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Evolution
A. Ganczarski, L. Barwacz
To cite this version:
A. Ganczarski, L. Barwacz.
Low Cycle Fatigue Based on Unilateral Damage Evolution.
Damage Evolution
A. GANCZARSKI* ANDL. BARWACZ
Institute of Applied Mechanics Cracow University of Technology Al. Jana Pawia II 37, 31-864 Krako´w, Poland
ABSTRACT: This article deals with the modeling of the low cycle fatigue of AISI 316L stainless steel from the viewpoint of continuum damage mechanics. The concept of kinetic law of damage evolution is adapted and three models are presented: one in which the effect of crack closure/opening is excluded and two others where either classical or new continuous microcrack closure/opening effects are taken into account. The problem is described by a system of ordinary differential equations derived for the case of uniaxial stress, then extended to the three-dimensional (3D) state of stress accompanying strain localization by the use of approximate, axisymmetric 3D stress formulas by Davidenkov and Spridonova. The results of numerical simulation for all models are compared and verified in order to achieve the best agreement with the experimental data. Detailed quantitative and qualitative analysis of obtained solutions confirms the necessity and correctness of an application of continuous microcrack closure/opening effect.
KEY WORDS: damage, continuous microcrack closure/opening effect, low cycle fatigue.
INTRODUCTION
I
N THE CASEof low cycle fatigue (LCF), when the stress level is larger than the yield stress, damage develops together with the cyclic plastic strain, after the incubation period that precedes the nucleation and growth of microdefects is met. The mechanism of the ductile damage on the LCF tests is manifested through the transgranular slipband fields of plasticity developed in the large size grains. In other words, for LCF tests the plasticity-damage fields involve a large volume of the specimen with weakInternational Journal ofDAMAGEMECHANICS, Vol. 16—April 2007 159
1056-7895/07/02 0159–19 $10.00/0 DOI: 10.1177/1056789506064939
ß 2007 SAGE Publications
localization, such that the classical local continuum damage mechanics (CDM) approach is applicable as the objective method for the number of cycles to failure prediction, which customarily is supposed to be between 102 and 104(see fatigue classification by Dufailly and Lemaitre, 1995).
In the most frequent approach to the LCF, in a case when a loading is the periodic strain-controlled of constant amplitude, the following assumptions are made: the material becomes perfectly plastic during the first cycle, the variation of damage is neglected for the integration over one cycle and the strain–damage relations are identical both for tension and compression. These allow to simplify calculations of damage cumulation per one cycle and give linear dependency of damage with N, finally leading to the Manson– Coffin law of LCF (cf. Lemaitre and Chaboche, 1985; Skoczen´, 1996, 2002; Skrzypek and Ganczarski, 1999; Garion and Skoczen´, 2003).
existing theories developed by Ramtani (1990), Ju (1989), or Krajcinovic and Fonseka (1981) either nonsymmetries of the elastic stiffness or nonrealistic discontinuities of the stress–strain response may occur for general multiaxial nonproportional loading conditions. It is easy to show that if the unilateral condition does affect both the diagonal and the off-diagonal terms of the stiffness or compliance tensor, a stress discontinuity takes place when one of the principal strains changes sign and the other remains unchanged (cf. Skrzypek and Kuna-Ciskal, 2003). In the model proposed by Chaboche (1993) only the diagonal components corresponding to negative normal strains are replaced by the initial (undamaged) values. The consistent description of the unilateral effect was recently developed by Halm and Dragon (1996, 1998). These authors introduced a new fourth-rank damage parameter built upon the eigenvectors of second-order damage tensor that controls the crack closure effect with the continuity requirement of the stress–strain response fulfilled.
EXPERIMENTAL RESULTS
When a material is subjected to cyclic loading at high values of stress or strain, damage develops together with cyclic plastic strain. If the material is strain loaded, the damage induces a drop of the stress amplitude and the decrease of the elasticity modulus. Results of Dufailly’s experiment (cf. Lemaitre, 1992), for a tension–compression test for AISI 316L stainless steel at constant amplitude of strain performed at room temperature, are shown in Figure 1.
Detailed analysis of subsequent strain–stress loops reveals existence of two softening processes corresponding to damage growth. In the first stage (0<N<2000 cycles) damage growth is isotropic and exhibits decreasing tendency. This results in isotropic drop of both stress amplitude and modulus of elasticity. For higher number of cycles (2000<N<2400) another unilateral damage process is activated. The drop of stress amplitude and elasticity modulus becomes faster for the tensile stress than for the compressive one. Finally, in the third stage, directly preceding failure (2400<N<2556), the plastic strain localization mechanism appears. It is manifested by the formation of stress instability for tensile stress on one hand, and a characteristic point of the zero curvature on this part of hysteresis which corresponds to compressive stress, on the other.
MODELING OF STRAIN–DAMAGE COUPLING Double Scalar Damage Variable Concept
Damage exhibits, in general, an anisotropic character, even if the material is originally isotropic. Therefore a second-rank damage tensor is the type of variable most frequently used to express the anisotropy introduced by damage in initially isotropic materials. Although it corresponds to the minimum complexity of an anisotropic theory in some particular cases damage may be described by isotropic tensor of zeroth rank – a scalar. This simplified model is adequate only for some materials under conditions of proportional loading (cf. Betten, 2002). The main disadvantage of such a scalar damage variable formulation is that the damage does not affect the Poisson’s ratio. In order to eliminate such a behavior, Chow and Wei (1999) proposed two-scalar damage variable formulation. In the present study, the existence of two separate damage mechanisms requires application of two independent scalar damage variables: the isotropic damage variable Dsdescribing transgranular damage acting on the deviatoric stress, and the unilateral damage variable Dvdescribing integranular damage acting on the volumetric stress. In the simplest case of uniaxial stress the effective stress takes the form (cf. Ladeveze, 1983; Vereecke and Billardon, 2004)
remain open, and the case of compression, when they are partly closed. The effective stress takes the final form
~ ¼ s 1 Ds þ v 1 Dvh ð2Þ where h 2 [0, 1] stands for damage closure/opening parameter such that the closure phenomenon is stress controlled.
h ¼ 1 for >0 hc for <0
(
ð3Þ
The above model admits effective stress discontinuity when changes sign.
Damage Modified 1D Ylinen’s Model
Smooth transition of the strain–tress relation between the elastic range and the nonlinear plastic range suggests application of the nonlinear approximation for the initial hysteresis loop. Among usually applied models like Ramberg–Osgood, Prager’s tangent hyperbolic, or Ylinen’s approxima-tions, the last one gives the best fitting to the experimental data.
In the classical model of nonlinear elastic-plastic material suggested by Ylinen (1956), the following uniaxial stress–strain relation holds
" ¼1 E c ð1 cÞ0ln 1 0 ð4Þ
where c, 0denote material constants (0 c 1).
In the case of material subjected to damage, the original 1D Ylinen model (4) has to be modified by replacement of the ‘nominal’ stress by the effective stress ~(2). After differentiation with respect to stress, the damage modified Ylinen 1D approximation for loading cycles takes, therefore, the form
d d"¼ ~ E ~ 0 j j ~ 0 c j j loading ~ E unloading 8 > < > : ð5Þ
Inserting definitions of the effective stress (2), the effective damage modulae of elasticity ~E and the effective asymptotic yield stresses ~
Ylinen’s model become functions of damage ~ Eþ¼E ð1 DsÞð1 DvÞ 1 ð2=3ÞDð v ð1=3ÞDsÞ ~ E¼E ð1 DsÞð1 DvhcÞ 1 ð2=3ÞDð vhc ð1=3ÞDsÞ ~ þ 0 ¼0 1 Ds ð Þð1 DvÞ 1 ð2=3ÞDð v ð1=3ÞDsÞ ~ 0 ¼0 1 Ds ð Þð1 DvhcÞ 1 ð2=3ÞDð vhc ð1=3ÞDsÞ ð6Þ
Application of the above definitions of the effective damage modulus requires additional restriction
1 Ds 1 Dvhc > ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 1 þ r ð7Þ
in order to eliminate a peculiar material response, namely the elongation in the direction transverse to the direction of uniaxial tension (cf. Ganczarski and Barwacz, 2004).
Continuous 1D Damage Closure/Opening Effect
In the case of unloading path, introducing definition (2) of the effective stress accounting for the crack closure/opening effect leads to the linear relation between stress and strain whose slope is described by modulus
~
E Dð Þdefined by (6). The deficiency of such a model is the switch from the part of path described by ~Eþð ÞD to the part of path characterized by the
modulus of elasticity ~Eð Þ. The real materials, however, do not exhibitD
such bilinear paths (cf. Figure 1). Therefore, a more realistic model closer to real material behavior, seems to be a proposal of a curvilinear path, suggested by Hansen and Schreyer (1995), the slope of which depends on the percentage of microcrack closure (cf. Forys´ and Ganczarski, 2002) (Figure 2).
In the simplest 1D case, the concept of continuous crack closure/opening consists in the replacement of parameter h in (3) by a function h() such that for uniaxial stress one can get
h ð Þ ¼hcþð1 hcÞ
h i
b
One-dimensional Kinetic Theory of Damage Evolution with Crack Closure Effect Included
The choice of a damage evolution theory, able to model experiment, turns out to be easier if the assumption is made that, in case of LCF, the damage is related to the accumulated plastic strain. The thermodynamically based theory taking advantage of such assumption is proposed by Lemaitre and Chaboche (1985) and Lemaitre (1992).
In the present formulation, as in the original one, the dissipation potential F is a sum of two parts, the first referring to Huber–Mises–Hencky yield condition f, where damage coupling acts only by the effective stress, and the other FD associated with the kinetic law of damage evolution. In case of uniaxial state of stress the dissipation potential takes the form
F ¼ f , Dð Þ þFDðY, DÞ
f ¼ ~j j y
ð9Þ
where the back stress is neglected in order to simplify mathematical formulas, whereas the isotropic hardening variable is included into Ylinen’s model. Application of the classical formalism of associated plasticity leads to the uniaxial plastic strain expressed as
d"p¼@F @d ¼ @F @s d ¼ sign ð sÞ d 1 Ds ð10Þ
and the accumulated plastic strain is given by dp ¼ d"j pj ¼ d
1 Ds
ð11Þ From the thermodynamical analysis point of view the damage conjugate variable is the strain energy density release rate Y. For the simplest expression of dissipation potential FD it is based on kinetic damage of a double scalar variable developed by Ladeveze (1983) as follows
FD¼ Y2 s 2Ssð1 DsÞ þ Y 2 v 2Svð1 DvhÞ ð12Þ
where Ss, Svstand for the deviatoric or volumetric part of a damage strength material parameter, respectively, and the strain energy density release rate associated to the unilateral damage is equal to
Y ¼ YsþYv¼ ð2=3Þ2 2E 1 Dð sÞ2 þ ð1=3Þ 2 2E 1 Dð vhÞ2 ð13Þ
Since the damage rates are computed as follows dDs¼ @FD @Ys d dDv ¼ @FD @Yv d ð14Þ
taking advantage of (11), the final form of the kinetic damage law for 1D case is written as dDs dp ¼ 2=3 ð Þ2 2ESsð1 DsÞ2 g pð Þ dDv dp ¼ 1=3 ð Þ2 2ESvð1 DvhÞ2 1 Ds 1 Dvh H p pð DÞ ð15Þ
Approximate 3D Description of Damage Evolution at Strain Localization The proper description of the stage after strain localization requires analysis of the 3D state of stress. Simultaneously, a simplicity in numerical integration of the uniaxial model suggests application of a hybrid formulation, in which the stage before strain localization is modeled by uniaxial equations (5) and (15) whereas the stage after strain localization is modeled by an approximate solution for the 3D stress state, expressed by an uniform uniaxial stress beyond the localization zone, substituted into axial strain derived from 3D extension of constitutive law. This allows to perform the governing equations in a simple form.
The simplest approximate 3D solution of the stress state accompanying axisymmetric strain localization is the one proposed by Davidenkov and Spiridonova (1945). Assuming material incompressibility, equality of hoop, and radial logarithmic strain at the minimal cross section, and curvature of an eigenstress trajectory at the minimal cross section being in the form 1/ ¼ r/(r11), the authors prove that intensity of logarithmic strain is equal to the logarithmic axial strain. Next the following approximate formulas were derived for stress in the point belonging to the axis of symmetry at the minimal cross section
r¼t¼i r1 21 z¼i 1 þ r1 21 i¼ r r1 2 1 þ r1=ð41Þ ð16Þ where i¼zr stands for von Mises stress, whereas denotes an
uniform uniaxial stress beyond the localization zone. The state of stress depends on two geometrical ratios r=r1 and r1/1The ratio of the radius of specimen r to radius of neck r1(Figure 3) results from the assumption of incompressibility and definition of logarithmic strain r=r1¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ "h i
p
and it is approximated by unity for applied amplitude of strain "max¼2.1 103. On the contrary, the ratio of radius of neck to radius of curvature r1=1ffir=1 cannot be neglected and it is assumed in the form
r 1
¼bpffiffiffiffiffiffih i" ð17Þ Strain under root appears in McAuley brackets since only its positive magnitude induces neck. Substitution of approximations derived above to Equations (16) eventually gives approximate 3D formulas that describe axisymmetric stress state in necked zone by nominal stress .
In what follows, an extension of the 1D damage modified Ylinen’s constitutive law (4) for 3D state of stress is used
"ij¼c 1 2G sij 1 Ds þ3 K vij 1 Dvh ð1 cÞ0 2Gln 1 i 0 3sij 20ð1 DsÞ ð19Þ in which the first elastic term is compressible whereas the second inelastic term is incompressible and sij denotes deviatoric stress component. Hence,
restricting Equation (19) to the axial strain one can obtain approximate formula for "z that accounts for the 3D axisymmetric stress at the neck
" ¼ "z¼c 1 2G sz 1 Ds þ3 K 3 v 1 Dvh ð1 cÞ0 2Gln 1 i 0 3sz 20ð1 DsÞ ð20Þ Calculating sz and volumetric stress v with Equations (18) taken into
account, lead to sz¼ 2 z2 r 3 ¼ 2 3 1 þ 0:25b pffiffiffiffiffiffih i" v¼ 2 rþz 3 ¼ 3 1 þ 0:5bpffiffiffiffiffiffih i" 1 þ 0:25bpffiffiffiffiffiffih i" ð21Þ
Next substitution of (21) together with i in (18)–(20), when some
simplifications are used give an approximate extension of 1D Ylinen’s law (5) to 3D axisymmetric stress state
d d"¼ ^ E ^ 0 j j ^ 0 c j j loading ^ E unloading 8 < : ð22Þ
where the new effective damage modulae of elasticity ^E and the effective
asymptotic yield stresses ^
0 of Ylinen’s model (6) are defined by the
following relations ^ E¼ ~E1 þ 0:25b ffiffiffiffiffiffi " h i p 1 þ 0:5bpffiffiffiffiffiffih i" ^ 0 ¼~ 0 1 þ 0:25bpffiffiffiffiffiffih i" 1 þ 0:5bpffiffiffiffiffiffih i" ð23Þ Note that, when localization zone is not admitted, then following Equation (17) that combines the neck geometry (radii r and 1) with the
approximate axial strain " ¼ "zat the neck bottom, the formulas (23) yield
the pure 1D formulas (6).
Extending now Equation (9) to 3D case yield condition takes the form F ¼ f , Dð Þ þFDðY, DÞ
f ¼ i 1 Ds
y ð24Þ
whereas approximate 3D plastic strain components are defined as d "p z ¼ @F @z d ¼ d 1 Ds d "p r ¼d " p t ¼ @F @r d ¼ d 1 Ds ð25Þ whereas if the assumptions of Davidenkov and Spiridionova approach are used, the accumulated plastic strain is given by the formula
d p ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3d " p ijd " p ij r ¼ ffiffiffi 2 p d 1 Ds ð26Þ The strain energy density release rate associated to the unilateral damage takes the form
Consequently, the 3D extension of 1D kinetic damage law (15) is written as dDs d p ¼ 2 2pffiffiffi2ESsð1 DsÞ2 1 1 þ 0:25bpffiffiffiffiffiffih i" 2g ð Þp dDv d p ¼ 2 6pffiffiffi2ESvð1 DvhÞ2 1 Ds 1 Dvh 1 þ 1:5bpffiffiffiffiffiffih i" 1 þ 0:25bpffiffiffiffiffiffih i" 2 H ðp pDÞ ð28Þ
Eventually, in what follows, in order to account for both pre-necking and necking stages, one shall apply the 1D formulas (22) and (28) equivalent to the appropriate 3D stress state, respectively. This approximation enables to simulate experimental results by Dufailly for 1D reverse tension–compres-sion test, where the stage of necking is taken into account. It will be shown that this concept allows for predicting the full range LCF including instability due to necking.
PARAMETER IDENTIFICATION
The magnitudes of material parameters related to elasticity and plasticity, namely E, , and yare cited by Lemaitre (1992).
The material parameters of Ylinen’s model are calibrated in the following way: the magnitude of c is set in order to close the first hysteresis loop, whereas the magnitude of 0 is established by comparison of the first hysteresis loop with that of a numerical simulation and minimization of the respective differences (Figure 4).
Identification of the material parameters describing kinetic law of damage evolution hc, a, Sv, Ss, pDis based on the procedure proposed by Lemaitre (1992) in the case of isotropic damage and extended by Garion and Skoczen´ (2003) for the case of orthotropic damage. In particular, the critical magnitude of damage closure/opening parameter hc generally depends on the material and loading. In practice, however, it is considered to be constant in order to simplify algebraic calculations. Lemaitre (1992) recommends a value hc¼0.2 as universal for a wide class of materials.
The complete material data of AISI 316L stainless steel at room temperature is presented in Table 1.
RESULTS
is numerically integrated for strain from "min¼ 2.45 103 to "max¼ 2.1 103by odeint for routine using fourth-order Runge–Kutta technique with adaptive stepsize control (cf. Press et al., 1992).
Numerical simulation with sampling model of crack closure/opening effect excluded (h ¼ 1.0), originally proposed by Lemaitre (1992), is presented in Figure 5. The model under consideration exhibits isotropic damage softening for tension and compression which disqualifies it to proper modeling of experimental results.
Predictions based on a more advanced model of classical crack closure/ opening effect (h ¼ hc) give quantitatively good agreement with experimental data (Figure 6). The considered model properly maps unilateral nature of damage softening in this sense that dead center ordinates of subsequent hysteresis loops coincide exactly with appropriate points on experimental curves. However, the model exhibits @=@" discontinuity for ¼ 0 leading to drastic disagreement with experiment in case of stage after strain localization.
( )
( )
E
−0.002 −0.001 0 0.001 0.002 −300 −200 −100 0 100 200 300 ε 1 200 2000 2300 2400 2500 2600 2630 Failure σ (MPa)
Figure 6. Numerical simulation with model of classical crack closure/opening effect (h ¼ hc).
−0.002 −0.001 0 0.001 0.002 −300 −200 −100 0 100 200 300 ε σ (MPa) 0 200 2000 2300 2400
In contrast to the above models, the numerical simulation with model of continuous crack closure/opening effect included (h() according to (8)) exhibits not only quantitative but also qualitative proper correctness in comparison with experimental results (Figure 7).
The essential defect of the previous model, an existence of non smooth, bilinear material characteristic separating tensile and compression, mani-fested specially at point of zero ordinate lying on tensile unloading to compression reloading branch of hysteresis loop, is successfully eliminated. The condition (7) is satisfied since final magnitudes of damage variables Ds¼0.1, Dv¼0.91 lead to the fracture (1 Ds)/(1 Dvhc) ¼ 1.1 whereas calculation for stainless steel of Poisson’s ration ¼ 0.3 gives a value equal to 0.554. The main limitation of the model seems to be bad mapping of this part of hysteresis which corresponds to compressive stress and consequently lack of characteristic point of zero curvature. Use of damage deactivation expressed in terms of total plastic strain (cf. Chaboche, 1999) instead of presented stress formulation may improve solution, however it does not guarantee that the inflection point appears. Proper modeling of this phenomenon is more sophisticated and requires application of a fully 3D approach.
CONCLUSIONS
. Modeling of LCF by use of sampling model of crack closure/opening effect excluded leads to isotropic damage softening insufficiently mapping complex experimental data (cf. Figure 5).
. Model of the classical crack closure/opening effect (cf. Figure 6) with discontinuous þ
to
transition allows for quantitatively good agree-ment with experiagree-mental data only. However, it reveals physically unjustified discontinuities in elastic modulae Eþand E.
. Application of the continuous crack closure/opening effect (cf. Figure 7) gives both quantitatively and qualitatively good agreement with the experimental data, and confirms necessity and correctness of this approach in the modeling of LCF of AISI 316L stainless steel.
. It is to be pointed out that the above description is based on a simplified assumption that for the analysis of the 1D reverse loading cycles it is sufficient to take into account axial strain component exclusively. However, in a more general case when nonproportional reverse loading cycles are considered, this simplified approach is no more appropriate, such that the full, exact 3D FEM analysis is required.
. Additional further investigations concerning even more precise modeling of cyclic damage that allows for formation of characteristic point of zero curvature at compressive loading branch of hysteresis require application of the FEM code and will be the subject of a separate publication.
NOMENCLATURE x j j ¼absolute value of x x h i ¼MacAuley bracket xh i ¼0 if x<0 or xh i ¼xif x 0 d ¼ differential operator
@ ¼operator of partial derivative exp ¼ exponent function
H ¼Heaviside function ln ¼ Napier’s logarithm ij¼unit tensor of second rank
sign(x) ¼ (þ) or () as the sign of x
¼tilde over symbol denotes effective quantity
– ¼ bar over symbol denotes approximate quantity at the bottom of neck
¼superscripts refer to tension (þ) or compression (), respectively
s,v ¼ subscripts refer to deviatoric or volumetric part of a quantity, respectively
a ¼isotropic damage evolution parameter b ¼parameter related to the neck geometry c ¼constant of Ylinen’s model
E ¼Young’s modulus of elasticity f ¼yield function
FD¼dissipation function G ¼Kirchhoff’s modulus
h, hc¼damage closure/opening parameter and its critical magnitude
K ¼bulk modulus
N ¼number of loading cycle p ¼accumulated plastic strain pD¼damage threshold plastic strain
r, r1¼radius of specimen beyond the neck and at the bottom
of neck sij¼stress deviator
S ¼damage strength material parameter Y ¼damage conjugate force
", "p¼uniaxial total and plastic strain
"min, "max¼minimum and maximum magnitudes of strain cycle "p
r, " p
t, "pz¼plastic strain components at the bottom of neck
¼plastic multiplier
1¼radius of curvature at the bottom of neck ¼uniaxial stress
i, v¼von Mises stress and volumetric stress r, t, z¼stress components at the bottom of neck
y, 0¼yield stress and its asymptotic magnitude
ACKNOWLEDGMENTS
The authors gratefully acknowledge the support rendered by the State Committee for Scientific Research, KBN, Poland, Project PB-807/T07/2003/25.
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