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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 6—11 appear in color online: http://ijd.sagepub.com

International Journal of DAMAGE MECHANICS, Vol. 19—November 2010                    971
            1056-7895/10/08 0971—30 $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 immiscible—miscible 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 Willam—Warnke 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 ½0,1Š 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 Willam—Warnke Yield Criterion

  In the proposed model, the generalized Willam—Warnke 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

Willam—Warnke 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 Š2 ,
                                                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 Willam—Warnke 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 Àh 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 Newton—Raphson 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 Šnþ1 :     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 Newton—Raphson 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 Šw2 À 2½ðu0 þ v0 Þw À ðu þ vÞw0 Šww0
          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 Willam—Warnke 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 stress—strain curve of axial elastic-damage test problem.


and damage potential (5) reduces to the form:
                      dm ¼ 1 À exp½100ð0:005 À "ފ:                               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
Willam—Warnke 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 Willam—Warnke 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 "ފE0 "              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  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  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 stress—strain 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 6—11 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 Willam—Warnke 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.
(3) A three-step operator split algorithm in numerical modeling and com-
    putation for isotropic, chemo-elstoplastic-damage porous continuum is
    developed.
(4) Consistent tangent modulus matrices for coupled nonlinear chemo-
    thermo-hygro-mechanical constitutive modeling are derived to ensure
    the second order convergence rate of the global iterative solution
    procedure.
Chemo-Elastoplastic-Damage Constitutive Model for Plain Concrete                          999

                              ACKNOWLEDGMENTS

  The authors are pleased to acknowledge the support of this work by the
National Natural Science Foundation of China through contract/grant
numbers 10672033, 90715011, 50278012 and the National Key Basic
Research and Development Program (973 Program) through contract
number 2010CB731502. The first author would also like to acknowledge
the support of Education Ministry of Liaoning Province of China through
the College Research Program with contract number 2009A064.


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