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 Pergamon Int. J. Mech. Sci. Vol. 39, No. 4, pp. 473M.86, 1997
Copyright ~.', 1996 Elsevier Science Ltd
Printed m Gr~at Britain. All rights reserved
0020 7403/97 $17.00 + 0.00

PII:S0020-7403(96)000034-3

CONSTITUTIVE AND DAMAGE EVOLUTION EQUATIONS

O F ELASTIC-BRITTLE MATERIALS BASED O N
IRREVERSIBLE T H E R M O D Y N A M I C S

S. MURAKAMI and K. KAMIYA

Department of Mechanical Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-01, Japan

(Received 5 April 1995; and in revised form 19 February 1996)

ANtract--Unified descriptions of the constitutive and evolution equations of elastic-brittle damage materials
are developed on the basis of irreversible thermodynamic theory for constitutive equations. The Helmholtz free
energy is assumed to be a function of elastic strain tensor and second rank symmetric damage tensor. In order
to take account of the effects of unilateral condition of damage due to the opening and closure of mierocracks,
modified elastic strain tensor is introduced into the Helmholtz free energy. A damage dissipation potential
related to the entropy production rate is expressed in terms of damage conjugate force. The constitutive and the
damage evolution equations derived by these potentials were applied to an elastic-brittle damage material. The
anisotropic elastic-brittle damage behavior of high-strength concrete under uniaxial, proportional and non-
proportional combined loading was analysed to elucidate the utility and the limitations of the present theory.
Finally, the initial damage surfaces in the axial-shear and biaxial stress spaces are calculated. Copyright ~2) 1996
Elsevier Science Ltd.

Keywords: elastic-brittle material, damage, constitutive equation, evolution equation, irreversible thermo-
dynamics, Helmholtz free energy, dissipation potential, unilateral damage, combined loading, damage
surface.

NOTATION
mk thermodynamic conjugate force corresponding to V~:
B thermodynamic conjugate force of
D and Dij second rank symmetric damage tensor and its components
D~ principal value of the tensor D
Eo initial Young's modulus
F dissipation potential
g temperature gradient; g =- - grad T
J mechanical flux vector
Kd material constant
L material constant tensor of fourth rank
Bi unit vector of principal direction of the tensor D
q heat flux
S specific entropy
T absolute temperature
v~ internal state variable tensor of even rank
X thermodynamic conjugate force corresponding to J
Y and Yij damage conjugate force tensor and its components
r~q equivalent damage conjugate force
scalar variable prescribing further development of damage
g strain tensor
g and e~j elastic strain tensor and its components
ge
modified elastic strain tensor
gP plastic strain tensor
qt, q2, q3, 94 material constants
2 and ,u Lain+ constants
P0 initial Poisson's ratio
P mass density
a and alj Cauchy stress tensor and its components
¢ specific Helmholtz free energy
specific entropy production rate
material constant describing the crack closure effect

473
474 S. Murakamiand K. Kamiya

1. INTRODUCTION
Continuum Damage Mechanics (CDM) has made significant development in the past two decades,
and has proved to be a systematic and promising approach to the analysis of damage and fracture
process, or the process of the damage and deformation interaction in wide variety of materials [ 1-4]. The
most crucial problems for the further development of CDM, however, will be the proper and accurate
modeling of material damage, besides the numerical schemes for the solution of initial and boundary
value problems in computational CDM. For this purpose, a number of irreversible thermodynamics
theories of CDM [-2-19] have been developed so far as a systematic framework to the unified description
of the inelastic deformation and the related internal structural change due to material damage.
Lemaitre, Chaboche and their co-workers [2 6,19,20] in particular, developed a series of
irreversible thermodynamic theory for wide variety of problems ranging from elastic-brittle to
elastic-plastic-ductile, creep, and fatigue damage of metals, geological materials and composites.
Krajcinovic and others [4,7-10] on the other hand, developed thermodynamic theories of
anisotropic damage mainly by use of vector damage variables, and discussed in detail the damage
process of brittle and creep materials. Chow and his co-workers [I 1-14] proposed an energy-based
elastic-plastic damage model in order to describe the difference in the observed failure modes of
geological materials under compression and tension, by use of a damage tensor identified by the
fourth rank elasticity tensor and a fourth rank projection tensor. Simo and Ju [15, 16] and Voyiadjis
and Kattan [17, 18] furthermore, discussed the coupling between elastic-plastic deformation and
anisotropic damage by use of a second rank symmetric damage tensor.
In the present paper, we will develop the description of anisotropic damage behavior of
elastic-brittle materials under multiaxial state of stress by employing a second rank symmetric
damage tensor [21,22] in the irreversible thermodynamic frameworks. The Helmholtz free energy is
first expressed as a function of simultaneous invariants of the elastic strain tensor and the damage
tensor by taking account of the unilateral behavior of elastic-brittle material. A dissipation potential
is represented as a quadratic function of the conjugate force of the damage tensor to ensure the
necessary condition for a thermodynamic potential. Constitutive equations are applied to describe
the anisotropic elastic damage behavior of high-strength concrete under various loading histories.

2. DAMAGE VARIABLES
Damage in materials is usually induced by nucleation, growth and coalescence of certain
microscopic cavities. Since the development of these cavities is governed by the action of applied
stress and strain, material damage is essentially anisotropic; this feature is especially important in
brittle materials damaged by the development of distributed and oriented microscopic cracks
[23, 24]. Thus, a scalar damage variable often has a serious limitation to the description of actual
material damage, and a number of theories have been developed to model the anisotropic damage
state by means of damage variables ranging from a vector to higher rank tensors [21, 28].
Vector damage variables [7 10, 25], above all, would be straightforward to be applied to describe
the distributed plane cracks; i.e. to describe the normal direction of each plane crack and the crack
density of the specific plane. The vector damage variables, however, cannot represent the effect of
a set of plane cracks of different orientations by means of a simple addition of the corresponding
vectors. Moreover, odd rank tensors in general can be excluded from the set of internal variables,
since they do not conform to the invariance with respect to the rotation of the frame [26].
Thus, in spite of the complexity of their mathematical structures, more elaborate theories have
been proposed for the description of the state and the effect of the distributed cavities by use of
higher and even rank tensors. Among them, second rank symmetric damage tensors [21,22, 26 28]
are most commonly employed because they are mathematically simpler than the higher rank of
tensors, and yet can describe most essential features of anisotropic damage.
In order to develop a feasible damage theory in this paper, we will postulate that the anisotropic
damage state can be described by a damage tensor
3
D = ~ Dini@ni (1)
i-1

where Di and ni are the principal value and the unit vector of principal direction of the tensor D. The
damage tensor D of Eqn (1) was derived by Murakami and Ohno [21,22] by postulating that the
 Constitutive and damage evolution equations 475

principle effect of the material damage consists in the net area reduction due to three-dimensional (3D)
distribution of microscopic cavities in the material. Then, Di in Eqn (1) can be interpreted as the ratio of
area reduction in the plane perpendicular to n~ caused by the development of cavities. Though the
second rank symmetric damage tensor D cannot describe more complicated damage state than
orthotropy, it has been often employed in the development of anisotropic damage theories [-11-24].

3. I R R E V E R S I B L E T H E R M O D Y N A M I C T H E O R Y FOR M A T E R I A L D A M A G E

Irreversible thermodynamics furnishes rational frameworks for the unified formulation of consti-
tutive and evolution equations of damaged material caused by internal change of the material
[-2 20, 29, 30]. For the sake of the subsequent development of this paper, we will first make a brief
review of the constitutive theory of thermodynamics with internal variables [3, 4, 8, 28, 29].
In the case of infinitesimal deformation, the Clausius-Duhem inequality has the following form:

q
a:g-p(~+s7 ~)-~.grad T~>0 (2)

where a and ~ are stress and strain tensor, p, 4', s, T and q are mass density, specific Helmholtz free
energy, specific entropy, absolute temperature and heat flux, respectively.
We will first assume that the total strain tensor e can be decomposed into elastic and inelastic
strain tensor ~ and ~P:
= ee + ~p. (3)

The microstructural change in materials in the process of inelastic deformation and damage may be
induced by the changes in density and configuration of dislocations, by the development of
microscopic cavities and other imperfections. We then postulate that internal states of materials
characterized by these changes can be properly described by a set of tensor variables of even ranks:

{Vk;k = 1,2, ..-}. (4)

In view of Eqn (3), we have the Helmholtz free energy as follows:

4, = 4'(.e, v~, T) (5)

Substitution of Eqns (3) and (5) into Eqn (2) furnishes the following constitutive equations and the
equation of specific entropy production rate:

c~4' ?4' (6)

= P ~,~e S = --p ~T

g
pq' = ~:~/P +Ak" ~ + ~ ' q /> 0 (7)
1

where Ak and g are the thermodynamic conjugate forces corresponding to Vk and temperature
gradient:
c~4'
Ak----p~ (k= 1,2,...), g=-gradT. (8)

Now, if we define the mechanical flux vector J and their thermodynamic conjugate force vector
X as follows;
j = p {ip, ~,q~,T (9)

x = )
f 1) (10)

then Eqn (7) of entropy production has the alternative form:

p ~ - - x - s ~ > 0. (11)
476 S. M u r a k a m i and K. Kamiya

Rice [30] supposed that the rate at which any structural rearrangement occurs is entirely
determined by the forces associated with the rearrangement. In this case, if each component Jk of
mechanical flux vector J can be assumed to be a function of the corresponding components Xk of the
thermodynamic force vector X and the current state:

Jk = JktXk, T,J), (12)

then there exists a dissipation potential F(X) which is a homogeneous, convex function of the
corresponding conjugate forces as follows:

?:F
Jk - (13)
?Xk"

By use of Eqns (9), (10) and (13), each component of J can be given as follows:

~F ?F tTF
iv = ~ " ~ -
?Ak" q -
c~(g,/T)
. (14)

The advantage of the proposed procedure is quite obvious. Instead of separate formulation of
constitutive and damage evolution equations in the CDM theory, a unified description of CDM is
possible only by establishing a single potential. Furthermore, the damage theory established in this
way is formally similar to the common theory of plasticity. Extension of the present theory to the
case of finite deformation will not induce any essential difficulties.

4. A P P L I C A T I O N TO ELASTIC-BRITTLE MATERIALS

4.1. Formulation of Helmholtz free energy Jbr elastic-brittle materials

We will now apply the thermodynamic theory discussed in the previous section to the damage
process of unilateral elastic-brittle materials. By the elastic-brittle materials we imply a material
which is damaged by the development of distributed microscopic cracks and leads to the final
fracture by their coalescence without significant inelastic deformation. The orientation of the
distributed cracks saliently depends on the direction of the external load. Geological materials under
confined uniaxial compression, for example, will be damaged mainly by nucleation and growth of
microcracks almost parallel to the loading direction.
In order to represent the state of anisotropic damage characterized by these cracks, we will
employ a second rank symmetric damage tensor D discussed in Section 2. Besides the damage effects
represented by D, an additional effect may be also needed to identify the damage state which is
responsible mainly to the further development of damage, and thus we will introduce another scalar
damage variable [J for this purpose, which corresponds to the isotropic hardening variable r in
plastic deformation theory [19].
By restricting the present discussion to a purely mechanical process and by disregarding inelastic
strain, the internal state variables of Eqn (4) will be given by

vk = Io,[~I. (15)
In view of this relation, Eqn (8) furnishes the thermodynamic conjugate force Ak = {Y, -- B}
corresponding to Vk as follows:

v--p , (16)

where the minus sign of the conjugate force B is taken for the convenience of further formulation.
The second rank symmetric tensor Y given above, in particular, represents the energy release of
material due to the development of damage, and is equivalent to the energy release rate G in fracture
mechanics [-3, 14, 23].
We are now in a position to determine a proper form of Helmholtz free energy. It will be assumed
first that the free energy may consist of two parts; one is the free energy mainly related to elastic
 Constitutive and damageevolutionequations 477

deformation but affected by damage, and the other is the energy exclusively related to the damage
development. Namely, we will postulate the following expression for the Helmholtz free energy:

p~t(~e,D, fl) = D[//e(~¢ D) ~_ p@d([j). (17)

According to the representation theory of non-linear algebra [3 l, 32], the most general form of the
tensor valued scalar function pO~(ee, o) can be expressed by the combination of ten basic invariants
of two symmetric tensors e~ and D. At the initial undamaged state, the elastic-brittle material is
assumed to be isotopic and linear elastic, and the elastic strain energy ptpe(~,D) may decrease with
the process of damage D. Thus, the function pO~(t~,D) is quadratic in t" and linear in D, and can be
given as the linear combination of the following terms:

(tree)2, tr(~) 2, (tre~)2 trD, tr(~*)2 trD

tr~ e tr(e~D), tr[(t~)ZD]. (18)

One of the difficulties in the formulation of CDM theory for the elastic-brittle material is
concerned with the unilateral nature of damage. Thus, the anisotropic nature of damage and its
unilateral character have been often discussed [15,16,20,33-36]. Ramtani [36] for example,
employed a second rank symmetric damage tensor and strain decomposition in order to describe the
unilateral nature of damage. Ju and his co-workers [ 15, 16] used the fourth rank damage tensor and
fourth rank positive projection tensor to discuss this problem. Chaboche [34], on the other hand,
reviewed the possibilities of the stress-strain discontinuity for multiaxial strain paths in the preced-
ing unilateral damage theories, and showed that the discontinuous stress-strain response may occur
when unilateral condition affects both the diagonal and the non-diagonal terms of the compliance
(or stiffness) tensors. Recently, he proposed a more general theory which eliminates the possibility of
discontinuous stress-strain response [35].
In the present paper, we will employ the following modified elastic strain tensor g~ to represent
unilateral response of the damaged materials as already employed by Mazars et al. [20], Ladveze
and Lemaitre [33] and Ramtani [36]:

ge = ( e ~ ) __ ~ ( _ _ t ~ > (19)

[<~">]= o <~> , [<_~e>]= o <-~> (20)

o o <e;> o o <-~;>

where < ) is Macauley bracket, and e~ (i = 1,2, 3) and ~ are the principal values of e° and a material
constant describing the crack closure effect.
Since the unilateral effect is observed only for non-vanishing damage D, the strain tensor in the
invariants of Eqn (18) except for the first and second ones are assumed to be replaced by the modified
strain tensor of Eqns (19) and (20). By use of Eqns (18)-(20), and by taking account of the avoidance
of the stress-strain discontinuity [34, 35], the final form of the damaged strain energy p~p~(g, D) can
be given as follows:

pOe(~e,O) = ½)~(trge)2 +/~tr(g) 2 + qltrO(tr~e) 2 + tl2trOtr(~e) 2 + r/atr~tr(~eD) + q4tr[(g~)2D]

(21)
where symbols 2 and p are Lam6 constants, and q1-~/4 are newly introduced material constants.
The Helmholtz free energy p~kd(/~) has been introduced here to represent the effect of the damage
state on the further development of damage, and thus we will assume the following quadratic form as
a simplest expression in/~ as

pt~d(fl) = ½Kdfl 2 (22)

where Ka is a material constant.

478 s. Murakami and K. Kamiya

4.2. Constitutive and damage evolution equations of elastic-brittle materials

Substitution of the Helmholtz free energy (21) into Eqns (6) and (16) yields the following
constitutive equations together with the expression for the thermodynamic conjugate forces:

= [2(tre e) + 2t/l(trs c) (trD) + q3tr(sCD)]l

+ 2[~ + ~/2(trD)]~~ + t/3(tr~)D + 2~/4g°: \ ~ o D + D ~ ) (23)

y- aGo,k) a(p~, °)
~D ~D

= - [-ql(tr~e)2 + r/2tr(ge)2] I - ~/3(trg) - rt~g~i ~ (24)

~(pO) a(pOd)
B . . . . Ka[3. (25)
a~ a~
The damage evolution equation will be furnished, provided a proper expression of the dissipation
potential of Eqn (13) can be established. Furthermore, according to the experiments of acoustic
emission on geological materials [37], a finite region was observed in strain (stress) space within
which no brittle damage occurs for any loading path. By considering this experimental fact, we can
assume the existence of a damage criterion in the space of the thermodynamic conjugate forces
{ Y , - B}. This criterion will be associated with the dissipation potential F(X), which should be
a convex and homogeneous function in the space of thermodynamic conjugate forces. Similarly to
Chow et al. [11, 14], we will assume the following homogeneous function of degree one for the
surface of dissipation potential (damage surface):

F(F, B) = Y~q - (Bo + B) = 0 (26)

y¢q = V/~z y : L . y (27)

where Bo is a material constant and represents the initial threshold of the damage evolution. The
symbol L in Eqn (27) is a fourth rank tensor which may depend on the state of damage D in general.
However, according to the assumption of Eqn (27), L may be given as the following isotropic tensor:

Lijk I = l (6ik(~jl + 6il•jk). (28)

In view of Eqn (14), the evolution equations for/) and fl are given by the dissipation potential of
Eqn (26) as follows:

• PF (29)

• ~,F ,
= )-a ~( B ) - Za (30)

where )ta is a multiplier to be determined by the consistency condition on the damage surface•
Namely, in order that the damage may further develop, the conjugate forces should remain on the
surface, and thus we have

• 8F ./[SB\ L:Y :I ? (31)

~F
~=1, if F = 0 and ?-~:I?>0
(32)
~F
~=0, if F < 0 or =:-:Y~<0.
c¥
 Constitutive and damage evolution equations 479

5. E L A S T I C - B R I T T L E D A M A G E A N A L Y S I S F O R H I G H S T R E N G T H C O N C R E T E

5.1. Uniaxial compression

To examine the utility of the constitutive and damage evolution equations formulated in the
preceding section, the process of elastic deformation and brittle fracture of a high strength concrete
will be analysed.
In order to determine the material constants of the preceding equations, we will first analyse
uniaxial compression of the column specimen of a high strength concrete, and then compare the
results with those of the experiments [38].
Let us employ a rectangular coordinate system (xbx2,x3), where the axis xi is taken in the
direction of column axis, while x2 and x3 are perpendicular to x~. Furthermore, in case of uniaxial
loading, there exist components of the elastic strain tensor g] ~, e,~z and ~3- Because of the fact that
g]~ < 0 and 8~2, ~.~3 > 0, the components of modified elastic strain specified by Eqns (19) and (20)
are given as follows:
~!1 0 0 ]
[i°] = e,22 0 . (33)
le
0 e33

The state of stress, on the other hand, has the following components:

[a] =
oil
0

0
, (0-11 < 0 ) . (34}

In view of Eqns (33) and (34), the constitutive equation (23) can be expressed as follows:
0-11 = [2+ 2p + 2(ql + qe}trD + 2(r13 + q~t~2}Oll] g]l
+ [2 + 2qltrO + r/3(Dll + D22)] g~2 + [2 + 2~htrD +/'/3(Oll + D33)]~3 (35)

0"22 = 0 = [ ) . + 2r/ltrO + Y/3(Dll -{--D22)]E]l

+ [2 + 2p + 2(r/1 + qz)trO + 2(r/3 + ~/4.)D2218~2
+ [2 + 2qttrD + q3(022 + 033)]~,~3 (36)

a33 = 0 = [5~ + 2qitrD + ~/3(Olt + D3~j]e]l

+ [2 + 2qltrD + q3(D22 + D33)]e~2
+ [2 + 2p + 2011 + r/2)trD + 2(q3 + r/4)O33]e~3. (37)
By use of Eqns (21} and (24}, each component of the conjugate force Yof the damage tensor, on the
other hand, are given as follows:
Y l l = --~11 (tree) 2 -- r/2tr(ge) 2 -- q3(tree)~] 1 - ~ 4 ~ 2 ( F, ] 1) 2 (38)
Y22 = --/71{trg~) 2 -- r/ztr(/~e)2 -- r/3(trl~e)g~2 - - ~4(~;~2) 2 {39}
y33 = _ ~h(tr~)2 - - t12tr(8~)2 - - q~(trg~)e~3
. - - ~ 2
/ ] 4 ( 8 3 3 ) " (40)

Thus, Eqns (29}, (31), (32} and (38)-(40) furnish the following damage evolution equations:

D11 = ~ Y I l ( Y I 1 P l t + Y22'P22 -r Y33Y33)

2 K d ( y 2 1 + Y222 + Y]3) (41)

D22 = o~ Y 2 2 ( Y l l Y l l + Y221522 + Y33Y33) {42)

2Ka(y21 + y22 + Y~3}

D33 = ~ Y 3 3 ( Y l l l Y l l + Y22"Y22 4- ,Y33]/33} {43)

2Kd(Y121 + y22 ÷ Y.~3)
where ~ is a constant defined in Eqn (31).
480 s. Murakami and K. Kamiya

The value of undamaged initial Young's modulus E0 and Poisson's ratio v0 are determined from
literature [7]:

Eo=21.4GPa, v0=0.2. (44)

The material constants in Eqns (19) (22) and (26), on the other hand, are determined so that the
present theory can describe the elastic and the damage behavior of the high strength concrete under
uniaxial compression (shown by symbols © in Fig. 1):

r/1=-400MPa, O2=-900MPa,

r/3 = 100 MPa, ~/4 = - 23500 MPa, (45)

= 0.1, Ka = 0.04, Bo = 2.6 x 10 3 MPa.

The symbols and the solid lines in Fig. 1, respectively, show the experimental and the predicted
results by means of Eqns (34)-(45) for the stress-strain relation of high strength concrete under
uniaxial compression. As observed in this figure, the material constants of Eqns (44), (45) and the
constitutive and evolution equations (35) (43) describe the experimental results with sufficient
accuracy. The dashed line, on the other hand, shows the relation between stress and volumetric
strain. This result predicts the dilatancy of this material as observed in a usual geological materials.
This phenomena may be caused by the anisotropic development of damage in this material.
Figures 2 and 3 show the process of material damage predicted by the present analysis by use of
Eqns (35) (45). Figure 2 is the evolution of the components of damage variable in the above
high-strength concrete under uniaxial compression. Though the difference between Dll and
D22 = D33 observed in Fig. 2 is relatively small, this results is consistent with the commonly
observed experiments; i.e. both of internal microcracks perpendicular and parallel to the loading
direction develop dominantly under compression in the case of elastic-brittle damage material.
In actual process of damage, since final fracture usually occurs before the magnitude of the
damage variable IlDII attains to unity, a critical value of damage variable Dcr is often employed to
define an additional criterion of the fracture [3, 4]. However, the relevant experimental data on
Dcr are not available in this case, and hence no fracture criterion has been incorporated in the present
formulation. However, it will be observed that the damage state D22 = D33 ~ 0.4 may be quite
practical in the critical stage of this material.

-60

............................ "°" /
-50
,.." o /
-40 -- ;' C)

-30

/-
r.~
_

-20

,."'~
//
O Experimental[38]
o--0 .2
-10
- ~ e d i c t e c l (Uniaxial strain) 1
( ~ ~ " iPredicted(V°lulmetricstraln) i
0
0 -0.1 -0.2 -0.3 -0.4xl 0 .2
Strain ~ e

Fig. 1. Uniaxialcompressionof high-strength concrete;experimentaland predicted results.

 Constitutive and damage evolution equations 481

0.5 --

E 0 = 21.4 GPa
v =0.2
0 .s
0.4-

It Damage Dll /
0.3- ....... Damage D22 = D33 ...

0.2-

0.1 -

0 -0.1 -0.2 -0.3 -0.4xl 02

Strain E e

Fig. 2. Damage-strain relations of high-strength concrete under uniaxial compression.

25 r I I 0.8

20
e., 0.6

15 O

;= 0.4
o
E 10
o,0
...-, 0
;zl .~ ............................................................... ."°"°'°°"°°')°'°
O - 0.2
5
Young's modulus E Eo = 21.4 G P a
....... Poisson's ratio v v = 0.2
0
l I ,
0 0.2
0 -0.1 -0.2 -0.3 -0.4x10
Strain e e

Fig. 3. Predicted results of Young's modulus, E, and Poisson's ratio, v, under uniaxial compression.

Figure 3 is the prediction of Young's modulus-strain relation and the apparent Poisson's
ratio-strain relation. These results also represent the characteristic aspects of the elastic-brittle
d a m a g e materials. The decrease of Young's modulus due to the development of d a m a g e is clearly
shown in Eqn (35). The increase of apparent Poisson's ratio, on the other hand, will have close
relation with the dilatancy of elastic-brittle d a m a g e material as observed in c o m m o n experimental
results of the materials.

5.2. Uniaxial tension

In case of uniaxial tension, ~ 2 and e~3 perpendicular to the loading direction x, take negative
values. Hence, the modified elastic tensor of Eqns (19) and (20), the constitutive equation (23) and the
expression of the t h e r m o d y n a m i c force, respectively, have the following forms:

Lo o
O
o]
<~- e
~.C, 3 3

~11 = I-2 + 2p + 2(r/~ + r/2)trD + 20/3 + 114)D,~] el,

+[2+2rlttrD+qs(D11 +D22)]gCzz+[).+2tlltrD+t13(D11 + O33)]g~3 (47)

482 S. Murakami and K. Kamiya

0"22 = 0 = [2 + 2r/ttrD + Y/3(Dtt + D22)]c]1

+ [2 + 2p + 2(r/1 + qz)trD + 2013 + q4g~'2 ) D 2 2 ] ~ ; 2e2
+ [2 + 2~/atrD + r/3(D22 + D33)]c~3 (48)
~33 = 0 = [5. + 2~/ttrD + t/3(Dlt + D33)]c] 1
+ [2 + 2rhtrD + 1/3(D22 + D33)]c~2
+ [2 + 21t + 20h + r/2)trO + 2013 + q4~_
~'2 e
)D33]~33 (49)

g t l = - ~h(tr~¢) 2 - r/2tr(~¢)2 - rl3(tr~¢) c.] 1 - r/4(z] 1)2 (50)

Y22 = -~ll(tr~e) 2 - ~]2tr(~e)2 -- tl3(tr~¢)~:~2 - - ~ 4 ~ 2 ( E ~ 2 ) 2 (51)

Y33 = -- F/l(tree) 2 - q2tr(l~e)2 -- t/3(tr~e)e-;3 - - Y~4~2(1-;33)

.'e 2 (52)
The expressions of damage evolution equations under this condition are same as those under
compression, Eqns (41)-(43). However, the components of the damage conjugate force Y under
uniaxial tension (50)-(52) are different from those under uniaxial compression (38) (40), and thus we
have different damage development for these cases.
Figures 4 and 5 show the stress-strain and the damage evolution under uniaxial tension predicted
by Eqns (46) (52) by use of the material constants of Eqns (44) and (45) identified by the uniaxial
compression test. Unstable fracture in brittle material under uniaxial tension will occur no later than
the ultimate stress. This critical state in Figs 4 and 5 corresponds to (a)u = 12.3 MPa,
(e°)u = 7.8 x 10 "*, (Dll)u = 0.13 and (D22 = D33)u = 0.01, and hence the curves in Figs 4 6 beyond
this stage have been entered by thin lines.
Comparison between Figs 1 and 4 shows that the stress at the final fracture under uniaxial tension
is about four times smaller than that under uniaxial compression. This feature may be in contrast to
the usual observations in concrete in which the fracture stress in tension is usually smaller by five
times or more than than that of compression [20]. Thus, though the introduction of modified elastic
strain tensor ge gives considerably smaller strength for tension than that for compression and
describes the unilateral nature of elastic-brittle damage materials qualitatively; some quantitative
limitation is observed in these results. An improvement of this discrepancy will be facilitated by
employing the critical damage D , in the present theory, as already pointed out in Section 5.1.
As regards the unilateral behavior of different elastic-brittle materials, preceding theories have
assumed the different formulation, or different material constants for under a tensile and compres-
sive condition [7]. In the present paper, however, we can describe the different behavior in these
conditions by use of only single set of equations and material constants, without any additional
assumptions for each condition.

15
E o = 21.4 GPa
v =0.2

10
e~

(13

0
0 0.02 0.04 0.06 0.08 0.10x10 "2
Strain e e

Fig. 4. Predicted results of stress-strain relation under uniaxial tension.

 Constitutive and damage evolution equations 483

As observed in Fig. 5, damage component/)11 which represents the damage of loading direction is
much larger than the other components 0 2 2 and D33. This result is different from that under
compression observed in Fig. 2. These results show that the anisotropic behavior of damage under
uniaxial tension is more significant than under uniaxial compression. The different aspects of
damage progress under each condition are consistent with the actual behavior of elastic-brittle
damage materials.
The change in Young's modulus and the apparent Poisson's ratio of the high strength concrete
under uniaxial compression are shown in Fig. 6. By comparing Fig. 6 with Fig. 3, we can observe
much more rapid decrease of Young's modulus in Fig. 6 than in Fig. 3. On the other hand, the value
of Poisson's ratio under uniaxial compression increases as the strain grows, while that under
uniaxial tension decreases. These different aspects are accounted for by the difference between
constitutive Eqns (35)-(37) and (47)-(49). Though the first bracketed term on the right-hand side of
Eqn (47) and that of Eqn 135) play a significant role on the decrease in the value of Young's modulus,
a decrease in the value of the term of Eqn (47) due to damage development is much larger than that
of Eqn (35). This difference results from the introduction of the modified elastic strain tensor (19) into
the Helmholtz free energy (21).

0.4 Eovo=0"2=
21.4 GPa

!
Damage D I1
II 0.3 i
- -- Damage D22 = D33

- 0.2

ol I
0 0.02 0.04 0.06 0.08 0.10x10 a
Strain e e

Fig. 5. Predicted results of damage-strain relation under uniaxial tension.

25 I 0.3

2O
Ct.

0.2 ="
0
15

E 10
~0 0.1 ~.

5 ?- -- Young's modulus E
--- Poisson's ratio v

0 L
0
J __
0.02
r
0.04
__l
0.06
I
0.08
I
0.10x10 -2
Strain e e

Fig. 6. Predicted results of Young's modulus, E, and Poisson's ratio, v, under uniaxial tension.
484 S. Murakami and K. Kamiya

5.3. Strain path dependence on elastic-brittle materials

Let us now calculate the elastic-brittle damage behavior under non-proportional loading paths
1, 2 and 3 shown in Fig. 7.
Figure 8 shows the three strain paths corresponding to stress paths in Fig. 7. As observed in Fig. 8,
despite that each stress path in Fig. 7 leads to the same stress state P, the corresponding strain paths
strain to the slightly different strain state P~, P2 and P3. Furthermore, each strain path shows the
significant non-linearity due to the different way of development of anisotropic damage. In particu-
lar, as observed in the strain path 3 in Fig. 8, the axial strain is developed even though shear stress
only is applied (see, around point B). However, such a phenomenon can not be observed in strain
path 1. These features are attributable to the path dependence of the damage development [22].

5.4. Initial damage surface

The proposed damage surface of Eqn (26) is described in the space of thermodynamic conjugate
forces Y of damage tensor. The proposed constitutive equation (23) and expression (24) of Y enable
us to describe the shape of damage surface (26) in stress (or strain) space. In general, the damage
surface will depend on the stress (or strain) path which the elastic-brittle damage materials has
experienced. However, as far as the initial damage surface is concerned, it is independent from any
stress (or strain) path, because the proposed equation show a usual elastic material behavior without
damage.
Figures 9(a) and (b) show the initial damage surface in the axial stress-shear stress space and that
in biaxial stress space. These surfaces are obtained as the stress loci where the damage starts to

io- ..... ........ ;;., P

i .,°"'°**'*'°
/
--
Path 2,.°°".."
.S
°.-,o"
°,. "
o,*
..'" Path l I A
0
0 5 10
Axial stress (y ( M P a )

Fig. 7. Shear stress r-axial stress a paths.

0.15x10 2 -
I 1

,2,,o3l ...... -- .... P,!

--.~: P i........,;YP, i
B
-- °.,°"
0.10 .°."
..-"
..= .°, 0.050 0.052x10 -2
..°
.oj'*
r.o
0.05 °..,*
o..°.'* Path 1
.,, ,°°°°"
°** ....... Path 2
..." .... Path 3
,/O I I A I I _ I
0 0,02 0.04 0.06 0.08 0 . 1 0 x 1 0 .2
A x i a l strain e e

Fig. 8. Predicted results of shear strain ;,C-axial strain ce paths.

 Constitutive and damage evolution equations 485

20
20tr T
10-

e-
o

-10 -

-20 -
-I0~
-20 -

-30 I I J -30
-30 -20 -10 0 10 20 -30 -20 -10 0 10 20

(a) Axial stress o" (MPa) (b) Stress oll (MPa)

Fig. 9. Predicted initial damage surface: (a) shear stress r-axial stress ~r space; (b) biaxial stress space.

develop by various proportional loading paths in the corresponding stress spaces. Though the
unilateral nature of damage initiation is well observed in these figures, the quantitative discussion on
the validity of these predictions should have recourse to the experiments.
Further quantitative improvement of the present theory may be possible by introducing the terms
which depends on the sign of the volumetric strain besides the modified elastic strain tensor U of
Eqns (19) and (20) into the Helmholtz free energy of Eqn (21), if the influence of the stress
discontinuity stated in Section 4.1 [34, 35] is not so significant.

6. CONCLUSIONS
Irreversible thermodynamic theory of the constitutive and the damage evolution equations for
elastic-brittle damage materials was developed by employing a second rank symmetric damage
tensor in order to describe the state of anisotropic damage. The Helmholtz free energy was
represented in terms of the modified elastic strain tensor in order to represent the anisotropic and the
unilateral effects. The resulting theory was applied to describe the anisotropic and unilateral aspects
in elastic damage behavior, the change in elastic moduli in a high-strength concrete under uniaxial
monotonic tension and compression as well as under non-proportional combined axial-shear
loading. The damage surfaces in the axial-shear and biaxial stress space were also discussed.
Material constants for the resulting equations were identified by applying them to describe the
results of the uniaxial compression tests on the high-strength concrete, and then the response under
tensile test was predicted.
The prediction of uniaxial loading show that, the present formulation can describe the unilateral
nature of the elastic-brittle damage materials without assuming the specific independent equations,
or assuming different material constants for different loading conditions. However, further improve-
ment of the accuracy of the prediction of the fracture under tensile loading may be facilitated by
incorporating the critical damage value Dcr in the present modeling.
The application to the different loading paths reveals the path dependence of the damage progress
as well as the interaction of shear strain to axial strain under the shear loading path. The path
dependence, however, is not so significant in the present result.
Though the initial damage surface calculated in the axial stress-shear stress and biaxial stress
space can describe qualitatively the behavior of the high-strength concrete, further research will be
necessary from theoretical and experimental point of view to verify its validity.
The discussion of the behavior of the damage surface under different loading conditions will be
needed for the further development of the present formulation.

Acknowledgement The authors are grateful for the support in part for the present work by the Ministry of Education,
Science, Sports and Culture of Japan under the Grant-in-Aid for Scientific Research (BI [No. 05452125] for the fiscal years of
1994 and 1995, and the Grant-in-Aid for DevelopmentalScientific Research (B) [No. 7555346] for the fiscal year of 1995.
486 S. Murakami and K. Kamiya

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