In-plane and out-of-plane crack-tip constraint effects under biaxial nonlinear deformation

Engineering Fracture Mechanics 78 (2011) 1771–1783

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In-plane and out-of-plane crack-tip constraint effects under biaxial nonlinear deformation V.N. Shlyannikov ⇑, N.V. Boychenko, A.M. Tartygasheva Research Center for Power Engineering Problems of the Russian Academy of Sciences, Lobachevsky Street 2/31, Post-Box 190, Kazan 420111, Russia

a r t i c l e

i n f o

Article history: Available online 19 January 2011 Keywords: Biaxial loading In-plane and out-of-plane constraint parameters Higher-order terms Crack-tip field Plane strain Plastic and creeping materials

a b s t r a c t In-plane and out-of-plane constraint effects on crack-front stress fields under both elastic– plastic and creep conditions are studied by means of three-dimensional numerical analyses of finite thickness boundary layer models and plane strain reference solutions. This investigation is an extension of the plane strain solution obtained by Shlyannikov and Boychenko in 2008, with special attention on what constraint parameters existed in the nonlinear crack-tip fields in a finite thickness solid. Characterization of constraint effects is given by using the non-singular T-stress, the local triaxiality parameter, the factor of the stress-state in 3D cracked body and the second order term amplitude factor. The influence of nominal stress load biaxiality and creep time on the behavior of constraint factors is considered. Stresses and constraint factors from FEA at the crack-front on different planes in the thickness direction of the plate are compared with plane strain reference solutions. The results show that 3D-stress fields can be characterized in common with the local triaxiality parameter and factor of the stress-state in 3D solid by the three-term solution throughout the thickness even in the region near the free surface. It is found that there is a distinct relationship between the in-plane and the crack-front out-of-plane constraint factors which can be well captured using the relation between the second order term amplitude factor and remote boundary layer stress. Published by Elsevier Ltd.

1. Introduction Constraint effects in modern fracture mechanics are usually considered as specimen configuration and loading conditions influence on crack-tip fields. Therefore, fracture toughness dependence is referred to these factors and cannot be used as a constant of material. However, the general discussion of constraint effects requires to be defined more exactly. Constraint effects can be defined as specimen prevention from plastic strains depending on geometry and loading conditions. Constraint effects near the crack tip have been extensively studied for a long time. Most of the researches are referred to in-plane constraint since Williams [1] presented the asymptotic expansion of the stress field around the crack tip in elastic body that includes a non-singular in-plane normal stress component, the T-stress. Subsequently Larsson and Carlsson [2], and Rice [3] have shown that including the T-stress gave an improved representation of the elastic–plastic crack tip stress fields. Based on T-stress, a two-parameter approach J–T was proposed by Betegon and Hancock [4], which takes into account the in-plane constraint on crack-tip fields. O’Dowd and Shih [5] have introduced as alternative constraint methodology a two-parametrical approach on the base of J and a hydrostatic stress parameter Q. The J–A2 two-parameter three-term approach was also proposed to describe the stress field in the vicinity of the crack tip in a power hardening material [6,7]. Shlyannikov and Boychenko [8] have discussed the influence of biaxial loading on the higher-order both stress and strain rate ⇑ Corresponding author. Tel.: +7 843 231 90 20. E-mail address: [email protected] (V.N. Shlyannikov). 0013-7944/$ - see front matter Published by Elsevier Ltd. doi:10.1016/j.engfracmech.2011.01.010

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Nomenclature a crack length A1, A2, A3 higher-order term amplitude factors B inherent biaxiality parameter  B creep parameter C(t) creep amplitude factor E the Young’s modulus dimensionless elastic angular stress functions fij ðhÞ h local stress triaxiality parameter K elastic stress intensity factor m strain hardening exponent n creep exponent r normalized crack tip distance r,h polar coordinates centered at the crack tip s1, s2, s3 exponent of stress functions t creep time tT time for transition from small-scale creep to extensive creep T local non-singular stress Tappl remote boundary layer stress Tz factor of the stress-state in 3D cracked body displacement components ux, uy b crack angle r nominal stress in the Y-axis direction rij stress tensor components ry the yield stress r~ ðkÞ ðhÞ dimensionless elastic–plastic angular functions ij r~ e the Mises effective dimensionless stress r0 reference stress e_ 0 reference creep strain rate g nominal stresses biaxial ratio m the Poisson’s ratio

fields and the second order amplitude coefficients for idealized plane strain conditions. However, all these approaches can successfully describe the in-plane constraint effects, but they are limited to a planar case. The description of out-of-plane constraint should include specimen’s dimension such as thickness. Only a few researches have been done to describe thickness effect on the crack-tip constraint [9–12]. Some authors [13,14] showed that 3D crack-front constraint effect in a thin plate and in thick SENB specimens are well represented by J–A2 three-term solution under small-scale yielding and largescale yielding conditions. This study focuses on the finite element analysis of the plastic and creeping materials under different in-plane and out-ofplane constraint levels. The geometry considered in detailed three-dimensional finite element calculations is a biaxially loaded finite thickness plate. Different degree of in-plane constraint is given in form of the applied stress Tappl by combination of far-field stress level, biaxial stress ratio and initial crack angle. In our case the Tappl is a parameter describing solely loading conditions. Loadings and initial crack angle were applied related to a range (1, +1) of far-field biaxial stress ratio. For the purpose of comparison, full-field finite element analysis based on a modified layer approach wherein the T-stress is prescribed as remote boundary conditions is employed to model the effects of biaxial loading on nonlinear behavior under plane strain conditions. The present work is concerned with the application of triaxiality parameter h, Tz-factor and the second order term amplitude A2 to 3D crack-front stress field and the characterization of the interaction of in and out-of-plane constraint.

2. Determination of constraint factors It is well known that different traditional approaches, which successfully describe the in-plane constrain, are not accurate for 3D cracks. Thus, it is necessary to use others factors to describe the out-of-plane constraint. The T z factor introduced by Guo [9] is an important parameter to characterize the constraint effect accurately, which is essential to establish a three parameter dominated stress field, and offers a possibility to characterize the stress-state in a 3D cracked body

Tz ¼

rzz ; mðrxx þ ryy Þ

where m is the Poisson’s ratio, and

ð1Þ

rxx ; ryy ; rzz are the stress tensor components.

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Because the validity of all mentioned above concepts depends on the chosen reference field, a local parameter of crack-tip constraint and stress triaxiality was proposed by Henry and Luxmoore [15] as a secondary fracture parameter

rffiffiffiffiffiffiffiffiffiffiffiffi! 3 sij sij ; hðr; h; zÞ ¼ rkk = 3 2

ð2Þ

where rkk and sij are hydrostatic and deviatoric stresses, respectively. Being such a parameter a function of both first invariant of the stress tensor and the second invariant of the stress deviator, it is a local measure of in-plane and out-of-plane constraint effects that is independent of any reference field. The different combinations of load biaxiality and nominal stress level are characterized by the in-plane elastic non-singular term [16]

K

rij ¼ pffiffiffiffiffiffiffiffiffi fij ðhÞ þ Td1i d1j ; and T appl ¼ T=ry ¼ ðr=ry Þð1  gÞ cos 2b; 2pr

ð3Þ

where g is the nominal stresses biaxial ratio, ry is the yield stress, b is the inclined crack angle, K is the elastic stress intensity factor, r and h are the polar coordinates and fij ðhÞ is the dimensionless angular stress function. In the notation of Rice [3], the second term in the stress expansion is denoted as the T-stress and can be regarded as a stress parallel to the crack flanks. In this case the magnitude of the T-stress is defined through an inherent biaxiality parameter B, introduced by Leevers and Rapffiffiffiffiffiffi don B ¼ T pa=K, where K is the elastic stress intensity factor and a is the crack length. It is therefore necessary to distinguish between the local values of the T-stress and that in the remote boundary layer field, Tappl. In the current notation the mag1 nitude of the load biaxiality is described by remote nominal stress ratio g ¼ r1 xx =ryy : It is well known that the HRR-type one term singularity field generally dominates in a very small region or does not exist near a crack tip for real structures. Some attempts were made to improve the description of the near-tip stress field by introducing an additional parameter. Thus, a more mathematically rigorous approach for the introduction of a second fracture mechanics parameter was proposed on the base of the higher-order elastic–plastic asymptotic expansions. Yang et al. [17] and Nikishkov [6] performed a complete analysis of three-term asymptotic fields for mode I plane strain conditions. They have shown that the first three terms of the asymptotic expansion are sufficient to represent the stress field in the crack tip vicinity in a power-law hardening material. Some details of these solutions follow. It is assumed the stress components near a crack tip are separable and can be expressed as a series:

h

i

s2 ~ ð2Þ s3 ~ ð3Þ r ij ¼ A1 rs1 r~ ð1Þ ij ðhÞ þ A2 r rij ðhÞ þ A3 r rij ðhÞ þ . . . ;

ð4Þ

where r and h are polar coordinates centered at the crack tip, and the index 1,2,3 correspond to the first-order, second-order and third-order fields, respectively. The first term of Eq. (4) is the HRR solution, where dimensionless angular functions r~ ð1Þ ij ðhÞ have been determined by solving the nonlinear fourth-order differential equation and are well known in the literature. A1, A2, A3 are undetermined amplitude factors, and s1, s2, s3 are the exponents of stress functions. In the polar coordinate system, the equilibrium equations have the following form:

1 r

1 r

1 r

2 r

rrr;r þ rrh;r þ ðrrr  rhh Þ ¼ 0; rrh;r þ r#h;h þ rrh ¼ 0:

ð5Þ

Substitution of Eq. (4) into Eq. (5) yields the relationships among the angular stress functions ðkÞ

ðkÞ

ðkÞ

ðsk þ 1Þrrr  rhh þ rrh;h ¼ 0 ðk ¼ 1; 2; 3Þ ðkÞ ðkÞ rh;h þ ðsk þ 2Þrrh ¼ 0

)

:

ð6Þ

Eq. (6) are the final governing equations for the first-order, second-order and third-order asymptotic fields and the stress exponents. They are exactly the same as those given by Nikishkov [6] for power-law hardening materials. According to these solution properties, the amplitudes of the second-order and third-order fields are not independent of each other and have a simple relationship

A3 ¼ A22 ;

ð7Þ

and, for hardening exponent m P 3, the stress exponents are related as follows:

s1 ¼ 1=ðm þ 1Þ;

s3 ¼ 2s2  s1 :

ð8Þ

Note that the amplitude factor A2 cannot be determined in the asymptotic analysis. Therefore, the values of A2 are herein determined by matching the three-term stress solutions in Eq. (4) with known crack-tip fields, such as finite element results. The first term in Eq. (4) corresponds to the HRR field with the amplitude coefficient A1

 A1 ¼

J

ary ey In

1=ðmþ1Þ ;

ð9Þ

and dimensionless scaling integral In. To complete the three-term asymptotic solution, it is necessary to calculate the amplitude factor A2. A few methods have been proposed for this purpose, e.g. the point-matching technique by Chao [7,18], the

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least square approach by Nikishkov [6], the weight function approach by Chao and Zhu [19]. In the present study, the amplitude factor A2 in the higher-order asymptotic solution is determined using a point match technique as follows: the finite element stress value at a point (r, h) close to the crack-front is determined and then the finite element stress value is equalled to the three-term analytical result in order to yield the A2 value. The characteristic length parameter is chosen as the plate  rr and r  hh at h = 0° for different crack-front distance r related to an unloading thickness, b. The normal stress components r zone and a uniform stress distribution area are used to determine an average A2 value. In order to cover both the regions that are discussed later, these distances are r = 2 and r = 20 for plastic material, whereas distances r = 1  104 and r = 1  103 are chosen for creeping material. Both HRR dimensionless angular stress fields and the three-term solution are determined by the present authors [20] having used the Nikishkov approach [6]. One of the purposes of this work is to reveal the asymptotic results that can accurately characterize the mechanical fields near a crack tip in creeping materials. It should be noted that creep fracture mechanics has often relied on the straightforward application of the Hoff analogy [21] by which the HRR solution for power-law creeping materials can be obtained replacing the J-integral by C(t)-integral, and replacing strains and displacements by strain rates and displacement rates, respectively. Based on this argument, Nguyen et al. [22,23] and Chao et al. [24] extended the J–A2 three-term solution by Yang et al. [17] and Nikishkov [6] for elastic–plastic hardening materials to a three-term near-tip solution for a plane strain mode-I crack in power-law creeping materials with two parameters: C-integral and a constraint parameter A2. Hence, the three-term asymptotic fields under the steady-state creep are described by Eq. (4) as the fields for elastic–plastic state, ~ ðkÞ but with different amplitude factors A1, A2, A3. The angular functions of stress r ij ðhÞ in Eq. (4) for creeping materials are the same as those for power-law hardening materials. It means that the solution constructions are similar for such two kinds of materials. In the case of the creeping material, the first term amplitude factor A1 can be obtained by replacing appropriate parameters. Using the Hoff analogy to contrast the power-law creep relation with the power-law hardening relation, Riedel and Rice [25] presented the HRR-type singularity field for power-law creep materials with amplitude A1

A1 ðtÞ ¼



CðtÞ e_ 0 r0 In

1=ðnþ1Þ ð10Þ

:

where r0 is a reference stress, e_ 0 is a reference creep strain rate and n is the creep exponent. The amplitude factor C(t), which depends upon the applied time, the magnitude of the remote loading, the crack configuration and the material properties, was defined by Bassani and McClintock [26]. All radial distributions of constraint factors are represented with respect to normalized crack tip distance. It then follows, purely from dimensional considerations, that the distance (for elastic–plastic problems) from the crack tip r must scale by the yield stress ry when the loading is governed solely by the J-integral

r ¼ ðry r=JÞ ¼ ðr=aÞðry E=r2 pÞ;

ð11Þ

where r is the applied nominal stress and a is the crack length. In the case of creeping material, the loading governing parameter is the C-integral. A dimensionless radial coordinate is given by:

0

pffiffiffi r ¼ ðr0 e_ 0 r=C Þ ¼ ðr=aÞ=@p n

pffiffiffi 3 2

r r0

!nþ1 1 A;

ð12Þ

where r0 is a reference stress, e_ 0 is a reference creep strain rate and n is the creep exponent. Note that, for creeping material, the amplitude factors in the three-term solution (Eq. (4)) generally depend on the creep time, magnitude of the applied loading, crack geometry and material properties. Due to the complexities, we consider different creeping stages in the present work. It is useful to normalize a current creep time t by the characteristic time tT for transition from small-scale creep to extensive creep which can be expressed as follows [27]:

t tðn þ 1ÞEC ¼ ; tT ð1  m2 ÞK 2I

ð13Þ

where KI is the elastic stress intensity factor, m is the Poisson’s ratio, E is the Young’s modulus. The results for elastic–plastic and creeping materials of angular stress distributions for both general FEM-numerical solutions and asymptotic problems of the order 1, 2, 3 are normalized

r~ FEM e;max ¼

 1=2 3 FEM FEM sij sij ¼ 1; 2 max

ðkÞ r~ e;max ¼

 1=2 3 ðkÞ ðkÞ sij sij ¼ 1; 2 max

k ¼ 1; 2; 3:

ð14Þ

~ e is the Mises effective dimensionless stress, and Sij are the components of the deviatoric stress tensor. In Eq. (14), r 3. The modified boundary layer approach and models of materials behavior The geometry considered in this study is traditional for the modified boundary layer formulation. A biaxially loaded finite thickness cracked panel is assumed to contain an annular region. A circular disc (Fig. 1) containing the crack tip in a large panel is removed, modeling the near-tip region with the modified boundary layer formulation. The displacement compo-

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Fig. 1. Biaxially loaded plate and FE model of circular disc.

nents based on the first two terms of the elastic crack-tip field which were determined by Eftis and Subramonian [16] are prescribed on the outer boundary of the domain. These elastic displacements corresponding to the mode I are applied along the circular boundary of the R radius. The biaxial loads are applied through the displacement components accounting for the non-singular term and the elastic stress intensity factor

9 h i 2 h ð1gÞr h 2 = cos  ð1  2 m Þ þ sin ð1  m Þ½r cos h þ a E 2p 2 2 ;   pffiffiffiffi uy ¼ KG1 2rp sin 2h 2ð1  mÞ  cos2 2h þ ð1EgÞr mð1 þ mÞ½r sin h ;

ux ¼ KG1

pffiffiffiffi r

pffiffiffiffiffiffi

K1 ¼

r pa 2

½ð1 þ gÞ  ð1  gÞ cos 2b:

ð15Þ

The crack tip is located at the center of the disc. The panel is subjected to two perpendicular loads: one parallel to the y1 1 axis r1 yy and another parallel to the x-axis rxx . The magnitude of the traverse load rxx can also be described by the nominal 1 1 stress biaxial ratio g ¼ rxx =ryy . For both elastic–plastic and creeping materials, we consider three different biaxial stress states: equi-biaxial tension–compression (g ¼ 1), uniaxial tension (g ¼ 0) and equi-biaxial tension–tension (g ¼ þ1). For creeping material, each type of biaxial loading is analyzed at nine stages of creep time which is varied from t ¼ 0 up to t ¼ 5  104 h. Our computations terminate at t ¼ 5  104 h, which is the time when the creep zone radius equals about 80% of the plate outer boundary. All values of creep time t are normalized by the reference time tT . We define the reference time, tT as the time corresponding to the transition from the small-scale creep to the extensive creep. The ANSYS [28] finite element code is used to solve modified boundary layer problems. A large circular domain containing a notch along one of its radii is considered (Fig. 1). The FEA calculations are based on the J2 incremental theory of plasticity. A similar coordinate system is employed such that the x-axis lies in the crack plane and is normal to the straight crack-front; yaxis is orthogonal to the crack plane and z-axis lies on the thickness direction. The origin of the coordinate system is located at the crack tip on the center plane. Only a quarter of the plate is modeled due to symmetry with respect to the mid-plane (z/ b = 0.5) and the crack surface plane (y = 0). Along the thickness direction, the identical planar mesh is repeated from the symmetry plane (z/b = 0.5) to the free surface (z/b = 0). In order to catch the drastic change of the stress field near the free surface, the thickness of successive element layers is exponentially reduced from the mid-plane toward the free surface. In the circumferential direction, 40 equally sized elements are defined in the angular region from 0 to p. The size of each ring gradually increases with the radial distance from the crack tip. Radial sizes of elements are varied according to the geometric progression. A finite element mesh consisting of 1144530 nodes and 275784 twenty-node quadrilateral brick elements

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was used. To consider the details of large deformation and blunting of the crack tip, a typical mesh was used to model the region near the notch tip. The radius R (along which the modified boundary condition is given) of outermost boundary circular domain is about 103q, where q is the initial notch root radius. In the undeformed configuration, the center of curvature of the notch coincides with the center of the circular domain. The ratio between plate thickness b and outermost boundary circular domain radius R is about b/R = 0.05. Our attention is focused on three-dimensional mode-I stationary crack problems. In general, the material deformation behavior is described by additive decomposition of the elastic, plastic and creep strains. In our work two different strain treatments are used for calculating total strain. The first of them is obtained by combining elastic and isotropic hardening, whereas the second one is obtained by combining elastic, isotropic hardening and creep. The Ramberg–Osgood model



e r r ¼ þa ry ey ry

m ;

ð16Þ

was employed to define the stress–strain curve corresponding to the elastic–plastic material properties. In Eq. (16), ry is the yield stress, m is the hardening coefficient, and a and ey are two fitting parameters. The parameter ey is usually taken equal to ry/E. Two fitting parameters are determined by means of least squares minimization. It is well known that this fit type yields a good approximation of the true stress–strain curve for high plastic strains but, for low plastic strains, the stresses are lower than the experimental data. In our case, the typical relative difference between experimental and fitting values of the true strain is lower than 5%. For a purely power-law creeping material behavior governed by Norton constitutive relation and under uniaxial tension, the total strain rate is related to the stress by the following relationship:

e_ ¼ e_ 0



r r0

n

or

e_ ¼ Brn ;

ð17Þ

where r0 is a reference stress, e_ 0 is a reference creep strain rate and n is the creep exponent. Note that a combined constant  ¼ e_ 0 =rn is often used. Typically, the material constants E, m, ry, m, n, r0, e_ 0 or B  are obtained experimentally from uniaxial B 0 tests at the temperature of interest. For the present problem, Young’s modulus, Poisson’s ratio and yield stress are considered to be equal to 205 GPa, 0.3 and 380 MPa, respectively. The strain hardening exponent m is 4.96. The creep parameter and the  = 1.4  1010 and n = 3. The reference stress and reference creep strain rate are r0 = 100 MPa and creep exponent are B _e0 = 1.1107 h1. The Ramberg–Osgood model is introduced into the finite element code using a piecewise linear model available in ANSYS for describing material properties. 4. Results and discussion Full-field 3D finite element analyses are carried out to determine the elastic–plastic and creep stress fields along the through-thickness crack-front in a circular disc subjected to different biaxial loadings. Loadings and initial crack angle are applied, related to the range g (1, +1) of far-field biaxial stress ratio. First, the stress distribution ahead of the crack tip (h ¼ 0) under mode I at b ¼ p=2 is examined. In Fig. 2 the elastic–plastic radial distributions of the out-of-plane constraint factors along the crack-front in the thickness direction are plotted for different biaxial loading conditions described by the transverse stress Tappl. The left row in Fig. 2 shows the behavior of Tz-factor and stress triaxiality parameter h under pure mode I characterized by in-plane Tappl value equal to T = 0.53 under equi-biaxial tension–compression (g ¼ 1), while the right row in Fig. 2 displays equi-biaxial tension (g ¼ þ1) at T = 0. All the constraint parameters are plotted against the normalized distance, r ¼ ðry r=JÞ. As can be seen from these figures, the out-ofplane constraint factors decrease along the crack-front toward the plate free surface. Compared with both the Tz-factor and the triaxiality parameter h distributions at the center plane (z/b = 0.5) in Fig. 2, FEA results at the plane near the free surface (z/b = 0.0125) show great relaxation on the crack-front constraint. It should be noted that, in the stress triaxiality parameter h radial distributions, an unloading zone exists where finite strain effects depending on crack-front position dominate for the normalized radial distance 2 < rry =J < 10. In the region rry =J > 10, where stress relaxation due to the crack tip blunting does not exist, the near-tip stresses depend on the sign and amplitude of the Tappl. For this reason, it is necessary to analyze the constraint parameters behavior along the crack-front in the thickness direction in the two regions characterized by normalized crack tip distances rry =J ¼ 2 and r ry =J ¼ 20, respectively. Fig. 3 shows the variations of the out-of-plane constraint factors radial distributions under equi-biaxial tension–compression (g ¼ 1) and equi-biaxial tension (g ¼ þ1) loading characterized by in-plane constraint parameters equal to T = 0.44 and T = 0, respectively. These distributions relate to extensive creep conditions with dimensionless creep time t ¼ 9:25 when the radial distance r is normalized by C=ðr0 e_ 0 Þ. The corresponding constraint factors distributions for the reference plane strain problem obtained by Shlyannikov and Boychenko [8] are plotted for comparison purposes in Figs. 2 and 3. It is found that, under biaxial loading, the differences between the full three-dimensional both elastic–plastic and creep stress fields and the plane strain reference solutions appear to depend on the distance to the crack tip and to the free surface of the plate. Moreover, the plane strain solution for the out-of-plane constraint Tz-factor coincides with the 3D finite size solids distribution only in the mid-plane (half-thickness of plate z/b = 0.5) at short distance from the crack tip. When the crack-front distance is small, 3D FEA results are the same as the plane strain results, but they deviate from the 2D results as the distance

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Fig. 2. Variation of out-of-plane constraint factors with normalized radial distance for plastic material.

increases because of the relaxation of crack-front constraint. Since the radial distributions for the creeping material under biaxial loading show similar trends to those of the plastic material, the behavior of the constraint parameters in the unloading zone is characterized by the normalized distance r ¼ C=ðr0 e_ 0 Þ ¼ 1  104 , whereas the distance is equal to r ¼ C=ðr0 e_ 0 Þ ¼ 1  103 for a region of uniform distributions. Through-thickness variations of the Tz-factor, the triaxiality parameter h and the second order term amplitude A2 at the crack-front for plastic material at r ¼ 2 and r ¼ 20 for different biaxial loading conditions are shown in Fig. 4. For crack tip distance r ¼ 2, inlying unloading zone out-of-plane constraint parameters Tz and h are nearly constant through most part of the thickness and then drastically decrease toward the free surface. At the same time, amplitude A2 remains relatively constant through the thickness. FEA results indicate that all constraint parameter distributions are nearly the same for different biaxial loading characterized by in-plane Tappl. It implies that the effect of load biaxiality on crack-front constraint can be ignored within unloading zone. Through-thickness variations of all constraint factors at the radial distance r ¼ 20 ahead of the crack tip are shown in Fig. 4 for equi-biaxial tension–compression (g ¼ 1), uniaxial tension (g ¼ 0) and equi-biaxial tension (g ¼ þ1) characterized by in-plane Tappl values T = 0.53, T = 0.26, T = 0, respectively. It is remarkable that, compared with the out-of-plane parameter distributions within unloading zone at r ¼ 2, FEA results in uniform area at r ¼ 20 show great influence of load biaxiality g on the crack-front constraint, especially on the triaxiality parameter h and the second order term amplitude A2 behavior. It can be seen in Fig. 4 that out-of-plane constraint parameters Tz and h values slightly decrease through most part of the thickness toward the mid-plane. For the crack tip distance of r ¼ 20, the distributions along the crack-front in Fig. 4

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Fig. 3. Variation of out-of-plane constraint factors with normalized radial distance for creeping material.

show that A2 slightly increases from the free surface to the quarter plate (z/b = 0.25) and then remains relatively constant through most part of the thickness. Similar to the plastic material, Fig. 5 plots the through-thickness variations of out-of-plane constraint parameters for the large-scale (t ¼ 9:25) creep conditions. These distributions are shown for the unloading zone at r ¼ C=ðr0 e_ 0 Þ ¼ 1  104 and the uniform area at r ¼ C=ðr0 e_ 0 Þ ¼ 1  103 . Fig. 5 depicts the typical behavior of the out-of-plane constraint Tz-factor and the stress triaxiality parameter h along plate thickness direction for creeping material under different biaxial pure mode I loading as a function of a given crack tip distance. These figures show that, for the some distance behind the unloading zone, the out-of-plane constraint factors are a monotonic increasing function of the in-plane increasing constraint factor Tappl, but nearly independent of the load biaxiality at short distance from the crack tip. It should be noted that the represented numerical solutions are accounting for border effect near the free surface of plate. The through-thickness behavior of all constraint parameters for the small-scale creep conditions (t ¼ 0:0925), which are not shown here, represents the same trend as that of the large-scale creep conditions (t ¼ 9:25). According to Figs. 4 and 5, for both elastic–plastic and creeping materials, the second order term amplitude A2 is the outof-plane constraint factor which is more sensitive to the in-plane constraint and load biaxiality. It is found that at the distance behind the unloading zone, the maximum values of amplitude A2 correspond to equi-biaxial tension (g ¼ þ1) characterized by in-plane Tappl-stress T = 0. Moreover, as the Tappl is proportional to the applied biaxial load, A2 is close to zero at the equi-biaxial tension (g ¼ þ1), and the stress field at the mid-plane corresponds to the HRR solution.

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Fig. 4. Variation of constraint factors through the thickness for plastic material.

In order to check the validity of the three-term expansion (4–6) containing amplitude A2 as out-of-plane constraint parameter, the full-field finite element results, the HRR-type fields and the three-term asymptotic solutions are considered and compared for creeping material. As has been shown by the present authors [8], the tangential stress behavior in the ra-

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Fig. 5. Variation of constraint factors through the thickness for creeping material.

dial direction within r=½C=ðr0 e_ 0 Þ < 0:0004 is characterized as an evident unloading state, while the radial stress distributions are approximately uniform beyond r=½C=ðr0 e_ 0 Þ > 0:0008. Moreover, when the crack distance scaled by r=½C=ðr0 e_ 0 Þ increases from 0.0008 up to 0.01, the tangential stress level monotonically decreases as the creep time increases. Fig. 6 shows ~ rr ; r ~ hh and shear r ~ rh stress components for the mid-plane at the position from crack tip angular distributions of the normal r

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Fig. 6. Angular stress distributions under different biaxial stress states.

r=½C=ðr0 e_ 0 Þ ¼ 1  104 (Fig. 6a–c) and r=½C=ðr0 e_ 0 Þ ¼ 1  103 (Fig. 6d–f), for different biaxial stress state (g ¼ þ1, g ¼ 0, g ¼ 1) at long creep time t=tT ¼ 9:25. The full-field finite element solution for the mid-plane in Fig. 6 is indicated by the symbols, while the HRR field is indicated by the thin solid line and the three-term solution by the thick line. It can be seen that, near the crack tip within the finite strain or unloading zone at r=½C=ðr0 e_ 0 Þ ¼ 1  104 , neither the HRR field nor the

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Fig. 7. Dependencies between in-plane and out-of-plane constraint factors for: (a) plastic and (b) creeping materials.

three-term asymptotic solution coincide with the finite element results. However, at a radial distance more than the unloading zone, the three-term solutions are in good agreement with the full-field FEM results for the time considered here t=t T ¼ 9:25. The HRR-type fields deviate from both the FEM results and the higher-order term solutions for all the time except for the equi-biaxial tension at g ¼ þ1 when the crack tip distance is more than r=½C=ðr0 e_ 0 Þ ¼ 8  104 . For extensive creep, when 9:2 < t=tT < 46:2, the same biaxial loading and crack distance effects similar to those related to the creep time t=t T ¼ 9:25 were observed for the dimensionless stress components angular distributions. From Fig. 6, we can remark that the regions of dominance of the HRR solution exist only in the case of equi-biaxial tension (g ¼ þ1), T ¼ 0 and A2  0. Other biaxial loading types do not fall under determinancy domain of the HRR solution because, in these cases, the deviation of the HRR field with respect to the three-term asymptotic solution is quite large. Good agreement between analytical findings and finite element results confirms again that HRR solution corresponds only to the equibiaxial tension g ¼ þ1 which is a particular case of the general biaxial loading. The results and analyses of polar stress distributions indicate that the three-term solution (4–6) is correct for all times from the small-scale creep to the extensive creep. Although not shown here, for plastic material, the same biaxial loading effects similar to the creep time t=t T ¼ 9:25 were observed for the dimensionless plastic stress components angular distributions. Generally, it can be concluded that the second order term amplitude A2 efficiently represents the out-of-plane constraint effect on the elastic–plastic and creep crack-front stress fields for plate throughout the thickness. The variation in the second order term amplitude A2 for the different discrete crack-front positions as a function of the inplane applied T-stress accounting for load biaxiality is shown in Fig. 7. These distributions are related to plastic material behavior for the crack tip distance r ¼ 20 (Fig. 7a), and to extensive creep conditions with dimensionless creep time t ¼ 9:25, when the radial distance is equals r ¼ 1  103 (Fig. 7b). The corresponding constraint factors distributions for the reference plane strain problem obtained by Shlyannikov and Boychenko [8] are plotted for comparison purposes in Fig. 7. It is found that, under biaxial loading characterized by Tappl, the differences between the full three-dimensional both elastic–plastic and creep out-of-plane constraint and the plane strain reference solutions appear to depend on the distance to the free surface of the plate. In literature the change of the parameters A2 and T-stress from zero to some negative values is related to the constraint loss. The T-stress influence is treated as the development of in-plane constraint effects, whereas the thickness influence in the full three-dimensional problem is connected to out-of-plane constraint effects. The present results show that the reduction of the parameter A2 (or constraint loss) takes place in case of the departure from the mid-plane to the free surface of plate or in case of the change of the biaxial stress ratio from +1 to 1. Therefore, the amplitude A2 combines constraint effects due to in-plane effects and out-of-plane effects. In Fig. 7 the amplitude A2 gradually decreases with decreasing of the inplane applied T-stress. There is a distinct relationship between the in-plane and the crack-front out-of-plane constraint factors which can be well captured using the A2–Tappl approach. Both stress triaxiality h and Tz factors have been found to be inappropriate for describing the interaction between in-plane and out-of-plane constraint because, as is mentioned above, they are not sensitive to load biaxiality. This is consistent with the recent findings of some authors [29]. 5. Conclusions A combined effect of in-plane and out-of-plane constraint is numerically examined considering the interaction between load biaxiality and plate thickness. Three constraint concepts are analyzed including the Tz-factor, the stress triaxiality

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parameter h and the second order term amplitude factor A2. Discrepancies in constraint parameters distributions along crack-front against the plate thickness have been observed under different biaxial loading conditions of elastic–plastic and creeping power-law hardening materials. It is found that the out-of-plane constraint factors behavior for the reference plane strain solution and the finite size 3D solids significantly differ with increasing the crack-front distance. Special attention is paid on which constraint parameters exist in the nonlinear crack-tip fields in a finite thickness solid. However, negative applied in-plane T-stress in the far biaxial field reduces constraint near the mid-plane in three-dimensional plate in the same manner as in two-dimensional plane strain fields. FEA dimensionless angular stress distributions are compared with the three-term solution to check the validity of the parameter A2 and to quantify the constraint effect for 3D cracks. Numerical results indicate that there is a distinct relationship between the in-plane and out-of-plane crack-front constraint which can be described by using A2 and Tappl parameters. 3D crack-front constraint quantified by A2–Tappl approach reveals the relationship between constrain, load biaxiality and plate thickness. Acknowledgment The authors gratefully acknowledge the financial support of the Federal Agency on Science and Innovation of Russia under the Project 02.740.11.0205. References [1] Williams ML. On the stress distribution at the base of stationary crack. J Appl Mech 1957;24:111–4. [2] Larsson SG, Carlsson AJ. Influence of non-singular stress terms and specimen geometry on small-scale yielding at crack tips in elastic–plastic materials. J Mech Phys Solids 1973;21:263–72. [3] Rice JR. Limitations to the small scale yielding approximation for crack tip plasticity. J Mech Phys Solids 1974;22:17–26. [4] Betegon C, Hancock JW. 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