed into an elastic and a plastic part, U = U e + Up (1) In analogy to classical plasticity, a region of admissible stress states is dened in the stress space by m failure and yield surfaces fk , respectively, that intersect in a non-smooth fashion: E = {b | fk (b; qk )60; k = 1; : : : ; m} (2) where qk is a stress-like internal variable associated with the damage (yield) surface fk related to a strain-like conjugate variable k by the relation 1 qk (k ) = ? @k S(k ) (3) is the density of the material and S(k ) is the part of the free energy associated with microstructural deterioration and slip processes in the material. The parameter qk (k ) determines the damage-dependent size of the damage surface fk in the stress space. The degradation of the elastic moduli C and the growth of inelastic strains Up associated with the damage (yield) surface fk are not regarded as independent processes. They are both controlled by a single scalar internal variable k . The function of free energy is dened as (Ue ; C; k ) = W (Ue ; C) + S(k ) = 1 e;T e U CU + S(k ) 2 (4) Restricting the present considerations to the purely mechanical theory, the Clausius?Duhem inequality requires ? = bT U? ? �? D = P ? (5) with P denoting the stress power. From (3) ? (5), considering C?D = ?CD? ? 1998 John Wiley & Sons, Ltd. (6) Int. J. Numer. Meth. Engng. 42, 703?727 (1998) 706 G. MESCHKE, R. LACKNER AND H. A. MANG follows D = bT U?p + 12 bT D?b + qk (k )?k � (7) In analogy to classical plasticity theory, the evolution of the compliance tensor D, of the inelastic strains Up and of the internal variables k is obtained from exploiting the postulate of maximum dissipation.22 For softening materials this hypothesis is replaced by the postulate of stationarity of the functional (7). Hence, for given admissible state variables (b, qk ) ? E, the rates D?, U?p and ?k are those which yield a stationary point of the dissipation D. To nd the solution of this constrained optimization problem, the method of Lagrange multipliers is used, introducing the Lagrangean functional L(b; qk ) = ?D + m P k=1 ? k fk (b; qk ) m P 1 ? k fk (b; qk ) = ?bT U?p ? bT D?b ? qk (k )?k + 2 k=1 (8) where ? k �are m Lagrange multipliers. From the associated optimality conditions @b L = 0; @qk L = 0 (9) follows U?p + D?b = U?p + U?d = m P k=1 ? k @b fk (b; qk ); ?k = m P k=1 ? k @qk fk (b; qk ) (10) where U?d are dierential strains associated with the degradation of the compliance matrix and the loading=unloading conditions are given as fk 60; ? k � ? k fk = 0 (11) Dening U?pd = U?p + U?d (12) (10)1 can be rewritten in a form analogous to classical associative plasticity theory as U?pd = m P k=1 ? k @b fk (b; qk ) (13) Introducing a scalar paramater , 0661, the plastic and the damage strains are given as U?p = (1 ? ) U?d = D?b = m P k=1 m P k=1 ? k @b fk (b; qk ) ? k @b fk (b; qk ) (14) The parameter allows a simple partitioning of eects associated with inelastic slip processes, resulting in an increase of inelastic strains Up and deterioration of the microstructure, resulting in an increase of the compliance moduli D. ? 1998 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 42, 703?727 (1998) 707 AN ANISOTROPIC ELASTOPLASTIC-DAMAGE MODEL The following considerations are restricted to homogeneous functions of degree one characterized by the general format: fk (b; qk ) = k (b) ? f + qk (k )60 (15) with f denoting the failure strength of the material and (b) = (bT Pb)1=2 + pT b (16) P is a constant projection matrix and p is a constant projection vector. From premultiplying (14) by bT follows m P bT D?b = ? k bT @b fk (b; qk ) k=1 = = m P k=1 m P k=1 ? k [bT @b fk (b; qk )][@b fkT (b; qk )b] @b fkT (b; qk )b ? k bT [@b fk (b; qk )@b fkT (b; qk )] b @b fkT (b; qk )b (17) Equation (17) results in the evolution law for the compliance tensor D? = m P k=1 ? k @b fk (b; qk )@b fkT (b; qk ) @b fkT (b; qk )b (18) Remark: Equation (17) also has the solution D?b = m P k=1 ? k @b fk (b; qk )@b fkT (b; qk ) b @b fkT (b; qk )b (19) From the symmetry relations Dijmn = Dmnij and from (19) follows @fk [email protected] = @fk [email protected] (20) This condition, however, is not compatible with the format of damage functions dened in (15). 3. ALGORITHMIC FORMULATION In the context of an incremental-iterative nite element scheme, the objective within each integration point is to compute, for a given set of state variables Un , k; n , Dn and a prescribed increment of strain U associated with the time interval [tn ; tn+1 ], the respective updated state variables at the end of the time step tn+1 . The stress?strain relation at tn+1 is written as tr ? Cn Up + CUen+1 bn+1 = Cn+1 Uen+1 = bn+1 tr tr = Cn Ue;n+1 , bn+1 tr Ue;n+1 (21) Upn . with = Un+1 ? Analogously, the trial with the trial stress state dened as tr ; qk;tr n+1 (k;tr n+1 ))60 then value of the internal variable k can be dened as k;tr n+1 = k; n . If fk (bn+1 the trial state is admissible according to the discrete counterpart of the loading=unloading conditions (11) at tn+1 fk; n+1 60; k � k fk; n+1 = 0 (22) tr fk (bn+1 ; qk;tr n+1 (k;tr n+1 ))� further evoluand is, therefore, the nal solution at tn+1 . In case that tion of damage and plasticization has to be accounted for within the time interval. The discrete ? 1998 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 42, 703?727 (1998) 708 G. MESCHKE, R. LACKNER AND H. A. MANG counterpart to the evolution equations (14) is obtained by employing a backward Euler integration scheme as m P Up = (1 ? ) k @b fk (bn+1 ; qk; n+1 ) k=1 (23) m P d U = Dbn+1 = k @b fk (bn+1 ; qk; n+1 ) k=1 From multiplying Cn+1 Dn+1 = 5 by Uen+1 and inserting (23)2 follows m P CUen+1 = ?Cn Ud = ?Cn k @b fk (bn+1 ; qk; n+1 ) k=1 (24) Inserting (23)1 and (24) into (21) yields the updated stresses at tn+1 in a form analogous to plasticity theory as m P tr bn+1 = bn+1 ? Cn k @b fk (bn+1 ; qk; n+1 ) (25) k=1 The updated values of the parameter qk are computed from (3) and from implicit integration of the hardening law (10)2 as 1 qk; n+1 = ? @k S(k; n+1 ) with k; n+1 = k; n + m P k=1 k @qk fk (bn+1 ; qk; n+1 ) (26) Due to the formally identical structure of the set of non-linear algebraic equations (25), (26) and the loading=unloading conditions (22) with the respective equations obtained for classical multisurface plasticity theory, the now standard return map algorithm (see Reference 23) can be employed without any change to compute the consistency parameter k . The updated state variables bn+1 , k; n+1 and Dn+1 are computed from (25), (26) and from implicit integration of (18), giving m P @b fk (b; qk )@b fkT (b; qk ) Dn+1 = Dn + k (27) @b fT (b; qk )b k=1 k n+1 The set of active damage (yield) surfaces Jact = {k ? 1; : : : ; m | fk � (28) is determined according to the following procedure:19 tr 1. Initialize the set of active yield surfaces in the predictor step: Jact = Jact = {k ? 1; : : : ; m | ftr �. 2. Perform the return mapping iteration and evaluate k ; bn+1 ; qk; n+1 ; k ? Jact . 3. Check the sign of k . If k � then reduce the predictor set by the respective yield surface fk . tr 4. Check if any yield surface fk; n+1 not contained in the predictor set Jact is violated. If fk �tr for k 6? Jact , then include fk in the set Jact and go to 2. The algorithmic elastoplastic-damage tangent moduli are obtained in the standard fashion known from multisurface plasticity theory, considering the class of damage functions specied in (15), as23 P P ?1 Aepd [g ]kl (@b fk @b flT )|n+1 (29) n+1 = ? k?Jact l?Jact ? 1998 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 42, 703?727 (1998) 709 AN ANISOTROPIC ELASTOPLASTIC-DAMAGE MODEL with ?1 = Dn + P k?Jact k @2bb fk ; gij = @b fiT @b fj + @q fi (@i qj )@q fj (30) 4. 1-D MODEL PROBLEM To provide more insight into the mechanism of the model, this section is concerned with a 1-D study of the proposed damage model for concrete including a comparison of a damage model ( = 1), taking only stiness degradation into account, with an elastoplastic model, obtained from setting = 0, considering only plastic slip. For the 1-D case, the evolution equations (18) for the damage model ( = 1) and for the plasticity model (14)1 ( = 0) can be integrated analytically to obtain the respective uniaxial stress?strain relation for the pre- and the postcracking response of a bar loaded uniaxially in tension. A linear, an exponential and a hyperbolic softening law are investigated. The uniaxial failure criterion for tensile stresses has the form f(; ) = () ? f + q() = ? q()60 (31) with q() = f ? q() and () = . 4.1. Plasticity theory ( = 0) The stress?strain relation, the associative ow and softening rule and the consistency condition are summarized below for 1-D plasticity theory: = E0 ?p = ? @ f = ? (32) ? = ? @q f = ? f?(; ) = ? + @ q()? = 0 The rate of dissipation is computed from (7), considering the yield condition f = 0, and inserting (32)2 and (32)3 , as Dp = ?p + q? = ( + q)? = f ? (33) According to (5), the rate of external work is obtained as ? Pp = Dp + (34) The total external work required to increase the stress from = 0 at t0 up to f and subsequently drive the residual stresses to zero at t = tu is computed from (34), noting that ? = Ee ?e + @ S ? = ?e ? q? (35) as Z tu t0 Pp = = ? 1998 John Wiley & Sons, Ltd. Z tu Dp + t0 Z u 0 Z tu t0 f d + ?= Z =0 =0 Z 0 u Z f d + d ? E0 Z 0 u u 0 de ? Z 0 u q d q d Int. J. Numer. Meth. Engng. 42, 703?727 (1998) 710 G. MESCHKE, R. LACKNER AND H. A. MANG Z = u 0 = 0 Z u 2 (f ? q) d + = () d = Gf Af =Vf 2E0 = 0 0 (36) Gf is the fracture energy per unit of fracture area Af of concrete and Vf is volume of the fracture zone. 4.2. Damage theory ( = 1) The stress?strain relation, the damage evolution law for the compliance modulus D, the evolution law for the internal variable and the consistency condition are summarized below for the 1-D damage model: = E 1 ? = E? ? = ? @q f = ? D? = (37) f?(; ) = ? + @ q()? = 0 From dierentiation of (37)1 follows ? = E? + E ? (38) The dissipation is computed from (7) and (15), considering the yield condition f = 0, as ? (39) Dd = 12 D? + q? = 12 ? + ? q = ? 12 (f ? q) + ? q = (f + q) 2 Remark: In Reference 15, a modied damage evolution law which diers from (37)2 by a factor of 2 is used. This ad hoc assumption yields the dissipation in the format Dd = ? f (40) which is identical to the respective form of classical plasticity theory. However, as will be shown below, the total dissipation required to separate both crack surfaces diers for both theories. Therefore, no attempt is made to achieve a formal agreement of the rate equations (33) and (39). The total external work required to increase the stress from = 0 at t0 up to f and subsequently drive the residual stresses to zero at t = tu is computed as Z tu Z tu Z tu d d ? P = D + (41) t0 where Z t0 tu t0 and, according to (4), Z tu t0 ? = Z d D = tu t0 Z t0 u 0 1 d ? 2 1 (f + q) d 2 Z tu t0 2 (42) Z dD ? 0 u q d Using integration by parts for the second term on the right-hand side of (43) gives tu Z tu Z tu Z tu 1 1 2 dD = 2 D ? D d = 0 ? d 2 t0 2 t0 t0 t0 ? 1998 John Wiley & Sons, Ltd. (43) (44) Int. J. Numer. Meth. Engng. 42, 703?727 (1998) AN ANISOTROPIC ELASTOPLASTIC-DAMAGE MODEL Inserting (44) into (43), observing that Z tu Z d + t0 yields tu t0 Z tu t0 t (45) q d (46) d = 2|tu0 = 0 Z ? =? u 0 711 The total external work Pd is obtained from inserting (42) and (46) into (41), considering (31), as Z tu Z u Z tu Z tu Z u Z u 1 1 d d ? f d + q d ? P = D + = q d t0 t0 t0 0 2 0 2 0 Z u Z u 1 1 (f ? q) d = () d (47) = 2 0 2 0 Obviously, the ratio between the total external work Pd invested in a process to drive the postcracking stresses to zero according to the damage theory to the respective work Pp for plasticity theory is Pd = 0�Pp This ratio is independent of the chosen softening law. (48) 4.3. Linear softening law A linear softening law characterized by q() = f = Hs ; u q() = f ? q() = f 1? u = f ? Hs (49) see Figure 1, is considered rst. The softening modulus Hs �may be related to the fracture energy and to the characteristic length (see Section 5). From inserting (49) and (38) into (37)4 follows 1 ? (50) ? = ? [E? + E ] Hs Inserting (37)2 into D? = ? E?D E (51) yields E? (52) E2 Inserting (37)1 into (52), substituting the result into (50) and separation of variables gives 1 ? Hs = E? (53) ? E2 E ? = ? ? 1998 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 42, 703?727 (1998) 712 G. MESCHKE, R. LACKNER AND H. A. MANG Figure 1. Uniaxial stress?strain curve obtained from the damage model ( = 1) for a linear softening law Integration of both sides of (53) yields Z E Z 1 1 ? E Hs 1 = ln dE = Hs ? ln = ? ? 2 E E E E E 0 0 0 0 E0 (54) From inserting (37)1 , noting that f = E0 0 , and solving for , the uniaxial post critical stress?strain law is obtained as ? for �0 ? 0 ? =E 1 f 1 (55) = for �0 ? ? H ln + E s 0 see Figure 1. It is characterized by a progressively decreasing ? curve followed by a supercritical softening branch. The critical strain c is computed from (55) as c = f Hs exp eHs E0 (56) At this strain level E = Hs . For E縃s (Es ), a subcritical (supercritical) softening characteristics is predicted by the damage model based on a linear softening law. From this observation follows, that a linear softening law is not suitable for the proposed damage model for concrete. The rate of dissipation per unit volume is computed from (39), using (37)4 together with (49) as ? ? (2f ? ) (57) Dd = (f + q) = ? 2 2Hs The total dissipation required to drive the residual postcracking stresses to zero is calculated from integrating (57) as Z 0 3 2 3 Dd = f = f u (58) 4H 4 s f The total external work Pd for damage theory ( = 1) is computed from (47) as Pd = ? 1998 John Wiley & Sons, Ltd. f2 4Hs (59) Int. J. Numer. Meth. Engng. 42, 703?727 (1998) AN ANISOTROPIC ELASTOPLASTIC-DAMAGE MODEL 713 Figure 2. Linear softening law: dissipation per unit of volume in uniaxial tension for damage theory ( = 1) and for plasticity theory ( = 0) For plasticity theory, Pp is obtained from (36) as Pp = f2 = 2 Pd 2Hs (60) Hence, the calibration of u according to the fracture energy concept requires u = 4Gf lc f for = 1; u = 2Gf lc f for = 0 (61) where lc = Vf =Af is the so-called characteristic length.24 Figure 2 shows a comparison between the dissipation per unit of volume obtained from damage theory and from plasticity theory for uniaxial tensile loading. 4.4. Exponential softening law An exponential softening law q() = f 1 ? exp ? u (62) is considered now (Figure 3). Inserting @ q() = f ? q u (63) into (37)4 and considering (31) and (38) yields ? = ? u [E? + E ] ? (64) From (64) and (52) follows after separation of variables u E? = ? E 2 ? u ? 1998 John Wiley & Sons, Ltd. (65) Int. J. Numer. Meth. Engng. 42, 703?727 (1998) 714 G. MESCHKE, R. LACKNER AND H. A. MANG Figure 3. Uniaxial stress?strain curve obtained from the damage model ( = 1) for an exponential softening law Integrating (65) and inserting (37)1 yields a linear stress?strain relation = f ? u 0 ? u (66) (Figure 3). A complete loss of residual strength is predicted at = u . The total dissipation required to drive the material to failure is computed next. From (37)2 one obtains ? = ? E? E2 (67) Inserting (65) into (67) and considering (37)4 yields ? = ? u ? ? u (68) From inserting (68) and (66) into (39), considering the yield condition f = 0, the rate of dissipation follows as u 1 u ? 1 D=? 2f ? f ? ? (69) ? = f u 2( ? u ) u ? 0 u ? 2(u ? 0 ) The total dissipation per unit volume required to drive the material to failure is obtained from integrating (69) as Z Z u 1 1 ? d D= f u u ? 2(u ? 0 ) 0 0 ? 0 u ? (70) = f u ? ln ? u ? 0 2(u ? 0 ) R Observe, that for ? u , the total Dissipation D ? ? (Figure 4). Hence, the exponential softening law is not suitable to be used in conjunction with the the proposed damage theory. This is particularly important in the context of coupled thermomechanical analyses, where the mechanical dissipation is used as the input for solving the thermal subproblem. ? 1998 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 42, 703?727 (1998) AN ANISOTROPIC ELASTOPLASTIC-DAMAGE MODEL 715 Figure 4. Exponential softening law: dissipation per unit of volume in uniaxial tension for damage theory ( = 1) and for plasticity theory ( = 0) Remark: In case of the modied evolution equations proposed by Simo et al.,15 the total dissipation is Z u 2u ? D = f u ? ln (71) 2u ? 0 0 The dissipation also becomes innite as ? u = 2u . 4.5. Hyperbolic softening law A softening law of the form q() = f 1 ? 1 (1 + )2 (72) where = =u , is investigated next (see Figure 5). From the failure condition (31) follows 1 (73) () = f (1 + )2 Inserting (73) into (37)2 and integrating the so-obtained result, using the initial conditions D = 1=E0 at = 0, yields Z Z u 1 D= dD = (1 + )2 d = [(1 + )3 ? 1] + 1=E0 (74) 3 f f 0 0 may be expressed in terms of by inserting (73) and (74) into (37)1 as = with 1 [ + 2 a?1=3 + a1=3 ] ? 1 u p b2 ? 6 ; (75) u (30 ? u ) (76) 2 Note, that the expression for a in (76) only has real values for arbitrary values of if u �0 . a=b + ? 1998 John Wiley & Sons, Ltd. b = 3 ? Int. J. Numer. Meth. Engng. 42, 703?727 (1998) 716 G. MESCHKE, R. LACKNER AND H. A. MANG Figure 5. Uniaxial stress?strain curve obtained from the damage model ( = 1) for a hyperbolic softening law Figure 6. Hyperbolic softening law: dissipation per unit of volume in uniaxial tension for damage theory ( = 1) and for plasticity theory ( = 0) The ? relation contained in Figure 5 is obtained from inserting (75) into (73) as ? ? E0 if �0 f u2 = if �0 ? [ + 2 a?1=3 + a1=3 ]2 (77) The form of this curve is well suited for describing the post-cracking softening behaviour of plain concrete, provided that u is properly calibrated to the uniaxial fracture energy Gf . The total dissipation is computed from inserting (72) into (39) and integrating the so-obtained result as Z f u 1 + 2 (78) Dd = 2 1+ 0 For nite values of , it follows from (75) and (77) that the dissipation is also nite (Figure 6). 5. A 2-D COMPOSITE PLASTIC-DAMAGE MODEL FOR CONCRETE Concrete cracking is modelled in the framework of the smeared crack concept.25 To account for the brittle material behaviour of concrete in tension, the maximum principle stress (Rankine) criterion ? 1998 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 42, 703?727 (1998) AN ANISOTROPIC ELASTOPLASTIC-DAMAGE MODEL 717 is used. After crack initiation the residual stresses are gradually decreasing. Considering plane stress conditions, the damage surface is dened as r 1 1 fRK (b; qRK ) = (x + y ) + (x ? y )2 + 2xy ? [ftu ? qRK (RK )]60 (79) 2 4 with ftu as the uniaxial tensile strength of concrete. The softening behaviour is accounted for by the hyperbolic law (72). According to the fracture energy concept for softening materials,26 the parameter RK; u is adjusted to the specic fracture energy of concrete in mode-I, Gf , and to the characteristic length lc . From integrating (73) from 0 to ? and setting the result equal to Gf =lc follows RK; u = Gf =(lc ftu ) (80) For the determination of the characteristic length lc an approach suggested by Oliver 24 is employed. The ductile behaviour of concrete subjected to compressive loading is described by a hardening= softening Drucker?Prager plasticity model. The yield function is dened as ? fcy ? qDP 60 fDP (b; qDP ) = J2 + DP I1 ? DP (81) with fcy as the uniaxial elastic limit, qDP (DP ) is the hardening=softening parameter dependent on the internal plastic variable DP and I1 = x + y and J2 = 16 [(x ? y )2 + y2 + x2 ] + 2xy (82) The material parameters DP and DP are adjusted according to the ratio between the uniaxial and the biaxial tensile strength of concrete, fcb =fcu ,27 to obtain ? 1 2fcb =fcu ? 1 fcb =fcu ? 1 ; DP = 3 (83) DP = ? fcb =fcu 3 2fcb =fcu ? 1 This ratio is assumed as fcb =fcu = 1�: The non-linear behaviour of concrete in compression is governed by the hardening=softening law, ? fcu ? fcy ? ? (DP; u ? DP )2 ? (fcu ? fcy ) for DP �DP; u ? 2 ? ? DP; u ? ? fcu (84) qDP (DP ) = (DP ? DP; u )2 ? (fcu ? fcy ) for DP; u 6DP �DP; c ? ? 2 ? (DP; c ? DP; u ) ? ? ? ? fcy for DP; c 6DP where DP; u and DP; c are material parameters, see Figure 7. DP; u is determined from the total strain level u at = fcu in uniaxial loading as DP; u = c(u ? fcu =Ec ) ? 1998 John Wiley & Sons, Ltd. with c = 1 ? DP (1= 3 + DP ) (85) Int. J. Numer. Meth. Engng. 42, 703?727 (1998) 718 G. MESCHKE, R. LACKNER AND H. A. MANG Figure 7. Hardening=softening law for modelling of concrete in compression The critical value DP; c is obtained from the uniaxial fracture energy of concrete Gc and the characteristic length lc by setting Z DP; c (fcy ? qDP ) dDP = Gc =lc (86) 0 as 3 1 DP; c = 2 fcu 1 Gc ? fcy DP; u c lc 3 (87) 5.1. Selected algorithmic issues The algorithmic formulation of the composite damage=plastic model for concrete described in Section 5 is completed by addressing specic computational aspects that may cause convergence problems of the return map iteration. 5.1.1. Apex region of the Rankine surface. In the case of vanishing shear stresses xy = 0, the Rankine failure surface (79) becomes non-smooth, constituting an apex in the x ?y space. The gradient of fRK is not dened uniquely in this region. Therefore, making use of the fact that the nal stress state bn+1 at tn+1 is a principal stress state, the damage function fRK at tn+1 is formulated in terms of the principal stresses x; n+1 and y; n+1 in the form f1; RK; n+1 = x; n+1 ? [ftu ? qRK (RK )]; f2; RK; n+1 = y; n+1 ? [ftu ? qRK (RK )] (88) The condition xy = 0 in the apex is accounted for by means of an additional damage function as f3; RK; n+1 = xy; n+1 = 0 (89) The nal stresses are obtained according to (25), setting the set of active damage surfaces to Jact = {f1; RK ; f2; RK ; f3; RK }, using (14)1 with @Tb f1; RK = [1; 0; 0]; @Tb f2; RK = [0; 1; 0] and @Tb f3; RK = [0; 0; 1]. The consistency parameters 1 , 2 and 3 are computed from the consistency conditions at tn+1 ; fi; RK; n+1 (i ; RK ) = 0; i = 1; : : : ; 3, with RK; n+1 = RK; n + 3 P i=1 i @qRK fi; RK; n+1 = RK; n + 1 + 2 (90) by means of a Newton iteration procedure. ? 1998 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 42, 703?727 (1998) AN ANISOTROPIC ELASTOPLASTIC-DAMAGE MODEL 719 Figure 8. Illustration of the Newton algorithm: selection of a starting iterate for The algorithmic moduli are computed according to (29) as 3 P 3 P Aepd [g?1 ]kl (Cn @b fk; RK @b fl;TRK Cn ) n+1 = Cn ? k=1l=1 with ? Cn; 11 + @ qRK; n+1 ? g = ? Cn; 21 + @ qRK; n+1 Cn; 31 Cn; 12 + @ qRK; n+1 Cn; 22 + @ qRK; n+1 Cn; 32 ? Cn; 13 ? Cn; 23 ? Cn; 33 (91) (92) 5.1.2. Starting iterate for return map algorithm. The standard return map algorithm used to compute the consistency parameter takes the elastic trial state, characterized by (0) = 0; as its starting iterate. In the present case of a hyperbolic softening law (72), however, the function fRK = 0 is not convex in . As an illustration of this fact, the 1D model problem investigated in Section 4.5 is used as an example. Inserting (72) into (31), considering p ) n+1 = En (n+1 ? np ? n+1 (93) and using (32)2 and (32)3 yields fRK = n+1 ? np ? ? ftu u2 =En =0 (u + n + )2 (94) Figure 8 contains a plot of fRK as a function of . fRK = 0 has three solutions. Taking = 0 as the starting iterate of the Newton iteration scheme would converge to the wrong (negative) solution for . To ensure that the iteration converges to the correct solution, a plastic trial state, assuming total p = n+1 ), is adopted as the starting value for instead of the elastic trial plastic response (n+1 state. The idea is illustrated in Figure 8. As ? �, the quadratic term in (94) vanishes, giving a linear relation between and fRK : fRK = n+1 ? np ? = 0 (95) (0) This linear relation is used to compute an explicit starting iterate . ? 1998 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 42, 703?727 (1998) 720 G. MESCHKE, R. LACKNER AND H. A. MANG 6. NUMERICAL EXAMPLES The proposed multisurface damage-plastic model for plain concrete has been implemented into the multi-purpose nite element program MARC and was used for numerical analyses of two plain concrete structures. The purpose of this section is to demonstrate the eectiveness of the model and to compare numerical results with experimental results and results from other material models for concrete. In Section 6.1, a notched beam subjected to cyclic loading is analysed numerically by means of the combined plastic-damage model. Section 6.2 contains results from comperative analyses of a cylinder splitting test. 6.1. Notched concrete beam subjected to cyclic loading In this example, a notched plain concrete beam subjected to cyclic quasistatic loading is analysed numerically. The respective test results are documented in Reference 28. The geometrical specications and the material data are taken from the test C2-D1-S3-R2.28 Figure 9 contains the dimensions and the nite element discretization of the beam; 492 bilinear plane stress nite elements are employed in the analyses. The specimen was tested under displacement control according to the following procedure: As soon as the peak load was reached, the beam was unloaded, then reloaded up to a value of the crack mouth opening displacement (CMOD) twice the one previousely attained at the peak load CMODp and unloaded again. Subsequently, the beam was unloaded and reloaded at CMOD = 5CMODp and CMOD = 10CMODp . The model parameters are summarized in Table I. The parameters RK; u , DP; c and DP; u , have been calibrated from the fracture energy in uniaxial tension (Gf ) and compression (Gc ), respectively. Gf was reported as Gf = 0�95 N mm=mm2 ;28 Gc was assumed as Gc = 50Gf . Figure 10 contains a comparison of the load?displacement diagrams obtained from the experiment28 and from two nite element analyses based on the combined plastic-damage model ( = 0� and on an elastoplastic model ( = 0). The peak load and the postpeak regime of the curve are replicated fairly well by both analyses. In contrast to the results from the elastoplastic model, the un- and reloading paths obtained from the composite plastic-damage model also agree well with the test results. Figure 11 illustrates the distribution of the plastic tensile strains in the vicinity of the notch at dierent loading stages according to the composite plastic-damage model. Figure 9. Notched concrete beam: dimensions and nite element discretization (units in mm) ? 1998 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 42, 703?727 (1998) AN ANISOTROPIC ELASTOPLASTIC-DAMAGE MODEL 721 Table I. Re-analysis of a notched concrete beam: model parameters Material property Young?s modulus Poisson?s ratio Uniaxial tensile strength Uniaxial compressive strength Fracture energy in tension Fracture energy in compression Post-cracking softening law in tension Hardening=softening law in compression Critical value of DP : Plasticity-damage partitioning factor Material parameter E = 43 600 N=mm2 = 0�ftu = 4�N=mm2 fcu = 63�N=mm2 Gf = 0�95 N mm=mm2 Gc = 5�50 N mm=mm2 RK; u = 8�5 � 10?3 DP; u = 5�7 � 10?2 DP; c = 7�9 � 10?4 = 0� Figure 10. Re-analysis of a notched concrete beam: load?displacement diagrams obtained from the experiment, from an elastoplastic model ( = 0) and from a combined plastic-damage model ( = 0� 6.2. Cylinder splitting test Cylinder splitting tests are frequently used to determine the tensile strength of concrete. In this section, this test is analysed numerically by means of the proposed plastic-damage model for plain concrete. The dimensions of the cylindrical concrete specimen and the plywood loading platen are shown in Figure 12. The material properties used for the analyses are summarized in Table II. These data have been taken from a Round Robin test documented in Reference 29. 184 bilinear plane stress nite elements (see Figure 14) have been used in the analyses. Three dierent models of cracked concrete were investigated. Results from two special cases of the proposed plastic-damage composite model, an elastoplastic model ( = 0) and a composite plastic-damage model ( = 1) were compared with respective results from a xed crack model. According to the xed crack model,30 cracks will open normal to the direction of the maximum principal stress if this principal stress exceeds the tensile strength ftu . Secondary cracks are restricted to the direction perpendicular to the rst crack. After crack initiation, tensile stresses are gradually released according to a linear postpeak stress?strain relationship. This ?tension-softening? approach considers the descending part of the load?displacement curve of the localized crack zones, averaged over the respective part of the element related to an integration point. The slope of the ? 1998 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 42, 703?727 (1998) 722 G. MESCHKE, R. LACKNER AND H. A. MANG Figure 11. Re-analysis of a notched concrete beam by means of the composite plastic-damage model: distribution of plastic strains in the vicinity of the notch at dierent loading stages: (a) u = 0� mm (P = 8�kN) and (b) u = 0� mm (P = 2�kN) (200-fold magnication of displacements) Figure 12. Cylinder splitting test: dimensions of the specimen (units in mm) descending part of the ? diagram of concrete in tension is related to the specic fracture energy Gf and the characteristic length lc . The energy release rate Gf is identical for all models. The hardening=softening characteristics qDP (DP ) in the compressive regime were modelled identically in all three analyses. ? 1998 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 42, 703?727 (1998) AN ANISOTROPIC ELASTOPLASTIC-DAMAGE MODEL 723 Table II. Cylinder splitting test: material parameters Material property Young?s modulus Poisson?s ratio Uniaxial compressive strength Uniaxial tensile strength Fracture energy in tension Fracture energy in compression Average characteristic length Material parameter E = 26200�(N=mm2 ) = 0�fcu = 30�(N=mm2 ) ftu = 3�(N=mm2 ) Gf = 0�(N mm=mm2 ) Gc = 5� (N mm=mm2 ) lc = 2� (mm) Figure 13. Cylinder splitting test: load?displacement diagram Figure 13 contains the relation between the applied load and the displacement of the top of the loading platen relative to the horizontal symmetry axis of the cylinder obtained from the three constitutive models. Expectedly, the xed crack model predicts the largest value for the peak load (Pmax = 171�kN). As far as the elastoplastic model and the composite plastic-damage model are concerned, it should be noted that both models are based on an isotropic form of the yield function and the damage function. The ultimate load resulting from both models is smaller than the one following from the xed crack model. They are obtained as Pmax = 154�kN for the elastoplastic model and Pmax = 161�kN for the composite plastic-damage model. The cylinder splitting test is strongly inuenced by the compression in the vicinity of the loading platens. Since the behaviour of concrete in compression is accounted for in the same way by all three models, and since the behaviour of concrete in compression is relevant for the overall structural behaviour, the dierent crack models only have a small inuence on this behaviour. Figures 14 and 15 contain one-quarter of the concrete cylinder. Figure 14 shows the distribution with the Drucker?Prager yield surface (Figure 14(a)), and crack of plastic strains Up , associated R strains, dened as Ud = D(t) db, associated with the Rankine damage criterion (Figure 14(b)), obtained from the composite plastic-damage model ( = 1) at u = 0�mm. In the vicinity of the loading platens, large inelastic compressive strains are induced. Along the symmetry line large horizontal tensile crack strains, attributed to the degradation of the stiness along this localized fracture zone, are predicted. For comparison, the distribution of the plastic strains Up obtained from the elastoplastic model ( = 0) is illustrated in Figure 15. Both Figures show large inelastic compressive strains induced in the vicinity of the loading platen. Large horizontal tensile strains developing along the symmetry line are predicted by both models. Note, however, that these strains result from dierent physical mechanisms. According to the plastic-damage composite model ( = 1), these strains are attributed ? 1998 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 42, 703?727 (1998) 724 G. MESCHKE, R. LACKNER AND H. A. MANG Figure 14. Cylinder splitting test: distribution of plastic strains and crack strains at u = 0�mm obtained from the composite plastic-damage model ( = 1): (a) plastic strains and; (b) crack strains (15-fold magnication of displacements) to the stiness degradation along this fracture zone, whereas according to the elastoplastic model ( = 0), these strains are of inelastic nature. A mesh sensitivity study is performed to ensure that a converged solution is obtained for sucessively rened meshes. Figure 16 illustrates three meshes used for this study containing 41 (Figure 16(a)), 184 (Figure 16(b)) and 352 elements (Figure 16(c)). Figure 17 contains the respective load?displacement curves obtained from the composite plastic-damage model ( = 1). The observed mesh dependence is within the range of the expected discretization error. It should be noted that the postpeak branch is primarily governed by the softening behavior of the model in compression. ? 1998 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 42, 703?727 (1998) AN ANISOTROPIC ELASTOPLASTIC-DAMAGE MODEL 725 Figure 15. Cylinder splitting test: distribution of plastic strains at u = 0�mm obtained from the elastoplastic model ( = 0) (15-fold magnication of displacements) Figure 16. Cylinder splitting test: dierent meshes used for the investigation of the discretization inuence: (a) 41 elements; (b) 184 elements; (c) 352 elements Figure 17. Cylinder splitting test: load?displacement diagrams obtained from the composite plastic-damage model ( = 1) for three dierent meshes ? 1998 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 42, 703?727 (1998) 726 G. MESCHKE, R. LACKNER AND H. A. MANG 7. CONCLUSIONS In this paper, an anisotropic damage evolution law based on the assumption of maximum dissipation has been used to develop a constitutive model for plain concrete in the framework of multisurface damage-elastoplasticity. A 1-D study of the proposed damage model has revealed limitations for the choice of the softening characteristics after opening of a crack. A linear softening law leads to a supercritical branch of the stress?strain diagram. An exponential softening law results in a linear stress?strain diagram. The dissipation predicted by this model, however, becomes innite at nite values of strains. These ndings exclude both formulations from being chosen as softening laws to represent the postcracking characteristics of concrete. As a consequence, a hyperbolic softening law has been proposed in the paper that by-passes both shortcomings and that is easily calibrated to the fracture energy release rate. Damage evolution is associated with crack propagation while the nonlinear behavior of concrete in compression and mixed tension-compression is represented by means of an elastoplastic constitutive model. Finite element modelling of cracks is based on the smeared crack concept. To account for permanent deformations in addition to the degradation of the stiness after a crack has opened, a partitioning factor was introduced to allow for modelling of both phenomena. The paper also addressed the algorithmic formulation of the model. Due to the formally indentical structure of the damage model and of classical multisurface elastoplastic models, the standard projection algorithm could be used. Additional computational issues were concerned with the convergence of the return map algorithm and with the algorithmic treatment of the apex region of the Rankine damage criterion. The algorithmic tangent is formulated for the regular case as well as for the apex region. 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