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