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.     

A treatment of interfacial cracks in the presence of friction

Frictional sliding on interface crack surfaces results in weak crack tip stress singularity and zero strain energy release rate. A fracture criterion based on finite extension strain energy release rate, is proposed to capture the intrinsic fracture toughness. The finite extension strain energy release rate is shown to represent the magnitude of the singular stress field. Numerical simulations of a center crack in a bimaterial infinite medium under remote shear as well as fiber pull-out and push-out in composite materials are presented to illustrate the frictional effect in both small and large scale contacts near the crack tip.     

界面具有垂直裂纹的热障涂层的失效模拟

采用有限元分析软件ANSYS构造了一个在热生长氧化层（TGO）与陶瓷层界面具有一个垂直裂纹的纳米结构热障涂层的有限元模型。并计算了在热震过程中裂纹处的应力分布图,及裂纹尖端的应力场强度因子K1变化图。计算结果表明：裂纹处存在应力集中现象,且裂纹尖端的应力场强度因子K1在热障涂层热循环的冷却过程中随着时间的延长而减小,且在冷却最开始阶段,温度梯度变化最大,K1值也变化最大,裂纹在冷却的初始具有最大的扩展可能性。且涂层最有可能发生开裂失效。     

Three-dimensional thermal analysis of an infinite solid containing an elliptical surface crack

This paper deals with the thermal problem of an infinite solid with an elliptical insulated surface crack subjected to a uniform heat flow. Using conformal mapping technique, the elliptical crack region is first mapped conformally onto a penny-shaped crack for which the solution on the crack surface is available. The complete solution of the temperature field of the entire solid studied is then obtained by the inverse Fourier transform technique and the singularity of temperature gradient on the crack surface near the crack front can be found. To explore the temperature gradient along and around the crack front further, a three-dimensional finite element model with collapsed quarterpoint singular elements around the crack front is employed. Several examples with various crack aspect ratios are solved analytically and numerically. The influence of the elliptical insulated surface crack on the local intensification of temperature gradient and heat flow is also illustrated.     

Thermal stress singularities at tips of a crack in a semi-infinite medium under uniform heat flow

In the framework of plane thermoelastic problems is discussed the thermal stress field near the tips of an arbitrarily inclined crack in an isotropic semi-infinite medium with the thermally insulated edge surface under uniform heat flow. The crack is replaced by continuous distributions of quasi-Volterra dislocations corresponding to line heat sources and edge dislocations, and we obtain a set of simultaneous singular integral equations for dislocation density functions, whose solution is given in the forms of series in terms of Tchebycheff polynomials of the first kind. By means of this method, the thermal stress singularities at the crack tips are estimated exactly and the stress intensity factors can be readily evaluated. Numerical results are given for the particular case where the surface of the inclined crack is maintained at constant temperature and the heat supplied across the surface of the crack vanishes as a whole. The effects of the distance from the crack tip to the edge surface of the semi-infinite medium and the angle of inclination of the crack on the stress intensity factors and the initial direction of crack extension are shown graphically.     

The thermal effect of piezoelectric medium for anti-plane problem under high electrical impact loading

In present paper, the anti-plane problem of thermal effect near crack tip region of piezoelectric material subjected to electrical impact loading is investigated by means of the integral transforms and the singular integral equations. By introducing the thermal power, the temperature field near crack tip is finally obtained on the basis of the hypotheses of the uncoupling between the thermal field and the electro-mechanical fields and the adiabatic approximation. The numerical results indicate that a high temperature field of small region near crack tip is deduced when high electrical impact load is applied. Moreover, the results show that the temperature field strongly depends on crack size. However, the thermal effect of mechanical impact comparing with electrical impact may almost be neglected.     

Analysis of temperature distribution near the crack tip under constant amplitude loading

An analytical/numerical method has been developed to find the temperature rise near the crack tip under fatigue loading. The cyclic plastic zone ahead of the crack tip is assumed to be the shape of the source of heat generation and some fraction of plastic work done in cyclic plastic zone as heat generation. Plastic work during fatigue load was found by obtaining stress and strain distribution within the plastic zone by Hutchinson, Rice and Rosengren (HRR) crack tip singularity fields applied to small scale yielding on the cyclic stress strain curve. A two‐dimensional conduction heat transfer equation, in moving co‐ordinates, was used to obtain temperature distribution around the crack tip. Temperature rise was found to be a function of frequency of loading, applied stress intensity factor and thermal properties of the material. A power–law relation was found between the rise in temperature at a fixed point near the crack tip and range of stress intensity factor.     

Plane Elastic Problem of a Crack Normal to and Terminating at Bimaterial Interface of Isotropic and Orthotropic Half Plates

In this paper, the displacement and stress fields for a crack normal to and terminating at a bimaterial interface of isotropic and orthotropic half planes are studied as a plane problem. The eigenequation, by which the order of stress singularity is determined, is given in an explicit form. A discriminant function is presented to judge whether the stress singularity at the crack tip is greater than -1/2 or not. An explicit closed form expression is derived for the displacement and stress distribution near the crack tip.     

Magnetoelastic coupling on soft ferromagnetic solids with an interface crack

Summary Soft ferromagnetic materials with an interface crack in a strong magnetic field are considered. The complex potentials for the plane problem are deduced based on Pao and Yeh's linear model of magnetoelasticity. The solution of coupling fields is obtained by solving Reimann continuation problems. Finally, the effects of the applied magnetic field and the material constants on the stress singularity near the crack tip are discussed by examples for some representative soft ferromagnetic materials.     

Increase of stress intensity near interface edge of elastic-creep Bi-material under a sustained load

The transition from small-scale creep to large-scale creep ahead of a crack tip or an interface edge with strong elastic stress singularity at the loading instant causes stress relaxation and the decrease of stress intensity in general. However, this study shows that the stress near the interface edge of bi-material with no or weak elastic stress singularity increases after the loading instant and brings about the stress concentration during the transition. In addition, the creep strain distribution of this bi-material after the loading instant is different from that occurred in the transition of an interface edge with strong elastic stress singularity or a crack tip (notch root). The criterion for the increase or decrease of stress intensity near the interface edge proved by the finite element method is proposed in this study. The stress intensity near the interface edge increases when the elastic stress singularity is lower than the creep stress singularity (λ^el < λ^cr) and vice versa.     

Dynamic crack growth along an elastoplastic bimaterial interface

Summary The near-tip stress and deformation rate fields of a crack dynamically propagating along an interface between dissimilar elastic-plastic bimaterials are presented in this paper. The elastic-plastic materials are characterised by theJ ₂-flow theory with linear plastic hardening. The solutions are assumed to be of variable-separable form with a power-law singularity in the radial direction. Two distinct solutions corresponding to the tensile and shear solutions exist with slightly different singularity strengths and very different mixities at the crack tip. The phenomenon of discrete and determinate mixities at the interfacial crack tip is confirmed in dynamic crack growth. This is not an artifact of the variable-separable solution assumption, arising from the linear-hardening material model. The dynamic crack analysis shows that the mixity of the near-tip field is mainly determined by the given material parameters and affected slightly by the crack propagation velocity. A significant variation of the mixity is observed near to the coalescing point of the tensile and shear solutions. The strength of the singularity is almost determined by the smaller strain-hardening alone, and dynamic inertia decreases the stress intensity. The asymptotic solutions reveal that the crack propagation velocity changes only the stress field of the tensile mode significantly. With increasing the crack propagation velocity, the stress singularity of the tensile solutions decreases obviously and the stress triaxiality at the tip (=0) falls considerably at the unity effective stress. These observations imply that the fracture toughness of the interface crack under tensile mode may be significantly higher than that under quasi-static conditions.     

Singularities of an inclined crack terminating at the bi-material interface in a Reissner plate

On the basis of Reissner’s plate theory, the stress singularities at the tip of an arbitrarily inclined semiinfinite crack terminating at the interface of two dissimilar materials are investigated in the present paper. Using the eigenfunction expansion method, the eigenequation of the corresponding problem is derived explicitly by directly solving the governing equations of Reissner’s plate theory in terms of three generalized displacement components. In this paper, the focus is on the calculation of the singularity order as a fundamental quantity in fracture mechanics. The singularity orders of the moments and shear force at the crack tip are determined by the dominant eigenvalues whose real parts lie between 0 and 1. The influences of the bi-material parameters and the crack inclination angle on the moment and shear force singularity orders are discussed in detail. Specifically, the variations of the shear force singularity order with the bi-material parameter and the crack inclination angle are examined in detail. It is proved that the shear force singularity order is a completely monotonic function of the bi-material parameter and the inclination angle. Some numerical results are given in order to prove the validity of the present study.     

Numerical analysis of plane cracks in strain-gradient elastic materials

The classical linear elastic fracture mechanics is not valid near the crack tip because of the unrealistic singular stress at the tip. The study of the physical nature of the deformation around the crack tip reveals the dominance of long-range atomic interactive forces. Unlike the classical theory which incorporates only short range forces, a higher-order continuum theory which could predict the effect of long range interactions at a macro scale would be appropriate to understand the deformation around the crack tip. A simplified theory of gradient elasticity proposed by Aifantis is one such grade-2 theory. This theory is used in the present work to numerically analyze plane cracks in strain-gradient elastic materials. Towards this end, a 36 DOF C¹ finite element is used to discretize the displacement field. The results show that the crack tip singularity still persists but with a different nature which is physically more reasonable. A smooth closure of the structure of the crack tip is also achieved.     

Mode-I crack problem for functionally graded layered structures

This paper deals with two bonded functionally graded finite strips with two collinear cracks. Different layers may have different nonhomogeneity properties in the structure. A bi-parameter exponential function was introduced to simulate the continuous variation of material properties. The problem is solved by using the integral transform, singular integral equation methods and the theory of residues. Various internal cracks and edge crack and crack crossing the interface configurations are investigated, respectively. The asymptotic stress field near the tip of a crack crossing the interface is examined and it is shown that, unlike the corresponding stress field in piecewise homogeneous materials, in this case the “kink” in material property at the interface does not introduce any singularity. Numerical calculations are carried out, and the influences of nonhomogeneity constants, geometric parameters and crack interactions on the stress intensity factors are investigated.     

含界面裂纹圆形夹杂的十二次对称二维准晶热应力分析

针对点热源作用下,无限大十二次对称二维准晶基体和圆形弹性夹杂界面之间含多条裂纹的问题进行了研究。基于复变函数分区全纯理论、留数定理、广义 Liouville 定理、Riemann-Schwarz 解析延拓定理及复应力函数奇性主部分析方法,获得了集中热源作用于准晶基体内任意一点时,准晶基体和圆形弹性夹杂内外温度场、声子场热应力的一般复势解。由此获得了含一条界面裂纹和两条界面裂纹时温度场以及声子场热应力的封闭形式解答,将所得结果与已有结果进行了对比,验证了该方法的有效性。最后通过数值算例分析了夹杂半径、点热源强度及裂纹角度对热应力和裂纹尖端热应力强度因子的影响规律。结果表明：随着热源强度的增大,裂纹尖端的声子场热应力也逐渐增大;随着裂纹角度的增大,裂纹尖端的声子场热应力强度因子变大;随着半径的增大,热应力强度因子的变化趋势越来越明显,并且取得的峰值越高,即裂纹角度和夹杂半径的增加,促进了裂纹的扩展。这些结论为准晶材料的结构设计和使用提供了科学依据。     

Effect of thermomechanical loads on the propagation of crack near the interface brittle/ductile

The purpose of this paper is to understand the combined effect of thermal and mechanical loading on the initiation and behaviour of sub-interface crack in the ceramic. In this study a 2D finite element model has been used to simulated mixed mode crack propagation near the bimaterial interface. The assembly ceramometalic is subjected simultaneously to thermomechanical stress field. The extent of a plastic zone deformation in the vicinity of the crack-tip has a significant influence on the rate of its propagation. The crack growth at the joint specimen under four-point bending (4PB) loading and the influence of residual stresses was also evaluated by the maximum tensile stress criterion. The J-integral at the crack tip is generally expressed by the thermomechanical local stresses. The results obtained show the effect of the temperature gradient ΔT, the size of the crack and the applied stresses on the J-integral.     

Elastic green's function for a bimaterial composite solid containing a crack inclined to the interface

An analytical closed-form expression is derived for the elastic Green's function of a bimaterial composite solid containing a planar interface and a straight crack inclined at an arbitrary angle with the interface. The crack tip is assumed to be at the interface. Both the constituent materials of the composite are assumed to anisotropic. The Green's function satisfies the interfacial boundary conditions of continuous tractions and displacements, and zero tractions at the crack surfaces. The boundary conditions are satisfied by using the virtual force technique. The determination of the virtual forces requires solutions of a Hilbert problem which is obtained by using an orthogonal complex transform. The method is illustrated by applying it to a copper/nickel composite. The Green's function should be useful in the boundary-element method of calculating the stress and the displacement field in the solid.     

Variable power singular interface elements for a crack normal to the bimaterial interface

Zero thickness crack tip interface elements for a crack normal to the interface between two materials are presented. The elements are shown to have the desired r^λ−1 (0 < λ < 1) singularity in the stress field at the crack tip and are compatible with other singular elements. The stiffness matrices of the quadratic and cubic interface element are derived. Numerical examples are given to demonstrate the applicability of the proposed interface elements for a crack perpendicular to the bimaterial interface.     

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.