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A Coupled Chemo-Elastoplastic- Damage Constitutive Model for Plain Concrete Subjected to High Temperature RONGTAO LI1,2 AND XIKUI LI1,* 1 The State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, People’s Republic of China 2 College of Civil and Architectural Engineering, Dalian University, Dalian 116622, People’s Republic of China ABSTRACT: A coupled elastoplastic-damage constitutive model taking into account chemo-induced elastoplastic-damage effects for the modeling of coupled chemo- thermo-hygro-mechanical behavior of concretes at high temperature is proposed in this article. A three-step operator split algorithm for the integration of the rate coupled constitutive equations is developed. Consistent tangent modulus matrices for coupled chemo-thermo-hygro-mechanical analysis are derived to preserve the quadratic rate of convergence of the global Newton iterative procedure. Numerical results demonstrate the validity of the presented algorithm and illustrate the capability of the proposed constitutive model in reproducing coupled chemo-thermo- hygro-mechanical behavior in concretes subjected to fire and thermal radiation. KEY WORDS: chemo-elastoplastic-damage, coupled constitutive model, consistent algorithm, concrete, high temperature. INTRODUCTION high performance concrete (HPC) has been widely used I N RECENT YEARS, in engineering practices such as tall buildings, tunnels, offshore oil *Author to whom correspondence should be addressed. E-mail: xikuili@dlut.edu.cn Figures 611 appear in color online: http://ijd.sagepub.com International Journal of DAMAGE MECHANICS, Vol. 19—November 2010 971 1056-7895/10/08 097130 $10.00/0 DOI: 10.1177/1056789509359667 ß The Author(s), 2010. Reprints and permissions: http://www.sagepub.co.uk/journalsPermissions.nav 972 R. LI AND X. LI production platforms, and nuclear engineering applications due to its intrinsic properties of high strength, high durability, etc. However, increases in concrete strength and durability have been accompanied by an increasing tendency towards explosive thermal spalling of HPC at high temperatures due to its high density and low permeability as well as ther- mally induced pore pressure buildup and restrained thermal dilatation (Brite Euram, 1999; Gawin et al., 1999, 2003). The fundamental study of the fail- ure phenomena characterized by the thermal spalling of concrete is required to acquire a deep understanding of its physical origin observed on concrete exposed to rapid heating. The results of the research can help one to better assess the serviceability and strength of civil engineering facilities during fires. To numerically study the complex multi-chemo-physical process, which occurs in concretes subjected to fire, a hierarchical mathematical model for analyses of coupled chemo-thermo-hygro-mechanical behavior of concretes at high temperature was developed (Li et al., 2006) based on the previous work of Gawin et al. (2003). The concretes are modeled as unsaturated deforming reactive porous media (Lewis and Schrefler, 1998) filled with pore fluids in two immisciblemiscible levels. In the primary level, the two pore fluids, that is, the gas mixture and the liquid mixture flow through the pore channels in an immiscible pattern. In the secondary level, dry air and vapor within the gas mixture phase, and the dissolved matrix components and pore water within the liquid mixture phase are homogeneously miscible between each other. The thermo-induced dehydration and desalination pro- cesses are integrated into the model. The chemical effects of both dehydration and desalination on the material damage and the degradation of the material strength are taken into account. The mathematical model consists of a set of coupled, partial differential equations governing the mass balance of the dry air, the mass balance of the water species, the mass balance of the matrix components dissolved in the liquid phases, the enthalpy (energy) balance and momentum balance of the whole medium mixture. Owing to the limitation of space the details of the coupled mathematical model (Li et al., 2006), which are not directly related to the present coupled chemo-elastoplastic-damage constitutive model, will not be re-described in this article. The present article aims to develop a relevant constitutive model, in the frame of the mathematical model mentioned above, which can quantita- tively describe the interrelated multi-chemo-physical process and progressive failure phenomena in concrete members subjected to fire. Many models have been developed for modeling the non-linear constitutive behavior of plain concrete subjected to different types of loading at room temperature. Among them are the elastoplastic models of Willam and Warnke (1975), Ohtani and Chen (1988) and the models given in Chen (1994), Chemo-Elastoplastic-Damage Constitutive Model for Plain Concrete 973 the damage models of Mazars (1984), Simo and Ju (1987a,b), Mazars and Pijaudier-Cabot (1989), Oliver et al. (1990), Zhang et al. (2008), based on continuum damage mechanics, the coupled elastoplastic-damage models of Ju (1989), Yazdani and Schreyer (1990), Wu et al. (2005), Jason et al. (2006), Abu Al-Rub and Voyiadjis (2009), Voyiadjis et al. (2009), etc. The above models, as phenomenological models, are based on the thermodynamics of an irreversible process and the internal state variable theory. To extend the constitutive models developed at room temperature to the cases at high temperature, thermally induced chemical effects on strength or/and stiffness loss of concrete have to be integrated into the models. Ulm et al. (1999) extended the Willam and Warnke model (1975) of plasticity to take into account the chemo-plastic softening effect due to high temperature, that is, the dehydration effect on the cohesion parameter of concrete. They applied the model to finite element analysis of the tunnel rings of the Channel Tunnel connecting England to France exposed to the fire, which occurred on November 18, 1996 and lasted 10 h with temperatures up to 700 C. Based on the damage model of Mazars (1984), Gawin et al. (2003) pre- sented a thermally induced chemo-elastic damage model, in which the effect of the irreversible dehydration process with increasing temperature on the concrete damage is taken into account. To account for the complexity of coupled chemo-thermo- hydro-mechanical behavior in concrete subjected to fire and the fact that occurrence and evolution of the micro-crack or micro-void growth are accom- panied by the plastic flow process observed in concrete material, a coupled elastoplastic-damage constitutive model with consideration of chemo- induced elastoplastic-damage effects is proposed in the present work to model the realistic failure phenomena, that is, loss of both the strength and stiffness, characterized by thermal spalling. The effects of both dehy- dration and desalination in concrete members exposed to high temperature on the material strength and stiffness are considered in the proposed coupled model. The model is developed on the basis of the damage model by Mazars (1984) and the WillamWarnke elastoplastic yield criterion (Willam and Warnke, 1975) for concrete at room temperature. The chemical softening and chemical damage, in addition to plastic strain hardening/ softening, suction hardening and mechanical damage, are taken into account in the model. Based on the previous work for constitutive modeling of coupled elastoplastic-damage by Ju (1989), coupled chemo-plasticity by Sercombe et al. (2000) and previously defined return mapping schemes for this type of coupled problems (Li et al., 1994; Li, 1995), a three-step operator split algo- rithm for the proposed coupled chemo-elastoplastic-damage model is derived. 974 R. LI AND X. LI Consistent tangent modulus matrices with consideration of the fully coupled effects are formulated. Numerical results demonstrate not only the capability of the proposed constitutive model in reproducing coupled chemo-thermo-hygro-mechanical behavior and failure phenomena in concretes subjected to fire and thermal radiation, but also the validity and capability of the derived algorithm in numerical modeling and computation for isotropic, chemo-elastoplastic- damage porous continuum. COUPLED CHEMO-ELASTOPLASTIC-DAMAGE MODEL Chemical and Mechanical Damage Models The Mazars Mechanical Damage Model The damage criterion can be postulated as: f d ðt ",t rÞ ¼ t " À t r ~ ~ 0, ð1Þ where t " is the equivalent strain and t r is the damage threshold at current ~ time t. The equivalent strain proposed by Mazars is defined as: vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u 3 uX À Á t ~¼t " t "m,e 2 , i ð2Þ i¼1 where t "m,e represents the i-th principal value of the elastic strain due to i mechanical effect at current time t and: & t m,e t m,e "i if t "m,e 4 0 i "i ¼ : ð3Þ 0 otherwise If 0 r denotes the initial damage threshold of the concrete, similar to the initial yield strength in the yield criterion, we must have for any current time t that t r ¼ maxð0 r,max "Þ, ~ ð4Þ where max " is the maximum value of " over the time period from 0 to t. ~ ~ Since the damaging mechanisms are different in uniaxial tension and uniaxial compression, mechanical damage parameter dm at current time t is taken as the weighted average of a uniaxial tension damage parameter dm,t and a uniaxial compression damage parameter dm,c, that is: dm ¼ t dm,t þ c dm,c , ð5Þ Chemo-Elastoplastic-Damage Constitutive Model for Plain Concrete 975 in which 0 rð1 À At Þ À À ÁÁ dm,t ¼ 1 À tr À At exp Bt 0 r À t r , ð6aÞ 0 rð1 À Ac Þ À À ÁÁ dm,c ¼ 1 À tr À Ac exp Bc 0 r À t r , ð6bÞ X "m,e "m,e À 3 i,t i Á t t ¼ t "2 ~ "40 , ð6cÞ i¼1 ~ X "m,e "m,e 3 i,c i c ¼ t "2 ðt " 4 0Þ, ~ ð6dÞ i¼1 ~ where At, Bt, Ac, Bc, are material damage parameters, "m,e , "m,e represent the i,t i,c i-th principal values of the mechanical elastic strain due to positive and negative effective principal stresses, respectively, and "m,e ¼ "m,e þ "m,e . i i,t i,c In uniaxial tension, at ¼ 1 and ac ¼ 0. In uniaxial compression, at ¼ 0 and ac ¼ 1. The damage evolution can be defined by the rate equation in the form: _ @dm _ ~ _ dm ¼ Hð",dm Þ ¼ , ð7Þ ~ @" _ where ! 0 is the damage consistency parameter. According to the damage condition (1), the definitions (2) and (5), we have: 1 À m,e ÁT m,e _ ¼ e _ e , ð8Þ " þ ~ where em,e is the positive elastic strain vector due to mechanical effect, which þ can be expressed in terms of its eigen-pairs determined with the spectral decomposition: X 3 em,e ¼ þ "m,e L0i , i ð9Þ i¼1 with Â ÃT L0 i ¼ n2 i,1 n2 i,2 n2 i,3 ni,1 ni,2 ni,2 ni,3 ni,3 ni,1 ði ¼ 1,2,3Þ, ð10Þ Â ÃT where ni,1 ni,2 ni,3 denotes the i-th principal direction associated with the i-th principal elastic strain "m,e due to mechanical effect. i 976 R. LI AND X. LI According to the damage parameters defined in terms of expressions (5) and (6), we may have: @dm ~ Hð",dm Þ ¼ ~ @" ! 0 rð1 À At Þ À À ÁÁ ¼ t þ At Bt exp Bt 0 r À " ~ "2 ~ ! 0 rð1 À Ac Þ À À0 ÁÁ þ c þ Ac Bc exp Bc r À " ~ , ð11Þ "2 ~ provided a radial loading is assumed (Mazars, 1984), that is, dt ¼ dc ¼ 0. The Chemical Damage Model The desalination of concretes with increasing temperature causes the reduction of the mechanical stiffness of the concrete material. Such an effect can be described with the chemical damage parameter ds, defined by Saetta et al. (1998, 1999), the evolution of which depends on the mass con- centration of the dissolved matrix component cp in terms of the following relationship: 1 ds ¼ ð1 À ’Þ 1 À , ð12Þ 1 þ ð2Rc Þ4 where the relative concentration Rc ¼ cp/cref represents the degree of chem- ical reaction (the desalination) defined as a ratio between the actual concen- tration of the dissolved matrix component cp with its reference value cref, that specifies the concentration for which the chemical degradation process reaches its maximum effect, u is the relative (normalized) residual strength of concrete due to the desalination achieved when the desalination is comple- tely developed. To quantitatively describe the damage effect due to the dehydration, the chemical damage parameter dh, proposed by Lackner et al. (2002), is intro- duced as: pﬃﬃﬃ dh ¼ 1 À , ð13Þ where n is the hydration degree defined as a ratio between current hydration mass msk and initial hydration mass m0 of chemically bound water depend- ing on temperature T and is expressed in the form (Gawin et al., 1999, 2003): msk Hh Ah ðTÞ ¼ ¼1þ ðT À Th0 Þ: ð14Þ m0 m0 Chemo-Elastoplastic-Damage Constitutive Model for Plain Concrete 977 In Equation (14), Hh ¼ Hh(T À Th0) is a Heaviside function and Th0 ¼ 393.15 K is the threshold temperature above which the dehydration of concrete may occur, the dehydration coefficient Ah ¼ À(0.04 À 0.08) kg/m3K (Gawin et al., 1999). With combination of the definitions (12) and (13), we denote dc as the chemical damage parameter due to both the dehydration and desalination effects given by: dc ¼ 1 À ð1 À ds Þð1 À dh Þ: ð15Þ It is remarked that with consideration of degradation of the elastic mod- ulus of the concrete skeleton due to the dehydration and desalination caused by increasing high temperatures, the growing thermal damage due to chem- ical deterioration of concrete is taken into account in the proposed model. Coupled Chemical and Mechanical Damage Model With further combination of both mechanical and chemical damage para- meters defined in terms of expressions (5) and (15), the total damage para- meter d, which synthesizes both chemically and mechanically induced damage effects can be defined as below: d ¼ 1 À ð1 À dm Þð1 À dc Þ ¼ 1 À ð1 À dm Þð1 À ds Þð1 À dh Þ, ð16Þ _ with d 2 ½0,1 and d ! 0. Then the net stress tensor p00 linked to the effective stress tensor p00 taking into account the damage effect can be written as: " p00 ¼ ð1 À d Þp00 , " ð17Þ where the net stress tensor r00 commonly used in geo-materials can also be expressed as: p00 ¼ p þ pg I, ð18Þ in which r is the total stress tensor, pg represents the pore gas pressure and I the identity matrix. Throughout this article, the stress is defined as tension positive whereas the pressure is compressive positive. Generalized WillamWarnke Yield Criterion In the proposed model, the generalized WillamWarnke yield criterion (Willam and Warnke, 1975; Ulm et al., 1999) depending on the three stress invariants is employed to describe the coupled chemo-elastoplastic behavior occurring in concrete, with consideration of mechanically induced strain softening/hardening, suction hardening and chemo-plastic softening due to the dehydration and desalination. The failure surface of the generalized 978 R. LI AND X. LI WillamWarnke yield criterion is shown in Figure 1 and the yield function is given by: " 00 " f ¼ q þ rðÞðm À cð"p , pc , , cp ÞÞ, ð19Þ " 00 where ðm , q, Þ stand for the three stress invariants of the effective stress " 00 "ij " 00 " 00 " 00 tensor ij ¼ s00 þ m ij with the three principal stresses 1 ! 2 ! 3 , in " 00 "00 " 00 which sij are denoted as the deviatoric part of ij and ij is the Kronecker delta. The three stress invariants are defined as: rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 00 00 1 00 " 00 " 00 2 00 À 2 À 3 " q¼ sij sij , m ¼ ii , cosðÞ ¼ 1 pﬃﬃﬃﬃﬃ " " " 00 " : ð20Þ 2 3 12q The frictional coefficient r() depends on Lode angle : uþv rðÞ ¼ , ð21Þ w varying between the frictional coefficient rt ¼ r( ¼ 0 ) on the tensile merid- ian and the frictional coefficient rc ¼ r( ¼ 60 ) on the compressive meridian, in which: u ¼ 2rc ðr2 À r2 Þ cos , c t 1 v ¼ rc ð2rt À rc Þ½4ðr2 À r2 Þ cos2 þ 5r2 À 4rt rc 2 , c t t ð22Þ 2 w¼ 4ðr2 c À r2 Þ cos2 t þ ðrc À 2rt Þ : q Drucker–Prager Willam–Warnke rc 1 Uniaxial σ′′ σ′′ 3 2 compression √3 Rc /√2 1 c σ′′ m R=cst R (q) Rt /√2 Uniaxial tension 1 √3 /2 1 rt Biaxial compression σ′′ 1 q " 00 Figure 1. Generalized WillamWarnke yield criterion: in (q, m ) plane; (b) in deviatoric stress plane. Chemo-Elastoplastic-Damage Constitutive Model for Plain Concrete 979 " cð"p , pc , , cp Þ is the cohesion pressure depending on the equivalent plastic " strain "p , the capillary pressure pc, the hydration degree n and the concen- tration cp of the dissolved matrix component. With the piecewise linear " hardening/softening assumption, the cohesion parameter cð"p , pc , , cp Þ of concrete can be expressed in the form: " " cð"p , pc , , cP Þ ¼ c0 þ hp "p þ hs pc þ h ð1 À Þ þ hc cp , ð23Þ where c0 is the initial cohesion pressure and hp, hs, hn, hc are material hard- ening/softening parameters. As uniaxial tensile and compressive strengths ft, fc and biaxial compressive strength fbc are defined, c0, rt, and rc can be expressed as: fbc ft pﬃﬃﬃ fbc À ft pﬃﬃﬃ ð fbc À ft Þ fc c0 ¼ , rt ¼ 3 , rc ¼ 3 : ð24Þ fbc À ft 2fbc þ ft 3fbc ft þ fbc fc À ft fc As hn < 0, hc < 0, the cohesion will decrease due to both the dehydration and the desalination developed with increasing temperature. Thus, the grow- ing thermal de-cohesion due to chemical deterioration of concrete can be taken into account in the proposed model. CONSTITUTIVE MODELING FOR COUPLED CHEMO-ELASTOPLASTIC-DAMAGE MODEL: COMPUTATIONAL ALGORITHM _ Let us assume that the total strain rate vector e is additively decomposed as (Wu et al., 2004): e ¼ em,e þ eT,e þ es,e þ e,e þ ec,e þ em,p , _ _ _ _ _ _ _ ð25Þ m,e m,p T,e s,e ,e c,e _ _ _ _ _ _ where e , e , e , e , e , e are denoted, respectively, as elastic and plas- _ tic portions of the strain rate e due to mechanical effect, strain rates due to thermal expansion, capillary pressure (suction), the hydration and the desal- ination effects. For the case of a 3D solid, the effective stress vector p00 and the strain " vector e take the form: Â 00 Ã " " 00 " 00 "00 p00 ¼ x y z xy yz zx , " "00 "00 T ð26Þ Â ÃT e ¼ "x "y "z xy yz zx : ð27Þ Â 00 00 00 00 00 00 ÃT The net stress vector p00 ¼ x y z xy yz zx linked to p00 and " m,e further e can be expressed as: p00 ¼ ð1 À d Þp00 ¼ ð1 À d ÞDem,e , " ð28Þ 980 R. LI AND X. LI where D is the elastic modulus matrix. The plastic strain rate em,p due to _ mechanical effect is given according to the plastic flow rule: _ @g em,p ¼ l 00 , _ ð29Þ " @p in which g is the plastic potential function and the non-associated plasticity is considered. One can obtain its associated form by simply assuming g ¼ f. " The accumulated equivalent plastic strain "p can be calculated by the " integration of incremental equivalent plastic strain Á"p as: Z " "p ¼ Á"p ," ð30Þ where the incremental equivalent plastic strain Á"p is defined as: " rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ! u 2 p p u2 @g @g 1 @g @g " Á"p ¼ Áe Áe ¼ l t À , ð31Þ 3 ij ij " 00 " 00 3 @ij @ij 3 @kk @ll " 00 " 00 in which ep is denoted as the deviatoric part of the plastic strain tensor "m,p . ij ij The elastic strain rate es,e due to the suction variation is given by (Alonso _ et al., 1990): s 1 es,e ¼ hs,e pc ¼ À _ _ _ m pc , ð32Þ ð1 þ eÞ ð pc þ pat Þ in which e is the void ratio, pat is the atmospheric pressure, s is elastic stiffness parameter due to the suction, and the unit main pressure vector Â ÃT is defined as m ¼ 1 1 1 0 0 0 . It is noted that in the hygroscopic region of concretes, where the satura- tion degree of the liquid phase Sw is lower than the solid saturation point Sssp, there exists no capillary water but bound water. Under this condition there exists neither capillary water nor capillary water pressure. As a con- sequence, the definition of the capillary pressure pc defined as the difference of the pressures in capillary water and gas mixture phases will no longer be valid. The capillary pressure is only formally used to denote the water adsorption potential describing thermodynamic equilibrium of the bound water with the vapor. Hence, in the proposed mathematical model the same symbol pc is used for both capillary pressure as Sl > Sssp and adsorption potential as Sl Sssp (Li et al., 2006) and the suction-water saturation degree curve (Bourgeois et al., 2002) with a unified formula: Sw ¼ ð1 þ ðapc Þb Þc , ð33Þ is employed (Li et al., 2006), in which a ¼ 2.35 Â 10À8 PaÀ1, b ¼ 1.83, c ¼ À0.58. In this way, the model can calculate the shrinkage strain due to Chemo-Elastoplastic-Damage Constitutive Model for Plain Concrete 981 the suction variation, including the transition from the capillary region to the hygroscopic region. The thermal strain rate eT,e is evaluated by: _ _ 1 _ eT,e ¼ Te T ¼ s mT, _ ð34Þ 3 where s depends on the type of aggregates in concrete, for concretes with calcareous or silicious aggregates we have s ¼ 1.8 À 2.1 Â 10À5/K or 3.6 À 3.9 Â 10À5/K, respectively (Ulm et al., 1999). The strain rates due to variations of the hydration degree and the desalination degree are written in the forms: _ _ _ e,e ¼ D,e ¼ Àh m, ec,e ¼ Dc,e cp ¼ p mcp , _ _ _ ð35Þ where h and p are chemical expansion coefficients of concrete due to the hydra- tion and the desalination, respectively. In the present work, we take h ¼ 4.286 Â 10À4 (Sercombe et al., 2000) and p is taken as the function of the concentration of the dissolved matrix component of concrete (Hueckel, 1997): Â À ÁÃ 1 p ¼ F0 0 exp 0 1 À cp þ ln cp À1 , ð36Þ cp where F0 > 0 and 0 > 0 are material parameters depending on the property of the dissolved matrix component. The Computational Algorithm for the Coupled Chemo-elastoplastic-damage Model The coupled chemo-elastoplastic-damage analysis of concretes at high temperature is a strongly nonlinear problem. Based on the work of Ju (1989), Sercombe et al. (2000), Li (1995) and Li et al. (1994) a three-step operator split algorithm for the chemo-elastoplastic-damage analysis is developed. The known stress vector and state variables at time t: ðt p00 t m,p e t dm Þ, ð37Þ are updated to those at time t + Át as: ðtþÁt p00 tþÁt m,p e tþÁt dm Þ: ð38Þ It is noted that the variables T, pc, n, and cp do not appear in the variable sets (37) and (38) to be updated in a NewtonRaphson iterative procedure as they are given by the solutions of the global equations governing the coupled chemo-thermo-hygro-mechanical process for the current incremen- tal step (Li et al., 2006) and fixed at the stage of integrating the rate 982 R. LI AND X. LI constitutive equations. The three-step operator split algorithm for the inte- gration of the rate constitutive equations for an incremental step from time tn ¼ t to tn+1 ¼ t + Át can be described as shown below. Elastic Predictor (1) Strain update. Given the incremental displacement field un+1 at a quad- rature point, the strain tensor is updated at gauss points as: enþ1 ¼ en þ runþ1 , ð39Þ where ruT ¼ ½ux,x nþ1 uy,y uz,z ux,y þ uy,x uy,z þ uz,y uz,x þ ux,z nþ1 : ð40Þ (2) Elastic trial stress. " trial p00 nþ1 ¼ Dðenþ1 À ðem,p þ eT,e þ es,e þ e,e þ ec,e ÞÞ: n nþ1 nþ1 nþ1 nþ1 ð41Þ Plastic Corrector n o È É " trial p00 nþ1 , dn , em,p ) p00 nþ1 , dn , em,p : n " nþ1 ð42Þ (3) Check for yielding & trial 0 elastic ) go to step ð5Þ f ðp00 nþ1 Þ " : ð43Þ 4 0 plastic ) return mapping (4) Plastic return mapping corrector. As the constitutive equation S ¼ p00 À Dem,e ¼ 0 and the elastoplastic " nþ1 nþ1 yield criterion f ðp00 Þ ¼ 0 are simultaneously fulfilled at the current iteration k: " nþ1 Sk ðp00 ,ÁlÞ ¼ SkÀ1 þ dS ¼ 0, " ð44Þ 00 " fk ðp ,ÁlÞ ¼ fkÀ1 þ df ¼ 0: ð45Þ The NewtonRaphson iteration for the update of the effective stress vector and the plastic multiplier at the integration point can be given as: 2 3 @S @S & ' & ' 6 @p00 @l 7 Áp00 " " ÀSkÀ1 4 @f @f 5 ¼ , ð46Þ Ál ÀfkÀ1 @p00 @l " Chemo-Elastoplastic-Damage Constitutive Model for Plain Concrete 983 in which @S @2 g 00 ¼ I þ lD 00 2 " @p @" "p 1 @r @cos T 1 1 ¼ I þ lD m þ PÀ " T Pp00 ðPp00 Þ " 3 @cos @p00 " 6q 36q3 2 T # 1 @r @cos T 00 @2 r @cos @cos T Ã @ cos þ " m þðm À cÞ þX , 3 @cos @p00 " @ðcosÞ2 @p00 " @p00 " @p00 2 " ð47Þ @S @g @2 g ¼ D 00 þ lD 00 @l @p" " @p @l ! 1 1 Àð21 À 2 À 3 Þ 1 Pp00 þ 2q2 ð2L1 À L2 À L3 Þ " 00 " 00 " 00 3 " ¼ D rðÞm þ Pp00 þ XÃ" pﬃﬃﬃﬃﬃ 3 6q 2 12q3 " @c d"p @r @ cos À lD , ð48Þ @"p dl @cos @p00 " " @f 1 1 Àð21 À 2 À 3 Þ 1 Pp00 þ 2q2 ð2L1 À L2 À L3 Þ " 00 " 00 " 00 3 " 00 ¼ rðÞm þ Pp00 þ XÃ " pﬃﬃﬃﬃﬃ , @p" 3 6q 2 12q3 ð49Þ @f " @f d"p ¼ , ð50Þ " @l @"p dl where I is the n00 Â n00 identity matrix and " " 2 3 2 À1 À1 0 0 0 6 À1 2 À1 0 0 07 6 7 6 À1 À1 2 0 0 07 P¼6 6 0 7, ð51Þ 6 0 0 6 0 077 4 0 0 0 0 6 05 0 0 0 0 0 6 @r ðu0 þ v0 Þw À ðu þ vÞw0 ¼ , ð52Þ @ cos w2 @2 r ½ðu00 þ v00 Þw À ðu þ vÞw00 w2 À 2½ðu0 þ v0 Þw À ðu þ vÞw0 ww0 2 ¼ ; ð53Þ @ðcos Þ w4 @ cos ð21 À 2 À 3 Þ 1 Pp00 À 2q2 ð2L1 À L2 À L3 Þ " 00 " 00 " 00 3 " 00 ¼À pﬃﬃﬃﬃﬃ , ð54Þ " @p 2 12q3 984 R. LI AND X. LI @2 cos ð21 À 2 À 3 Þ 1 P " 00 " 00 " 00 3 Pp00 " 00 2 ¼À pﬃﬃﬃﬃﬃ À pﬃﬃﬃﬃﬃ ð2L1 À L2 À L3 ÞT @p" 2 12q 3 6 12q3 " 00 " 00 " 00 ð21 À 2 À 3 Þ 00 ð2L1 À L2 À L3 Þ þ pﬃﬃﬃﬃﬃ " " T Pp ðPp00 Þ À pﬃﬃﬃﬃﬃ " T ðPp00 Þ , ð55Þ 12 12q 5 6 12q3 ðu0 þ v0 Þw À ðu þ vÞw0 " 00 XÃ ¼ ðm À cÞ , ð56Þ w2 @c ¼ hp , ð57Þ " @"p @f ¼ ÀrðÞhp , ð58Þ @"p" vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ! u d"p u2 @g @g 1 @g @g " t ¼ À , ð59Þ dl " 00 " 00 3 @ij @ij 3 @kk @ll " 00 " 00 with w0 2 u0 ¼ 2rc ðr2 À r2 Þ, v0 ¼ c t r ð2rt À rc Þ2 , w0 ¼ 8ðr2 À r2 Þ cos , ð60Þ 2v c c t Â ÃT Li ¼ n2 i,1 n2 i,2 n2 i,3 2ni,1 ni,2 2ni,2 ni,3 2ni,3 ni,1 ði ¼ 1, 2, 3Þ: ð61Þ Damage Corrector È 00 É È É pnþ1 , dn , em,p ) p00 , dnþ1 , em,p : " nþ1 nþ1 nþ1 ð62Þ (5) Damage evolution. vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u 3 D E2 uX ~ "nþ1 ¼t "m,e nþ1,i , ð63Þ i¼1 & dn ~ if "nþ1 À rn 0 dnþ1 ¼ , ð64Þ ~ ~ dn þ ð"nþ1 À "n ÞHnþ1 otherwise ~ rnþ1 ¼ maxðrn ,"nþ1 Þ, ð65Þ p00 ¼ ð1 À dnþ1 Þp00 : nþ1 " nþ1 ð66Þ Chemo-Elastoplastic-Damage Constitutive Model for Plain Concrete 985 Consistent Tangent Chemo-thermo-elastoplastic-damage Modulus Matrices According to the backward-Euler return algorithm, the effective stresses at time t +Át can be expressed as: @g " trial À ÁlD p00 ¼ p00 " , ð67Þ @p00 " where p00 trial is defined in Equation (41). Differentiation of both sides of (67) " nþ1 with respect to time gives: @g @2 g _ _ _ _ p00 ¼ D e À l 00 À Ál 00 2 p00 À eT,e À es,e À e,e À ec,e : " " _ _ _ _ ð68Þ @p" @p" Substitution of Equations (32), (34), and (35) into Equation (68) results in: @g @2 g _ _ " _ _ _ _ p00 ¼ D e À l 00 À Ál 00 2 p00 À Te T À hs,e pc À D,e À Dc,e cp , " _ _ ð69Þ @p" @p" That is, @g _ _ _ _ _ p00 ¼ . e À l 00 À Te T À hs,e pc À D,e À Dc,e cp , " _ _ ð70Þ @p" where À1 À1 @2 g . ¼ D þ Ál 00 2 : ð71Þ @p" The consistency condition of the generalized WillamWarnke yield criterion (19) can be given as: @f _ @f _ @f @f _ @f f_ ¼ 00 p00 þ l þ " _ pc þ þ _ cp ¼ 0: ð72Þ @p" @l @pc @ @cp Substitution of Equation (70) into (72) gives: " # _ @f T À e _ s,e ,e _ c,e Á @f _ @f @f l¼ _ _ _ . e À T T À h pc À D À D cp þ þ _ cp þ _ pc @p00 " @ @cp @pc " # @f T @g " @f d"p Ä . 00 À : ð73Þ @p00 " " @p " @"p dl Then, substitution of Equation (73) into Equation (70) results in: _ "m _ "T _ "s _ "c _ p00 ¼ Dedpc e þ Dedpc T þ Dedpc pc þ Dedpc cp , " ð74Þ 986 R. LI AND X. LI where the consistent chemo-thermo-elastoplastic tangent modulus matrices are given as: @g @f T . 00 . " m ¼ . À p @p@" " 00 Dedpc @f T @g @f d"p " . 00 À @p00 " @p" " @"p dl 2 3 @g @f T e 6 . 00 .T 7 "T 6 @p @p00 " " 7 Dedpc ¼ À6.Te À T 7 4 @f @g @f d"p 5 " . 00 À @p00 " @p" " @"p dl 2 " T #3 @g @f @f ,e 6 . 00 À .D 7 6 ,e " @p @ @p00 " 7 Hh Ah À66.D þ 7 4 @f T @g @f d"p 7 m0 " 5 ð75Þ . 00 À @p00 " @p " " @"p dl 2 " T #3 @g @f @f s,e 6 . 00 À .h 7 s 6 s,e " @p @pc @p00 " 7 Dedpc ¼ À6.h þ T " 6 7 4 @f @g @f d"p 7 " 5 . 00 À @p00 " @p" " @"p dl 2 " T #3 @g @f @f c,e 6 . 00 À .D 7 c 6 c,e " @p @cp @p00 " 7 Dedpc ¼ À6.D þ T " 6 7: 4 @f @g @f d"p 7 " 5 . 00 À @p00 " @p " " @"p dl From Equation (28) the rate net stress vector may then be written as: _ _" p00 ¼ ð1 À d Þp00 À dp00 _ " ð76Þ _ 00 À ð1 À dm Þ @dc þ @dc cp Dem,e " ¼ Med p _ _ @ @cp where H m,e À m,e ÁT À1 Med ¼ ð1 À d ÞI À ð1 À dc Þ De eþ D : ð77Þ ~ " Finally, substitution of Equation (74) into (76) results in: _ p00 ¼ Dm e þ DT T þ Ds pc þ Dc cp , _ edpc _ edpc edpc _ edpc _ ð78Þ Chemo-Elastoplastic-Damage Constitutive Model for Plain Concrete 987 where the consistent chemo-thermo-elastoplastic-damage tangent modulus matrices are given as: @g @f T Med . 00 . m @p @p00 " " Dedpc ¼Med . À T @f @g @f d"p " . 00 À @p00 " " @p @"p dl " 2 3 @g @f T e 6 Med . 00 .T 7 6 @p @p00 " " 7 DT ¼ À6Med .Te À T edpc 7 4 @f @g @f d"p 5" . 00 À @p00 " " @p @"p dl " 2 " T # @g @f @f ,e 6 Med . 00 À .D 6 " @p @ @p00 " À66Med .D,e þ T 4 @f @g @f d"p " Ä 00 À @p00 " " @p @"p dl " 3 7 @dc m,e 7 Hh Ah þ ð1 À dm Þ De 7 7 m0 @ 5 2 " T #3 @g @f @f s,e 6 Med . 00 À .h 7 6 " @p @pc @p00 " 7 Dedpc ¼ À6Med .h þ s 6 s,e T 7 7 4 @f @g " @f d"p 5 . 00 À @p00 " " @p @"p dl " 2 " T # 3 @g @f @f c,e 6 Med . 00 À .D 7 6 " @p @cp @p00 " @dc 7 Dc ¼ À6Med .Dc,e þ edpc 6 T þ ð1 À dm Þ Dem,e 7: 7 4 @f @g @f d"p" @cp 5 . 00 À @p00 " " @p @"p dl " ð79Þ NUMERICAL EXAMPLES The first two examples concern a bar carrying an axial load as depicted in Figure 2 under axial displacement control. As simple analytical solutions may be derived the two 1D problems are tested to demonstrate the validity 988 R. LI AND X. LI P 1 10 Figure 2. Axial cycling load problem of test bar in plane strain: geometry and boundary conditions. of the proposed algorithm for the coupled elastoplastic-damage analysis. The third example presented by Gawin et al. (1999, 2003) is performed to illustrate the capability of the proposed constitutive model in reproducing coupled chemo-thermo-hygro-mechanical behavior as well as in modeling deteriorations in both strength and stiffness in concretes subjected to fire and thermal radiation. In passing, as the pore fluid effect is not involved in the first two exam- ples, the (total) stresses, instead of the effective stresses, will be directly linked to the strains throughout the two examples. Elastic Damage The first example considers a bar subjected to an axial cycle load. The purpose of this example is to exhibit the capability of the present model in the simulation of the elastic damage response. The bar is cycled in tension over two cycles with an increasing peak tensile displacement prescribed at the top of the bar Cycle 1: displacement 0 ! 0:2 ! 0; Cycle 2: displacement 0 ! 0:4 ! 0. The elastic and damage parameters are set as: E0 ¼ 2000 Pa, ¼ 0, 0 r ¼ 0:005, At ¼ Ac ¼ 1, Bt ¼ Bc ¼ 100. The axial elastic stress r is then given by: ¼ "ð1 À dm ÞE0 , ð80Þ Chemo-Elastoplastic-Damage Constitutive Model for Plain Concrete 989 14 12 10 Stress Numerical 8 Analytical 6 4 2 0 0.00 0.01 0.02 0.03 0.04 0.05 Strain Figure 3. The stressstrain curve of axial elastic-damage test problem. and damage potential (5) reduces to the form: dm ¼ 1 À exp½100ð0:005 À "Þ: ð81Þ The onset of damage occurs when the initial threshold is breached at a strain level of e ¼ 0.005. Figure 3 illustrates the limiting stress, at increasing levels of strain, enforced by Equations (80) and (81) as well as the degrada- tion in the elastic stiffness (shown by the decrease in gradient during elastic unloading) following further damage. Exact agreement was achieved using the algorithm developed in this article with Equations (80) and (81), as shown in Figure 3. Pressure Dependent Elastoplastic Damage The purpose of this example is to demonstrate the validity of the derived algorithm in the integration of the constitutive behavior described by the proposed coupled elastoplastic damage model. The geometry used for this example is the same as the one used in the previous example. Regarding pressure dependent elastoplastic response to be modeled, the frictional coefficient r() defined by Equation (21) for the WillamWarnke yield function is assumed not to vary with the Lode angle since with this simplified assumption one may derive the analytical solutions for the example under the coupled elastoplastic damage. From the definitions of rt and rc, that is, rt ¼ r( ¼ 0 ) and rc ¼ r( ¼ 60 ), and from Equations (21) and (22), it is noted that the frictional coefficient r() will be independent from the Lode angle provided rt ¼ rc. Based on this equality and Equation (24), we have: 2ft fc fbc ¼ : ð82Þ 3ft À fc 990 R. LI AND X. LI Substitution of Equation (82) into Equation (24) leads to: 2ft fc pﬃﬃﬃ fc À ft c0 ¼ , r ¼ rt ¼ rc ¼ 3 : ð83Þ 3ð fc À ft Þ fc þ ft Finally, inserting Equation (83) into Equation (19) gives the degraded form of the WillamWarnke yield function: " f ¼ q þ rðm À cÞ, ð84Þ where pﬃﬃﬃ fc À ft r¼ 3 : ð85Þ fc þ ft The cohesion pressure c can be assumed to be a linear function of the " equivalent plastic strain "p and takes the following form: " c ¼ c 0 þ hp " p : ð86Þ The model parameters of the present example are chosen as E0 ¼ 2000 Pa, ¼ 0, 0 r ¼ 0:001, At ¼ Ac ¼ 1, Bt ¼ Bc ¼ 100, ð87Þ ft ¼ 40 Pa, fc ¼ 60 Pa, fbc ¼ 80 Pa, hp ¼ 100, That is, pﬃﬃﬃ 3 r¼ , c0 ¼ 80 Pa: ð88Þ 5 The bar is in tension under displacement control: the displacement pre- scribed at the top of the bar 0 ! 1. During the loading phase, damage is initiated at the strain level of e ¼ 0.01, followed by elastoplastic yielding as the axial tensile effec- tive stress level exceeds the threshold of tensile strength ft at e ¼ 0.02. The variation of axial stress for the loading section before plastic yielding is given by: ¼ E0 " ð" 0:01Þ, ð89Þ ¼ exp½100ð0:01 À "ÞE0 " 0:02Þ: ð0:01 " ð90Þ pﬃﬃﬃ " From Equations (84) and (86) and noticing q ¼ j j= 3 for this example, " we may calculate the level of equivalent plastic strain increasing with as soon as the threshold of plastic yielding is exceeded. ! 1 1 1 " "p ¼ pﬃﬃﬃ þ À c0 : " " ð91Þ hp 3r 3 Chemo-Elastoplastic-Damage Constitutive Model for Plain Concrete 991 In the context of associated plasticity, according to the definition (31) the relation of the incremental equivalent plastic strain Á"p with incremental " axial plastic strain Á"p can be given by: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 @f @f 1 @f @f 2 @f 2 l ¼ jÁ"p j: " Á"p ¼ l À ¼ ð92Þ " " 3 @ @ 3 @ @ " " 3 @ 3 " The variation of axial strain for the loading section after plastic yielding is given by: ! ! e 3 1 1 e 3 1 1 e "¼" þ pﬃﬃﬃ þ À c0 ¼ " þ " pﬃﬃﬃ þ E0 " À c0 , ð93Þ 2hp 3r 3 2hp 3r 3 which results in: 3 " þ 2hp c0 "e ¼ : ð94Þ 1 þ 2hp p1ﬃﬃr þ 1 E0 3 3 3 Therefore, the variation of axial stress for the section of the loading sec- tion after plastic yielding is given by: 2 0 13 3 3 " þ 2hp c0 " þ 2hp c0 ¼ exp4100@0:01 À A5E0 ð" 4 0:02Þ: 1 þ 2hp p1ﬃﬃr þ 1 E0 3 3 3 1 þ 2hp p1ﬃﬃr þ 1 E0 3 3 3 ð95Þ Figure 4 shows that exact agreement of the numerical results obtained by means of the proposed model to the analytical solutions was obtained. Square Concrete Column Exposed to Fire The example deals with a square concrete column of cross section 40 cm Â 40 cm exposed to fire. The example is modeled as a transient plane strain problem coupled with porous water and gas mixture flows as well as heat flow. The first 16 min of the coupled thermo-hydro-mechanical behavior of the column in fire is simulated. The time step size taken from its initial value Át ¼ 1 s varies up to Átmax ¼ 20 s in the heating period. The 15 Â 15 eight-node serendipity element mesh along with boundary conditions is shown in Figure 5 in which, by symmetry, only one quarter with sizes 20 cm Â 20 cm is taken and discretized. The column is heated at surface A according to the curve T ¼ 298 + 0.5Át [K] up to Tmax ¼ 798 K at tmax ¼ 1000 s The boundary conditions of the example are listed in Table 1, while the main material parameters employed in this example are shown in Table 2. 992 R. LI AND X. LI 25 20 15 Stress 10 Numerical Analytical 5 0 0.00 0.02 0.04 0.06 0.08 0.10 Strain Figure 4. The stressstrain curve of axial elastoplastic damage test problem. A B A Y X C Figure 5. Diagram of the FEM model for the heated column. Chemo-Elastoplastic-Damage Constitutive Model for Plain Concrete 993 Table 1. Boundary conditions for the heated column. Side Variables Values and coefficients A pg pg ¼ 101,325 Pa pc pv ¼ 1000 Pa, c ¼ 0.044 m/s T T ¼ 298 + 0.5Át (K), t (s), c ¼ 2 W/m2 K er0 ¼ 5.1 Â 10À8 W/m2 K4 cp qp ¼ 0 B ux ux ¼ 0 pg qg ¼ 0 pc qv ¼ qw ¼ 0 T qT ¼ 0 cp qp ¼ 0 C uy uy ¼ 0 pg qg ¼ 0 pc qv ¼ qw ¼ 0 T qT ¼ 0 cp qp ¼ 0 Table 2. Material parameters of concrete. Parameter Symbol Values Young’s modulus E0 42 GPa Poisson’s cofficient 0.2 Uniaxial tensile strength ft 6.4 MPa Uniaxial compressive strength fc 80 MPa Biaxial compressive strength fbc 88 MPa Material hardening/softening parameters hp À1 Â 107 MPa Material hardening/softening parameters hs 1 Â 10À3 Material hardening/softening parameters hn À2.76 Â 106 MPa Material hardening/softening parameters hc À1 Â 107 MPa Initial damage threshold 0 r 1 Â 10À4 Chemical expansion coefficient F0 1.6 Â 10À3 Chemical expansion coefficient 0 1.1 Thermal dilatation coefficient s 3.0 Â 10À5 KÀ1 Initial porosity n0 0.1 Absolute permeability coefficient k 3.7 Â 10À21 m2 994 R. LI AND X. LI Initial conditions over the whole domain to be analyzed for the exam- ple are given as follows: the initial saturation degree of liquid phase S0 ¼ 0:44, the initial pressure of gas mixture p0 ¼ 101:325 kPa correspond- l g ing to atmospheric pressure. Initial mass of chemically bound water m0 ¼ 80 kg/m3. Figures 611 show the results obtained for the distributions of tempera- ture, vapor pressure, equivalent plastic strain, mechanical damage parame- ter, chemical damage parameter due to both the dehydration and desalination effects, and total damage parameter, respectively, at discrete time levels t ¼ 4, 8, 12, and 16 min. Table 3 shows the convergence in the force norm of the equilibrium equations. It is observed that the convergence behavior is quadratic. It is observed that moisture content decreases rapidly with increasing temperature (Figure 6(a)(d)) in the zone close to the heated surface, where rapid evaporation develops and causes an increase of vapor pressure up to 0.7 MPa at the time t ¼ 16 min, as shown in Figure 7(a)(d). The physical process described above, together with thermal dilation of the outer part of the sample, while the internal remains still in the initial temperature, causes high tensile stresses and material degradation charac- terized by the plastic failure and the mechanical damage as illustrated by Figures 8 and 9. This material failure effect can be particularly observed at the corner of the column. The phenomenon possesses a typical significance to concrete members with prismatic shapes (Gawin et al., 1999, 2003; Brite Euram, 1999). It is noted that chemical damage parameter due to coupled dehydration and desalination takes a significant portion up to 16% of total damage parameter at the corner at time t ¼ 16 min as illustrated in Figure 10, and therefore the effect of chemical damage considerably deteriorates the mate- rial failure of the zone close to the heated surface as shown in Figure 11. High values of both the vapor pressure and total damage parameter at the corner zone close to the heated surface constitute a potential source to trigger the explosive spalling. Numerous experimental tests (Phan et al., 1997; Brite Euram, 1999) of concrete prism sample subjected to fire indicate that spalling often occurs at the corner close to the heated surface, that demonstrates the capability of the proposed constitutive model in reproducing coupled chemo-thermo-hygro-mechanical behavior in concretes subjected to fire and thermal radiation. Chemo-Elastoplastic-Damage Constitutive Model for Plain Concrete 995 (a) 0.20 298 (b) 0.20 298 315 333 333 367 0.16 350 0.16 402 368 436 385 471 403 505 0.12 420 0.12 540 0.08 0.08 0.04 0.04 0.00 0.00 0.00 0.04 0.08 0.12 0.16 0.20 0.00 0.04 0.08 0.12 0.16 0.20 (c) 0.20 298 (d) 0.20 298 355 370 413 441 0.16 470 0.16 513 528 585 585 657 643 728 0.12 700 0.12 800 0.08 0.08 0.04 0.04 0.00 0.00 0.00 0.04 0.08 0.12 0.16 0.20 0.00 0.04 0.08 0.12 0.16 0.20 Figure 6. Temperature distributions in concrete column at time levels: (a) 4 min; (b) 8 min; (c) 12 min; (d) 16 min. (a) 0.20 0 (b) 0.20 0 1.7E4 7.1E4 3.4E4 1.4E5 0.16 5.1E4 0.16 2.1E5 6.9E4 2.9E5 8.6E4 3.6E5 1.0E5 4.3E5 0.12 1.2E5 0.12 5.0E5 0.08 0.08 0.04 0.04 0.00 0.00 0.00 0.04 0.08 0.12 0.16 0.20 0.00 0.04 0.08 0.12 0.16 0.20 (c) 0.20 0 (d) 0.20 0 8.6E4 1.0E5 1.7E5 2.0E5 0.16 2.6E5 0.16 3.0E5 3.3E5 4.0E5 4.3E5 5.0E5 5.1E5 6.0E5 0.12 6.0E5 0.12 7.0E5 0.08 0.08 0.04 0.04 0.00 0.00 0.00 0.04 0.08 0.12 0.16 0.20 0.00 0.04 0.08 0.12 0.16 0.20 Figure 7. Vapor pressure distributions in concrete column at time levels: (a) 4 min; (b) 8 min; (c) 12 min; (d) 16 min. 996 R. LI AND X. LI (a) 0.20 0 (b) 0.20 0 7.9E–5 2.3E–4 1.6E–4 4.6E–4 0.16 2.4E–4 0.16 6.9E–4 3.1E–4 9.1E–4 3.9E–4 1.1E–3 4.7E–4 1.4E–3 0.12 5.5E–4 0.12 1.6E–3 0.08 0.08 0.04 0.04 0.00 0.00 0.00 0.04 0.08 0.12 0.16 0.20 0.00 0.04 0.08 0.12 0.16 0.20 (c) 0.20 0 (d) 0.20 0 2.5E–4 2.9E–4 5.0E–4 5.7E–4 0.16 1.0E–3 0.16 1.1E–3 2.0E–3 2.3E–3 4.0E–3 8.0E–3 6.0E–3 1.4E–2 0.12 7.0E–3 0.12 1.6E–2 0.08 0.08 0.04 0.04 0.00 0.00 0.00 0.04 0.08 0.12 0.16 0.20 0.00 0.04 0.08 0.12 0.16 0.20 Figure 8. Equivalent plastic strain distributions in concrete column at time levels: (a) 4 min; (b) 8 min; (c) 12 min; (d) 16 min. (a) 0.20 0% (b) 0.20 0% 5.0% 5.0% 10% 10% 0.16 15% 0.16 20% 20% 25% 25% 40% 30% 50% 0.12 35% 0.12 60% 0.08 0.08 0.04 0.04 0.00 0.00 0.00 0.04 0.08 0.12 0.16 0.20 0.00 0.04 0.08 0.12 0.16 0.20 (c) 0.20 0% (d) 0.20 0% 5.0% 5.0% 10% 10% 0.16 20% 0.16 20% 25% 25% 40% 40% 60% 60% 0.12 0.12 80% 70% 0.08 0.08 0.04 0.04 0.00 0.00 0.00 0.04 0.08 0.12 0.16 0.20 0.00 0.04 0.08 0.12 0.16 0.20 Figure 9. Mechanical damage distributions in concrete column at time levels: (a) 4 min; (b) 8 min; (c) 12 min; (d) 16 min. Chemo-Elastoplastic-Damage Constitutive Model for Plain Concrete 997 (a) 0.20 0% (b) 0.20 0% 0.01% 0.86% 0.03% 1.7% 0.16 0.05% 0.16 2.6% 0.1% 3.4% 0.2% 4.3% 0.5% 5.1% 0.12 0.12 6.0% 1% 0.08 0.08 0.04 0.04 0.00 0.00 0.00 0.04 0.08 0.12 0.16 0.20 0.00 0.04 0.08 0.12 0.16 0.20 (c) 0.20 0% (d) 0.20 0% 1.6% 2.3% 3.1% 4.6% 0.16 4.7% 0.16 6.9% 6.3% 9.1% 7.9% 11% 9.4% 14% 0.12 0.12 16% 11% 0.08 0.08 0.04 0.04 0.00 0.00 0.00 0.04 0.08 0.12 0.16 0.20 0.00 0.04 0.08 0.12 0.16 0.20 Figure 10. Chemical damage distributions in concrete column at time levels: (a) 4 min; (b) 8 min; (c) 12 min; (d) 16 min. (a) 0.20 0% (b) 0.20 0% 5.0% 5.0% 10% 10% 0.16 15% 0.16 20% 20% 25% 25% 40% 30% 50% 0.12 35% 0.12 60% 0.08 0.08 0.04 0.04 0.00 0.00 0.00 0.04 0.08 0.12 0.16 0.20 0.00 0.04 0.08 0.12 0.16 0.20 (c) 0.20 0% (d) 0.20 0% 5.0% 5.0% 10% 10% 0.16 20% 0.16 20% 25% 25% 40% 40% 60% 60% 0.12 0.12 80% 80% 0.08 0.08 0.04 0.04 0.00 0.00 0.00 0.04 0.08 0.12 0.16 0.20 0.00 0.04 0.08 0.12 0.16 0.20 Figure 11. Total damage distributions in concrete column at time levels: (a) 4 min; (b) 8 min; (c) 12 min; (d) 16 min. 998 R. LI AND X. LI Table 3. Force norm in the equilibrium equations for 57 steps. Step Force norm 6 1.20 Â 102 2.01 8.36 Â 10À4 27 1.66 8.33 Â 10À2 7.36 Â 10À4 57 5.42 Â 10À1 3.62 Â 10À2 6.21 Â 10À4 CONCLUSIONS A coupled elastoplastic-damage constitutive model with consideration of chemo-induced elastoplastic-damage effects for the modeling of coupled chemo-thermo-hygro-mechanical behavior of concretes at high temperature is proposed in this article. The main features of the proposed constitutive model can be summarized as follows: (1) To account for the complex behavior of concrete subjected to fire, the Mazars isotropic damage model and the WillamWarnke elastoplastic yield criterion have been effectively combined to form the coupled con- stitutive model. As compared to the existing models for the same pur- pose the developed coupled model is capable of simulating occurrence and evolution of the micro-crack or micro-void growth measured by damage parameters as well as accompanied plastic yielding observed in concretes in terms of effective plastic strain. (2) The thermally-induced chemical effects of both the desalination and the dehydration processes on the material damage such as the deterioration of the Young’s modulus and the loss of the material strength are taken into account. 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