Theoretical study on damage bifurcation of unstable failure process of quasi-brittle materials
WANG Zhong-chang(王忠昶)1, 2, ZHAO De-shen(赵德深)1, YANG Qing(杨 庆)2
1. Key Laboratory for Prediction & Control on Complicated Structure System of Education Department of
Liaoning Province, Dalian University, Dalian 116622, China;
2. Institute of Geotechnical Engineering, School of Civil and Hydraulic Engineering,
Dalian University of Technology, Dalian 116024, China
Received 10 August 2009; accepted 15 September 2009
Abstract: To obtain the damage effect in the process of elasto-plasticity deformation of quasi-brittle materials, the isotropic damage loading-unloading function and damage variable were introduced to non-continuous bifurcation. The critical bifurcation orientation and its corresponding hardening modulus for quasi-brittle materials were derived considering the effect of stiffness degradation and volumetric dilatancy under the assumption of isotropic damage. The relationships of localized orientation angle and maximal hardening modulus dependent on degree of damage and initial Poisson’s ratio of rock were explored. Comparative analyses were conducted to study the bifurcation of uniaxial tension-compression samples under the conditions of plane stress and plane strain. It is shown that as the initial Poisson’s ratio or degree of damage increases, the localization orientation angle of the plane uniaxial compression sample tends to be initiated to decrease. However, the localization orientation angle of the plane uniaxial tension sample tends to be initiated to increase. The sum of orientation angle under tension and compression conditions is 90?. There are plane stress and plane strain cases of the maximum hardening modulus that is independent of the uniaxial compression and tension.
Key words: isotropic damage; orientation of localization; maximum hardening modulus; bifurcation; uniaxial sample
1 Introduction
A bifurcation occurs when a small smooth change made to the parameter values of a system causes a sudden ‘qualitative’ or topological change in its behaviour[1]. It is characterized by the phenomenon that after the material undergoes a definite amount of equal deformation, suddenly it gets into a stage of deformation when the deformation of high localization takes place. HILL[2], RUDNICKI and RICE[3-4], OTTOSEN and RUNESSON[5], NEILSEN[6] studied on bifurcation and instability of materials from the viewpoint of elasto-plasticity. However, the effects of damage of materials on bifurcation and instability were not taken into consideration. LI and YE[7], ZEND et al[8], ZHAO et al[9] and L? et al[10] analyzed the damage localization bifurcation model for rock-like materials under different conditions. LIANG et al[11] gave the characteristics of fractal and percolation of rock subjected to uniaxial compression during their failure process. CHAI et al[12] obtained the characteristics of stability-losing of post-failure rock. In fact, there is also dilatancy of the quasi-brittle material along with presentation of a large quantity of microcosmic defects. At this time, Poisson’s ratio is increased and strength and stillness of the material are decreased. Therefore, in analyses of bifurcation and instability of quasi-brittle materials, the two nonlinear deformation features of elasto-plasticity and damage should be taken into consideration simultaneously. In this work, the isotropic damage loading-unloading function and damage variable are introduced into non-continuous bifurcation. The critical bifurcation orientation and its corresponding hardening modulus for quasi-brittle materials are derived considering the effect of stiffness degradation and volumetric dilatancy under the assumption of isotropic damage. The relationships of localized orientation angle and maximal hardening modulus depended on degree of damage and initial Poisson’s ratio of rock are explored.
Comparative analyses are conducted to study the bifurcation of uniaxial tension-compression samples under the conditions of plane stress and plane strain.
2 Theory derivation of damage constitutive functions
Traditionally, the constitutive relationship for isotropic damage may be expressed as
(1)
(2)
where C denotes flexibility tensor of material; I4 is fourth-order unit tensor; εkl is total second order strain tensor; 
, is elastic damage modulus of material which is progressively degraded along with the development of microcosmic defects[13].
Their derivatives can be written as
, 
(3)
where 
is initial elastic damage modulus; 
denotes initial flexibility tensor of material; and d is the isotropic variable depended only on the plastic strain history.
There is no change of damage modulus in the process of elastic loading and unloading and neutral loading. 
1/(1-d). Therefore,
(4)
where εp is the characteristic plastic strain.
The damage loading-unloading function is introduced:
(5)
where λ is an inner variable in elastic domain. The material will further be damaged and degraded when the inner stress level of the material satisfies yield function F=0. Let G be damage potential function of material, and 
be damage strain due to the degradation of material. The incremental constitutive relationship of material is
(6)
When potential function differs from yield function and using non-associated flow rule, it gives
, 
(7)
where 
is gradient function of damage potential function G in stress domain.
According to the identity condition of plasticity, the rate-format of Eq.(5) is
(8)
where 
,
, are hardening and softening moduli, respectively; and 
is damage gene.
Eq.(2) may be expressed by rate format as
(9)
By introducing flexibility tensor, the flow rule is expressed as
, 
(10)
where Mijkl is the initial flexibility tensor.
From Eqs.(7) and (10), the following equation is attained:
(11)
From Eqs.(6), (7) and (8), the following equations are attained:
(12)
＞0 (13)
Substitution of Eqs.(12) and (13) into Eqs.(6) and (7), the isotropic damage constitutive relation in incremental form can be written as
(14)
where 
is elastic-damage tangential modulus matrix.
(15)
The damage loading-unloading criterion follows Kuhn-Tucker reciprocal condition:
≤0,
≥0,
(16)
where damage gene, 
, and initial flexibility tensor, 
, are proposed by CAROL et al[14].
Then gij of Eq.(11) can be rewritten as
(17)
Thus, the general expression of hardening modulus is
(18)
And the gradient function of loading function is
(19)
where damage loading-unloading function, F=f(w0,
, is proposed by CAROL et al[14]; 
and 
are equivalent extending and driving force of cracks, respectively. is representative radius of yield plane of damage. This denotes that the cracking and yielding of material depend on current accumulative damage.
Substituting Eqs.(17)-(19) into Eq.(13), and considering Eq.(3), it gives
(20)
Finally, the elasto-plastic damage tangential modulus is expressed as
(21)
3 Damage localized bifurcation
The condition of non-continuous bifurcation of damaged material is depended on singularity of the following damage localized tensor 
, and is defined as
(22)
where nj (j=1, 2, 3) is outer unit normal vector of characteristic plane of localized zone. The necessary condition when non-continuous bifurcation appears in the material is
(23)
The condition can be regarded as necessary condition that nonlinear uniform algebraic function of unit normal vector of characteristic plane of localization obtains non singular solution. Therefore, corresponding algebraic functions of uniform linearity should be considered.
Thus, the following eigen-value problem is to be considered:
(i=1, 2, 3) (24)
Considering that is symmetric and positively definite, it can be proved that is positive as well. Its inverse matrix is 
. So, Eq.(24) may be rewritten as
(i=1, 2, 3) (25)
where 
,
,
.
Considering
and are non-singular, and the singularity of Bjl is nonsingular, it is indicated that λ=1 is an eigen-value with a multiplicity of two for the eigen-value problem in Eq.(25). Associated this with the properties of matrix, the third eigen-value can be attained as
(26)
To make Bji singular, at least one eigen-value should be equal to zero. Then λ(3)=0. Considering Eq.(13), the hardening modulus in accordance with occurrence of localization can be attained as
(27)
Assuming 
where 
is isotropic elastic tensor of material, from Eq.(3) it gives
(28)
where Gd=(1-d)G and 
are post-
damage shear modulus and Poisson’s ratio, respectively, of material[14]. G and 
are prior-damage shear modulus and Poisson’s ratio, respectively, of material.
(29)
Among the orientations along which localization is possible to occur, only the orientation corresponding to the maximum hardening modulus firstly satisfies the condition of localization, so it is the orientation along which localization occurs first. Therefore, the orientation that makes hardening modulus attain the maximum value Hd= maxH(ni) is the critical localized orientation.
Arrangement of the above equations reaches the following expression:
(30)
where 
.
fv, fij and gv, gij are the first invariables and deviatoric tensors of gradient function of yield function and
damage function, respectively; 
. Assuming 
(i=1, 2, 3) to be the main values of 
and
, 
(31)
Substituting above equations into Eq.(30), the unitary hardening modulus can be rewritten as
(32)
and 
in Eqs.(31) and (32) are
(33)
respectively.
Considering geometric restrained conditions of unit
vector 
, to obtain the maximum value of
hardening modulus for the extremum problem with restrained conditions, the Lagrange’s multiplier method is employed to establish the extremum problem without restrained conditions:
(34)
where β is the Lagrangian multiplier to be determined.
The extremum of hardening modulus H in Eq.(34) is to be determined by the following extremum conditions:
(35)
where
, 
(36)
The symmetric Hessian Hij of L is
(37)
where 
(i=1, 2, 3).
Considering ni≠0, it can be seen from the first condition of Eq.(36) that 
, i.e.
(38)
If 
it gives Hij= 
Assuming 
≥
≥
and
it can be obtained from Eq.(36) that
(39)
(40)
1) When 
, Eq.(39) and Eq.(40) are not able be satisfied simultaneously. For this case, no maximum of Eq.(34) exists.
2) When 
, for this case, from Eqs.(39) and (40), obviously 
. When 
and 
, from assumptions before and Eq.(38), the following equation is obtained:
(41)
From Eq.(41) and to associate with the equation 
, it gives
(42)
As 0≤
≤1, the following equations are obtained:
≤0, 
≥0 (43)
3) When 
, for this case, as well, then
(44)
≤0, ≥0 (45)
The expression of maximum hardening modulus for occurrence of non-continuous localization bifurcation can be derived from the above equations as
(46)
As shown in Fig.1, if r≤0 and 
and assuming n2=0, the orientation angle θcr along which localized damage occurs may be represented by the angle formed by the outer normal vector n of localized characteristic plane and the direction of x1 axis, and is given by
Fig.1 Orientation angle of localization for plane problems
(47)
4 Analysis of localization of uniaxial sample
The theories of analyses on isotropic damage localized bifurcation illustrated above are used to concretely calculate and analyze the localization of uniaxial compression-tension testing samples under plane strain and plane stress conditions. The damage loading and unloading function 
proposed by JU[15] is used in the following analysis. The results are shown in Table 1.
The dependence of orientation angle of the normal vector of localized characteristic plane on degree of damage and initial Poisson’s ratio are obtained by calculations, as shown in Figs.2 and 3, respectively.
1) Whether under plane stress or strain condition, variation of orientation angle of localization is depended on initial Poisson’s ratio and degree of damage. Under the condition of uniaxial compression, the larger the initial Poisson’s ratio or the degree of damage is, the smaller the orientation angle of localization is, and the slower the tendency of progressive decrease of orientation angle becomes. Under the condition of uniaxial tension, the larger the initial Poisson’s ratio or the degree of damage is, the larger the orientation angle of localization is, and the slower the tendency of progressive increase of orientation angle becomes. When d=1 or ν=0.5, the orientation angle keeps unchanged and reaches the minimum value under the condition of uniaxial compression.
2) When d=1 or ν=0.5, the orientation angle keeps unchanged and reaches the maximum value under the condition of uniaxial tension. Under plane strain uniaxial compression condition, the orientation angle of localization θcr=45?; and under plane stress uniaxial compression condition, the orientation angle of localization θcr=54.7?; under plane strain uniaxial tension condition, the orientation angle of localization θcr=45?; and under plane stress uniaxial tension condition, the orientation angle of localization θcr=35?.
The dependence of maximum hardening modulus on degree of damage and initial Poisson’s ratio are obtained by calculations, as shown in Fig.4. It can be seen that:
1) Whether under plane stress or plane strain condition, when ν＞0.35, the maximum hardening modulus decreases and reaches zero with the increase of degree of damage. The smaller the initial Poisson’s ratio is, the slower the tendency of progressive decrease of maximum hardening modulus becomes. When ν≤0.35, at the beginning, maximum hardening modulus increases, then decreases and reaches zero with the increase of degree of damage. The smaller the initial Poisson’s ratio is, the smaller the maximum hardening modulus becomes.
2) When ν＜0.20, the extension of effective value of the degree of damage is different with the change of initial poisson’s ration. The larger the initial poisson’s ratio is, the large the extension of effective value is. Under different degrees of damage, the effective extension of Poisson’s ratio is different. The larger the degree of damage is, the larger the extension is. When ν=0.50, the maximum hardening modulus keeps unchanged and reaches zero.
For comparing characteristic of bifurcation under plane strain and plane stress condition simultaneously, the change relation of maximum hardening modulus and orientation angle of the normal vector of localized characteristic plane on degree of damage under two conditions are given in Figs.5-8 for ν0=0.3 or d=0.5. It can be seen that the declining rate of localized orientation angle is lower under compression condition
Table 1 Result statistics of damage localization of uniaxial sample under plane condition
Fig.2 Dependency of orientation of localization on degree of damage for different Poisson’s ratios: (a) Plane-strain uniaxial tension; (b) Plane-strain uniaxial compression; (c) Plane-stress uniaxial tension; (d) Plane-stress uniaxial compression
Fig.3 Dependency of orientation of localization on Poisson’s ratios for different degrees of damage: (a) Plane-strain uniaxial tension; (b) Plane-strain uniaxial compression; (c) Plane-stress uniaxial tension; (d) Plane-stress uniaxial compression
Fig.4 Dependency of maximum hardening modulus ratio on damage degree and Poisson’s ratio: (a) Plane strain condition; (b) Plane stress condition; (c) Plane strain condition; (d) Plane stress condition
Fig.5 Comparison of dependency of localization orientation on Poisson’s ratio under different conditions
than that under tension condition. And the localized orientation angle will be larger with the same degree of damage or initial Poisson’s ratio. Under condition of plane strain, the localized orientation angle will increase, if initial Poisson’s ratio or the degree of damage increases. Under tension condition, the increasing rate of localized orientation angle is higher than that under compression condition. And the localized orientation angle will increase with the same degree of damage or initial Poisson’s ratio. The sum of localized orientation angle under tension and compression conditions is 90?. Secondly, there are plane stress and plane strain cases of the maximum hardening modulus that is independent of compression and tension. Maximum hardening modulus will increase when degree of damage or initial Poisson’s ratio increases. Under plane stain condition, the increasing rate of maximum hardening modulus is higher than that under plane stress condition. And maximum hardening modulus will increase with the same degree of damage or initial Poisson’s ratio, so does the effective range of Poisson’s ratio.
Fig.6 Comparison of dependency of localization orientation on degree of damage under different conditions
Fig.7 Comparison of dependency of maximum hardening modulus on Poisson’s ratio under different conditions
Fig.8 Comparison of dependency of maximum hardening modulus on degree of damage under different conditions
5 Conclusions
Taking damage degradation and dilatancy into consideration, the maximum hardening modulus and orientation angle at occurrence of bifurcation and instability of rock-like materials are deduced.
1) The orientation angle of localization at occurrence of instability is depended on degree of damage and initial Poisson’s ratio. The sum of localized orientation angle under tension and compression conditions is 90?.
2) There are plane stress and plane strain cases of the maximum hardening modulus that is independent of the uniaxial compression and tension.
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(Edited by YANG Bing)
Foundation item: Project(50574016) supported by the National Natural Science Foundation of China; Project(2009S006) supported by Science and Technology Research Foundation of Key Laboratory of Education Department of Liaoning Province, China
Corresponding author: WANG Zhong-chang; Tel: +86-13591122956; E-mail: wazoch@163.com
