The effect of T-stress on plane strain dynamic crack growth in elastic–plastic materials

In this paper, the effects of T‐stress on steady, dynamic crack growth in an elastic–plastic material are examined using a modified boundary layer formulation. The analyses are carried out under mode I, plane strain conditions by employing a special finite element procedure based on moving crack tip coordinates. The material is assumed to obey the J₂ flow theory of plasticity with isotropic power law hardening. The results show that the crack opening profile as well as the opening stress at a finite distance from the tip are strongly affected by the magnitude and sign of the T‐stress at any given crack speed. Further, it is found that the fracture toughness predicted by the analyses enhances significantly with negative T‐stress for both ductile and cleavage mode of crack growth.     

Steady-state crack growth and fracture work based on the theory of mechanism-based strain gradient plasticity

Mode I steady-state crack growth is analyzed under plane strain conditions in small scale yielding. The elastic-plastic solid is characterized by the mechanism-based strain gradient (MSG) plasticity theory [J. Mech. Phys. Solids 47 (1999) 1239, J. Mech. Phys. Solids 48 (2000) 99]. The distributions of the normal separation stress and the effective stress along the plane ahead of the crack tip are computed using a special finite element method based on the steady-state fundamental relations and the MSG flow theory. The results show that during the steady-state crack growth, the normal separation stress on the plane ahead of the crack tip can achieve considerably high value within the MSG strain gradient sensitive zone. The results also show that the crack tip fields are insensitive to the cell size parameter in the MSG theory. Moreover, in the present research, the steady-state fracture toughness is computed by adopting the embedded process zone (EPZ) model. The results display that the steady-state fracture toughness strongly depends on the separation strength parameter of the EPZ model and the length scale parameter in the MSG theory. Furthermore, in order for the results of steady crack growth to be comparable, an approximate relation between the length scale parameters in the MSG theory and in the Fleck-Hutchinson strain gradient plasticity theory is obtained.     

Fracture analysis in the conventional theory of mechanism-based strain gradient (CMSG) plasticity

In a remarkable series of experiments, Elssner et al. (1994) and Korn et al. (2002) observed cleavage cracking along a bimaterial interface between Nb and sapphire. The stress required for cleavage cracking is around the theoretical strength of the material. Classical plasticity models fall short to reach such a high stress level. We use the conventional theory of mechanism-based strain gradient plasticity (Huang et al., 2004) to investigate the stress field around the tip of an interface crack between Nb and sapphire. The tensile stress at a distance of 0.1 m to the interface crack tip reaches 13.3_Y, where _Y is the yield stress of Nb. This stress is nearly 4 times of that predicted by classical plasticity theory (3.6_Y) at the same distance to the crack tip, and is high enough to trigger cleavage cracking in materials and interfaces. This is consistent with Elssner et al.'s (1994) and Korn et al.'s (2002) experimental observations.     

Analysis of experimental factors in elastic–plastic small specimen mixed-mode I-II fracture mechanical testing

The quality of experimental fracture resistance testing results in mixed-mode I-II loading is more than questionable in several cases. This study describes efforts to gain consistent results with respect to mixed-mode I-II fracture resistance curve determination when working with elastic–plastic materials. The entire mixed-mode I-II field is evaluated, i.e. both numerical and experimental factors are considered the focus being on the asymmetric four-point bend set-up. Several error prone features are presented, and minimization of their effects on quantitative fracture resistance assessment considered. The results indicate that through a careful evaluation of the stages in experimental testing and numerical analysis valid, material characterizing, fracture toughness results can be obtained.     

Interface crack problems with strain gradient effects

In this paper, the strain gradient theory proposed by Chen and Wang (2001a, 2002b) is used to analyze an interface crack tip field at micron scales. Numerical results show that at a distance much larger than the dislocation spacing the classical continuum plasticity is applicable; but the stress level with the strain gradient effect is significantly higher than that in classical plasticity immediately ahead of the crack tip. The singularity of stresses in the strain gradient theory is higher than that in HRR field and it slightly exceeds or equals to the square root singularity and has no relation with the material hardening exponents. Several kinds of interface crack fields are calculated and compared. The interface crack tip field between an elastic-plastic material and a rigid substrate is different from that between two elastic-plastic solids. This study provides explanations for the crack growth in materials by decohesion at the atomic scale.     

Size effects in phenomenological strain gradient plasticity constitutively involving the plastic spin

In the small deformation range, we consider and discuss the phenomenological (or isotropic) “higher-order” theory of strain gradient plasticity put forward in Section 12 of Gurtin [1], which includes the dissipation due to the plastic spin through a material parameter called χ. In fact, χ weighs the square of the plastic spin rate into the definition of an effective measure of plastic flow peculiar of the isotropic hardening function. Such a model has been identified by Bardella [2] as a good isotropic approximation of a crystal model to describe the multislip behaviour of a single grain, provided that χ be set as a specific function of other material parameters involved in the modelling, including the length scales. The main feature of the underlying gradient approach is the accounting for both dissipative and energetic strain gradient dependences, with related size effects. The dissipative strain gradients enter the model through the definition of the above mentioned effective measure of plastic flow, whereas the energetic strain gradients are involved in the modelling by defining the defect energy, a function of Nye’s dislocation density tensor added to the free energy to account for geometrically necessary dislocations (see, e.g., Gurtin [1]). By exploiting the deformation theory approximation, we apply the model to a simple boundary value problem so that we can discuss the effects of (a) the criterium derived by Bardella [2] for choosing χ and (b) non-quadratic forms of the defect energy. We show that both χ and the nonlinearity chosen for the defect energy strongly affect quality and magnitude of the energetic size effect which is possible to predict.     

Analytic and numerical studies on mode I and mode II fracture in elastic-plastic materials with strain gradient effects

This paper presents a study on fracture of materials at microscale (∼1 μm) by the strain gradient theory (Fleck and Hutchinson, 1993; Fleck et al., 1994). For remotely imposed classical K fields, the full-field solutions are obtained analytically or numerically for elastic and elastic-plastic materials with strain gradient effects. The analytical elastic full-field solution shows that stresses ahead of a crack tip are significantly higher than their counterparts in the classical K fields. The sizes of dominance zones for mode I and mode II near-tip asymptotic fields are 0.3l and 0.5l,while strain gradient effects are observed within land 2l to the crack tip, respectively, where l is the intrinsic material length in strain gradient theory and is on the order of microns in strain gradient plasticity (Fleck et al., 1994; Nix and Gao, 1998; Stolken and Evans, 1997). The Dugdale–Barenblatt type plasticity model is obtained to provide an estimation of plastic zone size for mode II fracture in materials with strain grain effects. The finite element method is used to investigate the small-scale-yielding solution for an elastic-power law hardening solid. It is found that the size of the dominance zone for the near-tip asymptotic field is the intrinsic material lengthl. For mode II fracture under the small-scale-yielding condition, transition from the remote classical K _IIfield to the near-tip asymptotic field in strain gradient plasticity goes through the HRR field only when K _IIis relatively large such that the plastic zone size is much larger than the intrinsic material length l. For mode I fracture under small-scale-yielding condition, however, transition from the remote classical K _I field to the near-tip asymptotic field in strain gradient plasticity does not go through the HRR field, but via a plastic zone. This revised version was published online in July 2006 with corrections to the Cover Date.     

Finite elements for materials with strain gradient effects

A finite element implementation is reported of the Fleck–Hutchinson phenomenological strain gradient theory. This theory fits within the Toupin–Mindlin framework and deals with first‐order strain gradients and the associated work‐conjugate higher‐order stresses. In conventional displacement‐based approaches, the interpolation of displacement requires C¹‐continuity in order to ensure convergence of the finite element procedure for higher‐order theories. Mixed‐type finite elements are developed herein for the Fleck–Hutchinson theory; these elements use standard C⁰‐continuous shape functions and can achieve the same convergence as C¹ elements. These C⁰ elements use displacements and displacement gradients as nodal degrees of freedom. Kinematic constraints between displacement gradients are enforced via the Lagrange multiplier method. The elements developed all pass a patch test. The resulting finite element scheme is used to solve some representative linear elastic boundary value problems and the comparative accuracy of various types of element is evaluated. Copyright © 1999 John Wiley & Sons, Ltd.     

Constraint effects on multiple void interaction in pressure sensitive plastic solids

In this work, effects of pressure sensitive yielding and plastic dilatancy on void growth and void interaction mechanisms in fracture specimens displaying high and low constraint levels are investigated. To this end, large deformation finite element simulations are carried out with discrete voids ahead of the notch. It is observed that multiple void interaction mechanism which is favored by high initial porosity is further accelerated by pressure sensitive yielding, but is retarded by loss of constraint. The resistance curves predicted based on a simple void coalescence criterion show enhancement in fracture resistance when constraint level is low and when pressure sensitivity is suppressed.     

C0 solid elements for materials with strain gradient effects

Experiments conducted by various researchers in the past few decades have shown that materials display strong size effects when the material and characteristic length scales associated with non‐uniform plastic deformation are of the same order at micron and submicron levels. The state of stress under such a condition was observed to be a function of both strain and strain gradient. The meso‐scale constitutive relation taking into account Taylor dislocation theory is briefly described. The conventional theory of mechanism‐based strain‐gradient (CMSG) plasticity incorporating the intrinsic material length scale is adopted in the formulation of a series of C⁰ solid elements of 20–27 nodes. The model is implemented in ABAQUS, a finite element package via a user subroutine. Convergent studies have been carried for the series of elements with classical as well as CMSG plasticity theories. Numerical results on a bar under constant body force and indentation at submicron level reinforce the observation that materials are significantly strengthened for deformation at micron and submicron levels and the effects of strain gradient cannot be ignored without significant loss of the accuracy of the results. Copyright © 2005 John Wiley & Sons, Ltd.     

Stress based fracture envelope for damage plastic solids

Recent development of damage plasticity theory shows the critical plastic strain at fracture for ductile solids depends on the pressure and the Lode angle on the octahedral plane along the loading path. The determination of the fracture strain envelope is usually a difficult and time consuming process. This is due to the experimental difficulties in maintaining a constant pressure and Lode angle at the fracture site, which is further complicated by the coupled nature of the parameters to be calibrated and the geometrical localization of the deformation. The fracture strain envelope is one of the key ingredients of the damage plasticity theory and relates to the accuracy of predicted results. In the present paper, the Lode angle dependence and the pressure sensitivity functions for the fracture strain envelope are derived from the hardening rule of the matrix using Tresca type fracture condition and Drucker-Prager formula, respectively. Quantitative analyses of Clausing’s and Bridgman’s test data are presented. Then a pressure modified maximum shear stress condition is adopted as fracture initiation condition to examine their joint effects on the fracture strain envelope. The relationship of the strain hardening, the pressure sensitivity and the Lode angle dependence are examined and verified by existing experimental results. We show that within the moderate range of stress triaxiality, the pressure modified maximum shear condition can be used as the fracture stress envelope for ductile metals within the framework of damage plasticity. The present method reduces significantly the amount of work to calibrate the material parameters for ductile fracture.     

Compositional effects on the fracture behaviour of alkali–silicate glasses

The two-point bend strength and the fracture toughness of a series of soda–potassia–silicate and soda–potassia–calcia–silicate glass fibres have been measured. There is a clear variation of mean strength with composition for the soda–potassia–silicate glasses, however, there is much less variation of mean strength with composition for the soda–potassia–calcia–silicate glasses. There is also a greater variation of fracture toughness with composition for the soda–potassia–silicate glasses than for the soda–potassia–calcia–silicate glasses. The mean strength, fracture toughness and inferred flaw sizes for the soda–potassia–calcia–silicate glasses are all less than the equivalent values for the soda–potassia–silicate glasses. These results are related to the structural models and durability of the glasses tested.     

A note on fracture criteria for interface fracture

Several criteria for interface fracture are examined and compared to test results obtained from glass/epoxy specimens. These include two energy release rate criteria, a critical hoop stress criterion and a critical shear stress criterion. In addition, approximate plastic zone size and shape within the epoxy are determined for these tests.     

The boundary-layer effect on the crack tip field in mechanism-based strain gradient plasticity

The theory of mechanism-based strain gradient (MSG) plasticity involves two material length parameters, namely the intrinsic material length land the mesoscale cell size l , which are on the order of a few microns and 0.1 m, respectively. Prior studies suggest that l has essentially no effect on the macroscopic quantities, but it may affect the local stress distribution. We demonstrate in this paper that there is a boundary layer effect associated with l in MSG plasticity, and the thickness of the boundary layer is on the order of   l ² big/l. By neglecting this boundary layer effect, a stress-dominated asymptotic field around a crack tip in MSG plasticity is obtained. This asymptotic field is valid at a distance to the crack tip between l and l(i.e., from 0.1 m to a few microns). The stress in this asymptotic field has an approximate singularity of r ^–2/3, which is more singular than not only the HRR field in classical plasticity but also the classical elastic Kfield (r ^–1/2). The stress level in this asymptotic field is two to three times higher than the HRR field, which provides an alternative mechanism for cleavage fracture in ductile materials observed in experiments.     

Constraint effects on plastic crack-tip fields for plane strain mode-I, II and III cracks in non-hardening materials

To explore constraint effects on fully plastic crakc-tip fields, analytical solutions are examined for mode-I, II and III loading in non-hardening materials under plane strain conditions. The results reveal that under mode-II and III loading the crack-tip stress fields are unique, and thus can be characterized by a `single parameter'. Under mode-I loading, however, the crack-tip stress field is non-unique but can be characterized by two sets of solutions or `two parameters'. One set of the solutions is the well-known Prandtl field and the other is a plastic T-stress field. This conclusion corroborates the observation of McClintock (1971) that the slip-line field is non-unique for plane strain tensile cracks. A two-term plastic solution which combines the Prandtl field and the plastic T-stress field with two parameters B ₁ and B ₂ can then characterize the crack-tip stress field of plane strain mode-I crack over the plastic region and quantify the magnitude of crack-tip constraints. These characters are similar to those for hardening materials. Analyses and examples show that the two-term plastic solution can match well with the slip-line field or finite element results over plastic region. Thus the parameters B ₁ and B ₂ can be used to characterize the constraint level for mode-I finite-sized crack specimens in non-hardening materials under plane strain conditions.     

Mode III crack growth in linear hardening materials with strain gradient effects

The flow-theory version of couple stress strain gradient plasticity is adopted for investigating the asymptotic fields near a steadily propagating crack-tip, under Mode III loading conditions. By incorporating a material characteristic length, typically of the order of few microns for ductile metals, the adopted constitutive model accounts for the microstructure of the material and can capture the strong size effects arising at small scales. The effects of microstructure result in a substantial increase in the singularities of the skew-symmetric stress and couple stress fields, which occurs also for a small hardening coefficient. The symmetric stress field turns out to be non-singular according to the asymptotic solution for the stationary crack problem in linear elastic couple stress materials. The performed asymptotic analysis can provide useful predictions about the increase of the traction level ahead of the crack-tip due to the sole contribution of the rotation gradient, which has been found relevant and non-negligible at the micron scale.     

The mode III full-field solution in elastic materials with strain gradient effects

The near-tip asymptotic field and full-field solution are obtained for a mode III crack in an elastic material with strain gradient effects. The asymptotic analysis shows that, even though the near-tip field is governed by a single parameter B (similar to the mode III stress intensity factor), the near-tip field is very different from the classical K_III field; stresses have r ^-3/2 singularity near the crack tip, and are significantly larger than the classical K _III field within a zone of size l to the crack tip, where l is an intrinsic material length, depending on microstructures in the material. This high-order stress singularity, however, does not violate the boundness of strain energy around a crack tip. The parameter B of the near-tip asymptotic field has been determined for two anti-plane shear loadings: the remotely imposed classical K _III field, and the arbitrary shear stress tractions on crack faces. The mode III full-field solution is obtained analytically for an elastic material with strain gradient effects subjected to remotely imposed classical K _III field. It shows that the near-tip asymptotic field dominates within a zone of size 0.5 l to the crack tip, while strain gradient effects are clearly observed within 5l. It is also shown that the conventional way to evaluate the crack tip energy release rate would lead to an incorrect, infinite value. A new evaluation gives a finite crack tip energy release rate, and is identical to the J-integral. This revised version was published online in August 2006 with corrections to the Cover Date.     

Correlation of material constraint with fracture toughness of interface regions in a dissimilar metal welded joint

In this work, the constraint parameter A_p based on crack‐tip equivalent plastic strain was calculated by finite element analyses for the cracks located at different locations in two interface regions in a dissimilar metal weld joint (DMWJ). The capabilities of the parameter A_p for characterizing material constraint and establishing correlation of material constraint with fracture toughness of the interface region cracks have been examined. The results show that the parameter A_p can characterize material constraint effect caused by material mismatch and initial crack positions in the interface regions. Based on the A_p, the correlation lines and formulae of material constraint with fracture toughness of the interface region cracks in the DMWJ can be established, and they may be used for obtaining material constraint‐dependent fracture toughness for the interface region cracks. The results in this work combining with those in the previous studies indicate that the parameter A_p may be a unified constraint parameter that can characterize both geometry constraint (including in‐plane and out‐of‐plane constraints) and material constraint, and it may be used in accurate fracture assessments of welded components with different geometry and material constraints.     

微电子材料在微塑成形中的尺度效应

微电子材料在微机电系统（MEMS）的发展中越来越受到青睐，但是其工艺加工的不足限制了实际应用的步伐。微塑性成形可以成形微电子器件，由于其尺寸微型化，在微塑性成形中存在一个不可避免的“尺度效应”问题，尺度效应表现在材料的流动行为、成形中摩擦效应和实验结果的分散性上。在介绍尺度效应的基础上对其进行了分类，给出了判断标准，并从流动应力、晶粒尺度、摩擦效应和温度效应等方面综述了尺度效应对微塑性成形的影响。由于基于连续介质的传统塑性力学理论无法解释微塑性成形过程中的尺度效应，因此引入了非均匀介质的塑性应变梯度理论并进行了探讨，最后指出了尺度效应的研究发展方向，从而促进微电子材料的开发应用。