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Effects of characteristic material lengths on mode III crack propagation in couple stress elastic–plastic materials
International Journal of Plasticity 23 (2007) 1439–1456 www.elsevier.com/locate/ijplas
Effects of characteristic material lengths on mode III crack propagation in couple stress elastic–plastic materials Enrico Radi * Dipartimento di Scienze e Metodi dell’Ingegneria, Universita` di Modena e Reggio Emilia, Via Amendola, 2-I-42100 Reggio Emilia, Italy Received 29 October 2006; received in final revised form 17 December 2006 Available online 20 January 2007
Abstract The asymptotic fields near the tip of a crack steadily propagating in a ductile material under Mode III loading conditions are investigated by adopting an incremental version of the indeterminate theory of couple stress plasticity displaying linear and isotropic strain hardening. The adopted constitutive model is able to account for the microstructure of the material by incorporating two distinct material characteristic lengths. It can also capture the strong size effects arising at small scales, which results from the underlying microstructures. According to the asymptotic crack tip fields for a stationary crack provided by the indeterminate theory of couple stress elasticity, the effects of microstructure mainly consist in a switch in the sign of tractions and displacement and in a substantial increase in the singularity of tractions ahead of the crack-tip, with respect to the classical solution of LEFM and EPFM. The increase in the stress singularity also occurs for small values of the strain hardening coefficient and is essentially due to the skew-symmetric stress field, since the symmetric stress field turns out to be non-singular. Moreover, the obtained results show that the ratio g introduced by Koiter has a limited effect on the strength of the stress singularity. However, it displays a strong influence on the angular distribution of the asymptotic crack tip fields. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Crack propagation; Couple stress plasticity; Strain gradient; Elastic–plastic material; Asymptotic analysis
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0749-6419/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2007.01.002
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1. Introduction Due to the lack of a length scale, classical plasticity theories are not able to characterize the constitutive behavior of ductile materials at the micron scale. This lack is expected to be particularly significant for the analysis of the stress and deformation fields very near the tip of a crack propagating in a ductile metal, which are expected to be strongly influenced by the presence of the microstructure. In particular, experimental observations performed by Elssner et al. (1994) found that during crack propagation the interface between a ductile crystal of niobium and a sapphire single crystal remained sharp and not blunted up to the atomic scale. They also found that the stress level required to produce atomic decohesion of the lattice turns out to be about 10 times the tensile yield stress, whereas fracture mechanics analyses based on classical plasticity theories provide a maximum stress level near a crack tip not larger than 4–5 times the tensile yield stress (Drugan et al., 1982). Therefore, for the investigations of the crack tip fields at the micron scale it becomes necessary to adopt enhanced constitutive models, which account for the nonlinear behavior of the material as well as for the presence of microstructure and dislocations. A way of doing that consists in the inclusion of one or more characteristic lengths within the framework of elastic–plastic rate constitutive relations. These lengths are typically of the same order of the compositional grain size, namely few microns, for polycrystalline metals (Fleck et al., 1994). Several strain gradient (SG) plasticity models (Hutchinson, 2000; Fleck and Hutchinson, 1993, 1997, 2001; Chen and Wang, 2002; Ristinmaa and Vecchi, 1996; Ottosen et al., 2000) based on phenomenological theories have been proposed for this purpose. These refined constitutive theories can describe the mechanical behavior of many ductile materials with microstructure, such as metals at the micron scale, liquid crystal, rocks, granular materials, cellular solids, fibrous composites and laminates. Inside these materials, moments may be transmitted through grains, fibres, or in the cell ribs or walls. In a first attempt, Fleck and Hutchinson (1993) presented a couple stress (CS) flow-theory of plasticity which only involves strain rotation gradients. This model includes an intrinsic characteristic material length ‘, which specifies the range where strain gradients are dominant. A further elastic length scale ‘e was artificially introduced in order to partition the curvature rate tensor into elastic and plastic contributions. A generalization of this model within the framework of micropolar solids was presented by Chen and Wang (2002) by treating the micro-rotation field as an independent kinematical quantity, with no direct dependence upon the displacement field. A flow-theory version of CS plasticity based on the indeterminate theory of CS elasticity developed by Koiter (1964) was also developed by Ristinmaa and Vecchi (1996) and Ottosen et al. (2000). Their model involves two material additional material parameters, namely ‘ and g, which can be related to the material characteristic lengths in bending and in torsion and whose effects on the ductile crack propagation are almost unexplored. Several analyses have been carried out to investigate the effects of the aforementioned CS constitutive models in fracture mechanics. Xia and Hutchinson (1996) and Huang et al. (1997, 1999) examined mode I and mode II asymptotic and full-field solutions for a stationary crack within the CS deformation theory of plasticity presented by Fleck and Hutchinson (1993). They found that the symmetric stress field dominates the couple stress field near the crack tip and then the traction level ahead of the crack-tip does not increase substantially. Therefore, mode I crack-tip fields seem to be more significantly influenced
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by stretch gradients rather than by rotation gradients. The latter are expected to have a major effect on the tractions ahead of a mode III crack tip. It must be observed that the CS models failed to estimate the results of tests where also stretch gradients come into play, as for instance micro-indentation and void growth, since the strain gradients considered by these models derive uniquely from rotation gradients. Enhanced SG theories, which incorporate several material length scales, have been proposed to supply this lack (Fleck and Hutchinson, 1997, 2001; Gao et al., 1999). They may provide a link between the mesoscale SG plasticity level and the microscale dislocation mechanics. However, investigations of the stationary crack tip fields performed by adopting these refined constitutive models give physically unacceptable results. In particular, Chen et al. (1999) and recently Wei (2006) found that the normal stresses predicted by the SG theory of plasticity under mode I loading conditions turn out to be compressive within a small region near the crack tip, which is however much smaller beyond the concern of the continuum theory. Shi et al. (2000) showed that if a particular theory of SG plasticity is adopted, namely the mechanism-based strain gradients (MSG) theory, then the stress fields ahead of the crack tip is tensile. Unfortunately, the crack-tip fields obtained for the MSG theory do not admit a separable-variable form at very small distance from the crack-tip, due to the higher-order boundary conditions, so that only numerical or analytical full-field investigations can be performed. However, Huang et al. (2004) clearly showed that the higherorder boundary conditions are not so important and they consequently developed a simpler conventional theory of mechanism-based strain gradient (CMSG) plasticity. Recent analyses carried out to investigate the effects of microstructure on crack propagation in ductile materials showed that the incorporation of couple stress and rotation gradients in the constitutive description, by using the flow theories of CS or SG plasticity, considerably improves the estimation of the tractions level ahead of the crack-tip and may explain the occurring of cleavage or atomic decohesion during the crack growth process, experimentally observed in ductile metals also (Elssner et al., 1994). In particular, the analyses of a steadily propagating crack-tip under mode I loading conditions (Wei and Hutchinson, 1997; Radi, 2003; Qiu et al., 2003) found that the couple stress field is dominant near to the crack-tip and produces a remarkable increase in the stress singularity, even for small strain hardening, whereas former investigations of crack propagation performed by using classical elastic–plastic constitutive models displaying linear strain hardening (Amazigo and Hutchinson, 1977; Ponte Castan˜eda, 1987), predicted an extremely weak stress singularity for the small values of the strain hardening coefficient generally adopted for ductile metals. Moreover, the investigations performed by Wei and Hutchinson (1997) and Radi (2003) under mode I loading conditions showed that the contribution of the elastic strain gradients strongly affects the asymptotic crack-tip fields through the artificial elastic length scale ‘e. Recently, Wei and Xu (2005) extended the investigation of the fracture toughness and shielding effects produced by plastic deformations and discrete dislocations during mode I and mode II crack growth by using a multiscale approach. The problem of mode III crack propagation within the classical J2-flow theory was first analysed by Amazigo and Hutchinson (1977) and Ponte Castan˜eda (1987) for materials which display linear and isotropic hardening and extended by Bigoni and Radi (1996) to mixed isotropic/kinematic hardening. These investigations found that the strength of the stress singularity turns out to be extremely weak for the typically small values of the strain hardening for ductile metals. The effects of elastic strain gradients on the stress concentration at the tip of a mode III stationary crack were considered by Vardoulakis
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et al. (1996); Zhang et al. (1998); Unger and Aifantis (2000); Fannjiang et al. (2002) and Paulino et al. (2003). While in (Vardoulakis et al. (1996)) and (Fannjiang et al. (2002); Paulino et al. (2003)) strain gradients are introduced through the second gradient of displacement, in (Zhang et al., 1998) rotation gradients are considered, then adopting a couple stress theory which is the specialisation of the model adopted here when plasticity is absent and g ¼ 0. The results of Zhang et al. (1998) for the indeterminate theory of CS elasticity indicate that the effects of rotation gradients increase the stress singularity at the crack-tip from r1/2, as obtained for classical linear elasticity, to r3/2, with no need to take stretch gradients into consideration, essentially due to the skew-symmetric stress components. Although this singularity is much stronger than the conventional square-root singularity, it does not violate the boundness of strain energy surrounding the crack tip and leads to a finite energy release rate. The effects of strain rotation gradients on mode III crack propagation in ductile metals have been recently investigated by Radi and Gei (2004) by using the flow theory version of CS plasticity developed by Fleck and Hutchinson (1993). By performing an asymptotic analysis of the crack-tip fields, these authors showed that strain gradients become significant for r < ‘, where r is the distance from a crack-tip, and negligible for r ‘, with a gradual transition in the intermediate region, where r ‘. A qualitative analysis proved that the leading order term of the velocity field turns out to be rotational, so that the leading order term of the strain rotation gradients does not vanish. The obtained results confirmed that the contribution of the elastic strain gradients, related to the elastic length scale ‘e, strongly affects the asymptotic crack-tip fields, according to the results obtained for mode I crack propagation. Moreover, the skew-symmetric stress field asymptotically dominates the symmetric stress and couple stress fields, displaying a much stronger singularity than that predicted by the classical J2-flow theory. In particular, a strong increase in the singularity is observed if the elastic strain gradients are kept sufficiently small, namely for ‘e ‘. In the present work, the effects of strain rotation gradients on ductile steady-state crack propagation under mode III loading conditions are investigated by performing an asymptotic analysis of the crack-tip fields derived from the flow theory of the indeterminate CS elastoplasticity with two characteristic material lengths developed by Ristinmaa and Vecchi (1996) and Ottosen et al. (2000). Strain gradient effects have been also considered for purely elastic response. According to the results obtained by Radi (2003) and Radi and Gei (2004) for a single characteristic material length, the skew-symmetric stress field dominates the asymptotic field, producing thus a remarkable increase of traction level at the crack tip, almost independently of the value of the strain hardening coefficient. The influence of the characteristic lengths on the crack tip fields is explored by numerical investigations. As already observed by Dillard et al. (2006) the inclusion of two or more characteristic lengths provides more realistic predictions on the tractions level ahead of the propagating crack-tip then the classical solution obtained for the J2-flow theory and gives more accurate results then the CS theory of plasticity with a single characteristic length, allowing the detailed mechanisms by which fracture may grow and propagate in ductile metals to be understood in more depth, up to the micron scale. The following notation is used throughout the paper. The components of a generic vector v or second order tensor A with respect to an orthogonal unit vectors basis {ei} are defined as vi ¼ v ei and Aij ¼ ei Aej , respectively, where a dot denotes the natural inner product of two vectors a and b or two second-order tensors A and B. The prefix tr indi-
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cates the trace of a second order tensor and I is the second order identity tensor. The symbols $, div and curl indicate the gradient, the divergence and the curl of a vector or tensor field, respectively, defined componentwise as ðrvÞij ¼ vi;j ; ðdiv AÞi ¼ Aij;j and ðcurl AÞij ¼ eikl Ajl;k , where eikl is the third-order alternator and ( )j denotes the partial derivative with respect to the jth spatial coordinate. Moreover, let denote the third-order permutation tensor, defined as (AÞi ¼ eikl Akl . A superimposed dot denotes a material time derivative. Finally, the MacAuley brackets hi define the function hxi ¼ 12ðx þ jxjÞ, for every real value x. 2. Governing equations The flow-theory version of the CS elatoplasticity developed by Ristinmaa and Vecchi (1996) and Ottosen et al. (2000) is adopted in the present study. This constitutive model derives from the indeterminate theory of CS elasticity developed by Koiter (1964) and assumes the following kinematical compatibility conditions between the displacement vector u, rotation vector h, strain tensor e and deformation curvature tensor v 1 e ¼ ðru þ ruT Þ; 2
h¼
1 curl u; 2
v ¼ rh:
ð2:1Þ
Therefore, rotations are derived from displacements and the tensor field v turns out to be irrotational, being v ¼ curl e
ð2:2Þ
According to the CS theory (Koiter, 1964), the non-symmetrical Cauchy stress tensor t can be decomposed into a symmetric part r and a skew-symmetric part s, namely t ¼ r þ s. In addition, the couple stress tensor l is introduced as the work-conjugated quantity of vT. In absence of body forces and body couples and for a smooth boundary, the principle of virtual work for a body B writes (Radi and Gei, 2004) Z Z ðr de þ l dvT Þ dV ¼ ðp du þ q dhÞ dS; ð2:3Þ oB
B
where 1 p ¼ tT n þ rlnn n; q ¼ lT n lnn n; ð2:4Þ 2 are the reduced surface traction vector and couple stress traction vector, respectively, n denotes the outward unit normal and lnn ¼ n l n. The principle of virtual work (2.3) provides the conditions of quasi-static equilibrium of forces and moments within the body, namely div tT ¼ 0;
div lT þ s ¼ 0:
ð2:5Þ
Within the context of small deformations incremental theory, the total strain rate e_ is the sum of elastic e_ e and plastic e_ p parts. Similarly, the total deformation curvature rate v_ is the sum of elastic v_ e and plastic v_ p contributions. Both elastic parts are related to stress and couple stress rates through the following rate constitutive relations e_ e ¼
1 m _ ½r_ ðtr rÞI; 2G 1þm
v_ eT ¼
l_ gl_ T ; 2G‘2 ð1 g2 Þ
ð2:6Þ
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where G is the elastic shear modulus, m the Poisson ratio, ‘ and g the SG parameters introduced by Koiter (1964), with 1 < g < 1. Both material parameters ‘ and g depend on the microstructure and can be related to the material characteristic lengths in bending and in torsion, namely pffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ‘b ¼ ‘= 2; ‘t ¼ ‘ 1 þ g ð2:7Þ Typical values of ‘b and ‘t for some classes of materials with microstructure can be found in Lakes (1986, 1995). The yield condition is assumed in the form f ðR; Y Þ ¼ R Y ¼ 0;
ð2:8Þ
where Y is the uniaxial flow stress defining isotropic hardening behavior and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 3 l l gl lT SSþ 2 R¼ 2 ‘ ð1 g2 Þ
ð2:9Þ
is the overall effective stress, S being the deviatoric part of the symmetric stress tensor r. Then, associative flow rules write e_ p ¼ K
of 3K ¼ S; or 2R
v_ pT ¼ K
of 3K ðl glT Þ ¼ ; ol 2R ‘2 ð1 g2 Þ
ð2:10Þ
as they follow from (2.8) and (2.9), where K is the plastic multiplier. If the material displays linear and isotropic hardening then Y_ ¼ KH ;
ð2:11Þ
where H is the constant hardening modulus. It may be written in terms of the ratio a ¼ Gt / G between the tangent plastic shear modulus Gt and the elastic shear modulus G ð0 < a < 1Þ, according to 3a G: ð2:12Þ 1a The consistency condition f_ ¼ 0, or equivalently R_ ¼ Y_ , gives the non-negative plastic multiplier K as ( _ hRi=H if f ðR; Y Þ ¼ 0; K¼ ð2:13Þ 0 if f ðR; Y Þ < 0; H¼
where
T _R ¼ 3 S S_ þ ðl gl Þ l_ : 2R ‘2 ð1 g2 Þ
ð2:14Þ
Finally, the elastic–plastic rate constitutive equations resulting from (2.6) and (2.10) are 1 m 3K _ ðtrrÞI þ S; r_ e_ ¼ 2G 1þm 2R 1 1 3K 2 T T T ðl_ gl_ Þ þ ðl gl Þ : ‘ v_ ¼ 1 g2 2G 2R
ð2:15Þ ð2:16Þ
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These equations hold when the stress and couple stress fields obey the yield condition (2.8). Otherwise, the plastic multiplier K is set equal to 0. In this case, isotropic elastic behavior with couple stress is recovered. Therefore, the present constitutive model provides strain gradient effects for purely elastic response also. It is worth noting that the constitutive equations of the indeterminate CS theory do not define the skew-symmetric part s of the total stress tensor t, which instead follows from the equilibrium equation (2.5)2. Moreover, within the couple stress theory v, ve, vp and l are purely deviatoric tensors. 3. Mode III crack propagation problem The problem of a semi-infinite plane crack propagating at constant velocity V along a rectilinear path is considered in the present section by performing an asymptotic analysis of the crack tip fields. A cylindrical coordinate system (r, h, x3) centred at the crack-tip and moving with it towards the h ¼ 0 direction is considered. The x3-axis is assumed to coincide with the straight crack front. The condition of steady-state propagation yields the following time derivative rule for any arbitrary scalar function /: V /_ ¼ ½/;h sin h r/;r cos h; ð3:1Þ r which asymptotically holds also for not uniform, but quasi-static, crack propagation. For antiplane problems, the non-vanishing stress and couple stress components with respect to a cylindrical coordinate system in cylindrical coordinates are rr3 ; rh3 ; sr3 ; sh3 ; lrr ; lrh ; lhr and lhh ¼ lrr . Accordingly, the strain and deformation curvature components are er3 ; eh3 ; vhr ; vrh ; vrr ; andvhh ¼ vrr . The equilibrium conditions (2.5) write ðr tr3 Þ;r þ th3;h ¼ 0; ðrlrr Þ;r þ lhr;h lhh þ 2 r sh3 ¼ 0;
ð3:2Þ
ðrlrh Þ;r þ lhh;h þ lhr 2 r sr3 ¼ 0; and the kinematical compatibility conditions (2.1)1 and (2.2) in rate form write v3 ¼ u_ 3 ; v3;r ¼ 2 e_ r3 ; v3;h ¼ 2 r e_ h3 ; v_ rr ¼ e_ 3h;r ; v_ rh ¼ e_ 3h;h þ e_ 3r =r; v_ hr ¼ _e3r;r :
ð3:3Þ ð3:4Þ
Eqs. (3.2)–(3.4) together with rate constitutive equations (2.11), (2.15) and (2.16) form a system of first order PDEs that governs the problem of the crack propagation. The asymptotic crack-tip fields are sought in a separable variable form, of the type /ðr; hÞ ¼ rp F ðhÞ, where the exponent p defines the asymptotic behavior of the generic function / as r ! 0. A qualitative analysis performed by Radi and Gei (2004) gives the relative order of singularity of the crack-tip fields, which is assumed to hold in the present context also. Therefore, the asymptotic crack-tip fields and their rates are assumed in the following form: r p uðhÞ; v3 ðr; hÞ ¼ V wðhÞ; R R V r p V r p e_ a3 ðr; hÞ ¼ d a ðhÞ; v_ ab ðr; hÞ ¼ 2 X ab ðhÞ; r R r R u3 ðr; hÞ ¼ r
r p
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V r p ha ðhÞ; R r R ‘2 r p ‘2 r p M ab ðhÞ; l_ ab ðr; hÞ ¼ GV 2 H ab ðhÞ; lab ðr; hÞ ¼ G r R r R ‘ r p ‘ r p Y ðr; hÞ ¼ G cðhÞ; Y_ ðr; hÞ ¼ GV 2 jðhÞ; ð3:5Þ r R r R ‘ r p ‘2 r p CðhÞ; sa3 ðr; hÞ ¼ G 2 ta ðhÞ; Rðr; hÞ ¼ G r R r R where the greek indices a and b stand for polar coordinates r and h. Since l, M and H are purely deviatoric tensors, the following conditions hold true: ra3 ðr; hÞ ¼ G
r p
M hh ¼ M rr ;
sa ðhÞ;
r_ a3 ðr; hÞ ¼ G
H hh ¼ H rr :
ð3:6Þ
The function C resulting from the introduction of (3.5)5,7,11 in Eq. (2.9) is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 M 2rh þ M 2hr 2 gM rh M hr ; C¼ ð1 gÞM rr þ 1 g2 2
ð3:7Þ
since near the crack-tip, namely for r < k, the couple stress field gives the most singular contribution to the effective stress and, thus, to the yield condition. The exponent p introduced in representations (3.5) defines the radial dependence of the asymptotic crack-tip fields. As r goes to zero, the symmetric stress, couple stress and skewsymmetric stress fields behave as rp ; rp1 and rp2 , respectively. Therefore, the latter field gives the most singular contribution to the total stress near to the crack tip, namely for r < ‘. Oppositely, for r > ‘, the couple stress and skew-symmetric stress fields become negligible with respect to the symmetric stress field, in agreement with the classical theories of elastic–plastic fracture mechanics. The constant R in the representations (3.5) plays the role of an undetermined amplitude factor for the leading order asymptotic fields. The solution of the homogeneous asymptotic problem can indeed be determined up to an amplitude factor, which depends on far-field loading and specimen geometry and can be estimated by matching the asymptotic solution with the far-field conditions. Nevertheless, the asymptotic analysis can capture the strength of the singularity of the crack tip fields, and their angular variations, once a normalisation condition is adopted. When the asymptotic fields (3.5)5,7,12 are introduced into equilibrium equations (3.2), by using (3.6), the following ODEs are derived at leading order t0h ¼ ð1 pÞ tr ; M 0hr ¼ ð1 þ pÞ M rr 2 th ; M 0rr ¼ M hr þ p M rh 2 tr ;
ð3:8Þ
where the apex denote the derivative with respect to the angular coordinate h. The components ha and Hab introduced in (3.5)6,8 follow from an application of the steady-state derivative rule (3.1) to representations (3.5)5,7 of the symmetric stress and couple stress fields: hr ¼ ðs0r sh Þ sin h p sr cos h;
hh ¼ ðs0h þ sr Þ sin h p sh cos h;
ð3:9Þ
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and H rr ¼ ðM 0rr M hr M rh Þ sin h þ ð1 pÞ M rr cos h; H rh ¼ ðM 0rh þ 2M rr Þ sin h þ ð1 pÞ M rh cos h;
ð3:10Þ
H hr ¼ ð1 pÞðM rr sin h þ M hr cos hÞ 2 th sinh: Note that the equilibrium condition (3.8)2 has been used in the latter equation. The definition of the function j, introduced in (3.5)10, follows from (3.5)9 and (3.1) as j ¼ c0 sin h þ ð1 pÞc cos h:
ð3:11Þ
In contrast with the mode I and mode II crack problems (Xia and Hutchinson, 1996; Huang et al., 1997; Radi, 2003), the dominant strain rate field for the antiplane crack problem turns out to be rotational. Therefore, the compatibility equations (3.4) admit non-vanishing deformation curvature rates at lowest order. In agreement with the asymptotic representations (3.5)2,3 of the velocity and strain rate fields, the kinematical compatibility conditions (3.3) can be written as d r ¼ p w=2;
w0 ¼ 2 d h :
ð3:12Þ
Moreover, a substitution of (3.5)3,4 and (3.15)1 into Eqs. (3.4) gives X rr ¼ ðp 1Þd h ;
X hr ¼ ð1 pÞp w=2;
X rh ¼ d 0h þ p w=2:
ð3:13Þ
After the definitions (3.5)9,11 of the effective stress and flow stress, the yield condition (2.8) writes C ¼ c:
ð3:14Þ
The introduction of the asymptotic fields (3.5) in the incremental constitutive relationships (2.15), (2.16) and (2.11) yields the following set of equations: 1 d ¼ h þ hkis; 2 1 1 XT ¼ ½ ðH gHT Þ þ hkiðM gMT Þ; 1 g2 2 2 j ¼ hkihC; 3
ð3:15Þ
where h ¼ H /G is the non-dimensional hardening modulus and the function k(h) is defined such that k ¼ 0 when C < c, and k¼
1 ½2ð1 gÞM rr H rr þ ðM rh gM hr ÞH rh þ ðM hr gM rh ÞH hr ; h
ð3:16Þ
when C ¼ c, in agreement with (2.13) and (2.14), being 2
h ¼ hð1 g2 Þð2C=3Þ :
ð3:17Þ
Eqs. (3.15) hold true when the effective stress and flow stress obey the yield condition (3.14). Under elastic unloading or neutral loading k 6 0 and thus Eq. (3.15)1,2 reduce to the rate constitutive relations of linear isotropic elasticity with strain gradient effects. Moreover, Eq. (3.15)3 becomes equivalent to the condition Y_ ¼ 0.
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3.1. Explicit form of the ODEs system An explicit form of the ODEs system is now derived by rearranging the equations presented in the previous subsection. In particular, by using (3.12)1 and (3.13)3, Eqs. (3.15)1,2 give hr þ 2hkisr ¼ pw;
hh þ 2hkish ¼ 2d h ;
ð3:18Þ
and H rr þ 2hkiM rr ¼ 2ð1 þ gÞX rr ; ðH rh gH hr Þ þ 2hkiðM rh gM hr Þ ¼ 2ð1 g2 ÞX hr ; ðH hr gH rh Þ þ 2hkiðM hr gM rh Þ ¼ ð1 g
2
Þð2d 0h
ð3:19Þ
þ p wÞ:
The introduction of (3.16) in Eqs. (3.19)1,2, after some algebraic manipulations, allows to obtain the following explicit expression for the functions Hrr and Hrh: " # 2 H rr 1 h þ 2ðM rh gM hr Þ 2M rr ðM rh gM hr Þ zr ¼ ; ð3:20Þ D 4ð1 gÞM rr ðM rh gM hr Þ H rh zh h þ 4ð1 gÞM 2rr where D ¼ h ½h þ 2ðM rh gM hr Þ2 þ 4ð1 gÞM 2rr ; zr ¼ 2h ð1 þ gÞX rr 2M rr ðM hr gM rh ÞH hr ;
2
ð3:21Þ
zh ¼ 2h ð1 g ÞX hr þ ½h g 2ðM rh gM hr ÞðM hr gM rh ÞH hr : Note that D is always positive for h > 0, being 1 < g < 1. Since the derivatives of the unknown angular functions do not enter the algebraic expressions (3.20), (3.10)3 and (3.13)1,2 forHrr, Hrh, Hhr, Xrr and Xhr, then the explicit expressions (3.20) and (3.10)3 for Hrr, Hrh and Hhr can be introduced in (3.16) to obtain k in explicit form. Moreover, from (3.10)1,2 the derivatives with respect to h of Mrr and Mrh are M 0rr ¼ M hr þ M rh ð1 pÞM rr cot h þ H rr = sin h; ð3:22Þ M 0rh ¼ 2M rr ð1 pÞM rh cot h þ H rh = sin h: Finally, the derivatives of the symmetric stress components sr and sh follow from (3.9) as s0r ¼ sh þ ðhr þ p sr cos hÞ= sin h; s0h ¼ sr þ ðhh þ p sh cos hÞ= sin h;
ð3:23Þ
where hr and hh are reported in (3.18). Similarly, from (3.11) and (3.15)3 one obtains 2 c0 ¼ ðp 1Þc cot h þ hkihC= sin h: 3
ð3:24Þ
The use of the steady-state condition (3.1) and representation (3.5)1,2 in (3.3)1 yields u0 ¼ ½w þ ðp þ 1Þu cos h= sin h:
ð3:25Þ
Eqs. (3.8)1,2, (3.12)2, (3.19)3, (3.22)–(3.25) form a system of 10 first order and homogeneous ODEs governing the near-tip stress and velocity fields for the antiplane crack propagation problem in couple stress elastic–plastic solids. This system may be written in the following explicit form:
E. Radi / International Journal of Plasticity 23 (2007) 1439–1456
0
y ðhÞ ¼
f p ðh; yðhÞÞ if C ¼ c and k > 0; f e ðh; yðhÞÞ
if C < c or ðC ¼ c and k 6 0Þ;
1449
ð3:26Þ
where the vector y collects the 10 unknown angular functions, i.e. ð3:27Þ
y ¼ fw; th ; sr ; sh ; M rr ; M rh ; M hr ; d h ; c; ug:
Note that tr and dr are not primary unknown functions. In fact, tr may be obtained from (3.8)3, by using (3.20) and (3.22)1, whereas dr follows from the algebraic equation (3.12)1 once all the functions collected in the vector y are known. The unknown exponent p can be determined as an eigenvalue of the nonlinear homogeneous problem (3.26), once mode III symmetry conditions, traction free conditions at crack flanks and a normalization condition are imposed on the asymptotic crack-tip fields. The flow-theory of CS plasticity adopted in the present work allows considering the occurring of elastic unloading. A generic material point near the trajectory of the cracktip experiences plastic loading ahead of the crack-tip, possibly followed by elastic unloading at h ¼ h1 and subsequent plastic reloading at h ¼ h2 . Elastic unloading may occur at the angle h1 defined by the condition R_ ¼ 0, namely kðh1 Þ ¼ 0. Plastic reloading may then occur at crack flanks at the angular coordinate h2 where the material point reaches a stress state obeying the yield condition (3.14) discarded at unloading. Alternatively, the zone surrounding the crack tip may result as fully plasticized. 4. Mode III boundary conditions Under Mode III crack propagation the integration may be restricted to the range 0 6 h 6 p by symmetry condition. Since a regular behavior of the angular functions at h ¼ 0 may be expected, then relations (3.9) and (3.10) imply hð0Þ ¼ p sð0Þ;
Hð0Þ ¼ ð1 pÞMð0Þ:
ð4:1Þ
By using condition (4.1)2, the function k defined in (3.16) attains at h ¼ 0 the value kð0Þ ¼
3 ð1 pÞ; 2h
ð4:2Þ
which is positive for p < 1. The skew-symmetry of the antiplane crack problem requires ahead of the crack-tip at h ¼ 0 uð0Þ ¼ 0;
wð0Þ ¼ 0;
M hr ð0Þ ¼ 0:
ð4:3Þ
The introduction of relations (4.1)–(4.3) into the constitutive equations (3.18) and (3.19) evaluated at h ¼ 0, after some algebraic manipulations, yields sr ð0Þ ¼ 0;
d h ð0Þ ¼ ½3 ð3 þ hÞp sh ð0Þ=ð2hÞ; 3 M rr ð0Þ ¼ ð1 þ gÞ p sh ð0Þ; M rh ð0Þ ¼ 0; 3þh
ð4:4Þ d 0h ð0Þ ¼ 0:
ð4:5Þ
Vanishing of reduced tractions (2.4)1 along the radial direction at h ¼ 0 requires 2 r tr3 lrr;h ¼ 0; ð4:6Þ which implies M 0rr ð0Þ ¼ 2 tr ð0Þ:
ð4:7Þ
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Hence, from (4.7) and the equilibrium condition (3.8)3 it turns out that tr ð0Þ ¼ 0;
M 0rr ð0Þ ¼ 0:
ð4:8Þ
A numerical procedure based on the Runge–Kutta method is used to integrate the system (3.26) (subroutine DIVPRK of the IMSL library). This procedure requires knowledge of the initial values y(0). In order to avoid the trivial solution, the following normalization condition for the stress field sh ð0Þ ¼ 1
ð4:9Þ
is adopted, since all the assigned boundary conditions (4.3), (4.4), (4.5) and (4.8) are homogeneous. Plastic loading is assumed to occur in the zone ahead of the crack-tip. Therein the yield condition cð0Þ ¼ Cð0Þ must hold true, where C(0) follows from conditions (4.3)3, (4.5)2 and (3.7) as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cð0Þ ¼ 3M 2rr ð0Þ=ð1 þ gÞ: ð4:10Þ Vanishing of reduced tractions (2.4) on the crack surfaces implies the conditions 2 th3 þ lhh;r ¼ 0;
lhr ¼ 0;
ð4:11Þ
which yield the following two boundary conditions at h ¼ p for the system (3.26) 2 th ðpÞ þ ð1 pÞ M rr ðpÞ ¼ 0;
M hr ðpÞ ¼ 0:
ð4:12Þ
By using (3.12)2, (4.4) and (4.5) in the derivatives with respect to h of (3.18), (3.9) and (3.25) evaluated at h ¼ 0, the following results can be found h 0 sr ð0Þ ¼ p þ ð4:13Þ sh ð0Þ; s0h ð0Þ ¼ 0; u0 ð0Þ ¼ 2d h ð0Þ=p: 3þh Similarly, differentiation of (3.10)2 and (3.19)2 and use of (3.8)2 and (4.5)2 yield M 0rh ð0Þ ¼
2ð1 pÞð1 þ gÞ½ð2 pÞð1 þ pÞ þ 2kð0Þðp þ gÞ d h ð0Þ 2gth ð0Þ: ½1 p þ 2kð0Þ½2 p þ 2kð0Þ
ð4:14Þ
By using (4.8)1 the equilibrium equation (3.8)1 evaluated at h ¼ 0 gives t0h ð0Þ ¼ 0:
ð4:15Þ
By taking the derivative of C(h) and k(h) from (3.7) and (3.16) and using (4.3)3, (4.5)2 and (4.8)2 it follows that C0 ð0Þ ¼ k0 ð0Þ ¼ 0. Then, the value of c0 ð0Þ can be obtained from the derivative of (3.24) evaluated at h ¼ 0 as c0 ð0Þ ¼ 0:
ð4:16Þ
The values of all the unknown functions at h ¼ 0 are defined in (4.3), (4.4), (4.5)1,2, (4.8)1, (4.9) and (4.10) as functions of p and th(0). Then, the values of p and th(0) can be calculated by an iterative procedure based on the achievement of the two boundary conditions (4.12) at h ¼ p. This iteration is performed by using the modified Powell hybrid method (subroutine DNEQNF of the IMSL library), until the conditions (4.12) on the vanishing of the reduced tractions are verified within a prescribed accuracy. The procedure employed for the integration of the governing ODE system (3.26) displays a numerical difficulty at h ¼ 0, due to the term sin h which multiplies the highest
E. Radi / International Journal of Plasticity 23 (2007) 1439–1456
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order derivative. This inconvenience can be bypassed by performing a Taylor series expansions of the unknown angular functions collected in the vector y, starting at h ¼ 0, namely: yðeÞ ¼ yð0Þ þ ey0 ð0Þ þ oðeÞ;
ð4:17Þ
where e 1. Therefore, the unknown functions can be evaluated at h ¼ e, since the values assumed by the derivatives of the unknown functions at h ¼ 0 are known from relations (3.8)1,2, (4.5)3, (4.7), (4.8)2, and (4.13)–(4.16). Once that y(e) has been determined by (4.17) within an error lower than e, the numerical integration of (3.26) can be performed by starting at h ¼ e, rather than at h ¼ 0. 5. Results Fig. 1 shows the variation of the exponent p with the ratio g for different values of the strain hardening coefficient a. It can be observed that the ratio g has a limited influence on p, since a small variation of p occurs as g ranges between 1 and 1. As it can be expected, the larger is the coefficient a, the lower is the exponent p and thus the stronger is the singularity of the couple stress and skew-symmetric stress fields, which asymptotically behave as rp 1 and rp 2, respectively. In particular, the curves plotted in Fig. 1 show that the exponent p tends to 1 as the strain hardening coefficient a becomes vanishing small, namely for perfectly plastic behavior, whereas p tends to 0.5 as a approaches the limit value of 1, in agreement with the CS elastic solution provided by Zhang et al. (1998). Note that the strain gradient effect also occurs for purely elastic response, namely for a ¼ 1. Since 0:5 < p < 1 for 0 < a < 1, the asymptotic radial variation of the out of plane displacement considered in (3.5)1 implies that the crack-tip profile is sharp, whereas crack tip blunting is expected for the conventional theories of LEFM and EPFM. Moreover, the symmetric stress field turns out to be non-singular, as already observed for stationary crack tip fields in strain gradient elasticity (Zhang et al., 1998) and plasticity (Huang et al., 1997, 1999), whereas the skew-symmetric stress field displays a strong singularity. It must be remarked that the latter field does not contribute to the effective stress R and to the strain-energy density, which are instead dominated by the weakly singular
1.0
α = 0.001 0.01
0.9
0.05 p
0.8
0.1
0.7
0.3
0.6
0.5 0.9
0.5 -1.0
-0.5
0.0
η
0.5
1.0
Fig. 1. Variation of the exponent p with the ratio g for different values of the strain hardening coefficient a.
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couple-stress field, so that the flux of energy toward the crack-tip remains finite. Moreover, the tractions level ahead of the crack tip increases with respect to the classical J2-flow theory due to the contribution of the skew-symmetric stress components, also for very small strain hardening. In fact, the investigations performed for conventional elastic–plastic response with linear hardening by Amazigo and Hutchinson (1977) and Ponte Castan˜eda (1987) predict a weak stress singularity for small strain-hardening, which disappears in the limit case of elastic–perfectly plastic behavior of the material, namely when a is vanishing small. Elastic unloading has been detected only for a large, namely a > 0:1, whereas the crack tip zone is fully plastic for a small, as due to the plastic distribution caused by the strain gradient effect, also observed by Wei and Hutchinson (1997) in their f.e. investigations. In particular, Figs. 2a and b display the variation of the elastic unloading angle h1 and plastic reloading angle h2 with the ratio g for the hardening coefficient a ¼ 0:3 and a ¼ 0:5, respectively. It may be observed that as the ratio g increases then the plastic zone tends to concentrate ahead of the crack tip, resulting thus in a more stable crack propagation, according to previous findings (Ponte Castan˜eda, 1987; Bigoni and Radi, 1996). Figs. 3 and 4 display the angular distributions of the asymptotic crack-tip fields for g ¼ 0:5 in the realistic case of small strain hardening, namely for a ¼ 0:01. For both values of g the zone surrounding the crack-tip at distance lower than the characteristic material length k is fully plastic. In particular, the angular variations of symmetric stresses (a), out-of-plane velocity and displacement (b), couple stresses (c) and skew-symmetric stresses (d) are plotted therein. All functions are normalized by condition (4.9). It can be observed that the ratio g produces remarkable changes in the angular distribution of the crack tip fields. The curves plotted in Figs. 3a and 4a show that the out-of-plane displacement around the crack tip does not monotonically increase with the angular coordinate h but it oscillates and reaches its maximum value at about 140° instead of at the crack flanks. Note that in their investigations on mode III crack problem within their framework of gradient elas180
180
a θ2
150
elastic unloading sector
150
elastic unloading sector
120
b
120
θ1 90
θ1
90
plastic loading sector
60
plastic loading sector
60
30
30
α = 0.5
α = 0.3 0
0 -1.0
-0.5
0.0
η
0.5
1.0
-1.0
-0.5
0.0
0.5
1.0
η
Fig. 2. Variation of the elastic unloading angle h1 and plastic reloading angle h2 with the ratio g for a ¼ 0:3 and a ¼ 0:5.
E. Radi / International Journal of Plasticity 23 (2007) 1439–1456 10
2
a
b
w
sθ
1
5
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sr
u 0
0
-5
-1
α = 0.01 η = 0.5
-2
-10 0
30
60
90
120
150
180
0
30
60
0.2
90
120
150
180
θ
θ 0.4
c
d
tr 0.1
0.2
0.0
0.0
M θr Mr θ
tθ
Mrr
-0.1
Mθθ
-0.2
-0.2
-0.4
0
30
60
90
θ
120
150
180
0
30
60
90
120
150
180
θ
Fig. 3. Angular variation of the crack-tip fields for a = 0.01 and g = 0.5.
ticity Unger and Aifantis (2000) found an oscillation of the displacement along the radial direction. Moreover, in the present analysis the displacements ahead and behind the cracktip turn out to be opposite in sign, unlike the classical mode III crack-tip fields in nonpolar materials (Amazigo and Hutchinson, 1977; Ponte Castan˜eda, 1987; Bigoni and Radi, 1996). Similar results have been also obtained for mode III crack in CS elastic and CS elastic–plastic materials (Zhang et al., 1998; Radi and Gei, 2004). This unusual aspect seems to be due to the presence of microstructures (compositional grains) introduced through the material parameters ‘ and g. During crack growth, in facts, the relative rotation of the particles currently at the crack tip yields opposite displacements ahead and behind the crack tip, thus producing a scissor effect. The angular variations of the skew-symmetric stress components are plotted in Figs. 3c and 4c. In agreement with the inversion of the out-of-plane displacement ahead of the crack-tip, the shear stress th is also negative therein and, thus, opposite to its counterpart
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2
a
b
w 5
sθ
1
sr
u 0
0
-5
-1
α = 0.01 η = −0.5
-10
-2 0
0.04
30
60
90
θ
120
150
180
0
0.10
c
60
90
θ
120
150
180
d Mθ r
tr
0.02
30
0.05
0.00
Mr θ
0.00
tθ
Mrr
-0.02
Mθθ
-0.05
-0.04
-0.10 0
30
60
90
θ
120
150
180
0
30
60
90
120
150
180
θ
Fig. 4. Angular variation of the crack-tip fields for a ¼ 0:01 and g ¼ 0:5.
in the classical mode III solution. The negative sign of the tractions very near to the crack tip indicates that the rotation gradients may shield the crack tip from fracture, in agreement with the results provided by Tong et al. (2005) and Wei and Xu (2005) for mode I and mode II fracture. Shielding can reasonably be expected since strain gradient tends to stiffen the material. For positive values of the ratio g the radial shear stress component tr also displays negative values in a small sector ahead of the crack-tip. The switch in the displacement and shear directions agrees with the results obtained by Zhang et al. (1998) and Radi and Gei (2004). These investigations confirm that the microstructure remarkably affects the solutions of fracture mechanics problems. 6. Conclusions The obtained results show that the use of the CS theory of plasticity with two characteristic lengths for the analysis of the stress field near the tip of a propagating mode III
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crack gives accurate predictions on the increase of the tractions level ahead of the crack-tip occurring at very small distances from it, comparable with the size of the compositional grains. Moreover, it sheds some light on the shielding mechanisms occurring during crack propagation due to the presence of a complex microstructure. Strain gradient effects are found to be relevant at very small distances ahead of the crack-tip namely for r < ‘. They produce a switch in the sign of tractions and displacement ahead of the crack-tip, with respect to classical LEFM and EPFM. Moreover, an increase in the singularity of tractions ahead of the crack-tip is observed, even for low strain hardening, essentially due to the skew-symmetric stress field, which does not contribute to the effective stress and strain energy density and thus does not violate the boundness of the energy flux towards the crack tip. The contribution to the effective stress is instead dominated by the slightly singular couple-stress field. The ratio g also displays a strong influence on the angular distribution of the asymptotic crack tip fields. Since the obtained asymptotic solution holds at a distance to the crack tip much smaller than the range of application of classical LEFM, then the present approach provides a mean to link scales in fracture mechanics, from atomistic to macroscopic fracture, allowing understanding of the detailed mechanisms by which fracture may grow and propagate in materials with complex microstructure, up to the micron scale. With this end in view, the results here presented can be usefully referred in further numerical and analytical investigations of multiscale fracture problems in ductile materials. Acknowledgments Financial support from the Italian Ministry of Education, University and Research (MIUR) in the framework of the Project PRIN-2004 ‘‘Problemi e modelli microstrutturali: applicazioni in ingegneria civile e strutturale” is gratefully acknowledged. References Amazigo, J., Hutchinson, J.W., 1977. Crack-tip fields in steady crack-growth with linear strain hardening. Journal of the Mechanics and Physics of Solids 25, 81–97. Bigoni, D., Radi, E., 1996. Asymptotic solution for Mode III crack growth in J2-elastoplasticity with mixed isotropic-kinematic strain hardening. International Journal of Fracture 77, 77–93. Chen, J.Y., Wei, Y., Huang, Y., Hutchinson, J.W., Hwang, K.C., 1999. The crack-tip fields in strain gradient plasticity: the asymptotic and numerical analyses. Engineering Fracture Mechanics 64, 625–648. Chen, S.H., Wang, T.C., 2002. A new deformation theory with strain gradient effects. International Journal of Plasticity 18, 971–995. Dillard, T., Forest, S., Ienny, P., 2006. Micromorphic continuum modelling of the deformation and fracture behaviour of nickel foams. European Journal of Mechanics – A/Solids 25 (3), 526–549. Drugan, W.J., Rice, J.R., Sham, T.L., 1982. Asymptotic analysis of growing plane strain tensile cracks in elasticideally plastic solids. Journal of the Mechanics and Physics of Solids 30, 447–473. Elssner, G., Korn, D., Ruehle, M., 1994. The influence of interface impurities on fracture energy of UHV diffusion bonded metal–ceramic bicrystals. Scripta Metallurgica et Materialia 31, 1037–1042. Fannjiang, A.C., Chan, Y.S., Paulino, G.H., 2002. Strain gradient elasticity for antiplane shear cracks: a hypersingular integrodifferential equation approach. SIAM Journal of Applied Mathematics 62, 1066–1091. Fleck, N.A., Hutchinson, J.W., 1993. A phenomenological theory for strain gradient effects in plasticity. Journal of the Mechanics and Physics of Solids 41, 1825–1857. Fleck, N.A., Hutchinson, J.W., 1997. Strain gradient plasticity. In: Hutchinson, J.W., Wu, T.Y. (Eds.), Advances in Applied Mechanics, vol. 33. Academic Press, New York, pp. 295–361.
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Fleck, N.A., Hutchinson, J.W., 2001. A reformulation of strain gradient plasticity. Journal of the Mechanics and Physics of Solids 49, 2245–2271. Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W., 1994. Strain gradient plasticity: theory and experiment. Acta Metallurgica et Materialia 42, 475–487. Gao, H., Huang, Y., Nix, W.D., Hutchinson, J.W., 1999. Mechanism-based strain gradient plasticity – I. Theory. Journal of the Mechanics and Physics of Solids 47, 1239–1263. Huang, Y., Chen, J.Y., Guo, T.F., Zhang, L., Huang, K.C., 1999. Analytic and numerical studies on Mode I and Mode II fracture in elastic–plastic materials with strain gradient effects. International Journal of Fracture 100, 1–27. Huang, Y., Qu, S., Hwang, K.C., Li, M., Gao, H., 2004. A conventional theory of mechanism-based strain gradient plasticity. International Journal of Plasticity 20 (4–5), 753–782. Huang, Y., Zhang, L., Guo, T.F., Hwang, K.C., 1997. Mixed Mode near-tip fields for crack in materials with strain-gradient effects. Journal of the Mechanics and Physics of Solids 45, 439–465. Hutchinson, J.W., 2000. Plasticity at the micron scale. International Journal of Solids and Structures 37, 225–238. Koiter, W.T., 1964. Couple-stresses in the theory of elasticity, I and II. Proc. Ned. Akad. Wet. (B) 67, 17–44. Lakes, R., 1986. Experimental microelasticity of two porous solids. International Journal of Solids and Structures 22, 55–63. Lakes, R., 1995. Experimental methods for study of Cosserat elastic solids and other generalized elastic continua. In: Mu¨hlhaus, H. (Ed.), Continuum Models for Materials with Micro-structure. Wiley, New York, pp. 1–22 (Chapter 1). Ottosen, N.S., Ristinmaa, M., Ljung, C., 2000. Rayleigh waves obtained by the indeterminate couple-stress theory. European Journal of Mechanics – A/Solids 19, 929–947. Paulino, G.H., Fannjiang, A.C., Chan, Y.S., 2003. Gradient elasticity theory for mode III fracture in functionally graded materials – Part I: crack perpendicular to the material gradation. Journal of Applied Mechanics 70 (4), 531–542. Ponte Castan˜eda, P., 1987. Asymptotic fields in steady crack growth with linear strain-hardening. Journal of the Mechanics and Physics of Solids 35, 227–268. Qiu, X., Huang, Y., Wei, Y., Gao, H., Hwang, K.C., 2003. The flow theory of mechanism-based strain gradient plasticity. Mechanics of Materials 35, 245–258. Radi, E., 2003. Strain-gradient effects on steady-state crack growth in linear hardening materials. Journal of the Mechanics and Physics of Solids 51, 543–573. Radi, E., Gei, M., 2004. Mode III crack growth in linear hardening materials with strain-gradient effects. International Journal of Fracture 130, 765–785. Ristinmaa, M., Vecchi, M., 1996. Use of couple-stress theory in elasto-plasticity. Computational Methods in Applied Mechanical Engineering 136, 205–224. Shi, M.X., Huang, Y., Gao, H., Hwang, K.C., 2000. Non-existence of separable crack-tip field in mechanismbased strain gradient plasticity. International Journal of Solids and Structures 37, 5995–6010. Tong, P., Lam, D.C.C., Yang, F., 2005. Mode I solution for micron-sized crack. Engineering Fracture Mechanics 72 (12), 1779–1804. Unger, D.J., Aifantis, E.C., 2000. Strain gradient elasticity theory for antiplane shear cracks. Part I: Oscillatory displacements. Theoretical and Applied Fracture Mechanics 34, 243–252. Vardoulakis, I., Exadaktylos, G., Aifantis, E., 1996. Gradient elasticity with surface energy: Mode-III crack problem. International Journal of Solids and Structures 33, 4531–4559. Wei, Y., 2006. A new finite element method for strain gradient theories and applications to fracture analyses. European Journal of Mechanics – A/Solids 25, 897–913. Wei, Y., Hutchinson, J.W., 1997. Steady-state crack growth and work of fracture for solids characterized by strain gradient plasticity. Journal of the Mechanics and Physics of Solids 45, 1253–1273. Wei, Y., Xu, G., 2005. A multiscale model for the ductile fracture of crystalline materials. International Journal of Plasticity 21 (11), 2123–2149. Xia, Z.C., Hutchinson, J.W., 1996. Crack-tip fields in strain gradient plasticity. Journal of the Mechanics and Physics of Solids 44, 1621–1648. Zhang, L., Huang, Y., Chen, J.Y., Hwang, K.C., 1998. The Mode III full-field solution in elastic materials with strain gradient effects. International Journal of Fracture 92, 325–348.
Effects of characteristic material lengths on mode III crack propagation in couple stress elastic–plastic materials Enrico Radi * Dipartimento di Scienze e Metodi dell’Ingegneria, Universita` di Modena e Reggio Emilia, Via Amendola, 2-I-42100 Reggio Emilia, Italy Received 29 October 2006; received in final revised form 17 December 2006 Available online 20 January 2007
Abstract The asymptotic fields near the tip of a crack steadily propagating in a ductile material under Mode III loading conditions are investigated by adopting an incremental version of the indeterminate theory of couple stress plasticity displaying linear and isotropic strain hardening. The adopted constitutive model is able to account for the microstructure of the material by incorporating two distinct material characteristic lengths. It can also capture the strong size effects arising at small scales, which results from the underlying microstructures. According to the asymptotic crack tip fields for a stationary crack provided by the indeterminate theory of couple stress elasticity, the effects of microstructure mainly consist in a switch in the sign of tractions and displacement and in a substantial increase in the singularity of tractions ahead of the crack-tip, with respect to the classical solution of LEFM and EPFM. The increase in the stress singularity also occurs for small values of the strain hardening coefficient and is essentially due to the skew-symmetric stress field, since the symmetric stress field turns out to be non-singular. Moreover, the obtained results show that the ratio g introduced by Koiter has a limited effect on the strength of the stress singularity. However, it displays a strong influence on the angular distribution of the asymptotic crack tip fields. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Crack propagation; Couple stress plasticity; Strain gradient; Elastic–plastic material; Asymptotic analysis
*
Tel.: +39 0522 522221; fax: +39 0522 522609. E-mail address: [email protected]
0749-6419/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2007.01.002
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1. Introduction Due to the lack of a length scale, classical plasticity theories are not able to characterize the constitutive behavior of ductile materials at the micron scale. This lack is expected to be particularly significant for the analysis of the stress and deformation fields very near the tip of a crack propagating in a ductile metal, which are expected to be strongly influenced by the presence of the microstructure. In particular, experimental observations performed by Elssner et al. (1994) found that during crack propagation the interface between a ductile crystal of niobium and a sapphire single crystal remained sharp and not blunted up to the atomic scale. They also found that the stress level required to produce atomic decohesion of the lattice turns out to be about 10 times the tensile yield stress, whereas fracture mechanics analyses based on classical plasticity theories provide a maximum stress level near a crack tip not larger than 4–5 times the tensile yield stress (Drugan et al., 1982). Therefore, for the investigations of the crack tip fields at the micron scale it becomes necessary to adopt enhanced constitutive models, which account for the nonlinear behavior of the material as well as for the presence of microstructure and dislocations. A way of doing that consists in the inclusion of one or more characteristic lengths within the framework of elastic–plastic rate constitutive relations. These lengths are typically of the same order of the compositional grain size, namely few microns, for polycrystalline metals (Fleck et al., 1994). Several strain gradient (SG) plasticity models (Hutchinson, 2000; Fleck and Hutchinson, 1993, 1997, 2001; Chen and Wang, 2002; Ristinmaa and Vecchi, 1996; Ottosen et al., 2000) based on phenomenological theories have been proposed for this purpose. These refined constitutive theories can describe the mechanical behavior of many ductile materials with microstructure, such as metals at the micron scale, liquid crystal, rocks, granular materials, cellular solids, fibrous composites and laminates. Inside these materials, moments may be transmitted through grains, fibres, or in the cell ribs or walls. In a first attempt, Fleck and Hutchinson (1993) presented a couple stress (CS) flow-theory of plasticity which only involves strain rotation gradients. This model includes an intrinsic characteristic material length ‘, which specifies the range where strain gradients are dominant. A further elastic length scale ‘e was artificially introduced in order to partition the curvature rate tensor into elastic and plastic contributions. A generalization of this model within the framework of micropolar solids was presented by Chen and Wang (2002) by treating the micro-rotation field as an independent kinematical quantity, with no direct dependence upon the displacement field. A flow-theory version of CS plasticity based on the indeterminate theory of CS elasticity developed by Koiter (1964) was also developed by Ristinmaa and Vecchi (1996) and Ottosen et al. (2000). Their model involves two material additional material parameters, namely ‘ and g, which can be related to the material characteristic lengths in bending and in torsion and whose effects on the ductile crack propagation are almost unexplored. Several analyses have been carried out to investigate the effects of the aforementioned CS constitutive models in fracture mechanics. Xia and Hutchinson (1996) and Huang et al. (1997, 1999) examined mode I and mode II asymptotic and full-field solutions for a stationary crack within the CS deformation theory of plasticity presented by Fleck and Hutchinson (1993). They found that the symmetric stress field dominates the couple stress field near the crack tip and then the traction level ahead of the crack-tip does not increase substantially. Therefore, mode I crack-tip fields seem to be more significantly influenced
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by stretch gradients rather than by rotation gradients. The latter are expected to have a major effect on the tractions ahead of a mode III crack tip. It must be observed that the CS models failed to estimate the results of tests where also stretch gradients come into play, as for instance micro-indentation and void growth, since the strain gradients considered by these models derive uniquely from rotation gradients. Enhanced SG theories, which incorporate several material length scales, have been proposed to supply this lack (Fleck and Hutchinson, 1997, 2001; Gao et al., 1999). They may provide a link between the mesoscale SG plasticity level and the microscale dislocation mechanics. However, investigations of the stationary crack tip fields performed by adopting these refined constitutive models give physically unacceptable results. In particular, Chen et al. (1999) and recently Wei (2006) found that the normal stresses predicted by the SG theory of plasticity under mode I loading conditions turn out to be compressive within a small region near the crack tip, which is however much smaller beyond the concern of the continuum theory. Shi et al. (2000) showed that if a particular theory of SG plasticity is adopted, namely the mechanism-based strain gradients (MSG) theory, then the stress fields ahead of the crack tip is tensile. Unfortunately, the crack-tip fields obtained for the MSG theory do not admit a separable-variable form at very small distance from the crack-tip, due to the higher-order boundary conditions, so that only numerical or analytical full-field investigations can be performed. However, Huang et al. (2004) clearly showed that the higherorder boundary conditions are not so important and they consequently developed a simpler conventional theory of mechanism-based strain gradient (CMSG) plasticity. Recent analyses carried out to investigate the effects of microstructure on crack propagation in ductile materials showed that the incorporation of couple stress and rotation gradients in the constitutive description, by using the flow theories of CS or SG plasticity, considerably improves the estimation of the tractions level ahead of the crack-tip and may explain the occurring of cleavage or atomic decohesion during the crack growth process, experimentally observed in ductile metals also (Elssner et al., 1994). In particular, the analyses of a steadily propagating crack-tip under mode I loading conditions (Wei and Hutchinson, 1997; Radi, 2003; Qiu et al., 2003) found that the couple stress field is dominant near to the crack-tip and produces a remarkable increase in the stress singularity, even for small strain hardening, whereas former investigations of crack propagation performed by using classical elastic–plastic constitutive models displaying linear strain hardening (Amazigo and Hutchinson, 1977; Ponte Castan˜eda, 1987), predicted an extremely weak stress singularity for the small values of the strain hardening coefficient generally adopted for ductile metals. Moreover, the investigations performed by Wei and Hutchinson (1997) and Radi (2003) under mode I loading conditions showed that the contribution of the elastic strain gradients strongly affects the asymptotic crack-tip fields through the artificial elastic length scale ‘e. Recently, Wei and Xu (2005) extended the investigation of the fracture toughness and shielding effects produced by plastic deformations and discrete dislocations during mode I and mode II crack growth by using a multiscale approach. The problem of mode III crack propagation within the classical J2-flow theory was first analysed by Amazigo and Hutchinson (1977) and Ponte Castan˜eda (1987) for materials which display linear and isotropic hardening and extended by Bigoni and Radi (1996) to mixed isotropic/kinematic hardening. These investigations found that the strength of the stress singularity turns out to be extremely weak for the typically small values of the strain hardening for ductile metals. The effects of elastic strain gradients on the stress concentration at the tip of a mode III stationary crack were considered by Vardoulakis
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et al. (1996); Zhang et al. (1998); Unger and Aifantis (2000); Fannjiang et al. (2002) and Paulino et al. (2003). While in (Vardoulakis et al. (1996)) and (Fannjiang et al. (2002); Paulino et al. (2003)) strain gradients are introduced through the second gradient of displacement, in (Zhang et al., 1998) rotation gradients are considered, then adopting a couple stress theory which is the specialisation of the model adopted here when plasticity is absent and g ¼ 0. The results of Zhang et al. (1998) for the indeterminate theory of CS elasticity indicate that the effects of rotation gradients increase the stress singularity at the crack-tip from r1/2, as obtained for classical linear elasticity, to r3/2, with no need to take stretch gradients into consideration, essentially due to the skew-symmetric stress components. Although this singularity is much stronger than the conventional square-root singularity, it does not violate the boundness of strain energy surrounding the crack tip and leads to a finite energy release rate. The effects of strain rotation gradients on mode III crack propagation in ductile metals have been recently investigated by Radi and Gei (2004) by using the flow theory version of CS plasticity developed by Fleck and Hutchinson (1993). By performing an asymptotic analysis of the crack-tip fields, these authors showed that strain gradients become significant for r < ‘, where r is the distance from a crack-tip, and negligible for r ‘, with a gradual transition in the intermediate region, where r ‘. A qualitative analysis proved that the leading order term of the velocity field turns out to be rotational, so that the leading order term of the strain rotation gradients does not vanish. The obtained results confirmed that the contribution of the elastic strain gradients, related to the elastic length scale ‘e, strongly affects the asymptotic crack-tip fields, according to the results obtained for mode I crack propagation. Moreover, the skew-symmetric stress field asymptotically dominates the symmetric stress and couple stress fields, displaying a much stronger singularity than that predicted by the classical J2-flow theory. In particular, a strong increase in the singularity is observed if the elastic strain gradients are kept sufficiently small, namely for ‘e ‘. In the present work, the effects of strain rotation gradients on ductile steady-state crack propagation under mode III loading conditions are investigated by performing an asymptotic analysis of the crack-tip fields derived from the flow theory of the indeterminate CS elastoplasticity with two characteristic material lengths developed by Ristinmaa and Vecchi (1996) and Ottosen et al. (2000). Strain gradient effects have been also considered for purely elastic response. According to the results obtained by Radi (2003) and Radi and Gei (2004) for a single characteristic material length, the skew-symmetric stress field dominates the asymptotic field, producing thus a remarkable increase of traction level at the crack tip, almost independently of the value of the strain hardening coefficient. The influence of the characteristic lengths on the crack tip fields is explored by numerical investigations. As already observed by Dillard et al. (2006) the inclusion of two or more characteristic lengths provides more realistic predictions on the tractions level ahead of the propagating crack-tip then the classical solution obtained for the J2-flow theory and gives more accurate results then the CS theory of plasticity with a single characteristic length, allowing the detailed mechanisms by which fracture may grow and propagate in ductile metals to be understood in more depth, up to the micron scale. The following notation is used throughout the paper. The components of a generic vector v or second order tensor A with respect to an orthogonal unit vectors basis {ei} are defined as vi ¼ v ei and Aij ¼ ei Aej , respectively, where a dot denotes the natural inner product of two vectors a and b or two second-order tensors A and B. The prefix tr indi-
E. Radi / International Journal of Plasticity 23 (2007) 1439–1456
1443
cates the trace of a second order tensor and I is the second order identity tensor. The symbols $, div and curl indicate the gradient, the divergence and the curl of a vector or tensor field, respectively, defined componentwise as ðrvÞij ¼ vi;j ; ðdiv AÞi ¼ Aij;j and ðcurl AÞij ¼ eikl Ajl;k , where eikl is the third-order alternator and ( )j denotes the partial derivative with respect to the jth spatial coordinate. Moreover, let denote the third-order permutation tensor, defined as (AÞi ¼ eikl Akl . A superimposed dot denotes a material time derivative. Finally, the MacAuley brackets hi define the function hxi ¼ 12ðx þ jxjÞ, for every real value x. 2. Governing equations The flow-theory version of the CS elatoplasticity developed by Ristinmaa and Vecchi (1996) and Ottosen et al. (2000) is adopted in the present study. This constitutive model derives from the indeterminate theory of CS elasticity developed by Koiter (1964) and assumes the following kinematical compatibility conditions between the displacement vector u, rotation vector h, strain tensor e and deformation curvature tensor v 1 e ¼ ðru þ ruT Þ; 2
h¼
1 curl u; 2
v ¼ rh:
ð2:1Þ
Therefore, rotations are derived from displacements and the tensor field v turns out to be irrotational, being v ¼ curl e
ð2:2Þ
According to the CS theory (Koiter, 1964), the non-symmetrical Cauchy stress tensor t can be decomposed into a symmetric part r and a skew-symmetric part s, namely t ¼ r þ s. In addition, the couple stress tensor l is introduced as the work-conjugated quantity of vT. In absence of body forces and body couples and for a smooth boundary, the principle of virtual work for a body B writes (Radi and Gei, 2004) Z Z ðr de þ l dvT Þ dV ¼ ðp du þ q dhÞ dS; ð2:3Þ oB
B
where 1 p ¼ tT n þ rlnn n; q ¼ lT n lnn n; ð2:4Þ 2 are the reduced surface traction vector and couple stress traction vector, respectively, n denotes the outward unit normal and lnn ¼ n l n. The principle of virtual work (2.3) provides the conditions of quasi-static equilibrium of forces and moments within the body, namely div tT ¼ 0;
div lT þ s ¼ 0:
ð2:5Þ
Within the context of small deformations incremental theory, the total strain rate e_ is the sum of elastic e_ e and plastic e_ p parts. Similarly, the total deformation curvature rate v_ is the sum of elastic v_ e and plastic v_ p contributions. Both elastic parts are related to stress and couple stress rates through the following rate constitutive relations e_ e ¼
1 m _ ½r_ ðtr rÞI; 2G 1þm
v_ eT ¼
l_ gl_ T ; 2G‘2 ð1 g2 Þ
ð2:6Þ
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where G is the elastic shear modulus, m the Poisson ratio, ‘ and g the SG parameters introduced by Koiter (1964), with 1 < g < 1. Both material parameters ‘ and g depend on the microstructure and can be related to the material characteristic lengths in bending and in torsion, namely pffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ‘b ¼ ‘= 2; ‘t ¼ ‘ 1 þ g ð2:7Þ Typical values of ‘b and ‘t for some classes of materials with microstructure can be found in Lakes (1986, 1995). The yield condition is assumed in the form f ðR; Y Þ ¼ R Y ¼ 0;
ð2:8Þ
where Y is the uniaxial flow stress defining isotropic hardening behavior and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 3 l l gl lT SSþ 2 R¼ 2 ‘ ð1 g2 Þ
ð2:9Þ
is the overall effective stress, S being the deviatoric part of the symmetric stress tensor r. Then, associative flow rules write e_ p ¼ K
of 3K ¼ S; or 2R
v_ pT ¼ K
of 3K ðl glT Þ ¼ ; ol 2R ‘2 ð1 g2 Þ
ð2:10Þ
as they follow from (2.8) and (2.9), where K is the plastic multiplier. If the material displays linear and isotropic hardening then Y_ ¼ KH ;
ð2:11Þ
where H is the constant hardening modulus. It may be written in terms of the ratio a ¼ Gt / G between the tangent plastic shear modulus Gt and the elastic shear modulus G ð0 < a < 1Þ, according to 3a G: ð2:12Þ 1a The consistency condition f_ ¼ 0, or equivalently R_ ¼ Y_ , gives the non-negative plastic multiplier K as ( _ hRi=H if f ðR; Y Þ ¼ 0; K¼ ð2:13Þ 0 if f ðR; Y Þ < 0; H¼
where
T _R ¼ 3 S S_ þ ðl gl Þ l_ : 2R ‘2 ð1 g2 Þ
ð2:14Þ
Finally, the elastic–plastic rate constitutive equations resulting from (2.6) and (2.10) are 1 m 3K _ ðtrrÞI þ S; r_ e_ ¼ 2G 1þm 2R 1 1 3K 2 T T T ðl_ gl_ Þ þ ðl gl Þ : ‘ v_ ¼ 1 g2 2G 2R
ð2:15Þ ð2:16Þ
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These equations hold when the stress and couple stress fields obey the yield condition (2.8). Otherwise, the plastic multiplier K is set equal to 0. In this case, isotropic elastic behavior with couple stress is recovered. Therefore, the present constitutive model provides strain gradient effects for purely elastic response also. It is worth noting that the constitutive equations of the indeterminate CS theory do not define the skew-symmetric part s of the total stress tensor t, which instead follows from the equilibrium equation (2.5)2. Moreover, within the couple stress theory v, ve, vp and l are purely deviatoric tensors. 3. Mode III crack propagation problem The problem of a semi-infinite plane crack propagating at constant velocity V along a rectilinear path is considered in the present section by performing an asymptotic analysis of the crack tip fields. A cylindrical coordinate system (r, h, x3) centred at the crack-tip and moving with it towards the h ¼ 0 direction is considered. The x3-axis is assumed to coincide with the straight crack front. The condition of steady-state propagation yields the following time derivative rule for any arbitrary scalar function /: V /_ ¼ ½/;h sin h r/;r cos h; ð3:1Þ r which asymptotically holds also for not uniform, but quasi-static, crack propagation. For antiplane problems, the non-vanishing stress and couple stress components with respect to a cylindrical coordinate system in cylindrical coordinates are rr3 ; rh3 ; sr3 ; sh3 ; lrr ; lrh ; lhr and lhh ¼ lrr . Accordingly, the strain and deformation curvature components are er3 ; eh3 ; vhr ; vrh ; vrr ; andvhh ¼ vrr . The equilibrium conditions (2.5) write ðr tr3 Þ;r þ th3;h ¼ 0; ðrlrr Þ;r þ lhr;h lhh þ 2 r sh3 ¼ 0;
ð3:2Þ
ðrlrh Þ;r þ lhh;h þ lhr 2 r sr3 ¼ 0; and the kinematical compatibility conditions (2.1)1 and (2.2) in rate form write v3 ¼ u_ 3 ; v3;r ¼ 2 e_ r3 ; v3;h ¼ 2 r e_ h3 ; v_ rr ¼ e_ 3h;r ; v_ rh ¼ e_ 3h;h þ e_ 3r =r; v_ hr ¼ _e3r;r :
ð3:3Þ ð3:4Þ
Eqs. (3.2)–(3.4) together with rate constitutive equations (2.11), (2.15) and (2.16) form a system of first order PDEs that governs the problem of the crack propagation. The asymptotic crack-tip fields are sought in a separable variable form, of the type /ðr; hÞ ¼ rp F ðhÞ, where the exponent p defines the asymptotic behavior of the generic function / as r ! 0. A qualitative analysis performed by Radi and Gei (2004) gives the relative order of singularity of the crack-tip fields, which is assumed to hold in the present context also. Therefore, the asymptotic crack-tip fields and their rates are assumed in the following form: r p uðhÞ; v3 ðr; hÞ ¼ V wðhÞ; R R V r p V r p e_ a3 ðr; hÞ ¼ d a ðhÞ; v_ ab ðr; hÞ ¼ 2 X ab ðhÞ; r R r R u3 ðr; hÞ ¼ r
r p
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V r p ha ðhÞ; R r R ‘2 r p ‘2 r p M ab ðhÞ; l_ ab ðr; hÞ ¼ GV 2 H ab ðhÞ; lab ðr; hÞ ¼ G r R r R ‘ r p ‘ r p Y ðr; hÞ ¼ G cðhÞ; Y_ ðr; hÞ ¼ GV 2 jðhÞ; ð3:5Þ r R r R ‘ r p ‘2 r p CðhÞ; sa3 ðr; hÞ ¼ G 2 ta ðhÞ; Rðr; hÞ ¼ G r R r R where the greek indices a and b stand for polar coordinates r and h. Since l, M and H are purely deviatoric tensors, the following conditions hold true: ra3 ðr; hÞ ¼ G
r p
M hh ¼ M rr ;
sa ðhÞ;
r_ a3 ðr; hÞ ¼ G
H hh ¼ H rr :
ð3:6Þ
The function C resulting from the introduction of (3.5)5,7,11 in Eq. (2.9) is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 M 2rh þ M 2hr 2 gM rh M hr ; C¼ ð1 gÞM rr þ 1 g2 2
ð3:7Þ
since near the crack-tip, namely for r < k, the couple stress field gives the most singular contribution to the effective stress and, thus, to the yield condition. The exponent p introduced in representations (3.5) defines the radial dependence of the asymptotic crack-tip fields. As r goes to zero, the symmetric stress, couple stress and skewsymmetric stress fields behave as rp ; rp1 and rp2 , respectively. Therefore, the latter field gives the most singular contribution to the total stress near to the crack tip, namely for r < ‘. Oppositely, for r > ‘, the couple stress and skew-symmetric stress fields become negligible with respect to the symmetric stress field, in agreement with the classical theories of elastic–plastic fracture mechanics. The constant R in the representations (3.5) plays the role of an undetermined amplitude factor for the leading order asymptotic fields. The solution of the homogeneous asymptotic problem can indeed be determined up to an amplitude factor, which depends on far-field loading and specimen geometry and can be estimated by matching the asymptotic solution with the far-field conditions. Nevertheless, the asymptotic analysis can capture the strength of the singularity of the crack tip fields, and their angular variations, once a normalisation condition is adopted. When the asymptotic fields (3.5)5,7,12 are introduced into equilibrium equations (3.2), by using (3.6), the following ODEs are derived at leading order t0h ¼ ð1 pÞ tr ; M 0hr ¼ ð1 þ pÞ M rr 2 th ; M 0rr ¼ M hr þ p M rh 2 tr ;
ð3:8Þ
where the apex denote the derivative with respect to the angular coordinate h. The components ha and Hab introduced in (3.5)6,8 follow from an application of the steady-state derivative rule (3.1) to representations (3.5)5,7 of the symmetric stress and couple stress fields: hr ¼ ðs0r sh Þ sin h p sr cos h;
hh ¼ ðs0h þ sr Þ sin h p sh cos h;
ð3:9Þ
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1447
and H rr ¼ ðM 0rr M hr M rh Þ sin h þ ð1 pÞ M rr cos h; H rh ¼ ðM 0rh þ 2M rr Þ sin h þ ð1 pÞ M rh cos h;
ð3:10Þ
H hr ¼ ð1 pÞðM rr sin h þ M hr cos hÞ 2 th sinh: Note that the equilibrium condition (3.8)2 has been used in the latter equation. The definition of the function j, introduced in (3.5)10, follows from (3.5)9 and (3.1) as j ¼ c0 sin h þ ð1 pÞc cos h:
ð3:11Þ
In contrast with the mode I and mode II crack problems (Xia and Hutchinson, 1996; Huang et al., 1997; Radi, 2003), the dominant strain rate field for the antiplane crack problem turns out to be rotational. Therefore, the compatibility equations (3.4) admit non-vanishing deformation curvature rates at lowest order. In agreement with the asymptotic representations (3.5)2,3 of the velocity and strain rate fields, the kinematical compatibility conditions (3.3) can be written as d r ¼ p w=2;
w0 ¼ 2 d h :
ð3:12Þ
Moreover, a substitution of (3.5)3,4 and (3.15)1 into Eqs. (3.4) gives X rr ¼ ðp 1Þd h ;
X hr ¼ ð1 pÞp w=2;
X rh ¼ d 0h þ p w=2:
ð3:13Þ
After the definitions (3.5)9,11 of the effective stress and flow stress, the yield condition (2.8) writes C ¼ c:
ð3:14Þ
The introduction of the asymptotic fields (3.5) in the incremental constitutive relationships (2.15), (2.16) and (2.11) yields the following set of equations: 1 d ¼ h þ hkis; 2 1 1 XT ¼ ½ ðH gHT Þ þ hkiðM gMT Þ; 1 g2 2 2 j ¼ hkihC; 3
ð3:15Þ
where h ¼ H /G is the non-dimensional hardening modulus and the function k(h) is defined such that k ¼ 0 when C < c, and k¼
1 ½2ð1 gÞM rr H rr þ ðM rh gM hr ÞH rh þ ðM hr gM rh ÞH hr ; h
ð3:16Þ
when C ¼ c, in agreement with (2.13) and (2.14), being 2
h ¼ hð1 g2 Þð2C=3Þ :
ð3:17Þ
Eqs. (3.15) hold true when the effective stress and flow stress obey the yield condition (3.14). Under elastic unloading or neutral loading k 6 0 and thus Eq. (3.15)1,2 reduce to the rate constitutive relations of linear isotropic elasticity with strain gradient effects. Moreover, Eq. (3.15)3 becomes equivalent to the condition Y_ ¼ 0.
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3.1. Explicit form of the ODEs system An explicit form of the ODEs system is now derived by rearranging the equations presented in the previous subsection. In particular, by using (3.12)1 and (3.13)3, Eqs. (3.15)1,2 give hr þ 2hkisr ¼ pw;
hh þ 2hkish ¼ 2d h ;
ð3:18Þ
and H rr þ 2hkiM rr ¼ 2ð1 þ gÞX rr ; ðH rh gH hr Þ þ 2hkiðM rh gM hr Þ ¼ 2ð1 g2 ÞX hr ; ðH hr gH rh Þ þ 2hkiðM hr gM rh Þ ¼ ð1 g
2
Þð2d 0h
ð3:19Þ
þ p wÞ:
The introduction of (3.16) in Eqs. (3.19)1,2, after some algebraic manipulations, allows to obtain the following explicit expression for the functions Hrr and Hrh: " # 2 H rr 1 h þ 2ðM rh gM hr Þ 2M rr ðM rh gM hr Þ zr ¼ ; ð3:20Þ D 4ð1 gÞM rr ðM rh gM hr Þ H rh zh h þ 4ð1 gÞM 2rr where D ¼ h ½h þ 2ðM rh gM hr Þ2 þ 4ð1 gÞM 2rr ; zr ¼ 2h ð1 þ gÞX rr 2M rr ðM hr gM rh ÞH hr ;
2
ð3:21Þ
zh ¼ 2h ð1 g ÞX hr þ ½h g 2ðM rh gM hr ÞðM hr gM rh ÞH hr : Note that D is always positive for h > 0, being 1 < g < 1. Since the derivatives of the unknown angular functions do not enter the algebraic expressions (3.20), (3.10)3 and (3.13)1,2 forHrr, Hrh, Hhr, Xrr and Xhr, then the explicit expressions (3.20) and (3.10)3 for Hrr, Hrh and Hhr can be introduced in (3.16) to obtain k in explicit form. Moreover, from (3.10)1,2 the derivatives with respect to h of Mrr and Mrh are M 0rr ¼ M hr þ M rh ð1 pÞM rr cot h þ H rr = sin h; ð3:22Þ M 0rh ¼ 2M rr ð1 pÞM rh cot h þ H rh = sin h: Finally, the derivatives of the symmetric stress components sr and sh follow from (3.9) as s0r ¼ sh þ ðhr þ p sr cos hÞ= sin h; s0h ¼ sr þ ðhh þ p sh cos hÞ= sin h;
ð3:23Þ
where hr and hh are reported in (3.18). Similarly, from (3.11) and (3.15)3 one obtains 2 c0 ¼ ðp 1Þc cot h þ hkihC= sin h: 3
ð3:24Þ
The use of the steady-state condition (3.1) and representation (3.5)1,2 in (3.3)1 yields u0 ¼ ½w þ ðp þ 1Þu cos h= sin h:
ð3:25Þ
Eqs. (3.8)1,2, (3.12)2, (3.19)3, (3.22)–(3.25) form a system of 10 first order and homogeneous ODEs governing the near-tip stress and velocity fields for the antiplane crack propagation problem in couple stress elastic–plastic solids. This system may be written in the following explicit form:
E. Radi / International Journal of Plasticity 23 (2007) 1439–1456
0
y ðhÞ ¼
f p ðh; yðhÞÞ if C ¼ c and k > 0; f e ðh; yðhÞÞ
if C < c or ðC ¼ c and k 6 0Þ;
1449
ð3:26Þ
where the vector y collects the 10 unknown angular functions, i.e. ð3:27Þ
y ¼ fw; th ; sr ; sh ; M rr ; M rh ; M hr ; d h ; c; ug:
Note that tr and dr are not primary unknown functions. In fact, tr may be obtained from (3.8)3, by using (3.20) and (3.22)1, whereas dr follows from the algebraic equation (3.12)1 once all the functions collected in the vector y are known. The unknown exponent p can be determined as an eigenvalue of the nonlinear homogeneous problem (3.26), once mode III symmetry conditions, traction free conditions at crack flanks and a normalization condition are imposed on the asymptotic crack-tip fields. The flow-theory of CS plasticity adopted in the present work allows considering the occurring of elastic unloading. A generic material point near the trajectory of the cracktip experiences plastic loading ahead of the crack-tip, possibly followed by elastic unloading at h ¼ h1 and subsequent plastic reloading at h ¼ h2 . Elastic unloading may occur at the angle h1 defined by the condition R_ ¼ 0, namely kðh1 Þ ¼ 0. Plastic reloading may then occur at crack flanks at the angular coordinate h2 where the material point reaches a stress state obeying the yield condition (3.14) discarded at unloading. Alternatively, the zone surrounding the crack tip may result as fully plasticized. 4. Mode III boundary conditions Under Mode III crack propagation the integration may be restricted to the range 0 6 h 6 p by symmetry condition. Since a regular behavior of the angular functions at h ¼ 0 may be expected, then relations (3.9) and (3.10) imply hð0Þ ¼ p sð0Þ;
Hð0Þ ¼ ð1 pÞMð0Þ:
ð4:1Þ
By using condition (4.1)2, the function k defined in (3.16) attains at h ¼ 0 the value kð0Þ ¼
3 ð1 pÞ; 2h
ð4:2Þ
which is positive for p < 1. The skew-symmetry of the antiplane crack problem requires ahead of the crack-tip at h ¼ 0 uð0Þ ¼ 0;
wð0Þ ¼ 0;
M hr ð0Þ ¼ 0:
ð4:3Þ
The introduction of relations (4.1)–(4.3) into the constitutive equations (3.18) and (3.19) evaluated at h ¼ 0, after some algebraic manipulations, yields sr ð0Þ ¼ 0;
d h ð0Þ ¼ ½3 ð3 þ hÞp sh ð0Þ=ð2hÞ; 3 M rr ð0Þ ¼ ð1 þ gÞ p sh ð0Þ; M rh ð0Þ ¼ 0; 3þh
ð4:4Þ d 0h ð0Þ ¼ 0:
ð4:5Þ
Vanishing of reduced tractions (2.4)1 along the radial direction at h ¼ 0 requires 2 r tr3 lrr;h ¼ 0; ð4:6Þ which implies M 0rr ð0Þ ¼ 2 tr ð0Þ:
ð4:7Þ
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Hence, from (4.7) and the equilibrium condition (3.8)3 it turns out that tr ð0Þ ¼ 0;
M 0rr ð0Þ ¼ 0:
ð4:8Þ
A numerical procedure based on the Runge–Kutta method is used to integrate the system (3.26) (subroutine DIVPRK of the IMSL library). This procedure requires knowledge of the initial values y(0). In order to avoid the trivial solution, the following normalization condition for the stress field sh ð0Þ ¼ 1
ð4:9Þ
is adopted, since all the assigned boundary conditions (4.3), (4.4), (4.5) and (4.8) are homogeneous. Plastic loading is assumed to occur in the zone ahead of the crack-tip. Therein the yield condition cð0Þ ¼ Cð0Þ must hold true, where C(0) follows from conditions (4.3)3, (4.5)2 and (3.7) as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cð0Þ ¼ 3M 2rr ð0Þ=ð1 þ gÞ: ð4:10Þ Vanishing of reduced tractions (2.4) on the crack surfaces implies the conditions 2 th3 þ lhh;r ¼ 0;
lhr ¼ 0;
ð4:11Þ
which yield the following two boundary conditions at h ¼ p for the system (3.26) 2 th ðpÞ þ ð1 pÞ M rr ðpÞ ¼ 0;
M hr ðpÞ ¼ 0:
ð4:12Þ
By using (3.12)2, (4.4) and (4.5) in the derivatives with respect to h of (3.18), (3.9) and (3.25) evaluated at h ¼ 0, the following results can be found h 0 sr ð0Þ ¼ p þ ð4:13Þ sh ð0Þ; s0h ð0Þ ¼ 0; u0 ð0Þ ¼ 2d h ð0Þ=p: 3þh Similarly, differentiation of (3.10)2 and (3.19)2 and use of (3.8)2 and (4.5)2 yield M 0rh ð0Þ ¼
2ð1 pÞð1 þ gÞ½ð2 pÞð1 þ pÞ þ 2kð0Þðp þ gÞ d h ð0Þ 2gth ð0Þ: ½1 p þ 2kð0Þ½2 p þ 2kð0Þ
ð4:14Þ
By using (4.8)1 the equilibrium equation (3.8)1 evaluated at h ¼ 0 gives t0h ð0Þ ¼ 0:
ð4:15Þ
By taking the derivative of C(h) and k(h) from (3.7) and (3.16) and using (4.3)3, (4.5)2 and (4.8)2 it follows that C0 ð0Þ ¼ k0 ð0Þ ¼ 0. Then, the value of c0 ð0Þ can be obtained from the derivative of (3.24) evaluated at h ¼ 0 as c0 ð0Þ ¼ 0:
ð4:16Þ
The values of all the unknown functions at h ¼ 0 are defined in (4.3), (4.4), (4.5)1,2, (4.8)1, (4.9) and (4.10) as functions of p and th(0). Then, the values of p and th(0) can be calculated by an iterative procedure based on the achievement of the two boundary conditions (4.12) at h ¼ p. This iteration is performed by using the modified Powell hybrid method (subroutine DNEQNF of the IMSL library), until the conditions (4.12) on the vanishing of the reduced tractions are verified within a prescribed accuracy. The procedure employed for the integration of the governing ODE system (3.26) displays a numerical difficulty at h ¼ 0, due to the term sin h which multiplies the highest
E. Radi / International Journal of Plasticity 23 (2007) 1439–1456
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order derivative. This inconvenience can be bypassed by performing a Taylor series expansions of the unknown angular functions collected in the vector y, starting at h ¼ 0, namely: yðeÞ ¼ yð0Þ þ ey0 ð0Þ þ oðeÞ;
ð4:17Þ
where e 1. Therefore, the unknown functions can be evaluated at h ¼ e, since the values assumed by the derivatives of the unknown functions at h ¼ 0 are known from relations (3.8)1,2, (4.5)3, (4.7), (4.8)2, and (4.13)–(4.16). Once that y(e) has been determined by (4.17) within an error lower than e, the numerical integration of (3.26) can be performed by starting at h ¼ e, rather than at h ¼ 0. 5. Results Fig. 1 shows the variation of the exponent p with the ratio g for different values of the strain hardening coefficient a. It can be observed that the ratio g has a limited influence on p, since a small variation of p occurs as g ranges between 1 and 1. As it can be expected, the larger is the coefficient a, the lower is the exponent p and thus the stronger is the singularity of the couple stress and skew-symmetric stress fields, which asymptotically behave as rp 1 and rp 2, respectively. In particular, the curves plotted in Fig. 1 show that the exponent p tends to 1 as the strain hardening coefficient a becomes vanishing small, namely for perfectly plastic behavior, whereas p tends to 0.5 as a approaches the limit value of 1, in agreement with the CS elastic solution provided by Zhang et al. (1998). Note that the strain gradient effect also occurs for purely elastic response, namely for a ¼ 1. Since 0:5 < p < 1 for 0 < a < 1, the asymptotic radial variation of the out of plane displacement considered in (3.5)1 implies that the crack-tip profile is sharp, whereas crack tip blunting is expected for the conventional theories of LEFM and EPFM. Moreover, the symmetric stress field turns out to be non-singular, as already observed for stationary crack tip fields in strain gradient elasticity (Zhang et al., 1998) and plasticity (Huang et al., 1997, 1999), whereas the skew-symmetric stress field displays a strong singularity. It must be remarked that the latter field does not contribute to the effective stress R and to the strain-energy density, which are instead dominated by the weakly singular
1.0
α = 0.001 0.01
0.9
0.05 p
0.8
0.1
0.7
0.3
0.6
0.5 0.9
0.5 -1.0
-0.5
0.0
η
0.5
1.0
Fig. 1. Variation of the exponent p with the ratio g for different values of the strain hardening coefficient a.
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couple-stress field, so that the flux of energy toward the crack-tip remains finite. Moreover, the tractions level ahead of the crack tip increases with respect to the classical J2-flow theory due to the contribution of the skew-symmetric stress components, also for very small strain hardening. In fact, the investigations performed for conventional elastic–plastic response with linear hardening by Amazigo and Hutchinson (1977) and Ponte Castan˜eda (1987) predict a weak stress singularity for small strain-hardening, which disappears in the limit case of elastic–perfectly plastic behavior of the material, namely when a is vanishing small. Elastic unloading has been detected only for a large, namely a > 0:1, whereas the crack tip zone is fully plastic for a small, as due to the plastic distribution caused by the strain gradient effect, also observed by Wei and Hutchinson (1997) in their f.e. investigations. In particular, Figs. 2a and b display the variation of the elastic unloading angle h1 and plastic reloading angle h2 with the ratio g for the hardening coefficient a ¼ 0:3 and a ¼ 0:5, respectively. It may be observed that as the ratio g increases then the plastic zone tends to concentrate ahead of the crack tip, resulting thus in a more stable crack propagation, according to previous findings (Ponte Castan˜eda, 1987; Bigoni and Radi, 1996). Figs. 3 and 4 display the angular distributions of the asymptotic crack-tip fields for g ¼ 0:5 in the realistic case of small strain hardening, namely for a ¼ 0:01. For both values of g the zone surrounding the crack-tip at distance lower than the characteristic material length k is fully plastic. In particular, the angular variations of symmetric stresses (a), out-of-plane velocity and displacement (b), couple stresses (c) and skew-symmetric stresses (d) are plotted therein. All functions are normalized by condition (4.9). It can be observed that the ratio g produces remarkable changes in the angular distribution of the crack tip fields. The curves plotted in Figs. 3a and 4a show that the out-of-plane displacement around the crack tip does not monotonically increase with the angular coordinate h but it oscillates and reaches its maximum value at about 140° instead of at the crack flanks. Note that in their investigations on mode III crack problem within their framework of gradient elas180
180
a θ2
150
elastic unloading sector
150
elastic unloading sector
120
b
120
θ1 90
θ1
90
plastic loading sector
60
plastic loading sector
60
30
30
α = 0.5
α = 0.3 0
0 -1.0
-0.5
0.0
η
0.5
1.0
-1.0
-0.5
0.0
0.5
1.0
η
Fig. 2. Variation of the elastic unloading angle h1 and plastic reloading angle h2 with the ratio g for a ¼ 0:3 and a ¼ 0:5.
E. Radi / International Journal of Plasticity 23 (2007) 1439–1456 10
2
a
b
w
sθ
1
5
1453
sr
u 0
0
-5
-1
α = 0.01 η = 0.5
-2
-10 0
30
60
90
120
150
180
0
30
60
0.2
90
120
150
180
θ
θ 0.4
c
d
tr 0.1
0.2
0.0
0.0
M θr Mr θ
tθ
Mrr
-0.1
Mθθ
-0.2
-0.2
-0.4
0
30
60
90
θ
120
150
180
0
30
60
90
120
150
180
θ
Fig. 3. Angular variation of the crack-tip fields for a = 0.01 and g = 0.5.
ticity Unger and Aifantis (2000) found an oscillation of the displacement along the radial direction. Moreover, in the present analysis the displacements ahead and behind the cracktip turn out to be opposite in sign, unlike the classical mode III crack-tip fields in nonpolar materials (Amazigo and Hutchinson, 1977; Ponte Castan˜eda, 1987; Bigoni and Radi, 1996). Similar results have been also obtained for mode III crack in CS elastic and CS elastic–plastic materials (Zhang et al., 1998; Radi and Gei, 2004). This unusual aspect seems to be due to the presence of microstructures (compositional grains) introduced through the material parameters ‘ and g. During crack growth, in facts, the relative rotation of the particles currently at the crack tip yields opposite displacements ahead and behind the crack tip, thus producing a scissor effect. The angular variations of the skew-symmetric stress components are plotted in Figs. 3c and 4c. In agreement with the inversion of the out-of-plane displacement ahead of the crack-tip, the shear stress th is also negative therein and, thus, opposite to its counterpart
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2
a
b
w 5
sθ
1
sr
u 0
0
-5
-1
α = 0.01 η = −0.5
-10
-2 0
0.04
30
60
90
θ
120
150
180
0
0.10
c
60
90
θ
120
150
180
d Mθ r
tr
0.02
30
0.05
0.00
Mr θ
0.00
tθ
Mrr
-0.02
Mθθ
-0.05
-0.04
-0.10 0
30
60
90
θ
120
150
180
0
30
60
90
120
150
180
θ
Fig. 4. Angular variation of the crack-tip fields for a ¼ 0:01 and g ¼ 0:5.
in the classical mode III solution. The negative sign of the tractions very near to the crack tip indicates that the rotation gradients may shield the crack tip from fracture, in agreement with the results provided by Tong et al. (2005) and Wei and Xu (2005) for mode I and mode II fracture. Shielding can reasonably be expected since strain gradient tends to stiffen the material. For positive values of the ratio g the radial shear stress component tr also displays negative values in a small sector ahead of the crack-tip. The switch in the displacement and shear directions agrees with the results obtained by Zhang et al. (1998) and Radi and Gei (2004). These investigations confirm that the microstructure remarkably affects the solutions of fracture mechanics problems. 6. Conclusions The obtained results show that the use of the CS theory of plasticity with two characteristic lengths for the analysis of the stress field near the tip of a propagating mode III
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crack gives accurate predictions on the increase of the tractions level ahead of the crack-tip occurring at very small distances from it, comparable with the size of the compositional grains. Moreover, it sheds some light on the shielding mechanisms occurring during crack propagation due to the presence of a complex microstructure. Strain gradient effects are found to be relevant at very small distances ahead of the crack-tip namely for r < ‘. They produce a switch in the sign of tractions and displacement ahead of the crack-tip, with respect to classical LEFM and EPFM. Moreover, an increase in the singularity of tractions ahead of the crack-tip is observed, even for low strain hardening, essentially due to the skew-symmetric stress field, which does not contribute to the effective stress and strain energy density and thus does not violate the boundness of the energy flux towards the crack tip. The contribution to the effective stress is instead dominated by the slightly singular couple-stress field. The ratio g also displays a strong influence on the angular distribution of the asymptotic crack tip fields. Since the obtained asymptotic solution holds at a distance to the crack tip much smaller than the range of application of classical LEFM, then the present approach provides a mean to link scales in fracture mechanics, from atomistic to macroscopic fracture, allowing understanding of the detailed mechanisms by which fracture may grow and propagate in materials with complex microstructure, up to the micron scale. With this end in view, the results here presented can be usefully referred in further numerical and analytical investigations of multiscale fracture problems in ductile materials. Acknowledgments Financial support from the Italian Ministry of Education, University and Research (MIUR) in the framework of the Project PRIN-2004 ‘‘Problemi e modelli microstrutturali: applicazioni in ingegneria civile e strutturale” is gratefully acknowledged. References Amazigo, J., Hutchinson, J.W., 1977. Crack-tip fields in steady crack-growth with linear strain hardening. Journal of the Mechanics and Physics of Solids 25, 81–97. Bigoni, D., Radi, E., 1996. Asymptotic solution for Mode III crack growth in J2-elastoplasticity with mixed isotropic-kinematic strain hardening. International Journal of Fracture 77, 77–93. Chen, J.Y., Wei, Y., Huang, Y., Hutchinson, J.W., Hwang, K.C., 1999. The crack-tip fields in strain gradient plasticity: the asymptotic and numerical analyses. Engineering Fracture Mechanics 64, 625–648. Chen, S.H., Wang, T.C., 2002. A new deformation theory with strain gradient effects. International Journal of Plasticity 18, 971–995. Dillard, T., Forest, S., Ienny, P., 2006. Micromorphic continuum modelling of the deformation and fracture behaviour of nickel foams. European Journal of Mechanics – A/Solids 25 (3), 526–549. Drugan, W.J., Rice, J.R., Sham, T.L., 1982. Asymptotic analysis of growing plane strain tensile cracks in elasticideally plastic solids. Journal of the Mechanics and Physics of Solids 30, 447–473. Elssner, G., Korn, D., Ruehle, M., 1994. The influence of interface impurities on fracture energy of UHV diffusion bonded metal–ceramic bicrystals. Scripta Metallurgica et Materialia 31, 1037–1042. Fannjiang, A.C., Chan, Y.S., Paulino, G.H., 2002. Strain gradient elasticity for antiplane shear cracks: a hypersingular integrodifferential equation approach. SIAM Journal of Applied Mathematics 62, 1066–1091. Fleck, N.A., Hutchinson, J.W., 1993. A phenomenological theory for strain gradient effects in plasticity. Journal of the Mechanics and Physics of Solids 41, 1825–1857. Fleck, N.A., Hutchinson, J.W., 1997. Strain gradient plasticity. In: Hutchinson, J.W., Wu, T.Y. (Eds.), Advances in Applied Mechanics, vol. 33. Academic Press, New York, pp. 295–361.
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Fleck, N.A., Hutchinson, J.W., 2001. A reformulation of strain gradient plasticity. Journal of the Mechanics and Physics of Solids 49, 2245–2271. Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W., 1994. Strain gradient plasticity: theory and experiment. Acta Metallurgica et Materialia 42, 475–487. Gao, H., Huang, Y., Nix, W.D., Hutchinson, J.W., 1999. Mechanism-based strain gradient plasticity – I. Theory. Journal of the Mechanics and Physics of Solids 47, 1239–1263. Huang, Y., Chen, J.Y., Guo, T.F., Zhang, L., Huang, K.C., 1999. Analytic and numerical studies on Mode I and Mode II fracture in elastic–plastic materials with strain gradient effects. International Journal of Fracture 100, 1–27. Huang, Y., Qu, S., Hwang, K.C., Li, M., Gao, H., 2004. A conventional theory of mechanism-based strain gradient plasticity. International Journal of Plasticity 20 (4–5), 753–782. Huang, Y., Zhang, L., Guo, T.F., Hwang, K.C., 1997. Mixed Mode near-tip fields for crack in materials with strain-gradient effects. Journal of the Mechanics and Physics of Solids 45, 439–465. Hutchinson, J.W., 2000. Plasticity at the micron scale. International Journal of Solids and Structures 37, 225–238. Koiter, W.T., 1964. Couple-stresses in the theory of elasticity, I and II. Proc. Ned. Akad. Wet. (B) 67, 17–44. Lakes, R., 1986. Experimental microelasticity of two porous solids. International Journal of Solids and Structures 22, 55–63. Lakes, R., 1995. Experimental methods for study of Cosserat elastic solids and other generalized elastic continua. In: Mu¨hlhaus, H. (Ed.), Continuum Models for Materials with Micro-structure. Wiley, New York, pp. 1–22 (Chapter 1). Ottosen, N.S., Ristinmaa, M., Ljung, C., 2000. Rayleigh waves obtained by the indeterminate couple-stress theory. European Journal of Mechanics – A/Solids 19, 929–947. Paulino, G.H., Fannjiang, A.C., Chan, Y.S., 2003. Gradient elasticity theory for mode III fracture in functionally graded materials – Part I: crack perpendicular to the material gradation. Journal of Applied Mechanics 70 (4), 531–542. Ponte Castan˜eda, P., 1987. Asymptotic fields in steady crack growth with linear strain-hardening. Journal of the Mechanics and Physics of Solids 35, 227–268. Qiu, X., Huang, Y., Wei, Y., Gao, H., Hwang, K.C., 2003. The flow theory of mechanism-based strain gradient plasticity. Mechanics of Materials 35, 245–258. Radi, E., 2003. Strain-gradient effects on steady-state crack growth in linear hardening materials. Journal of the Mechanics and Physics of Solids 51, 543–573. Radi, E., Gei, M., 2004. Mode III crack growth in linear hardening materials with strain-gradient effects. International Journal of Fracture 130, 765–785. Ristinmaa, M., Vecchi, M., 1996. Use of couple-stress theory in elasto-plasticity. Computational Methods in Applied Mechanical Engineering 136, 205–224. Shi, M.X., Huang, Y., Gao, H., Hwang, K.C., 2000. Non-existence of separable crack-tip field in mechanismbased strain gradient plasticity. International Journal of Solids and Structures 37, 5995–6010. Tong, P., Lam, D.C.C., Yang, F., 2005. Mode I solution for micron-sized crack. Engineering Fracture Mechanics 72 (12), 1779–1804. Unger, D.J., Aifantis, E.C., 2000. Strain gradient elasticity theory for antiplane shear cracks. Part I: Oscillatory displacements. Theoretical and Applied Fracture Mechanics 34, 243–252. Vardoulakis, I., Exadaktylos, G., Aifantis, E., 1996. Gradient elasticity with surface energy: Mode-III crack problem. International Journal of Solids and Structures 33, 4531–4559. Wei, Y., 2006. A new finite element method for strain gradient theories and applications to fracture analyses. European Journal of Mechanics – A/Solids 25, 897–913. Wei, Y., Hutchinson, J.W., 1997. Steady-state crack growth and work of fracture for solids characterized by strain gradient plasticity. Journal of the Mechanics and Physics of Solids 45, 1253–1273. Wei, Y., Xu, G., 2005. A multiscale model for the ductile fracture of crystalline materials. International Journal of Plasticity 21 (11), 2123–2149. Xia, Z.C., Hutchinson, J.W., 1996. Crack-tip fields in strain gradient plasticity. Journal of the Mechanics and Physics of Solids 44, 1621–1648. Zhang, L., Huang, Y., Chen, J.Y., Hwang, K.C., 1998. The Mode III full-field solution in elastic materials with strain gradient effects. International Journal of Fracture 92, 325–348.