Dynamic crack-tip field for tensile cracks propagating in power-law hardening materials

The near-tip field of a dynamically propagating crack in an incompressible power-law hardening material is studied using the asymptotic method as an extension of Leighton et al. (Journal of the Mechanics and Physics of Solids, 1987, Vol.35, pp.541–563). The crack is subjected to tensile loads and propagates steadily under plane strain conditions. The material deformation is described by the J ₂ flow theory of plasticity with infinitesimal displacement gradients. Results show that, in the present crack-tip field, (a) the angular variations of stresses, strains and particle velocities around the crack tip are fully continuous; (b) the stresses and strains at the crack tip are bounded; (c) there is a free parameter _eq0 which cannot be determined in the asymptotic analysis. Comparisons indicate that the present asymptotic solution matches well with full-field numerical results, and the parameter _eq0 can characterize the effects of the far field on the crack-tip field. Furthermore, the present solution approaches that of Leighton et al. (1987) in the limit as the material hardening exponent goes to infinity, but does not reduce to the accepted solution for quasi-statically growing cracks in the limit as the crack speed goes to zero.     

On the process zone of a quasi-static growing tensile crack with power-law elastic-plastic damage*

Parametric studies are made on the growing crack tip fields under quasi-static tensile loading with damage coupled power-law elastic-plastic constitutive equations. A two-sector demarcation scheme is used. The asymptotic orders and angular distributions of the stress and damage fields are obtained. The shapes of the process zone are also determined. The law of crack growth is formulated as well.     

Crack-tip fields in elastic-plastic material under plane stress mode I loading

New results on the crack-tip fields in an elastic power-law hardening material under plane stress mode I loading are presented. Using a generalized asymptotic expansion of the stress function, higher-order terms are found which have newly-discovered characteristics. A series solution is obtained for the elastic-plastic crack-tip fields. The expansion of stress fields contains both the and terms where t_i is real and t_k is complex; the terms σ⁽ⁱ⁾ _pq(θt_i) and σ^(k) _rsθt_k) are real and complex functions of θ respectively. Comparing the results with that for the plane strain mode I loading shows that: (1) the effect of higher-order solutions on the crack-tip fields is much smaller; and (2) the path-independent integral J also controls the second-order or third-order term in the asymptotic solutions of the crack-tip fields for most of the engineering materials (1 < n < 11) in plane stress, while the J-integral does not control the second and the third-order terms for the plane strain mode I case for n > 3. These theoretical results imply that the crack-tip fields can be well characterized by the J-integral, and can be used as a criterion for fracture initiation under plane stress mode I loading. This is in agreement with existing full-field solutions and experimental data that J at crack growth initiation is essentially independent of in-plane specimen geometry. The comparison confirms the theoretical asymptotic solutions developed in this study. This revised version was published online in August 2006 with corrections to the Cover Date.     

Near-tip dynamic fields for a crack advancing in a power-law elastic-plastic material: Modes I, II and III

Near-tip dynamic asymptotic stress fields of a crack advancing in an incompressible power-law elastic-plastic material are presented. It is shown that the stress- and strain-singularity are, respectively, of the order (In(R₀/r))^1/(n−1) and (In(R₀/r))^n/(n−1), where R₀ is a length parameter, r measures distance from the crack tip, and n is the power-law exponent. The angular variations of these fields are identical with those corresponding to dynamic crack growth in an elastic-perfectly-plastic material (Gao and Nemat-Nasser, 1983a,b).     

Plane Stress Steady Crack Growth in a Power-Law Hardening Material

The paper reports the results from an asymptotic analysis for a crack growing quasistatically under Mode I, plane stress conditions for a power-law hardening material. The asymptotic stress and deformation velocity fields near the growing crack tip are determined, comparisons to related work are discussed and some numerical results for aluminum are included. This revised version was published online in August 2006 with corrections to the Cover Date.     

Significance of morphology changes at a propagating crack edge

During dynamic crack propagation, several distinct changes in the morphology of the dissipative region near the crack edge occur, and they have a pronounced influence on the main propagation mechanism. There are also distinct morphological differences between mode I and mode II. For mode I crack expansion, four successive generations of localization may be observed: micro-separations coalescing with the main crack, protruding clusters of micro-separations, micro-branches, and finally (macro-)branches. The region of localizations is increasing laterally from the main crack plane with crack growth and velocity, as it appears, because of high normal stresses in planes normal to the crack direction. If sufficient space is available, an expanding mode I crack accelerates to a constant velocity, which appears to prevail even after branching and multiple-branching. This indicates an amazing self-similarity over the four generations of localization. The morphology changes during crack propagation depend both on the magnitude of the applied load and on the travelled length of the crack edge. For mode II, the energy dissipation seems generally to be much more concentrated to the crack plane than for mode I. A main reason appears to be that normal stresses in planes normal to the crack direction are comparatively small in front of the crack. Therefore, strong micro-separation localizations seem to appear mainly in shear planes parallel with the crack plane. The appearance of such localizations may be analogous to the remarkable flow velocity gradient discontinuity discovered in turbulent shear flow near a wall.As a consequence of the apparently stronger concentration of the dissipative region to the crack plane, a mode II crack can reach higher velocities than a mode I crack, and it may even reach intersonic velocities.     

Asymptotic analysis for a crack on interface of damaged materials

An asymptotic analysis for a crack lying on the interface of a damaged plastic material and a linear elastic material is presented in this paper. The present results show that the stress distributions along the crack tip are quite similar to those with HRR singularity field and the crack faces open obviously. Material constants n, μ and m0 are varied to examine their effects on the resulting stress distributions and displacement distributions in the damaged plastic region. It is found that the stress components σrr, σθ θ, σr θ and σe are slightly affected by the changes of material constants n, μ and m0, but the damaged plastic region are greatly disturbed by these material parameters. This revised version was published online in July 2006 with corrections to the Cover Date.     

Dynamic crack growth in anisotropic material

Dynamic propagation of a crack in an anisotropic material under uniform stress is studied. The crack is assumed to nucleate from an infinitesimal micro-crack and expands with a constant velocity. Explicit expressions for the stress intensity factor and the energy release rate are derived.     

Simulation of crack growth in composite material shell structures

This paper implements a domain integral energy method for modelling crack growth in composite material shell structures using the finite element method. Volume integral expressions to evaluate the dynamic energy release rate in a through‐thickness three‐dimensional crack are derived. Using the domain integral, the energy release rate computation is implemented in the DYNA3D explicit non‐linear dynamic finite element analysis program wherein crack propagation is modelled by releasing the constraints between initially constrained node pairs. The implementation enables the program to either determine the energy resistance response for the material (provided experimental data is available) or predict the rate of crack propagation in shell structures. The numerical implementation was verified by simulating mode I and mode III slow crack growth problems in semi‐infinite transversely isotropic media, for which analytic solutions are available. Oscillations of energy following the release of nodal constraints as the crack propagates in discrete increments were suppressed using light mass proportional damping and a moving averaging scheme. Copyright © 2004 John Wiley & Sons, Ltd.     

Transient analysis of dynamic crack propagation in piezoelectric materials

Abstract

In this paper, the transient analysis of semi‐infinite propagating cracks in piezoelectric materials subjected to dynamic anti‐plane concentrated body force is investigated. The crack surface is assumed to be covered with an infinitesimally thin, perfectly conducting electrode that is grounded. In analyzing this problem, it has characteristic lengths and a direct attempt towards solving this problem by transform and Wiener‐Hopf techniques (Noble, 1958) is not applicable. In order to solve this problem, a new fundamental solution for propagating cracks in piezoelectric materials is first established and the transient response of the propagating crack is obtained by superposition of the fundamental solution in the Laplace transform domain. The fundamental solution to be used is the responses of applying exponentially distributed traction in the Laplace transform domain on the propagating crack surface. Taking into account the quasi‐static approximation, exact analytical transient solutions for the dynamic stress intensity factor and the dynamic electric displacement intensity factor are obtained by using the Cagniard‐de Hoop method (Cagnard, 1939; de Hoop, 1960) of Laplace inversion and are expressed in explicit forms. Numerical calculations of dynamic intensity factors are evaluated and the results are discussed in detail. The transient solutions for stationary cracks have been shown to approach the corresponding static values after the shear wave of the piezoelectric material has passed the crack tip.     

Dynamics steady crack propagation of antiplane shear in an elastic perfectly plastic material

Abstract

A theoretical analysis of steady‐state crack growth in an elastic ideally‐plastic material under small‐scale yielding conditions has been carried out for anti‐plane shear. Asymptotic expansion method is used to construct the solutions for the region near the crack line. Exact solutions for the distribution of strain on the crack line within the primary active plastic zone is obtained. It is shown that the solution reduces to the correct asymptotic form as the crack speed approaches zero (quasi‐static) for any point on the crack line. The results are used to discuss the applicability of quasi‐static solutions to moving steady‐state situations. It is found that if the crack propagation speed is less than 0.1 of the shear wave speed, the quasi‐static solutions can be accurately approximated for the steady state solutions.     

Simulation of dynamic crack curving and branching under biaxial loading by a time domain boundary integral equation method

Bifurcation and trifurcation of a fast running crack under various biaxial loading conditions is investigated numerically. The solution procedure for the 2D model in the framework of linear elastodynamics employs a time-domain boundary element method and allows for arbitrary curvilinear crack propagation. Branching events are controlled by the criterion of a critical mode I stress intensity factor while the propagation direction and growth rate of each branch are determined from the criterion of maximum circumferential stress. Numerical results are compared with experimental findings and are discussed with respect to macroscopic and microscopic aspects of dynamic fracture.     

Fatigue crack growth simulation in heterogeneous material using s-version FEM

A fully automatic fatigue crack growth simulation system is developed using the s-version Finite Element Method (s-FEM). This system is extended to fractures in heterogeneous materials. In a heterogeneous material, the crack tip stress field has a mixed-mode condition, and the crack growth path is affected by inhomogeneous materials and mixed-mode conditions. Stress intensity factors (SIFs) in the mixed-mode condition are evaluated using the virtual crack closure method (VCCM). The criteria for the crack growth amount and crack growth path are based on these SIFs, and the growing crack configurations are obtained. At first, the basic problem is solved, and the results are compared with some results available in the literature. It is shown that this system gives an adequate accurate estimation of the SIFs. Then, two-dimensional fatigue crack growth problems are simulated using this system. The first example is a plate with an interface between hard and soft materials. The cracks tend to grow in soft materials through the interface. A second example is a plate with distributed hard inclusions. The crack takes a zig-zag path by propagating around the hard inclusions. In each case, the crack growth path changes in a complicated manner. Changes of the SIFs values are also shown and discussed.     

Computational dynamic fracture mechanics

This paper provides a review on the state of the art in computational dynamic fracture mechanics. The following important essential ingredients in computational dynamic fracture mechanics are included: (i) fundamental aspects of dynamic fracture mechanics, (ii) types of fracture simulation, (iii) computational models of dynamic crack propagation, and (iv) use of dynamic J-integral in computational models. In the item (i), a special attention is focused on the asymptotic eigen fields for various states of dynamic crack tips, which provide the foundation of dynamic fracture mechanics as Williams' asymptotic eigen solutions provided the foundation of static linear fracture mechanics. In the item (ii), a new concept of mixed-phase simulation is presented for general nonself-similar crack propagation, in addition to the generation-phase and application-phase simulations. A comprehensive summary of computational models for dynamic crack propagation is given in the item (iii). Finally, in the item (iv) several attractive features of the dynamic J-integral are presented. This revised version was published online in July 2006 with corrections to the Cover Date.     

双悬臂梁试件裂纹动态扩展的准静态数值分析

本文采用双悬臂梁（DCB）试件研究了复合材料层合板层间插入韧性胶膜（Interleaf）层的Ⅰ型断裂行为。试验结果表明,含和不含Interleaf层试件分别呈现脆性非稳态和脆性稳态分层扩展特性。针对非稳定裂纹扩展问题,依据动态断裂力学中应变能释放率与动能变化率的关系,提出了以断裂韧性值G_IC变化来抵消动能变化对裂纹扩展过程影响的准静态分析方法,根据试验中裂纹扩展的韧性变化,推导出适用于准静态裂纹扩展模拟的等效韧性G_IC*,利用ABAQUS平台和虚裂纹闭合技术（VCCT）建立了三维有限元计算模型;实现了从起裂到止裂的整个裂纹动态扩展过程的数值模拟,揭示了非稳定裂纹扩展过程中一些复杂的力学现象。      

A perturbation solution for a crack in a power-law material under gross yielding

Abstract In order to develop an analytical method for quantifying the plastic-blunting behaviour of a short crack embedded in a notch plastic zone, the perturbation solution of He and Hutchinson is extended to include the effect of strain gradient. An edge-cracked plate subjected to a linearly varying remote strain is considered in this work to simulate the plastic deformation associated with a small crack at a notch root. The strain hardening of the material is assumed to obey a power-law. Comparison with finite-element (FE) computations demonstrates that this perturbation solution provides accurate values for the crack-tip opening displacement (CTOD) under gross-yielding conditions for a range of hardening parameters.     

Interface crack problem in elastic dielectric/piezoelectric bimaterials

By considering an isotropic elastic dielectric material as a transversely isotropic piezoelectric material with little piezoelectricity, the interface crack problem in elastic/piezoelectric bimaterials is treated in this paper based on Stroh's complex potential theory (1958) with the impermeable crack model. In order to obtain universal results, Numerical results of the near tip stress field and the electric field for 35 kinds of dissimilar bimaterials constructed by five kinds of elastic dielectric materials, namely Epoxy, Polymer, Al₂O₃, SiC and Si₃N₄, and seven kinds of piezoelectric ceramics, namely PZT-4, BaTiO₃, PZT-5H, PZT-6B, PZT-7A, P-7, and PZT-PIC151, are presented. It is concluded that all the combinations lead to the same results: in which the first crack tip singularity parameter does not vanish whereas the second parameter always vanishes. From the physical point of view, an interface crack in such a dissimilar material shows a similar oscillating singularity as that in dissimilar elastic bimaterials. Moreover, by using a maximization value technique, the regular inverse square root singularity r ^–1/2 of the stress and the electric field at the crack tip can be realized, although, theoretically, an interface crack in such bimaterials possesses the well-known oscillating singularity r ^{–1/2± i}. Of great significance is that, in the absence of mechanical loadings, a purely electric loading can induce relative large model I or II stress intensity factor for a interface crack in some elastic/piezoelectric bimaterials, which implies that the electric-induced failure may be realized in such bimaterials.     

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.     

Transient response of a piezoelectric material with a semi-infinite mode-III crack under impact loads

The problem of a semi-infinite impermeable mode-III crack in a piezoelectric material is considered under the action of impact loads. For the case when a pair of concentrated anti-plane impact loads and electric displacements are exerted symmetrically on the upper and lower surfaces of the crack, the asymptotic electroelastic field ahead of the crack tip is determined in explicit form. The dynamic intensity factors of electroelastic field and dynamic mechanical strain energy release rate are obtained. The obtained results can be taken as fundamental solutions, from which general results may directly be evaluated by integration. The method adopted is to reduce the mixed initial-boundary value problem, by using the Laplace and Fourier transforms, into two simultaneous dual integral equations. One may be converted into an Abel's integral equation and the other into a singular integral equation with Cauchy kernel, and the solutions of both equations can be determined in closed-form, respectively. For some particular cases, the present results reduce to the previous results.     

Characteristics of three-dimensional stress fields in plates with a through-the-thickness crack

Three-dimensional finite element analyses were performed on plates with a through-the-thickness crack. Global-local finite element technique with sub-modeling was used to achieve the refinement required to obtain an accurate stress field. The existence of a weaker singularity was verified, and a model was proposed to explain the behavior of stresses in the boundary layer. This model is able to account for the competing interaction between the inverse square root singular term and the vertex singular term. The energy release rate was calculated using the modified crack closure method and energy balance. A simple technique without 3-D calculation was suggested for evaluating an approximate 3-D stress intensity factor at the mid-plane. The effect of plate thickness on the size of the three-dimensional region was studied, and the validity of two-dimensional linear elastic fracture mechanics was discussed.