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A micromechanics-based damage model for microcrack-weakened brittle solids
OF NAI'INlUU ELSEVIER
Mechanics of Materials 20 (1995) 59-76
A micromechanics-based damage model for microcrack-weakened brittle solids * Shou-Wen Yu, Xi-Qiao Feng Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China
Received 1 November 1993, revised version received 16 May 1994
Abstract In the present paper, a micromechanics-based, three-dimensional damage model for microcrack-weakened brittle solids is developed. In order to describe the evolutionary damage state and anisotropic properties of materials, the concept of domain of microcrack growth (DMG) is defined as the union of all possible orientations of propagated microcracks after a loading path. Based on modified mixed-mode growth criteria of microcrack, the evolution of DMG as well as the overall effective compliance tensor of damaged materials are formulated. Through a micromechanical analysis, the damage mechanisms and the complex constitutive behavior of materials are studied under complex loadings. The self-similar growth of open microcracks under tension, the mode-II growth and the kinking of closed microcracks under compression and the influences of these mechanisms on the mechanical properties of materials are all considered. It is explained that axial splitting may occur in a material only when the lateral normal stresse,; have positive or a small negative value. And the condition for axial splitting of a material is given. The evolutiona~:y damage model is illustrated by two examples of uniaxial tension and uniaxial compression and the theoretical re:~ults are compared with experimental data and theoretical results obtained by others. Keywords: Domain of microcrack growth; Damage mechanism; Effective compliance tensor; Growth and kinking of microcracks; Brittle solids; Compression loading condition; Anisotropic properties
I. Introduction Many brittle materials, such as concrete, rock, most ceramics and intermetallics, have a large number of preexisting microcracks. The nucleation, growth and coalescence of microcracks may cause a nonlinear degradation of materials and the failure of strucr~ures. So the macro- and micro-damage of materials is a popular and also
* Correspondence to: Prof. Shou-Wen Yu, Office of the President, Tsinghua University, Beijing 100084, P.R. China. Fax: 0861-2568116. Tel.: 0861-2595533.
important subject in solid mechanics and material science. In the current literature, there are two main classes of damage models, i.e., continuum damage models (CDM) and micromechanical damage models. Some fairly complex responses of materials have been simulated by CDM (see, e.g., Lemaitre and Chaboche, 1988; Murakami and Ohno, 1981; Krajcinovic, 1986, 1989). Models for the establishment of micromechanical damage have improved the understanding of physical process and microscopic mechanisms of damage in materials. In the case of stationary elliptical microcrack and isotropic damage, the self-consistent
0167-6636/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0167-6636(94)00046-J
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S.-W. Yu, X.-Q. Feng / Mechanics o f Materials 20 (1995) 5 9 - 7 6
method was employed by Budiansky and O'Connell (1976) and Horii and Nemat-Nasser (1983) in deriving effective moduli. For anisotropic damage with open penny-shaped microcracks, several important micromechanical evolutionary damage models can be found in Krajcinovic and Fanella (1986), Krajcinovic and Sumarac (1989), Ju and I~e (1991), Ortiz (1985), Nemat-Nasser and Obata (1988). Most of the damage models in these two classes were summarized and recapitulated in review articles (see, e.g., Krajcinovic, 1986, 1989; Bazant, 1986; Kachanov, 1992). The many phenomenological and micromechanical damage models of materials that have been based on different damage variables, have, at least, two evident drawbacks. First, the damage variables in the form of either a scalar, a vector or a tensor defined in the earlier models cannot describe the evolutionary complex anisotropic damage state of materials exactly. In order to improve the accuracy of damage description, some fourth- and even eighth-order tensors were suggested as damage variables (see Krajcinovic, 1989). But such a description often entailed complexities exceeding their advantages. Secondly, it is difficult for such models to handle a complex loading path. In this study, the usual damage variables of scalars and tensors are all abandoned, and instead the domain of microcrack growth (DMG) is defined to describe the constitutive behavior of damaged materials. The DMG, which is defined as one set, or a union of sets, is the orientation scope of all possible propagated microcracks after a loading path. In other words, all microcracks whose normal vectors are in the orientation scope of the DMG must have propagated after the loading path. By this concept, the damage state of microcrack-weakened solids can be described more exactly and the problem of complex loadings can be solved (Feng and Yu, 1993). Based on DMG and a micromechanical analysis, Feng and Yu (1994) have investigated damage localization and strain softening in brittle materials. In this paper, DMG is extended to brittle materials subjected to three-dimensional tensile or compressive loadings. In Section 2, the inelastic compliance tensor induced by a single open
penny-shaped microcrack is derived at first. Based on an appropriate growth criterion of microcrack, the DMG under a general three-dimensional proportional loading is formulated. Then, the evolution of the DMG during any complex loading path is considered, and the overall effective compliance tensor of damaged materials is given. Thus the anisotropic constitutive behaviors of materials can be simulated. In Section 3 the damage mechanisms of brittle materials under triaxial compression below the brittle-ductile transition points are studied through a micromechanical analysis. The self-similar model-II growth and kinking of closed microcracks under compression are considered and their effects on the mechanical properties of materials are introduced into the fourth-order compliance tensor. Finally, in Section 4, the three-dimensional damage model suggested here is applied to two simple examples of uniaxial tension and uniaxial compression and the results obtained are compared with the experimental data and theoretical results obtained by others. 2. The damage model of DMG under tensile loadings
2.1. Single open microcrack-induced inelastic compliance tensor Choose a representative volume element (RVE) containing a statistically valid sample of microstructures and assume that only small strains and small rotations occur. The virgin matrix material is assumed to be linearly elastic and isotropic. Then the volume-averaged strain tensor is
Eij = ~ej + ~-ii j ,
(2.1)
with --e
0
--
(2.2)
~'ij ~---S i j k l O ' k l ,
1% =
~ij
(2.3)
,
k=l
~/(.ot)_ tj
~i(ot)~.
-- L'ijklUkl
1
1
= -~ f s j ( b i n j + bjni) (~) dS,
(2.4)
where N c is the number of microcracks in the RVE occupying the volume V, ~ej and ~ij -i are the
S.-W. Yu, X.-Q. Feng / Mechanics o f Materials 20 (1995) 59-76
elastic strains and the microcrack-induced strains, respectively, ~'~) the strains induced by the ozth microcrack, S°.kt and ~i(~) Oijkt the undamaged elastic compliances and the a t h microcrack-induced compliances, respectively, and where b i = [u i] are the components of the displacement discontinuity vector, n~ those of the normal vector of the microcrack, and ~j the volume-average stresses which are assumed to be equal to the applied far-field stresses o-iy in the Taylor model. First consider the a t h single penny-shaped microcrack with radius a in an isotropic body uniformly loaded at far field (Fig. 1). Establish the global coordinate system ( O x ~ x z x 3) and its corresponding local coordinate system (Ox'~x'2x'3), as shown in Fig. 1, in which the x~-axis is parallel to the normal vector of the microcrack n, and the x~-axis is coplanar with the x~-axis and the x3-axis. Then the orientation of the microcrack can be expressed as (0, ~b). The basic vectors of the two coordinate systems are related by t
t
ei = g i j ~ : (2.5) where the transformation matrix g~j and its inverse matrix g i y = ( g ~ i ) - l = ( g ~ y ) T can be obtained from g~j = g T
ei -- g i j e j ,
=
COS 0 COS ~b - s i n O c o s 4' sin $
sin 0 cosO 0
- c o s 0 sin ~b] sinOsin~b [. cos 4, J
X 2 X 2
2
b i = (a 2 - r )
1/2
t
t
,
Btyo'2jgti,
(2.7)
where the crack opening displacement tensor B' depends on the compliance of the microcrackweakened solids. A.ssuming here that it depends only on the compliances of an isotropic elastic matrix, then the nonvanishing components of B' for the open microcrack are 16(1 -~,~2) 8(1 - z,2) B~I --- B 3,3 B~2 (2 - v),rrE ' ~E
(2.8)
-
-
-
-
J
I
X 3
x3/" F i g . 1. A m i c r o c r a c k corresponding
with global coordinate
local coordinate
system
and
its
s y s t e m in t h e R V E .
The stresses in the local coordinate system tr'j are given by
% = g~kg~ttrkt.
(2.9)
If the normal stress to the microcrack surface is compressive, assume in this part that the effective shear stress is smaller than the cohesive strength and no frictional sliding occurs. The closed microcracks and their influence on the compliance tensor will be discussed in Section 3. Then the displacement discontinuity vector b can be rewritten as
(2.6) For an active penny-shaped microcrack, the components of the displacement discontinuity vector b = b N i take the form (Budiansky and O'Connell, 1976)
61
2
b i = (a 2 - r )
1/2
t
t
t
t
t
t
B~sgjigzkgslO'kl(O'22), "
(2.10)
where the angle bracket is defined as (x) =
1( x)
1 + ~-~ .
(2.11)
Thus, from (2.4) and (2.10), the components of the inelastic compliance tensor induced by the a t h single open penny-shaped microcrack with radius a and orientation (0, ~b) are
L/jk,(a, 0,
ij)
T/-a 3
-
t
t
t
t
t
t
t
3 V Bm"g2kg"t(gminy +gmyni)(trstg2~g2t)' (2.12)
--i where Sijkl
are
simplified denotations of Oijkl" ~i(~)
S.-W.. Yu, X.-Q. Feng / Mechanics of Materials 20 (1995) 59-76
62
2.2. D M G under three-dimensional proportional tensile loading
l
In fact, it is extremely difficult to give a unified criterion for preexisting microcrack growth, if damage-induced anisotropic compliances and microcrack interaction are involved. Therefore, for practical applications, it is assumed that the microcrack fracture criterion is based on the Taylor model, i.e., no microcrack interaction is considered (Kachanov, 1992), and self-similar microcrack growth will happen when the average value of the energy release rate along the microcrack edge reaches a critical value. The mixed-mode fracture criterion for a penny-shaped crack may take the following modified form (Kanninen and Popelor, 1985; Ju and Lee, 1991) glc
+(~)2=1
[(tr:~l) 2 + (o',~3)2]1/2. (2.14)
Once the fracture criterion (2.13) has been satisfied, a microcrack will become unstable increasing its radius from the initial statistically averaged value a 0 to the final characteristic value a u instantaneously and will be arrested by energy barriers with higher strength (such as grain boundaries). Under a given stress state, the growth of a microcrack is related to only three components of stress tensor in the plane of the microcrack surfaces, i.e., o-~1, o-~2 and try3. By substituting (2.6), (2.9) and (2.14) into (2.13), we obtain
L~-K~]c ] g'2k g'2s -[- ~
l,t g ' ]k g'ls +g'agg'3~)
77> ( g 2' l g 2 t 'o ' k l O ' s t
:
(7 I
3
(2.13)
~22,
Kii = ~
"~l
Fig. 2. The principal stress coordinate system and its corresponding local coordinate system.
where K'I and K'n and Kic and g l i c are the mode-I and mode-II stress intensity factors and their critical values, respectively. K~ and K}I are defined as K'I = 2
O'.~
-4a -r °
iic,
(2.15)
which is the condition satisfied by the boundary of the orientation domain in the orientation space (0, ~b) of all microcracks having propagated under the given stress state. In other words, the orientation scope bounded by Eq. (2.15), which is defined as the domain of microcrack growth (DMG), is the scope of all possible orientations of microcracks with the characteristic radius a u. It is difficult, however, to give an analytical solution of (2.15) directly. For simplicity, we establish the principal stress coordinate system (O$15153), whose ~1, ~2 and ~3-axes have the same directions as the principal stresses trl, tr2 and tr3, and establish another local coordinate system (O$~$~$~), which corresponds to the principal stress coordinate system shown in Fig. 2. The basic vectors #i of the principal stress coordinate system are related to those of the global coordinate system in Fig. 1 by ei = gijey, where the components of trix gij a r e the cosines between ei and ei, and textbook on elasticity. In the principal stress
(2.16) the transformation maof the angle included can be found in any coordinate system and
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S.-W. Yu, X.-Q. Feng / Mechanics of Materials 20 (1995) 59-76
in the global coordinate system, the normal vector of the microcrack can be expressed as
The solution of Eq. (2.20) is
COS2t~ =
- A 2 _ i ( Z 2 ) 2 -- 4A1A 3 2A 1
(2.17) where g~j(0, 4;) are the components of the transformation matrix between the principal stress coordinate system an,5 the corresponding local coordinate system whiich can be obtained from (2.6) by replacing (0, 49) with (0, 4~). While in the global coordinate system, the same normal vector can be re-expressed as
n =ee' =
- s i n 0 cos 49 e l + c o s 0 e 2 + sin 0 sin 49 %.
(2.18)
By comparing (2.17) with (2.18), the transformation relationship between (0, 49) and (0, 4~) can be obtained as .,
cos
-
O=gzi(O, cb)gi2,
g2i(O, dP)gi3
tan 49=
gEl( O' c~)git
(2.19) In the coordinate system of principal stress, Eq. (2.15) is simplified as A x cos44~ + A 2 cos25 + A 3 = 0, where
A 1 = L~-~IC [ / K I I c I 2] - ( - ~ 2
)2]
\2 KIIC a 2 = [12~~ g l c ~1 (c°$20 0-2 -I-sin20
T/" --K2c" 4a0
1 >/
- A 2 + t ( A 2 ) 2 -- 4 A ~ A 3 2A1
>/0.
(2.23)
So for any value of d in the range (0 ~< 0 ~< 7r/2), the solution of (2.20) can be given by (2.22) if the conditions (2.23) are satisfied. In addition, if the conditions (2.23) are not satisfied for a given value of 0, the two following situations should be distinguished: (1) The left-hand side of (2.20) is negative for any value of 4~, then no microcrack propagates for this value of 0, and we set 4)(0) = 0.
(2.24)
(2) The left-hand side of (2.20) is positive for any value of 4~, then we set the orientation scope of propagated microeracks for this value of d as (2.25)
from (2.22) to (2.25), the D M G in the principal stress coordinate system under three-dimensional stresses is derived and can be written as
1"2(0, ~b, 0-ij)
=(0
= ( "IIC 12(c0s2~2 + sin 2-00"3)
-
(A2) 2 - 4AIA 3/> 0,
0-3)
X (0-1 - 0"3) sine0;
-
The conditions for the existence of solutions are
Thus
crI - cos 0 0-2 + cos 2/~ o"3
+ ~
(2.22)
0 ~<49(0) ~<27r.
(0-1--0-3)2 sin40;
r/
+
(2.20)
(0 < q~ ~<27r).
n(o, 49, 0-,j)
('re + 0"3)2 sin2/~ cos20 (2.21)
=
1
49-(0,
(2.27)
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S.-W. Yu, X.-Q. Feng / Mechanics of Materials 20 (1995) 59-76
As the externally applied stresses increase, more and more microcracks propagate, increasing their radius from a 0 to a, and being arrested at the edges of the weak planes by energy barriers with higher strength. Under a certain load level, some of the arrested microcracks will become unstable again and pass through the energy barriers when the following secondary fracture criterion is satisfied
g--~cc ]
(2) If solutions of (2.20) exist, express the D M G under the stresses trq + Atriy as
~ ( orij -~- morij ) = {0 ~ 0 ~ 17T, ~ - ( 0, Orij -[- mo'ij ) ~<6 ~<~b+(0, ~rij+ A~rij)}. (2.30) So the D M G at time t + At is
J2(t+At)=O(t)Ug2(crij+Atriy ).
(2.31)
(3) Decide from (2.28) whether gross failure occurs in the material under the stresses ~ij + Atrij.
\ glICC
where Kic c and Kiicc are the critical values of mode-I and mode-II stress intensity factors of the energy barriers, respectively. Once the criterion (2.28) is reached by a microcrack in a preferred orientation, we take it that the stresses have caused the gross failure of material. In fact, the secondary growth of microcracks may cause the complex phenomena of damage localization, stress drop and strain softening, as was investigated by Feng and Yu (1994).
2.4. Compliance tensor of damaged materials The orientations and sizes of penny-shaped microcracks in the RVE can be viewed as random variables and represented by a probability density function p(a, O, qb), which must satisfy the following normalization condition
faamaXfo/2foe~P(a' O' c~) sin O ddp dO da = l" mill
2.3. Evolution of DMG under complex loadings Within the framework of the Taylor model, no microcrack interaction is considered, and the fracture criterion of microcrack is uninfluenced by the loading history. So the D M G under a complex loading path can be obtained by substituting every stress state of the loading path into (2.26) and (2.27) and making the union of the DMGs corresponding to all stress states. In order to show the evolution of DMG, assume that the D M G at time t can be expressed as O(t)={0~<0~<½~',
~b-(0, t ) ~ b ~ < ~ b + ( 0 , t)}. (2.29)
From time t to t + At, the stresses change f r o m Orij(t) to orij(t d- At) = a/y + Aorij. Then the D M G at time t + At can be calculated through the following procedures. (1) Determine the principal stresses trl(t + At), ~ 2 ( t + A t ) and t r a ( t + A t ) and substitute them into Eq. (2.20). If no solution of (2.20) exists and no microcrack propagates, then the D M G at time t + At is same as (2.29).
(2.32) In particular, in the case of penny-shaped microcracks with the same initial radius a 0 and uniform distribution in the orientation space, 1
p(a, 0, ~b) - 2~-"
(2.33)
Thus, the overall effective compliance tensor of damaged material is
Sijk l = SOkl + Sijk ,
(2.34)
where the microcrack-induced inelastic compliance tensor is Sijkl = frr/2 f 2zr
Jo
Jo Jvcp[a' O' ~b)
--i ×Sijkt(ao, O, dp, triy) sin 0 d~b d0
+ ffo(,)gcp(a, O, 4,)[S-/%(a., 0,
Orij)
--i --Sijkl(ao, O, ¢~, o'ij)] sin 0 d~b d0. (2.35)
S.-W. Yu, X.-Q. Feng / Mechanics of Materials 20 (1995) 59-76
3. The damage madei of DMG under triaxial compression In the preceding part, all microcracks are assumed to be open and the damage mechanisms of brittle materials under compressive loadings are not considered. However, some experiments shown that the micromechanical mechanisms of damage are rather complex in such brittle materials as concrete and rocks under compression (see, e.g., Horii and Nemat-Nasser, 1985, 1986; Krajcinovic, 1989; Deng and Nemat-Nasser, 1992). In this part, the case of brittle materials subjected to triaxial compressive stresses below the brittle-ductile transition points (and hence the materials exhibit evident brittle properties) is considered. The damage mechanisms of brittle materials under compression are mainly the frictional sliding, mode-II growth and kinking of closed microcracks. When a material is subjected to triaxial compression, all or part of the microcracks are closed. In a certain orientation scope of microcracks, which is called as the domain of frictional sliding, the microcracks are closed and experience frictional sliding. The domain of frictional sliding depends on the current stresses and the frictional coefficient of microcrack surfaces but not on the loading history and the sizes of microcracks. With increasing stresses, some closed microcracks in the domain of frictional sliding will grow along the weak plane (an interface or intergranular plane) in a self-similar fashion (Lee and Ju, 1991; Fanella and Krajcinovic, 1988) and even kind into the matrix material in a non-self-similar fashion (Horii and Nemat-Nasser, 1985). The kinked microcracks tend to line up in the direction parallel to the maximum compressive principal stress in the f~Lr field. Further, these microcracks grow continuously with increasing compressive loading and cause the ultimate axial splitting of the material. Due to the frictional sliding and kinking of microcracks, the compliance tensor of a brittle material under compression is unsymmetrical, anisotropic and has 36 independent components. Closure and frictional sliding of microcracks was considered by Horii and Nemat-Nasser (1983) in
65
the context of a two-dimensional stationary selfconsistent model. In the cases of uniaxial and axial-symmetric triaxial compression, Fanella and Krajcinovic (1988) gave an evolutionary damage model based on the Taylor model. Lee and Ju (1991) introduced microcrack interaction by a self-consistent method into the damage model of Fanella and Krajcinovic (1988). Using the concept of D M G proposed above, we further analyze here the microcrack damage mechanisms and the constitutive responses of the materials under threedimensional compression.
3.1. The closure and frictional sliding of microcracks Consider a microcrack with unit normal vector n subjected to the uniform far-field compressive stresses tro- in an RVE. Let 0-2 denotes the maximum compressive principal stress, 0-1 and 0-3 the lateral principal stresses in the direction of £l and x3, respectively. The absolute values of o-1 and 0-3 are assumed to be small enough so that the material behaves like an evidently brittle one. The orientation of the microcrack is (0, ~b) in the global coordinate system and (0, 4~) in the principal stress coordinate system. The microcracks can be classified into three classes as follows according to the deformation of their surfaces. (1) The microcracks are closed and no frictional sliding occurs, then 0-12 "~ 0,
--]£O"12 t> [(0-11) 2 + (O"13) 2] 1/2.
(3.1)
where /z is the coefficient of friction of microcrack surfaces. The conditions in (3.1) are formulated in the local coordinate system (OX'lX'2X'3) corresponding to the global one. In the local coordinate system ( O $ ~ ) corresponding to the principal stress coordinate system, the conditions of no frictional sliding for closed microcracks is similar to (3.1) but with 0-/'i replaced by a;j =~i~,j,~k,, where a l l = O"1, t~22 = 0"2, ~33 = 0"3 and all others ~ij = 0. The contact stresses transmitted across the microcrack surfaces 0-'~ are . =. 0-22, . . 0"22
c = O"21, . . 0-23 . . = 0"23 0-21
(3 2)
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S. - W. Yu, X. -Q. Feng / Mechanics o f Materials 20 (1995) 59-76
and the components of the displacement discontinuity microcrack vector are
b i = 0.
(3.3)
(2) The microcracks are closed and experiencing frictional sliding, then 0-22 < O,
--/.L0-~2<
[(0-~1) 2 "t- (0-~3)2] '/2
(3.4)
and tc t 0-22 ~---0-22,
r ! -- ]'~ 0-22 0-2,
tc 0-21 =
Als COS2~ + Azs <~O,
¢(0-;0 2 + (0"5) !
' nls cos4t~ --]-n2s c0s2(~ --bB3s ~ O,
t
--/-t.0-220"23
o-~ =
The orientation domain in which all microcracks satisfy inequality (3.4) is called the domain of frictional sliding. In other words, all microcracks in the domain of frictional sliding are closed and experience frictional sliding. For convenience, the domain of frictional sliding is calculated first in the principal stress coordinate system. Then from O~j--gikgjl~rkl, -' the conditions of microcrack frictional sliding are rewritten as
(3.5)
¢ ( 0 " ; 1 ) 2 nt- (0-;3) 2 '
b; = ~a z -- r2 B ij0-2j ' ,d
(3.6)
where the nonvanishing components of the displacement tensor for a closed microcrack B~j are
where AI~ = sin26(0-1 - 0-3), A2, = cos20 02 + sin20 0"3, B,, = sin2
(1 +
2)(0"1 - 0"3) 2,
B2s ~- sin2[~( 0", -- 0-3) [2]d'2( cOS2~ 0"2 "]-
16(1 - v 2) ' B ' 'I -- B33
(3.7)
r r E ( 2 - v)
and the driving stresses of deformation of microcrack surfaces are vd
l
tc ~ O,
vd
0"22 = 0"22 -- 0-22
v
vc
v
0"2, ----0-2, -- 0"21 =/~0-21
,d = 0-23 , -- 0-23 ,~ = ~0-23, 0-22 with
(3.9)
Using the transformation relations (2.5) between the global coordinate system and its corresponding local coordinate system, the components of the displacement discontinuity vector for the closed microcrack in the global coordinate system are (3.10)
b i = ( a 2 - r2)'/2flB~sg~ig'2kg'sl0-kl .
Then the inelastic compliance tensor induced by a single closed microcrack with radius a and orientation (0, ~b) can be obtained as
S~jkt( a, O, q~, 0-mn) ~a --
3 t
t
t
t
t
t
t
3V Blsg2kgsl(gtigzJ + gtjgzi)"
sin2ff 0"3)
+ 2 COS20 (0-2 - 0-3) - (0-1 - 0"3)], B3~ =/z2( c°s2ff 0-2 + sin26 o"3)2 - sin20 c°s20 (0-2 - 0"3)2.
(3 8)
/z 0-~2 /3 = 1 + ~/(0-;1) 2 + (0-5)
(3.12)
(3.11)
(3.13)
For any value of t~ in the range (0 ~<0 ~<~ / 2 ) , the range of ~ satisfying the conditions of frictional sliding can be obtained easily from (3.12) and expressed as ¢~1(0) ~<~ ~<4~2(0). So the domain of frictional sliding of microcracks can be written as /~={0~0~-12T/"
,
(~1(0) ~(]~(]~2(0)}.
(3.14)
(3) The microcracks are open, then o'~2 > O,
(3.15)
which can be recast as A l s COS2~ + A 2 2
> 0.
(3.16)
If 0-1 > 0 or 0-3 > 0, there will be a certain orientation scope of open microcracks, which can be calculated from (3.16). The influences of open microcracks on the constitutive relations of materials have been discussed above and will be omitted in this section.
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S.-W. Yu, X.-Q. Feng /Mechanics of Materials 20 (1995) 59-76
3.2. The mode-H growth of closed microcrack and the DMG All microcracks are assumed to be pennyshaped and have the same radius a 0 in the initial state before loading. With the increase of compressive stresses, seme microcracks in the domain of frictional sliding will grow in a self-similar fashion, and their radius increase from a 0 to a u. The transition from a 0 to au is unstable and instantaneous. However, the microcrack does become arrested at the edge of the interface by matrix materials that have higher strength than interface. When a microcrack is closed and undergoes frictional s][iding, the mode-I stress intensity factor K'I = 0, and the mixed-mode fracture criterion (2.13) red,aces to K~I = KHc ,
(3.17)
where the mode-II stress intensity factor K'n for the closed microcrack is
K]I= (-~4)~f~"
[/~o.;2"k-¢(O'21)2"F(o.23)21 •
Then under the condition of proportional loading, the D M G under the stresses trij, which is defined as same as that above, is obtained from (3.19) as the following form
=
{0
-
1
~< 0 ~< Un- ,
(3.21) where 4~3(0, O/j) and t~4(0, O/j) denote two functions of the angle 0 and the stresses trij. Since the mode-II self-similar growth occurs only for part of the microcracks experiencing frictional sliding, the D M G / ] is a subset of H under the condition of proportional stress loading. From (3.21) and the transformation relation (2.19) between (0, ~b) and (/~, q~), the D M G in the global coordinate system can be obtained as a(o/j) = {O
(3.18) Similarly to (3.12), the condition of growth of a closed microcrack can be re-expressed as C 1 cos44~ + C 2 cos24~ + C 3 >/O,
(3.19)
where C~, C 2 and C 3 are functions of 6 and C, = - sin40 (1 +/.[,2)(o. 1 -- 0"3) 2,
C2 = sin20( 0.' - °'s ) [ l't( 2 - v ) KI'c -
3.3. Evolution of DMG and calculation of compliance under complex loadings At time t, the material is subjected to the triaxial compressive stresses O-gj(t), and the evolved D M G is 120). From time t to t + At, the stresses change to another triaxial compressive stress state trij(t + At) = trij(t) + Ao.o. Substituting the stresses O/j + AO/j into (3.22), we obtain the D M G corresponding to this stress state as
1 , a(o.ij"]- Ao'ij) = {O < O ~ ~'w
2/z2( c°s2~ o.2 + sinZ0 o.3)
~)3(0, o.ij-[- mo'ij)
~<~b ~< 64(0, trij + AO/j)}. (3.23) ~ 2
C0S20( O.2-
+
(0.,
-
Then, the D M G at time t + At is obtained from the union of O(t) and O(O/j + AO/j), i.e.,
C3 = sin20 cos2t~ (oh -o'3) 2
O(t+At)
[ (2 -- 4) KIIC ~ f ~
- / x ( c°s2~ o.2 + sin2o o3 )]~ •
=O(t)
ua(o/ +ao/j).
(3.24)
The stress-strain relation in the global coordinate system is (3.20)
iO l q_ Sijk iS I ..}_Sijkl)o.kl Eij = SijklO.kl = ( SO.kl q_ Sijk iG (3.25)
68
S.-W. Yu, X.-Q. Feng/Mechanics of Materials 20 (1995) 59-76
where sii~°t is the inelastic compliance tensor due to open microcracks, si~St and Si~l are due to microcracks experiencing frictional sliding, and microcracks growing in a mode-II self-similar fashion, respectively, and
,s Sijkl
=ff. (trij)NcP(a,
onset of microcrack kinking. As an illustration, let us consider the special case of 0-1 0"3, then /eli =
= (4) ~-~ [sin2dcos2d(0-1 °"2) +/z( sin2~ 0-t
O, dp)
+ COS200-2)] "
(3.29)
×S~jkl(ao, O, ok, trij ) sin 0 d~b dO
By differentiating (3.29) with respect to 0, the orientation of the first kinking microcrack is given by
- ffa Ncp(a, O, 6 )
00 -- arctan(/x + ~ 2 ~ - ~ )
×~jkl(aO, O, q~, o'ij ) sijGl = ffa
sin 0 d~b dO; (3.26)
Ncp(a, O, 4,)
×$/ykt(au, 0, 6, 0-/j) sin 0 d~b dO,
(3.27)
which depends only upon the frictional coefficient but not on the value of 0-1 and 0"2. The corresponding kinking threshold value of the maximum compressive principal stress 0-2 is 1
0-2
where f2 o = O(t)(3 H(0-iy) is the orientation domain of frictional sliding microcracks with radius a u under stresses 0-ij.
F(ff0 )
[ v~-(2-/~)KICC~u +/~0-1]+°'1 × -
8 (3.31)
3.4. Kinking of microcracks under compression With a further increase in the compressive loadings, the higher tensile stresses induced at the tip of microcracks will drive some arrested microcracks in D M G to kink into the matrix material. Since the brittle matrix materials have low resistance to tension, we adopt the maximum circular stress at the microcrack tip (0"0)m~x as the kinking criterion of microcrack. Then from the K-fields of stresses at the near tip of mode-I and mode-II cracks, the kinking criterion of closed microcrack is derived as (Fanella and Krajcinovic, 1988) Kil = - ~ - K I c c.
(3.28)
From the mode-II stress intensity factor Kii in (3.18), we can obtain the orientation (0o, ~b0) at which Kii reaches its maximum value among all orientations. The microcrack kinking occurs at first in the microcracks with orientation (00, q~0). By substituting (00, 4~0) into (3.18) and (3.28), we further obtain the compressive stresses at the
(3.30)
where F(O 0) = sin200 cos200 - t~ cos200 . (3.31) agrees with the result of Fanella and Krajcinovic (1988). For a general triaxial compressive stress state, similarly to (3.12) and (3.19), we obtain the following inequality Clk
COS4~÷ C2k COS2~÷ C3k ~ O,
(3.32)
where the coefficients Clk , C2k and Cak can be obtained from C~, C 2 and C a in (3.20) by replacing Knc with V3 K i c c / 2 . For a given stress state, (3.32) gives the domain of kinked microcracks, which is called simply the kinking domain and denoted as /l(0"iy) in the principal stress coordinate system. The kinking domain A(0"i~) in the global coordinate system can be obtained from A(0"q) and (2.19).
3.5. The compliance tensor induced by a kinked microcrack For a closed, frictional sliding microcrack, the maximum circular tensile stress (0-o)max occurs at
S.-W. Yt~ X.-Q. Feng / Mechanics o f Materials 20 (1995) 59-76
the direction of an angle Ok = arcsin(2v~-/3) to the microcrack surfaces. So the initial direction of kinking is not the same as the direction of the ;2-axis (or the direction of the maximum compressive principal stress o-2 in the far field). The microcrack reaching the kinking criterion (3.28) propagates first into the matrix material in the
69
direction at the angle Ok to the microcrack surfaces, as shown in Fig. 3a. With the growing microcrack, the lengths emanating from both ends of the sliding microcrack increase; and they eventually align themselves parallel to the direction of o"2 and cause macroscopic axial splitting. When °'1 *o-3, the stress intensity factor reaches its
X2
O
Z/
(a)
(b)
t
a k
k
2~ k a . sinO
2a~
~ "~
I P = 2~a a k
(c)
(d)
(e)
Fig. 3. (a) A three-dimensional kinked microcrack; (b) an approximate three-dimensional kinked microcrack; (c) two-dimensional kinked microcrack system; (d) equivalent microcrack system (Fanella and Krajcinovic, 1988); (e) equivalent microcrack system (Zaitsev, 1983).
70
S.-W. Yu, X.-Q. Feng / Mechanics of Materials 20 (1995) 59-76
maximum value along the microcrack edge at A' and B' with a small angle /30 = arctan (¢T23//0~21) included between A'B' and the $~-axis, see Fig. 3a. Therefore, the positions on the microcrack edge from which the kinked microcrack emanates are not at the intersection points of the $~-axis and the microcrack edge, but at the points A' and B'. It is impossible to give a closed form of the solution of the displacements of the kinked, three-dimensional microcrack in Fig. 3a and their contributions to the compliance tensor. Kachanov (1982) and Fanella and Krajcinovic (1988) assumed that the three-dimensional kinked microcrack can be approximated by a series of two-dimensional kinked cross-sections. Here a similar method is used to deal with kinked microcracks. First, we assume that the microcrack kinks and grows in the same direction of the maximum compressive stress 0"2, as shown in Fig. 3b, and that the projection of the microcrack on the 22--3~; plane is an ellipse with its lengths of the semimajor and semiminor axes a, and a k. Secondly, the three-dimensional kinked microcrack in Fig. 3b is approximated by a series of two-dimensional cross-sections with angle/3 ( - ~ r / 2 ~
nience, they are expressed as functions of (0, 4;) and o7/'j in the form cotan PO ~
-
'
S0=
(/-1.~;2 + V/(t~l) 2 -1--(~3) 2 )
-
ot k
~ (j~O~2 -1- }/( 0~;1) 2 -t- (~;3) 2 )
a k sin (3.33) where a k is a dimensionless factor introduced to guarantee the equivalence of the stress intensity factors of the equivalent microcrack and the actual one, here ~g is taken to be 0.25 (Lee and Ju, 1991). The displacements of the kinked microcrack consist of two parts. The first part is the opening displacement 5(/3) perpendicular to the microcrack surface (kink opening) for any cross-section located at an angle /3. The second part is the average sliding displacement t~(/3) of the closed surfaces of microcrack for the /3 cross-section (Fanella and Krajcinovic, 1988). 4.8(1 - 1'2)
-, COS /3 F 1( a u , O'21
a k,
e) ,
4.8(1 - v 2) a(/3)
-
rrE
a;1 cos/3 F 2 ( a , , a k, 0), (3.34)
where
Fl(au, ak, if) COS 0 , - sin---2-~F2(au, ak, O) -
c°tan20[arcsin(all~(a2)Z_(al)2 ak [ ~ a2 I
,n(al l ] iazl] with a 1 = aka u sin 0 and a 2 = a 1 + a k. By averaging the displacement components along the interface microcrack rim ( - 7r/2 ~
S.-W. Yu, X.-Q. Feng /Mechanics of Materials 20 (1995) 59-76
kinked microcrack are determined by the following two parts.
"k, a u, ak, O, ~b, '~ij) Sijkl(
71
the other is the lateral confinement loading p* acting on the kinked microcrack surfaces due to the applied far-field stresses, p* = cos24~ or1 + sin24~ tr3.
0"6/3a"ak (gi~g2j + g#g2i)
-
Then the mode-I stress intensity factor at the kinked microcrack tip under the concentrated force P1 and the distributed force p* can be obtained as (Tada, 1973)
V .
.
.
.
.
(3.41)
,
Xg2kgtlBstFll au, ak,
~kz ijkl~(a u,. ak, O, ~ , ,Tij)
K~=
0 " 6 / 3 ( a , ) 2 / - , ~, +o~,~i,) -[ gis 2j
2au7"n cos/~ ~
+ ~
p*.
(3.42)
V
X~'2k~,;iB'stF2(au, ak, 0),
(3.36)
where 8(1 - t'z) rrE ,
B]I
gii =
and all other B~j = 0
01 co: 1
0
-cos4~
0
(3.37)
(3.38)
sin4~]"
Then the corre,;ponding inelastic compliance tensor in the global coordinate system induced by the kinked microcrack is obtained from (2.16) and (2.19) as
7S~fkt(au, a k, O, ~b, trij ) =[
[ ~k, mnst~[ a u , ak,
O,~,o'ij)
"k2 a u, ak, 19, qb, trij)] "[-Smnst( Xgmignjgskgi! l
(3.39)
where the length a k will be determined below.
3.6. Stability analysis of microcrack kinking In order to find the relation between the kinking length a k of a microcrack and the applied stresses, the simplified equivalent microcrack in Fig. 3e is adopted (Ziatsev, 1983). The loadings acting on the kinked microcrack surfaces actually have two parts, one is the concentrated force P1 due to the shear stress ~'n = / x ~ 2 + ~t
2
¢ ( ~ 1 ) 2 + (t r23) , i.e., P1 = 2a,~'n cos/~,
From (3.42) and the criterion of growth for the kinked microcrack K~ = Kicc, the kink length a k (at /3 = 0) can be derived as the following form
(3.40)
where sgn(p*) = 1 for p* > 0 and - 1 for p* < 0. For the special case of p* = 0, the kink length of the microcrack is
1 a k = -•-
(
2au7 n cos 0 (3.44) gic C
Notice that the stress intensity factor in (3.42) is in a good agreement with the numerical results in Horii and Nemat-Nasser (1985) only for big a,, and is invalid when ak tends to zero. For different cases of lateral confinement loading p*, the stability of kinking of microcrack is analyzed as follows. (1) p* = 0. An a,-K~ curve for p* = 0 is obtained from (3.42) and shown in Fig. 4a. Under a given stress state, the value of K~ decreases monotonically with increasing a, and tends to be zero when a k tends to infinity. Therefore, the microcrack stops kinking when its kink length ak reaches the length akC. So the kinking of microcrack is stable. With the applied stresses increasing proportionally, the stress intensity factor and also the corresponding kink length increase. As shown in Fig. 4a, if the stresses crib-2) aO'/(j1) with a > 1, the corresponding kink lengths satisfy a~k2) > aCk1). SO the gross axial splitting will occur ultimately in the material when the applied stresses reach a certain level. =
S.-W. Yu, X.-Q. Feng / Mechanics of Materials 20 (1995) 59-76
72
(2) p * < 0 . An a k - K ~ curve for p * < 0 is shown in Fig. 4b. For a given stress state, K~ decreases monotonically with the increase of a k and intersects the araxis. Therefore, the microcrack stops kinking when a k reaches a k c and hence the kinking of microcrack is also stable. For p* < 0, there is a critical length a k max satisfying K ' i ( a k max) = 0. From (3.42), we have a k max --
2 a u r n cos -- "wp*
t?
(3.45)
With the applied stresses increasing proportionally, a k max remains constant, and so the kink
t
length of microcrack increases also but cannot exceed the critical value a k max" If the lateral confinement load p* is small (and then a k max is big), it is still possible that the axial splitting occurs in the material. When p* is big, the microcrack kinking is limited by ak max" Therefore, the kinking of microcrack is not the mainly damage mechanism in materials when the confinement load is big enough. Instead of axial splitting, the faulting or ductile flow becomes the main failure mode of brittle material (Horii and Nemat-Nasser, 1986). (3) p * > 0. As shown in Fig. 4c, the a k - K ' I
t
KI
K x
Kicc
Klcc
-,.-.<_
I
(2)
ii
i I
0
(t)
(a)
akc
akC
a k
0)
(2)
akc akc
(a)
a kraax~
(b)
t
KI
Kmc
,
,, O)
(2)
a kC
a kC
(2)
I
m
'
a k
a kC
(c) Fig. 4. T h e ak-K ~ curves for: (a) p * = 0; (b) p * < 0; (c) p * > 0.
ak
S.-W. Yu, X.-Q. Feng /Mechanics of Materials 20 (1995) 59-76 curve for p* > 0 has a knee point, before which the curve decreases monotonically with increasing a k and after which the curve increases. Firstly, the kinked microcrack grows until the length a k reaches akC. akc increases with the increase of stresses, i.e., a~2) > a(k1) when 0"(2)> 0-/(jl). Even if the lateral tensile sl:ress p* is much smaller than 0-2, the kinking of rnicrocrack will become unstable with the externally applied stresses increasing. From (3.42), the critical kink length of microcrack growing unstably a~c can be obtained for the case p* > 0. Since the kin.kin.g occurs first in the microcrack orientation (00, ~b0), substituting (O0, 4~0) into (3.42) leads to
g'I(ak, 0o, ~0) =
73
where Si/J~t is the inelastic compliance tensor induced the kinked microcracks, and
iG _Sijkt
aNcP( a, O, c~) X~S[jkl(au, O, q~, 0"ij) sin 0 d~b dO
- ffaNcP( a, O, 4~) XS~jkt(a,,, O, dp, 0"ij) sin 0 d~b d0, siijK l =
ffANcP( a, O, 40 ×S~yk,(au, a k, 0, ok, tri~) sin 0 d~b d0.
2au'rn(dO, ~bo) cos Oo
(3.51)
+
(3.46) Then for a given stress state, K~(a k, d o, ~o) reaches its minimum value .
(gI)min =
[8au~'n(dO,
] 1/2
t ~ o ) P * ( t ~ O ) cos 00]
(3.47) when
2au~'n(Oo, qbo) COS do ak=a'~c=
4. Theoretical results and experimental validation
~'P* (4~0)
(3.48)
From (K~)mi n = g l c c , the condition for applied stresses under which the microcrack kinking becomes unstable is obtained as 8a,zn(0 o, ~o)p*(q~o) cos 0o = K2cc.
(3.49)
Once this condition is met, the macroscopic axial splitting occurs in tlhe material.
3. 7. Calculation of the compliance tensor After considering the closure, frictional sliding, self-similar growing and kinking of microcracks, the overall compliance tensor is obtained as iG + Sijkl iK Sijk, = siO.kl + siij2l + siijSl -J- Sijkl
(3.50)
In order to validate the proposed three-dimensional micromechanical model for brittle materials, two examples of concrete under uniaxial tension and uniaxial compression are considered in this section. The theoretical results obtained here are compared with the experimental data of Gopalaratnam and Shah (1985), and the results of Ju and Lee (1991). Some theoretical results from the model of DMG for a specimen under complex tensile loadings can be found in Feng and Yu (1993).
4.1. Example of uniaxial tension As an illustration, the damage model of D M G is applied first to a concrete specimen subjected to uniaxial tension loading, i.e., 0-1 = 0"3 ~--0, 0 2 = 0". Based on the analytical expressions derived in the previous sections, the D M G for uniaxial tension is
1 ~'~1 ~--"{0 -~< 0 ~.< 0ma x -~< g'/T,
0 ~< ~b < rr}
(4.1)
S.-W. Yu, X.-Q. Feng/Mechanics of Materials20 (1995)59-76
74
with
80 ~ Stress (MPa)
2
-B2-~B2-4B1B3
tan 0max = - - - - - ~ F - - - -
;
*~
(4.2)
*'~
*~
BI = -
/t/,
, ~N~Lateral 1K\~\ Volumetric
/ , '/~, Ax,al /~". //9"*
601" *\\
4-ao KII C , B2 = a r t -~- [ 2---~ ) , ( o'KIIc ) 2
B3 = B1+
~
.
(4.3)
The effective compliance of the specimen under uniaxial tension is 1 + P(10/Z
3v)
-~o
V
/5
1
---
,
Resultsin thispaper Experimentaldata ,1991)
- Results in Lee a n d Ju
1,0
0
for O < ~ r < o " A $2222 =
i 1 ~ / // }I ~ / /t/
20
=
Strain (x 10 -4)
Fig. 6. Comparison of the theoretical and experimental results for uniaxial compression.
p
~-+~[y(lO-3v)+(1-y)
where
× (10 COS30max -- 3v COS50max)] , for o-c ~
ire=
Kic ,
O'cc=
Kxcc;
(4.5)
(4.4) 16( 1 - v 2) Nc a3 P=
( au ]3. ' Y = k ao l
(4.6)
As in Ju and Lee (1991) and Hoenig (1978), we take K i c = 0.165 M N / m 2/2, K]I C = 0.33 M N / m 3/2, K1c c = 0.301 M N / m 3/2, a 0 = 0.34 cm, a u = 0.49 cm, E -- 34450 MPa and v = 0.3. Using the formula from Fanella and Krajcinovic (1986) and Ju and Lee (1991), the microcrack number density N c / V is taken to be 1.81 × 106 (1/m3). Then the uniaxial tensile stress-strain curve obtained from (4.4) is plotted in Fig. 5 and compared with the experimental data in Gopalaratnam and Shah (1985). It is observed that the analytical results are in a good agreement with the experimental results. The small error is due to microcrack interaction which has not been considered. If the self-consistent method is introduced in the damage model, the results will be better.
4 Stress (MPa)
3
2
1
*
45(2-v)V
Experimental data Strain ( x 10 - 6)
0
i
i
iI-
50 100 150 Fig. 5. Theoretical stress-strain curve for uniaxial tension compared with experimental results.
4.2. Example of uniaxial compression
0
The presented damage model is applied again to the concrete specimen subjected to uniaxial
S.-W. Yu, X.-Q. Feng / Mechanics of Materials 20 (1995) 59-76
compression. The experimental parameters reported in Lee and Ju (1991) are as follows: the initial values of the modulus of elasticity E and the Poisson's ratio v are 40020 MPa and 0.2, respectively; the critical stress intensity factors of the interface and matrix Kic = 0.165 M N / m 3/2, KII c = 0.33 M N / m 3/2, Kic c = 0.495 M N / m 3 / 2 ; the volume fraction of the coarse aggregate is given as 0.39; the statically averaged radius of microcracks are a 0 = 0.47 cm, a u = 0.81 cm. The theoretical results of the axial, lateral and volumetric strains for the case of uniaxial compression are compared with the experimental and theoretical results Jin Lee and Ju (1991). It was shown that these results correlate well with each other.
5. Conclusions Based on a sound and physically convincing model of microstructures, the micromechanicsbased damage theory of the domain of microcrack growth (DMG) is developed for microcrack-damaged brittle solids. The model describes the complex evolutionary damage process during loading and the anisotropic constitutive behaviors of materials under tensile a n d / o r compressive loadings. The concept of D M G is defined to describe the damage state of brittle materials and the evolution of the overall effective fourth-order compliance tensor for the case of complex loading path. Based on modified fracture criteria of mixed-mode cracks, the D M G for brittle materials under complex tensile or compressive loadings can be calculated. Mode-I growth of open rnicrocracks, frictional sliding, mode-II growth and kinking of closed microcracks were all studied and their influence on the nonlinear constitutive behaviors of brittle materials were introduced analytically. From micromechanical analyses, it is explained that the macroscopic axial splitting will occur in the material only when the lateral principal stresses o-~ and 0"3 have positive or sxlaall negative values. And the condition of axial splitting of materials is formulated. The analytical results from the theory developed here fit the experimental data and results
75
of others well, even though a few approximate assumptions are adopted.
Acknowledgement The support of the National Natural Science Foundation of China is gratefully acknowledged.
References Bazant, Z.B. (1986), Mechanics of distributed cracking, Appl. Mech. Rev. 39, 675-705. Budiansky, B. and R.J. O'Connell (1976), Elastic moduli of a cracked solid, Int. J. Solids Struct. 12, 81-97. Deng, H. and S. Nemat-Nasser (1992), Dynamic damage evolution in brittle solids, Mech. Mater 14, 83-103. Fannella, D. and D. Krajcinovic (1988), A micromechanical model for concrete in compression, Eng. Fract. Mech. 29, 59-66. Feng, X.Q. and S.W. Yu (1993), A new damage model for microcrack-weakened brittle solids, Acta Mech. Sinica 9, 251-260. Feng, X.Q. and S.W. Yu (1994), Micromechanical modelling of strain softening in microcrack-weakened quasi-brittle materials, 12th US National Congress of Applied Mechanics, Washington. Gopalaratnam V.S. and S.P. Shah (1985), Softening response of plain concrete in direct tension, ACI Mater J., M a y / June, 310-323. Hoenig, A. (1978), The behavior of a plat elliptical crack on an anisotropic elastic body, Int. J. Solids Struct. 14, 925934. Horii, H. and S. Nemat-Nasser (1983), Overall moduli of solids with microcracks: load-induced anisotropy, J. Mech. Phys. Solids 31, 155-171. Horii, H. and S. Nemat-Nasser (1985), Compressive-induced microcrack growth in brittle solids: axial splitting and shear failure, J. Geophys. Res. 90, 3105-3125. Horii, H. and S. Nemat-Nasser (1986), Brittle failure in compression: splitting, faulting and brittle--ductile transition, Philos. Trans. R. Soc. (London) A 319, 337-374. Ju, J.W. and X. Lee (1991), Micromechanical damage models for brittle solids, I: tensile loadings, J. Eng. Mech. 117, 1495-1514. Kachanov, M. (1982), A microcrack model of rock inelasticity, Mech. Mater. 1, 19-40. Kachanov, M. (1987), Elastic solid with many cracks: a simple method of analysis, Int. J. Solids Struct. 23, 23-43. Kachanov, M. (1992), Effective elastic properties of cracked solids: critical review of some basic concept, Appl. Mech. Rev. 47, 304-335. Kanninen, M.F. and C.P. Popelor (1985), Advanced Fracture Mechanics, Oxford University Press, London.
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Krajcinovic, D. (1986), Continuum damage mechanics, Appl. Mech. Rev. 37, 1-6. Krajcinovic, D. (1989), Damage mechanics, Mech. Mater 8, 117-197. Krajcinovic, D. and D. Fanella (1986), A micromechanical model for concrete, Eng. Fract. Mech. 25, 585-596. Krajcinovic, D. and D. Sumarac (1989), A mesomechanical model for brittle deformation processes, Part I and II, Z Appl. Mech. 56, 51-56, 57-62. Lee, X. and J.W. Ju (1991), Micromechanical damage models for brittle solids, II: compressive loadings, J. Eng. Mech. 117, 1515-1536. Lemaitre, J. and J.L. Chaboche (1988), Mechanics of Solids Materials, Cambridge University Press, London. Moss, W.C. and Y.M. Gupta (1982), A constitutive model
describing dilatancy and failure in brittle rock, J. Geophys.
Res. 87, 2985-2998. Murakami, S. and N. Ohno (1981), A continuum damage theory of creep and creep damage, in: A.R.S. Ponter, ed., Creep of Structures, IUTAM Symposium, Springer, Berlin, pp. 422-444. Nemat-Nasser, S. and M. Obata (1988), A microcrack model of dilatancy in brittle materials, J. Appl. Mech., 55 24-35. Ortiz, M. (1985), A constitutive theory for the inelastic behavior of concrete, Mech. Mater. 4, 67-93. Tada, H. (1973), The StressAnalysis of Cracks Handbook. Del Research Corporation., Paris. Zaitsev, Y.B. (1983), Crack propagation in a composite material, in: F.H. Wittmann, ed., Fracture Mechanics of Concrete, Elsevier, Amsterdam, pp. 31-60.
Mechanics of Materials 20 (1995) 59-76
A micromechanics-based damage model for microcrack-weakened brittle solids * Shou-Wen Yu, Xi-Qiao Feng Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China
Received 1 November 1993, revised version received 16 May 1994
Abstract In the present paper, a micromechanics-based, three-dimensional damage model for microcrack-weakened brittle solids is developed. In order to describe the evolutionary damage state and anisotropic properties of materials, the concept of domain of microcrack growth (DMG) is defined as the union of all possible orientations of propagated microcracks after a loading path. Based on modified mixed-mode growth criteria of microcrack, the evolution of DMG as well as the overall effective compliance tensor of damaged materials are formulated. Through a micromechanical analysis, the damage mechanisms and the complex constitutive behavior of materials are studied under complex loadings. The self-similar growth of open microcracks under tension, the mode-II growth and the kinking of closed microcracks under compression and the influences of these mechanisms on the mechanical properties of materials are all considered. It is explained that axial splitting may occur in a material only when the lateral normal stresse,; have positive or a small negative value. And the condition for axial splitting of a material is given. The evolutiona~:y damage model is illustrated by two examples of uniaxial tension and uniaxial compression and the theoretical re:~ults are compared with experimental data and theoretical results obtained by others. Keywords: Domain of microcrack growth; Damage mechanism; Effective compliance tensor; Growth and kinking of microcracks; Brittle solids; Compression loading condition; Anisotropic properties
I. Introduction Many brittle materials, such as concrete, rock, most ceramics and intermetallics, have a large number of preexisting microcracks. The nucleation, growth and coalescence of microcracks may cause a nonlinear degradation of materials and the failure of strucr~ures. So the macro- and micro-damage of materials is a popular and also
* Correspondence to: Prof. Shou-Wen Yu, Office of the President, Tsinghua University, Beijing 100084, P.R. China. Fax: 0861-2568116. Tel.: 0861-2595533.
important subject in solid mechanics and material science. In the current literature, there are two main classes of damage models, i.e., continuum damage models (CDM) and micromechanical damage models. Some fairly complex responses of materials have been simulated by CDM (see, e.g., Lemaitre and Chaboche, 1988; Murakami and Ohno, 1981; Krajcinovic, 1986, 1989). Models for the establishment of micromechanical damage have improved the understanding of physical process and microscopic mechanisms of damage in materials. In the case of stationary elliptical microcrack and isotropic damage, the self-consistent
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60
S.-W. Yu, X.-Q. Feng / Mechanics o f Materials 20 (1995) 5 9 - 7 6
method was employed by Budiansky and O'Connell (1976) and Horii and Nemat-Nasser (1983) in deriving effective moduli. For anisotropic damage with open penny-shaped microcracks, several important micromechanical evolutionary damage models can be found in Krajcinovic and Fanella (1986), Krajcinovic and Sumarac (1989), Ju and I~e (1991), Ortiz (1985), Nemat-Nasser and Obata (1988). Most of the damage models in these two classes were summarized and recapitulated in review articles (see, e.g., Krajcinovic, 1986, 1989; Bazant, 1986; Kachanov, 1992). The many phenomenological and micromechanical damage models of materials that have been based on different damage variables, have, at least, two evident drawbacks. First, the damage variables in the form of either a scalar, a vector or a tensor defined in the earlier models cannot describe the evolutionary complex anisotropic damage state of materials exactly. In order to improve the accuracy of damage description, some fourth- and even eighth-order tensors were suggested as damage variables (see Krajcinovic, 1989). But such a description often entailed complexities exceeding their advantages. Secondly, it is difficult for such models to handle a complex loading path. In this study, the usual damage variables of scalars and tensors are all abandoned, and instead the domain of microcrack growth (DMG) is defined to describe the constitutive behavior of damaged materials. The DMG, which is defined as one set, or a union of sets, is the orientation scope of all possible propagated microcracks after a loading path. In other words, all microcracks whose normal vectors are in the orientation scope of the DMG must have propagated after the loading path. By this concept, the damage state of microcrack-weakened solids can be described more exactly and the problem of complex loadings can be solved (Feng and Yu, 1993). Based on DMG and a micromechanical analysis, Feng and Yu (1994) have investigated damage localization and strain softening in brittle materials. In this paper, DMG is extended to brittle materials subjected to three-dimensional tensile or compressive loadings. In Section 2, the inelastic compliance tensor induced by a single open
penny-shaped microcrack is derived at first. Based on an appropriate growth criterion of microcrack, the DMG under a general three-dimensional proportional loading is formulated. Then, the evolution of the DMG during any complex loading path is considered, and the overall effective compliance tensor of damaged materials is given. Thus the anisotropic constitutive behaviors of materials can be simulated. In Section 3 the damage mechanisms of brittle materials under triaxial compression below the brittle-ductile transition points are studied through a micromechanical analysis. The self-similar model-II growth and kinking of closed microcracks under compression are considered and their effects on the mechanical properties of materials are introduced into the fourth-order compliance tensor. Finally, in Section 4, the three-dimensional damage model suggested here is applied to two simple examples of uniaxial tension and uniaxial compression and the results obtained are compared with the experimental data and theoretical results obtained by others. 2. The damage model of DMG under tensile loadings
2.1. Single open microcrack-induced inelastic compliance tensor Choose a representative volume element (RVE) containing a statistically valid sample of microstructures and assume that only small strains and small rotations occur. The virgin matrix material is assumed to be linearly elastic and isotropic. Then the volume-averaged strain tensor is
Eij = ~ej + ~-ii j ,
(2.1)
with --e
0
--
(2.2)
~'ij ~---S i j k l O ' k l ,
1% =
~ij
(2.3)
,
k=l
~/(.ot)_ tj
~i(ot)~.
-- L'ijklUkl
1
1
= -~ f s j ( b i n j + bjni) (~) dS,
(2.4)
where N c is the number of microcracks in the RVE occupying the volume V, ~ej and ~ij -i are the
S.-W. Yu, X.-Q. Feng / Mechanics o f Materials 20 (1995) 59-76
elastic strains and the microcrack-induced strains, respectively, ~'~) the strains induced by the ozth microcrack, S°.kt and ~i(~) Oijkt the undamaged elastic compliances and the a t h microcrack-induced compliances, respectively, and where b i = [u i] are the components of the displacement discontinuity vector, n~ those of the normal vector of the microcrack, and ~j the volume-average stresses which are assumed to be equal to the applied far-field stresses o-iy in the Taylor model. First consider the a t h single penny-shaped microcrack with radius a in an isotropic body uniformly loaded at far field (Fig. 1). Establish the global coordinate system ( O x ~ x z x 3) and its corresponding local coordinate system (Ox'~x'2x'3), as shown in Fig. 1, in which the x~-axis is parallel to the normal vector of the microcrack n, and the x~-axis is coplanar with the x~-axis and the x3-axis. Then the orientation of the microcrack can be expressed as (0, ~b). The basic vectors of the two coordinate systems are related by t
t
ei = g i j ~ : (2.5) where the transformation matrix g~j and its inverse matrix g i y = ( g ~ i ) - l = ( g ~ y ) T can be obtained from g~j = g T
ei -- g i j e j ,
=
COS 0 COS ~b - s i n O c o s 4' sin $
sin 0 cosO 0
- c o s 0 sin ~b] sinOsin~b [. cos 4, J
X 2 X 2
2
b i = (a 2 - r )
1/2
t
t
,
Btyo'2jgti,
(2.7)
where the crack opening displacement tensor B' depends on the compliance of the microcrackweakened solids. A.ssuming here that it depends only on the compliances of an isotropic elastic matrix, then the nonvanishing components of B' for the open microcrack are 16(1 -~,~2) 8(1 - z,2) B~I --- B 3,3 B~2 (2 - v),rrE ' ~E
(2.8)
-
-
-
-
J
I
X 3
x3/" F i g . 1. A m i c r o c r a c k corresponding
with global coordinate
local coordinate
system
and
its
s y s t e m in t h e R V E .
The stresses in the local coordinate system tr'j are given by
% = g~kg~ttrkt.
(2.9)
If the normal stress to the microcrack surface is compressive, assume in this part that the effective shear stress is smaller than the cohesive strength and no frictional sliding occurs. The closed microcracks and their influence on the compliance tensor will be discussed in Section 3. Then the displacement discontinuity vector b can be rewritten as
(2.6) For an active penny-shaped microcrack, the components of the displacement discontinuity vector b = b N i take the form (Budiansky and O'Connell, 1976)
61
2
b i = (a 2 - r )
1/2
t
t
t
t
t
t
B~sgjigzkgslO'kl(O'22), "
(2.10)
where the angle bracket is defined as (x) =
1( x)
1 + ~-~ .
(2.11)
Thus, from (2.4) and (2.10), the components of the inelastic compliance tensor induced by the a t h single open penny-shaped microcrack with radius a and orientation (0, ~b) are
L/jk,(a, 0,
ij)
T/-a 3
-
t
t
t
t
t
t
t
3 V Bm"g2kg"t(gminy +gmyni)(trstg2~g2t)' (2.12)
--i where Sijkl
are
simplified denotations of Oijkl" ~i(~)
S.-W.. Yu, X.-Q. Feng / Mechanics of Materials 20 (1995) 59-76
62
2.2. D M G under three-dimensional proportional tensile loading
l
In fact, it is extremely difficult to give a unified criterion for preexisting microcrack growth, if damage-induced anisotropic compliances and microcrack interaction are involved. Therefore, for practical applications, it is assumed that the microcrack fracture criterion is based on the Taylor model, i.e., no microcrack interaction is considered (Kachanov, 1992), and self-similar microcrack growth will happen when the average value of the energy release rate along the microcrack edge reaches a critical value. The mixed-mode fracture criterion for a penny-shaped crack may take the following modified form (Kanninen and Popelor, 1985; Ju and Lee, 1991) glc
+(~)2=1
[(tr:~l) 2 + (o',~3)2]1/2. (2.14)
Once the fracture criterion (2.13) has been satisfied, a microcrack will become unstable increasing its radius from the initial statistically averaged value a 0 to the final characteristic value a u instantaneously and will be arrested by energy barriers with higher strength (such as grain boundaries). Under a given stress state, the growth of a microcrack is related to only three components of stress tensor in the plane of the microcrack surfaces, i.e., o-~1, o-~2 and try3. By substituting (2.6), (2.9) and (2.14) into (2.13), we obtain
L~-K~]c ] g'2k g'2s -[- ~
l,t g ' ]k g'ls +g'agg'3~)
77> ( g 2' l g 2 t 'o ' k l O ' s t
:
(7 I
3
(2.13)
~22,
Kii = ~
"~l
Fig. 2. The principal stress coordinate system and its corresponding local coordinate system.
where K'I and K'n and Kic and g l i c are the mode-I and mode-II stress intensity factors and their critical values, respectively. K~ and K}I are defined as K'I = 2
O'.~
-4a -r °
iic,
(2.15)
which is the condition satisfied by the boundary of the orientation domain in the orientation space (0, ~b) of all microcracks having propagated under the given stress state. In other words, the orientation scope bounded by Eq. (2.15), which is defined as the domain of microcrack growth (DMG), is the scope of all possible orientations of microcracks with the characteristic radius a u. It is difficult, however, to give an analytical solution of (2.15) directly. For simplicity, we establish the principal stress coordinate system (O$15153), whose ~1, ~2 and ~3-axes have the same directions as the principal stresses trl, tr2 and tr3, and establish another local coordinate system (O$~$~$~), which corresponds to the principal stress coordinate system shown in Fig. 2. The basic vectors #i of the principal stress coordinate system are related to those of the global coordinate system in Fig. 1 by ei = gijey, where the components of trix gij a r e the cosines between ei and ei, and textbook on elasticity. In the principal stress
(2.16) the transformation maof the angle included can be found in any coordinate system and
63
S.-W. Yu, X.-Q. Feng / Mechanics of Materials 20 (1995) 59-76
in the global coordinate system, the normal vector of the microcrack can be expressed as
The solution of Eq. (2.20) is
COS2t~ =
- A 2 _ i ( Z 2 ) 2 -- 4A1A 3 2A 1
(2.17) where g~j(0, 4;) are the components of the transformation matrix between the principal stress coordinate system an,5 the corresponding local coordinate system whiich can be obtained from (2.6) by replacing (0, 49) with (0, 4~). While in the global coordinate system, the same normal vector can be re-expressed as
n =ee' =
- s i n 0 cos 49 e l + c o s 0 e 2 + sin 0 sin 49 %.
(2.18)
By comparing (2.17) with (2.18), the transformation relationship between (0, 49) and (0, 4~) can be obtained as .,
cos
-
O=gzi(O, cb)gi2,
g2i(O, dP)gi3
tan 49=
gEl( O' c~)git
(2.19) In the coordinate system of principal stress, Eq. (2.15) is simplified as A x cos44~ + A 2 cos25 + A 3 = 0, where
A 1 = L~-~IC [ / K I I c I 2] - ( - ~ 2
)2]
\2 KIIC a 2 = [12~~ g l c ~1 (c°$20 0-2 -I-sin20
T/" --K2c" 4a0
1 >/
- A 2 + t ( A 2 ) 2 -- 4 A ~ A 3 2A1
>/0.
(2.23)
So for any value of d in the range (0 ~< 0 ~< 7r/2), the solution of (2.20) can be given by (2.22) if the conditions (2.23) are satisfied. In addition, if the conditions (2.23) are not satisfied for a given value of 0, the two following situations should be distinguished: (1) The left-hand side of (2.20) is negative for any value of 4~, then no microcrack propagates for this value of 0, and we set 4)(0) = 0.
(2.24)
(2) The left-hand side of (2.20) is positive for any value of 4~, then we set the orientation scope of propagated microeracks for this value of d as (2.25)
from (2.22) to (2.25), the D M G in the principal stress coordinate system under three-dimensional stresses is derived and can be written as
1"2(0, ~b, 0-ij)
=(0
= ( "IIC 12(c0s2~2 + sin 2-00"3)
-
(A2) 2 - 4AIA 3/> 0,
0-3)
X (0-1 - 0"3) sine0;
-
The conditions for the existence of solutions are
Thus
crI - cos 0 0-2 + cos 2/~ o"3
+ ~
(2.22)
0 ~<49(0) ~<27r.
(0-1--0-3)2 sin40;
r/
+
(2.20)
(0 < q~ ~<27r).
n(o, 49, 0-,j)
('re + 0"3)2 sin2/~ cos20 (2.21)
=
1
49-(0,
(2.27)
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S.-W. Yu, X.-Q. Feng / Mechanics of Materials 20 (1995) 59-76
As the externally applied stresses increase, more and more microcracks propagate, increasing their radius from a 0 to a, and being arrested at the edges of the weak planes by energy barriers with higher strength. Under a certain load level, some of the arrested microcracks will become unstable again and pass through the energy barriers when the following secondary fracture criterion is satisfied
g--~cc ]
(2) If solutions of (2.20) exist, express the D M G under the stresses trq + Atriy as
~ ( orij -~- morij ) = {0 ~ 0 ~ 17T, ~ - ( 0, Orij -[- mo'ij ) ~<6 ~<~b+(0, ~rij+ A~rij)}. (2.30) So the D M G at time t + At is
J2(t+At)=O(t)Ug2(crij+Atriy ).
(2.31)
(3) Decide from (2.28) whether gross failure occurs in the material under the stresses ~ij + Atrij.
\ glICC
where Kic c and Kiicc are the critical values of mode-I and mode-II stress intensity factors of the energy barriers, respectively. Once the criterion (2.28) is reached by a microcrack in a preferred orientation, we take it that the stresses have caused the gross failure of material. In fact, the secondary growth of microcracks may cause the complex phenomena of damage localization, stress drop and strain softening, as was investigated by Feng and Yu (1994).
2.4. Compliance tensor of damaged materials The orientations and sizes of penny-shaped microcracks in the RVE can be viewed as random variables and represented by a probability density function p(a, O, qb), which must satisfy the following normalization condition
faamaXfo/2foe~P(a' O' c~) sin O ddp dO da = l" mill
2.3. Evolution of DMG under complex loadings Within the framework of the Taylor model, no microcrack interaction is considered, and the fracture criterion of microcrack is uninfluenced by the loading history. So the D M G under a complex loading path can be obtained by substituting every stress state of the loading path into (2.26) and (2.27) and making the union of the DMGs corresponding to all stress states. In order to show the evolution of DMG, assume that the D M G at time t can be expressed as O(t)={0~<0~<½~',
~b-(0, t ) ~ b ~ < ~ b + ( 0 , t)}. (2.29)
From time t to t + At, the stresses change f r o m Orij(t) to orij(t d- At) = a/y + Aorij. Then the D M G at time t + At can be calculated through the following procedures. (1) Determine the principal stresses trl(t + At), ~ 2 ( t + A t ) and t r a ( t + A t ) and substitute them into Eq. (2.20). If no solution of (2.20) exists and no microcrack propagates, then the D M G at time t + At is same as (2.29).
(2.32) In particular, in the case of penny-shaped microcracks with the same initial radius a 0 and uniform distribution in the orientation space, 1
p(a, 0, ~b) - 2~-"
(2.33)
Thus, the overall effective compliance tensor of damaged material is
Sijk l = SOkl + Sijk ,
(2.34)
where the microcrack-induced inelastic compliance tensor is Sijkl = frr/2 f 2zr
Jo
Jo Jvcp[a' O' ~b)
--i ×Sijkt(ao, O, dp, triy) sin 0 d~b d0
+ ffo(,)gcp(a, O, 4,)[S-/%(a., 0,
Orij)
--i --Sijkl(ao, O, ¢~, o'ij)] sin 0 d~b d0. (2.35)
S.-W. Yu, X.-Q. Feng / Mechanics of Materials 20 (1995) 59-76
3. The damage madei of DMG under triaxial compression In the preceding part, all microcracks are assumed to be open and the damage mechanisms of brittle materials under compressive loadings are not considered. However, some experiments shown that the micromechanical mechanisms of damage are rather complex in such brittle materials as concrete and rocks under compression (see, e.g., Horii and Nemat-Nasser, 1985, 1986; Krajcinovic, 1989; Deng and Nemat-Nasser, 1992). In this part, the case of brittle materials subjected to triaxial compressive stresses below the brittle-ductile transition points (and hence the materials exhibit evident brittle properties) is considered. The damage mechanisms of brittle materials under compression are mainly the frictional sliding, mode-II growth and kinking of closed microcracks. When a material is subjected to triaxial compression, all or part of the microcracks are closed. In a certain orientation scope of microcracks, which is called as the domain of frictional sliding, the microcracks are closed and experience frictional sliding. The domain of frictional sliding depends on the current stresses and the frictional coefficient of microcrack surfaces but not on the loading history and the sizes of microcracks. With increasing stresses, some closed microcracks in the domain of frictional sliding will grow along the weak plane (an interface or intergranular plane) in a self-similar fashion (Lee and Ju, 1991; Fanella and Krajcinovic, 1988) and even kind into the matrix material in a non-self-similar fashion (Horii and Nemat-Nasser, 1985). The kinked microcracks tend to line up in the direction parallel to the maximum compressive principal stress in the f~Lr field. Further, these microcracks grow continuously with increasing compressive loading and cause the ultimate axial splitting of the material. Due to the frictional sliding and kinking of microcracks, the compliance tensor of a brittle material under compression is unsymmetrical, anisotropic and has 36 independent components. Closure and frictional sliding of microcracks was considered by Horii and Nemat-Nasser (1983) in
65
the context of a two-dimensional stationary selfconsistent model. In the cases of uniaxial and axial-symmetric triaxial compression, Fanella and Krajcinovic (1988) gave an evolutionary damage model based on the Taylor model. Lee and Ju (1991) introduced microcrack interaction by a self-consistent method into the damage model of Fanella and Krajcinovic (1988). Using the concept of D M G proposed above, we further analyze here the microcrack damage mechanisms and the constitutive responses of the materials under threedimensional compression.
3.1. The closure and frictional sliding of microcracks Consider a microcrack with unit normal vector n subjected to the uniform far-field compressive stresses tro- in an RVE. Let 0-2 denotes the maximum compressive principal stress, 0-1 and 0-3 the lateral principal stresses in the direction of £l and x3, respectively. The absolute values of o-1 and 0-3 are assumed to be small enough so that the material behaves like an evidently brittle one. The orientation of the microcrack is (0, ~b) in the global coordinate system and (0, 4~) in the principal stress coordinate system. The microcracks can be classified into three classes as follows according to the deformation of their surfaces. (1) The microcracks are closed and no frictional sliding occurs, then 0-12 "~ 0,
--]£O"12 t> [(0-11) 2 + (O"13) 2] 1/2.
(3.1)
where /z is the coefficient of friction of microcrack surfaces. The conditions in (3.1) are formulated in the local coordinate system (OX'lX'2X'3) corresponding to the global one. In the local coordinate system ( O $ ~ ) corresponding to the principal stress coordinate system, the conditions of no frictional sliding for closed microcracks is similar to (3.1) but with 0-/'i replaced by a;j =~i~,j,~k,, where a l l = O"1, t~22 = 0"2, ~33 = 0"3 and all others ~ij = 0. The contact stresses transmitted across the microcrack surfaces 0-'~ are . =. 0-22, . . 0"22
c = O"21, . . 0-23 . . = 0"23 0-21
(3 2)
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S. - W. Yu, X. -Q. Feng / Mechanics o f Materials 20 (1995) 59-76
and the components of the displacement discontinuity microcrack vector are
b i = 0.
(3.3)
(2) The microcracks are closed and experiencing frictional sliding, then 0-22 < O,
--/.L0-~2<
[(0-~1) 2 "t- (0-~3)2] '/2
(3.4)
and tc t 0-22 ~---0-22,
r ! -- ]'~ 0-22 0-2,
tc 0-21 =
Als COS2~ + Azs <~O,
¢(0-;0 2 + (0"5) !
' nls cos4t~ --]-n2s c0s2(~ --bB3s ~ O,
t
--/-t.0-220"23
o-~ =
The orientation domain in which all microcracks satisfy inequality (3.4) is called the domain of frictional sliding. In other words, all microcracks in the domain of frictional sliding are closed and experience frictional sliding. For convenience, the domain of frictional sliding is calculated first in the principal stress coordinate system. Then from O~j--gikgjl~rkl, -' the conditions of microcrack frictional sliding are rewritten as
(3.5)
¢ ( 0 " ; 1 ) 2 nt- (0-;3) 2 '
b; = ~a z -- r2 B ij0-2j ' ,d
(3.6)
where the nonvanishing components of the displacement tensor for a closed microcrack B~j are
where AI~ = sin26(0-1 - 0-3), A2, = cos20 02 + sin20 0"3, B,, = sin2
(1 +
2)(0"1 - 0"3) 2,
B2s ~- sin2[~( 0", -- 0-3) [2]d'2( cOS2~ 0"2 "]-
16(1 - v 2) ' B ' 'I -- B33
(3.7)
r r E ( 2 - v)
and the driving stresses of deformation of microcrack surfaces are vd
l
tc ~ O,
vd
0"22 = 0"22 -- 0-22
v
vc
v
0"2, ----0-2, -- 0"21 =/~0-21
,d = 0-23 , -- 0-23 ,~ = ~0-23, 0-22 with
(3.9)
Using the transformation relations (2.5) between the global coordinate system and its corresponding local coordinate system, the components of the displacement discontinuity vector for the closed microcrack in the global coordinate system are (3.10)
b i = ( a 2 - r2)'/2flB~sg~ig'2kg'sl0-kl .
Then the inelastic compliance tensor induced by a single closed microcrack with radius a and orientation (0, ~b) can be obtained as
S~jkt( a, O, q~, 0-mn) ~a --
3 t
t
t
t
t
t
t
3V Blsg2kgsl(gtigzJ + gtjgzi)"
sin2ff 0"3)
+ 2 COS20 (0-2 - 0-3) - (0-1 - 0"3)], B3~ =/z2( c°s2ff 0-2 + sin26 o"3)2 - sin20 c°s20 (0-2 - 0"3)2.
(3 8)
/z 0-~2 /3 = 1 + ~/(0-;1) 2 + (0-5)
(3.12)
(3.11)
(3.13)
For any value of t~ in the range (0 ~<0 ~<~ / 2 ) , the range of ~ satisfying the conditions of frictional sliding can be obtained easily from (3.12) and expressed as ¢~1(0) ~<~ ~<4~2(0). So the domain of frictional sliding of microcracks can be written as /~={0~0~-12T/"
,
(~1(0) ~(]~(]~2(0)}.
(3.14)
(3) The microcracks are open, then o'~2 > O,
(3.15)
which can be recast as A l s COS2~ + A 2 2
> 0.
(3.16)
If 0-1 > 0 or 0-3 > 0, there will be a certain orientation scope of open microcracks, which can be calculated from (3.16). The influences of open microcracks on the constitutive relations of materials have been discussed above and will be omitted in this section.
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S.-W. Yu, X.-Q. Feng /Mechanics of Materials 20 (1995) 59-76
3.2. The mode-H growth of closed microcrack and the DMG All microcracks are assumed to be pennyshaped and have the same radius a 0 in the initial state before loading. With the increase of compressive stresses, seme microcracks in the domain of frictional sliding will grow in a self-similar fashion, and their radius increase from a 0 to a u. The transition from a 0 to au is unstable and instantaneous. However, the microcrack does become arrested at the edge of the interface by matrix materials that have higher strength than interface. When a microcrack is closed and undergoes frictional s][iding, the mode-I stress intensity factor K'I = 0, and the mixed-mode fracture criterion (2.13) red,aces to K~I = KHc ,
(3.17)
where the mode-II stress intensity factor K'n for the closed microcrack is
K]I= (-~4)~f~"
[/~o.;2"k-¢(O'21)2"F(o.23)21 •
Then under the condition of proportional loading, the D M G under the stresses trij, which is defined as same as that above, is obtained from (3.19) as the following form
=
{0
-
1
~< 0 ~< Un- ,
(3.21) where 4~3(0, O/j) and t~4(0, O/j) denote two functions of the angle 0 and the stresses trij. Since the mode-II self-similar growth occurs only for part of the microcracks experiencing frictional sliding, the D M G / ] is a subset of H under the condition of proportional stress loading. From (3.21) and the transformation relation (2.19) between (0, ~b) and (/~, q~), the D M G in the global coordinate system can be obtained as a(o/j) = {O
(3.18) Similarly to (3.12), the condition of growth of a closed microcrack can be re-expressed as C 1 cos44~ + C 2 cos24~ + C 3 >/O,
(3.19)
where C~, C 2 and C 3 are functions of 6 and C, = - sin40 (1 +/.[,2)(o. 1 -- 0"3) 2,
C2 = sin20( 0.' - °'s ) [ l't( 2 - v ) KI'c -
3.3. Evolution of DMG and calculation of compliance under complex loadings At time t, the material is subjected to the triaxial compressive stresses O-gj(t), and the evolved D M G is 120). From time t to t + At, the stresses change to another triaxial compressive stress state trij(t + At) = trij(t) + Ao.o. Substituting the stresses O/j + AO/j into (3.22), we obtain the D M G corresponding to this stress state as
1 , a(o.ij"]- Ao'ij) = {O < O ~ ~'w
2/z2( c°s2~ o.2 + sinZ0 o.3)
~)3(0, o.ij-[- mo'ij)
~<~b ~< 64(0, trij + AO/j)}. (3.23) ~ 2
C0S20( O.2-
+
(0.,
-
Then, the D M G at time t + At is obtained from the union of O(t) and O(O/j + AO/j), i.e.,
C3 = sin20 cos2t~ (oh -o'3) 2
O(t+At)
[ (2 -- 4) KIIC ~ f ~
- / x ( c°s2~ o.2 + sin2o o3 )]~ •
=O(t)
ua(o/ +ao/j).
(3.24)
The stress-strain relation in the global coordinate system is (3.20)
iO l q_ Sijk iS I ..}_Sijkl)o.kl Eij = SijklO.kl = ( SO.kl q_ Sijk iG (3.25)
68
S.-W. Yu, X.-Q. Feng/Mechanics of Materials 20 (1995) 59-76
where sii~°t is the inelastic compliance tensor due to open microcracks, si~St and Si~l are due to microcracks experiencing frictional sliding, and microcracks growing in a mode-II self-similar fashion, respectively, and
,s Sijkl
=ff. (trij)NcP(a,
onset of microcrack kinking. As an illustration, let us consider the special case of 0-1 0"3, then /eli =
= (4) ~-~ [sin2dcos2d(0-1 °"2) +/z( sin2~ 0-t
O, dp)
+ COS200-2)] "
(3.29)
×S~jkl(ao, O, ok, trij ) sin 0 d~b dO
By differentiating (3.29) with respect to 0, the orientation of the first kinking microcrack is given by
- ffa Ncp(a, O, 6 )
00 -- arctan(/x + ~ 2 ~ - ~ )
×~jkl(aO, O, q~, o'ij ) sijGl = ffa
sin 0 d~b dO; (3.26)
Ncp(a, O, 4,)
×$/ykt(au, 0, 6, 0-/j) sin 0 d~b dO,
(3.27)
which depends only upon the frictional coefficient but not on the value of 0-1 and 0"2. The corresponding kinking threshold value of the maximum compressive principal stress 0-2 is 1
0-2
where f2 o = O(t)(3 H(0-iy) is the orientation domain of frictional sliding microcracks with radius a u under stresses 0-ij.
F(ff0 )
[ v~-(2-/~)KICC~u +/~0-1]+°'1 × -
8 (3.31)
3.4. Kinking of microcracks under compression With a further increase in the compressive loadings, the higher tensile stresses induced at the tip of microcracks will drive some arrested microcracks in D M G to kink into the matrix material. Since the brittle matrix materials have low resistance to tension, we adopt the maximum circular stress at the microcrack tip (0"0)m~x as the kinking criterion of microcrack. Then from the K-fields of stresses at the near tip of mode-I and mode-II cracks, the kinking criterion of closed microcrack is derived as (Fanella and Krajcinovic, 1988) Kil = - ~ - K I c c.
(3.28)
From the mode-II stress intensity factor Kii in (3.18), we can obtain the orientation (0o, ~b0) at which Kii reaches its maximum value among all orientations. The microcrack kinking occurs at first in the microcracks with orientation (00, q~0). By substituting (00, 4~0) into (3.18) and (3.28), we further obtain the compressive stresses at the
(3.30)
where F(O 0) = sin200 cos200 - t~ cos200 . (3.31) agrees with the result of Fanella and Krajcinovic (1988). For a general triaxial compressive stress state, similarly to (3.12) and (3.19), we obtain the following inequality Clk
COS4~÷ C2k COS2~÷ C3k ~ O,
(3.32)
where the coefficients Clk , C2k and Cak can be obtained from C~, C 2 and C a in (3.20) by replacing Knc with V3 K i c c / 2 . For a given stress state, (3.32) gives the domain of kinked microcracks, which is called simply the kinking domain and denoted as /l(0"iy) in the principal stress coordinate system. The kinking domain A(0"i~) in the global coordinate system can be obtained from A(0"q) and (2.19).
3.5. The compliance tensor induced by a kinked microcrack For a closed, frictional sliding microcrack, the maximum circular tensile stress (0-o)max occurs at
S.-W. Yt~ X.-Q. Feng / Mechanics o f Materials 20 (1995) 59-76
the direction of an angle Ok = arcsin(2v~-/3) to the microcrack surfaces. So the initial direction of kinking is not the same as the direction of the ;2-axis (or the direction of the maximum compressive principal stress o-2 in the far field). The microcrack reaching the kinking criterion (3.28) propagates first into the matrix material in the
69
direction at the angle Ok to the microcrack surfaces, as shown in Fig. 3a. With the growing microcrack, the lengths emanating from both ends of the sliding microcrack increase; and they eventually align themselves parallel to the direction of o"2 and cause macroscopic axial splitting. When °'1 *o-3, the stress intensity factor reaches its
X2
O
Z/
(a)
(b)
t
a k
k
2~ k a . sinO
2a~
~ "~
I P = 2~a a k
(c)
(d)
(e)
Fig. 3. (a) A three-dimensional kinked microcrack; (b) an approximate three-dimensional kinked microcrack; (c) two-dimensional kinked microcrack system; (d) equivalent microcrack system (Fanella and Krajcinovic, 1988); (e) equivalent microcrack system (Zaitsev, 1983).
70
S.-W. Yu, X.-Q. Feng / Mechanics of Materials 20 (1995) 59-76
maximum value along the microcrack edge at A' and B' with a small angle /30 = arctan (¢T23//0~21) included between A'B' and the $~-axis, see Fig. 3a. Therefore, the positions on the microcrack edge from which the kinked microcrack emanates are not at the intersection points of the $~-axis and the microcrack edge, but at the points A' and B'. It is impossible to give a closed form of the solution of the displacements of the kinked, three-dimensional microcrack in Fig. 3a and their contributions to the compliance tensor. Kachanov (1982) and Fanella and Krajcinovic (1988) assumed that the three-dimensional kinked microcrack can be approximated by a series of two-dimensional kinked cross-sections. Here a similar method is used to deal with kinked microcracks. First, we assume that the microcrack kinks and grows in the same direction of the maximum compressive stress 0"2, as shown in Fig. 3b, and that the projection of the microcrack on the 22--3~; plane is an ellipse with its lengths of the semimajor and semiminor axes a, and a k. Secondly, the three-dimensional kinked microcrack in Fig. 3b is approximated by a series of two-dimensional cross-sections with angle/3 ( - ~ r / 2 ~
nience, they are expressed as functions of (0, 4;) and o7/'j in the form cotan PO ~
-
'
S0=
(/-1.~;2 + V/(t~l) 2 -1--(~3) 2 )
-
ot k
~ (j~O~2 -1- }/( 0~;1) 2 -t- (~;3) 2 )
a k sin (3.33) where a k is a dimensionless factor introduced to guarantee the equivalence of the stress intensity factors of the equivalent microcrack and the actual one, here ~g is taken to be 0.25 (Lee and Ju, 1991). The displacements of the kinked microcrack consist of two parts. The first part is the opening displacement 5(/3) perpendicular to the microcrack surface (kink opening) for any cross-section located at an angle /3. The second part is the average sliding displacement t~(/3) of the closed surfaces of microcrack for the /3 cross-section (Fanella and Krajcinovic, 1988). 4.8(1 - 1'2)
-, COS /3 F 1( a u , O'21
a k,
e) ,
4.8(1 - v 2) a(/3)
-
rrE
a;1 cos/3 F 2 ( a , , a k, 0), (3.34)
where
Fl(au, ak, if) COS 0 , - sin---2-~F2(au, ak, O) -
c°tan20[arcsin(all~(a2)Z_(al)2 ak [ ~ a2 I
,n(al l ] iazl] with a 1 = aka u sin 0 and a 2 = a 1 + a k. By averaging the displacement components along the interface microcrack rim ( - 7r/2 ~
S.-W. Yu, X.-Q. Feng /Mechanics of Materials 20 (1995) 59-76
kinked microcrack are determined by the following two parts.
"k, a u, ak, O, ~b, '~ij) Sijkl(
71
the other is the lateral confinement loading p* acting on the kinked microcrack surfaces due to the applied far-field stresses, p* = cos24~ or1 + sin24~ tr3.
0"6/3a"ak (gi~g2j + g#g2i)
-
Then the mode-I stress intensity factor at the kinked microcrack tip under the concentrated force P1 and the distributed force p* can be obtained as (Tada, 1973)
V .
.
.
.
.
(3.41)
,
Xg2kgtlBstFll au, ak,
~kz ijkl~(a u,. ak, O, ~ , ,Tij)
K~=
0 " 6 / 3 ( a , ) 2 / - , ~, +o~,~i,) -[ gis 2j
2au7"n cos/~ ~
+ ~
p*.
(3.42)
V
X~'2k~,;iB'stF2(au, ak, 0),
(3.36)
where 8(1 - t'z) rrE ,
B]I
gii =
and all other B~j = 0
01 co: 1
0
-cos4~
0
(3.37)
(3.38)
sin4~]"
Then the corre,;ponding inelastic compliance tensor in the global coordinate system induced by the kinked microcrack is obtained from (2.16) and (2.19) as
7S~fkt(au, a k, O, ~b, trij ) =[
[ ~k, mnst~[ a u , ak,
O,~,o'ij)
"k2 a u, ak, 19, qb, trij)] "[-Smnst( Xgmignjgskgi! l
(3.39)
where the length a k will be determined below.
3.6. Stability analysis of microcrack kinking In order to find the relation between the kinking length a k of a microcrack and the applied stresses, the simplified equivalent microcrack in Fig. 3e is adopted (Ziatsev, 1983). The loadings acting on the kinked microcrack surfaces actually have two parts, one is the concentrated force P1 due to the shear stress ~'n = / x ~ 2 + ~t
2
¢ ( ~ 1 ) 2 + (t r23) , i.e., P1 = 2a,~'n cos/~,
From (3.42) and the criterion of growth for the kinked microcrack K~ = Kicc, the kink length a k (at /3 = 0) can be derived as the following form
(3.40)
where sgn(p*) = 1 for p* > 0 and - 1 for p* < 0. For the special case of p* = 0, the kink length of the microcrack is
1 a k = -•-
(
2au7 n cos 0 (3.44) gic C
Notice that the stress intensity factor in (3.42) is in a good agreement with the numerical results in Horii and Nemat-Nasser (1985) only for big a,, and is invalid when ak tends to zero. For different cases of lateral confinement loading p*, the stability of kinking of microcrack is analyzed as follows. (1) p* = 0. An a,-K~ curve for p* = 0 is obtained from (3.42) and shown in Fig. 4a. Under a given stress state, the value of K~ decreases monotonically with increasing a, and tends to be zero when a k tends to infinity. Therefore, the microcrack stops kinking when its kink length ak reaches the length akC. So the kinking of microcrack is stable. With the applied stresses increasing proportionally, the stress intensity factor and also the corresponding kink length increase. As shown in Fig. 4a, if the stresses crib-2) aO'/(j1) with a > 1, the corresponding kink lengths satisfy a~k2) > aCk1). SO the gross axial splitting will occur ultimately in the material when the applied stresses reach a certain level. =
S.-W. Yu, X.-Q. Feng / Mechanics of Materials 20 (1995) 59-76
72
(2) p * < 0 . An a k - K ~ curve for p * < 0 is shown in Fig. 4b. For a given stress state, K~ decreases monotonically with the increase of a k and intersects the araxis. Therefore, the microcrack stops kinking when a k reaches a k c and hence the kinking of microcrack is also stable. For p* < 0, there is a critical length a k max satisfying K ' i ( a k max) = 0. From (3.42), we have a k max --
2 a u r n cos -- "wp*
t?
(3.45)
With the applied stresses increasing proportionally, a k max remains constant, and so the kink
t
length of microcrack increases also but cannot exceed the critical value a k max" If the lateral confinement load p* is small (and then a k max is big), it is still possible that the axial splitting occurs in the material. When p* is big, the microcrack kinking is limited by ak max" Therefore, the kinking of microcrack is not the mainly damage mechanism in materials when the confinement load is big enough. Instead of axial splitting, the faulting or ductile flow becomes the main failure mode of brittle material (Horii and Nemat-Nasser, 1986). (3) p * > 0. As shown in Fig. 4c, the a k - K ' I
t
KI
K x
Kicc
Klcc
-,.-.<_
I
(2)
ii
i I
0
(t)
(a)
akc
akC
a k
0)
(2)
akc akc
(a)
a kraax~
(b)
t
KI
Kmc
,
,, O)
(2)
a kC
a kC
(2)
I
m
'
a k
a kC
(c) Fig. 4. T h e ak-K ~ curves for: (a) p * = 0; (b) p * < 0; (c) p * > 0.
ak
S.-W. Yu, X.-Q. Feng /Mechanics of Materials 20 (1995) 59-76 curve for p* > 0 has a knee point, before which the curve decreases monotonically with increasing a k and after which the curve increases. Firstly, the kinked microcrack grows until the length a k reaches akC. akc increases with the increase of stresses, i.e., a~2) > a(k1) when 0"(2)> 0-/(jl). Even if the lateral tensile sl:ress p* is much smaller than 0-2, the kinking of rnicrocrack will become unstable with the externally applied stresses increasing. From (3.42), the critical kink length of microcrack growing unstably a~c can be obtained for the case p* > 0. Since the kin.kin.g occurs first in the microcrack orientation (00, ~b0), substituting (O0, 4~0) into (3.42) leads to
g'I(ak, 0o, ~0) =
73
where Si/J~t is the inelastic compliance tensor induced the kinked microcracks, and
iG _Sijkt
aNcP( a, O, c~) X~S[jkl(au, O, q~, 0"ij) sin 0 d~b dO
- ffaNcP( a, O, 4~) XS~jkt(a,,, O, dp, 0"ij) sin 0 d~b d0, siijK l =
ffANcP( a, O, 40 ×S~yk,(au, a k, 0, ok, tri~) sin 0 d~b d0.
2au'rn(dO, ~bo) cos Oo
(3.51)
+
(3.46) Then for a given stress state, K~(a k, d o, ~o) reaches its minimum value .
(gI)min =
[8au~'n(dO,
] 1/2
t ~ o ) P * ( t ~ O ) cos 00]
(3.47) when
2au~'n(Oo, qbo) COS do ak=a'~c=
4. Theoretical results and experimental validation
~'P* (4~0)
(3.48)
From (K~)mi n = g l c c , the condition for applied stresses under which the microcrack kinking becomes unstable is obtained as 8a,zn(0 o, ~o)p*(q~o) cos 0o = K2cc.
(3.49)
Once this condition is met, the macroscopic axial splitting occurs in tlhe material.
3. 7. Calculation of the compliance tensor After considering the closure, frictional sliding, self-similar growing and kinking of microcracks, the overall compliance tensor is obtained as iG + Sijkl iK Sijk, = siO.kl + siij2l + siijSl -J- Sijkl
(3.50)
In order to validate the proposed three-dimensional micromechanical model for brittle materials, two examples of concrete under uniaxial tension and uniaxial compression are considered in this section. The theoretical results obtained here are compared with the experimental data of Gopalaratnam and Shah (1985), and the results of Ju and Lee (1991). Some theoretical results from the model of DMG for a specimen under complex tensile loadings can be found in Feng and Yu (1993).
4.1. Example of uniaxial tension As an illustration, the damage model of D M G is applied first to a concrete specimen subjected to uniaxial tension loading, i.e., 0-1 = 0"3 ~--0, 0 2 = 0". Based on the analytical expressions derived in the previous sections, the D M G for uniaxial tension is
1 ~'~1 ~--"{0 -~< 0 ~.< 0ma x -~< g'/T,
0 ~< ~b < rr}
(4.1)
S.-W. Yu, X.-Q. Feng/Mechanics of Materials20 (1995)59-76
74
with
80 ~ Stress (MPa)
2
-B2-~B2-4B1B3
tan 0max = - - - - - ~ F - - - -
;
*~
(4.2)
*'~
*~
BI = -
/t/,
, ~N~Lateral 1K\~\ Volumetric
/ , '/~, Ax,al /~". //9"*
601" *\\
4-ao KII C , B2 = a r t -~- [ 2---~ ) , ( o'KIIc ) 2
B3 = B1+
~
.
(4.3)
The effective compliance of the specimen under uniaxial tension is 1 + P(10/Z
3v)
-~o
V
/5
1
---
,
Resultsin thispaper Experimentaldata ,1991)
- Results in Lee a n d Ju
1,0
0
for O < ~ r < o " A $2222 =
i 1 ~ / // }I ~ / /t/
20
=
Strain (x 10 -4)
Fig. 6. Comparison of the theoretical and experimental results for uniaxial compression.
p
~-+~[y(lO-3v)+(1-y)
where
× (10 COS30max -- 3v COS50max)] , for o-c ~
ire=
Kic ,
O'cc=
Kxcc;
(4.5)
(4.4) 16( 1 - v 2) Nc a3 P=
( au ]3. ' Y = k ao l
(4.6)
As in Ju and Lee (1991) and Hoenig (1978), we take K i c = 0.165 M N / m 2/2, K]I C = 0.33 M N / m 3/2, K1c c = 0.301 M N / m 3/2, a 0 = 0.34 cm, a u = 0.49 cm, E -- 34450 MPa and v = 0.3. Using the formula from Fanella and Krajcinovic (1986) and Ju and Lee (1991), the microcrack number density N c / V is taken to be 1.81 × 106 (1/m3). Then the uniaxial tensile stress-strain curve obtained from (4.4) is plotted in Fig. 5 and compared with the experimental data in Gopalaratnam and Shah (1985). It is observed that the analytical results are in a good agreement with the experimental results. The small error is due to microcrack interaction which has not been considered. If the self-consistent method is introduced in the damage model, the results will be better.
4 Stress (MPa)
3
2
1
*
45(2-v)V
Experimental data Strain ( x 10 - 6)
0
i
i
iI-
50 100 150 Fig. 5. Theoretical stress-strain curve for uniaxial tension compared with experimental results.
4.2. Example of uniaxial compression
0
The presented damage model is applied again to the concrete specimen subjected to uniaxial
S.-W. Yu, X.-Q. Feng / Mechanics of Materials 20 (1995) 59-76
compression. The experimental parameters reported in Lee and Ju (1991) are as follows: the initial values of the modulus of elasticity E and the Poisson's ratio v are 40020 MPa and 0.2, respectively; the critical stress intensity factors of the interface and matrix Kic = 0.165 M N / m 3/2, KII c = 0.33 M N / m 3/2, Kic c = 0.495 M N / m 3 / 2 ; the volume fraction of the coarse aggregate is given as 0.39; the statically averaged radius of microcracks are a 0 = 0.47 cm, a u = 0.81 cm. The theoretical results of the axial, lateral and volumetric strains for the case of uniaxial compression are compared with the experimental and theoretical results Jin Lee and Ju (1991). It was shown that these results correlate well with each other.
5. Conclusions Based on a sound and physically convincing model of microstructures, the micromechanicsbased damage theory of the domain of microcrack growth (DMG) is developed for microcrack-damaged brittle solids. The model describes the complex evolutionary damage process during loading and the anisotropic constitutive behaviors of materials under tensile a n d / o r compressive loadings. The concept of D M G is defined to describe the damage state of brittle materials and the evolution of the overall effective fourth-order compliance tensor for the case of complex loading path. Based on modified fracture criteria of mixed-mode cracks, the D M G for brittle materials under complex tensile or compressive loadings can be calculated. Mode-I growth of open rnicrocracks, frictional sliding, mode-II growth and kinking of closed microcracks were all studied and their influence on the nonlinear constitutive behaviors of brittle materials were introduced analytically. From micromechanical analyses, it is explained that the macroscopic axial splitting will occur in the material only when the lateral principal stresses o-~ and 0"3 have positive or sxlaall negative values. And the condition of axial splitting of materials is formulated. The analytical results from the theory developed here fit the experimental data and results
75
of others well, even though a few approximate assumptions are adopted.
Acknowledgement The support of the National Natural Science Foundation of China is gratefully acknowledged.
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Krajcinovic, D. (1986), Continuum damage mechanics, Appl. Mech. Rev. 37, 1-6. Krajcinovic, D. (1989), Damage mechanics, Mech. Mater 8, 117-197. Krajcinovic, D. and D. Fanella (1986), A micromechanical model for concrete, Eng. Fract. Mech. 25, 585-596. Krajcinovic, D. and D. Sumarac (1989), A mesomechanical model for brittle deformation processes, Part I and II, Z Appl. Mech. 56, 51-56, 57-62. Lee, X. and J.W. Ju (1991), Micromechanical damage models for brittle solids, II: compressive loadings, J. Eng. Mech. 117, 1515-1536. Lemaitre, J. and J.L. Chaboche (1988), Mechanics of Solids Materials, Cambridge University Press, London. Moss, W.C. and Y.M. Gupta (1982), A constitutive model
describing dilatancy and failure in brittle rock, J. Geophys.
Res. 87, 2985-2998. Murakami, S. and N. Ohno (1981), A continuum damage theory of creep and creep damage, in: A.R.S. Ponter, ed., Creep of Structures, IUTAM Symposium, Springer, Berlin, pp. 422-444. Nemat-Nasser, S. and M. Obata (1988), A microcrack model of dilatancy in brittle materials, J. Appl. Mech., 55 24-35. Ortiz, M. (1985), A constitutive theory for the inelastic behavior of concrete, Mech. Mater. 4, 67-93. Tada, H. (1973), The StressAnalysis of Cracks Handbook. Del Research Corporation., Paris. Zaitsev, Y.B. (1983), Crack propagation in a composite material, in: F.H. Wittmann, ed., Fracture Mechanics of Concrete, Elsevier, Amsterdam, pp. 31-60.