Weight functions for interface cracks in dissimilar anisotropic materials

Bueckner‘s work conjugate integral customarily adopted for linear elastic materials is established for an interface crack in dissimilar anisotropic materials. The difficulties in separating Stroh‘s six complex arguments involved in the integral for the dissimilar materials are overcome and then the explicit function representations of the integral are given and studied in detail. It is found that the pseudo-orthogonal properties of the eigenfunction expansion form (EEF) for a crack presented previously in isotropic elastic cases, in isotopic bimaterial cases, and in orthotropic cases are also valid in the present dissimilar arbitrary anisotropic cases. The relation between Bueckner‘s work conjugate integral and the J-integral in these cases is obtained by introducing a complementary stressdisplacement state. Finally, some useful path-independent integrals and weight functions are proposed for calculating the crack tip parameters such as the stress intensity factors.     

压电材料中的Bueckner功共轭积分及和J积分M积分的关系

研究横观各向同性压电材料中裂纹问题,提出了Bueckner功共轭积分在这类材料中的表达式：并通过引出两类辅助的应力-位移-电位移-电势场,证明功共轭积分和这类材料中的J积分和M积分仍然存在简单的两倍关系由此,各类在脆性材料断裂问题中已广泛应用的权函数方法可顺理成章地推广到压电材料的研究中来．这对独立地确定电位移强度因子和经典的I、II型应力强度因子提供了有力的数学上的工具．进而通过计算机械应变能释放率对压电材料中裂纹的稳定做出判断．     

Bueckner's work conjugate integrals and weight functions for a crack in anisotropic solids

The Bueckner work conjugate integrals are studied for cracks in anisotropic elastic solids. The difficulties in separating Lekhnitskii's two complex arguments involved in the integrals are overcome and explicit functional representations of the integrals are given for several typical cases. It is found that the pseudoorthogonal property of the eigenfunction expansion forms presented previously for isotropic cases, isotropic bimaterials, and orthotropic cases, are proved to be also valid in the present case of anisotropic material. Finally, Some useful path-independent integrals and weight functions are proposed. The project supported by the National Natural Science Foundation of China (19891180) and Doctorate Foundation of Xi'an Jiaotong University     

A new approach to the pseudo-orthogonal properties of eigenfunction expansion form of the crack-tip complex potential function in anisotropic and piezoelectric fracture mechanics

Over the past twenty years, the well-known weight function theory based on the Bueckner work conjugate integral has been widely used to calculate crack tip fracture dominant parameter such as the stress intensity factor, the energy release rate (or the J-integral) and the T-stress in various kinds of cracked materials (e.g. isotropic materials, anisotropic materials and piezoelectric materials). Meanwhile, the pseudo-orthogonal property of the eigenfunction expansion form of the crack tip stress complex potential function has been proved to play a very important role in the theory. In this paper, we provide a new approach to establish the pseudo-orthogonal properties for crack problems in anisotropic and/or piezoelectric materials. In the latter case associated with mechanical-electric coupling, the electrical boundary conditions under both impermeable and permeable crack models are considered. The approach developed is much simpler than the classical complex variable separation technique proposed by previous researchers and hence the cumbersome and lengthy manipulations are avoided. Moreover, it is shown that, unlike previous works, the orthogonal properties of the material characteristic matrices A and B induced by the Stroh theory are no longer necessary in establishing the pseudo-orthogonal properties of eigenfunction expansion form in cracked piezoelectric materials. The approach can be easily extended to treat many other different crack problems concerning the Bueckner integral involving several complex arguments.     

On the applicability of theI-integral in fracture mechanics

Summary The use of conservation integrals as fracture mechanics parameters has been extended since Rice's work on theJ-integral originally restricted to hyperelastic materials, to other classes of constitutive laws in the past few years. The so-calledI-integral being valid for any constitutive law seemeed to be suitable as a generalization of theJ-integral in the elastic-plastic regime.In this paper a proof is given that theI-integral vanishes under standard boundary conditions. Therefore, theI-integral has not the quality of a fracture mechanics parameter. A comparison betweenI andJ leads to a new representation of theJ-integral.

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Über die Anwendbarkeit desI-Integrals in der Bruchmechanik

Übersicht Die Verwendung von Erhaltungsintegralen als Bruchmechanikparameter wurde seit der Arbeit von Rice über dasJ-integral, das ursprünglich auf hyperelastisches Material beschränkt war, in den letzten Jahren auf andere Klassen von Materialgesetzen erweitert. Das sogenannteI-Integral, das für ein beliebiges Materialgesetz gültig ist, schien als Verallgemeinerung desJ-Integrals im elastisch-plastischen, Bereich in Frage zu kommen.In dieser Arbeit wird bewiesen, daß dasI-Integral unter Standardrandbedingungen stets verschwindet. Es ist daher kein Bruchmechanik-Parameter. Ein Vergleich zwischenI undJ führt zu einer neuen darstellung desJ-Integrals.

     

Energy release or absorption due to simultaneous rotation of two nano voids in plane elastic materials as influenced by both surface effect and interacting effect

In this paper, the L-integral analysis for two nano-sized voids in plane elasticity under uniaxial loading is present. Three surface parameters are considered including the surface tension and two surface Lamé constants. Attention is focused on the mutual influence on the L-integral from both the surface effect at voids’ rims and the interacting effect between voids. A close-form expression of L-integral for multiple nano voids is obtained. Comparing with those in macro fracture mechanics, the L-integral shows some different features when the surface effect is taken into account. It is concluded that under tensile loading and due to the mutual influence, the L-integral might be either positive or negative, depending on the loading level. The numerical results show that the surface tension is the dominant one in surface parameters on impacting the L-integral. It is also concluded that the surface effect shields the energy release (positive L-integral value) while enhances the energy absorption (negative L-integral value). The two-state L-integral analysis is performed to clarify the way that the surface effect impacts the L-integral. It is concluded that the contribution to L-integral from the voids’ configuration could either be negative or positive, while that from the surface effect is always negative. Besides, the size dependence in the present problem is studied in detail.     

A path-independent integral for fracture of solids under combined electrochemical and mechanical loadings

In this study, we first demonstrate that the J-integral in classical linear elasticity becomes path-dependent when the solid is subjected to combined electrical, chemical and mechanical loadings. We then construct an electro-chemo-mechanical J-integral that is path-independent under such combined multiple driving forces. Further, we show that this electro-chemo-mechanical J-integral represents the rate at which the grand potential releases per unit crack growth. As an example, the path-independent nature of the electro-chemo-mechanical J-integral is demonstrated by solving the problem of a thin elastic film delaminated from a thick elastic substrate.     

Arbitrarily oriented crack near interface in piezoelectric bimaterials

Arbitrarily oriented crack near interface in piezoelectric bimaterials is considered. After deriving the fundamental solution for an edge dislocation near the interface, the present problem can be expressed as a system of singular integral equations by modeling the crack as continuously distributed edge dislocations. In the paper, the dislocations are described by a density function defined on the crack line. By solving the singular integral equations numerically, the dislocation density function is determined. Then, the stress intensity factors (SIFs) and the electric displacement intensity factor (EDIF) at the crack tips are evaluated. Subsequently, the influences of the interface on crack tip SIFs, EDIF, and the mechanical strain energy release rate (MSERR) are investigated. The J-integral analysis in piezoelectric bimaterals is also performed. It is found that the path-independent of J₁-integral and the path-dependent of J₂-integral found in no-piezoelectric bimaterials are still valid in piezoelectric bimaterials.     

On Sanders' energy-release rate integral and conservation laws in finite elastostatics

Sanders showed in 1960, within the framework of two-dimensional elasticity, that in any body a certain integral I around a closed curve containing a crack is path-independent. I is equal to the rate of release of potential energy of the body with respect to crack length. Here we first derive, in a simple way, Sanders' integral I for a loaded elastic body undergoing finite deformations and containing an arbitrary void. The strain energy density need not be homogeneous nor isotropic and there may be body forces. In the absence of body forces, for flat continua, and for special forms of the strain energy density, it is shown that I reduces to the well-known vector and scalar path-independent integrals often denoted by J, L, and M.     

Analysis of L-integral and theory of the derivative stress field in plane elasticity

In this paper, analysis of the L-integral in plane elasticity is present. An infinite plate with any number of inclusions and cracks and with any remote tractions is assumed in analysis. Arbitrary forces are applied on the cracks, inclusions or at a point of the infinite medium. To study the problem, the concept of the derivative stress field is introduced, which is derived from a physical stress field. The mutual work difference integral (MWDI) is also introduced, which is defined as a difference of mutual works done by each other from the physical stress field and the derivative field. It is proved that the L(CR) (L-integral on a large circle) is equal to a particular MWDI. General expression for the L(CR) is obtained. For a given stress field, the variation of the L(CR) is studied when the coordinates have a translation or rotation. It is found that the L(CR) is an invariant with respect to the rotation of coordinates, and it has a variation when the coordinates have a translation.     

Fracture toughness evaluation of natural rubber

A method for estimating the fracture toughness of rubber-like materials is presented. Experimental data of a notched natural rubber sheet is analysed by application of the path-independent J-integral. A finite element code for large elastic deformations is used to evaluate the deformed shape of the rubber at crack growth initiation.Discussed is a stability criterion based on the existence of a critical value of J in relation with the experimental results.     

A domain independent integral expression for the crack extension force of a curved crack in three dimensions

An integral expression that is domain independent in curvilinear coordinates and compatible with zero divergence of Eshelby's (Phil. Trans. Roy. Soc. (London) 244 (1951) 87.) energy momentum tensor was obtained from the principle of virtual work. By applying Eshelby's definition of the force on a material defect a general expression of the crack extension force for a curved crack in three dimensions, here called the F-integral, was derived from the domain independent integral expression. The F-integral is given explicitly for a number of curved cracks and found to be in agreement with previously known solutions, when available. The influence of crack surface and crack front curvature upon the various forms of the F-integral is discussed. The F-integral presented in this work is a generalisation of the J-integral (Rice, J. Appl. Mech. 35 (1968) 379.) to curved cracks in orthogonal curvilinear coordinates.     

M-integral analysis for microcrack-damaged anisotropic composite materials

Summary  This paper presents an M-integral analysis for the microcracked anisotropic composite materials. By using an elementary solution derived for a single finite crack subjected to a concentrated force on crack faces, the problem of strong interacting, arbitrarily oriented and located microcracks in an anisotropic composite materials is reduced to a system of Fredholm integral equations. The crack-tip fracture parameters, such as the stress intensity factors, are evaluated from a numerical solution of the system of integral equations. Its dependence on the coordinate system, calculation, and physical interpretation of the M-integral are discussed in the interaction problem. Finally, a numerical example of the damage evaluation by the M-integral analysis is given. Received 24 September 1999; accepted for publication 8 February 2000     

Interaction between an interface crack and subinterface microcracks in metal/piezoelectric bimaterials

The J-integral analysis is presented for the interaction problem between a semi-infinite interface crack and subinterface matrix microcracks in dissimilar anisotropic materials. After deriving the fundamental solutions for an interface crack subjected to different loads and the fundamental solutions for an edge dislocation beneath the interface, the interaction problem is deduced to a system of singular integral equations with the aid of a superimposing technique. The integral equations are then solved numerically and a conservation law among three values of the J-integral is presented, which are induced from the interface crack tip, the microcracks and the remote field, respectively. The conservation law not only provides a necessary condition to confirm the numerical results derived, but also reveals that the microcrack shielding effect in such materials could be considered as a redistribution of the remote J-integral. It is this redistribution that does lead to the phenomenological shielding effect.     

Calculation of the J-integral for limited permeable cracks in piezoelectrics

Summary This paper deals with the calculation of the J-integral for electrically limited permeable cracks in piezoelectrics. The electromechanical J-integral is extended to account for electrical crack surface charge densities representing electric fields inside the crack. To avoid the costly implementation of the line integral along the crack faces, an alternative is proposed replacing the line integral by a simple jump term across the crack faces. Previous work by other authors related to the same subject is critically illuminated. The derivation was inspired by the Dugdale- Barenblatt cohesive zone model and yields an expression containing solely the local jump of displacements and electric potentials across the crack faces. This approach is shown to be exact for the Griffth crack.Numerical examples give evidence that the simplified approach works well for arbitrary crack configurations too.     

The surface-forming energy release rate based fracture criterion for elastic–plastic crack propagation

The J-integral based criterion is widely used in elastic–plastic fracture mechanics. However, it is not rigorously applicable when plastic unloading appears during crack propagation. One difficulty is that the energy density with plastic unloading in the J-integral cannot be defined unambiguously. In this paper, we alternatively start from the analysis on the power balance, and propose a surface-forming energy release rate (ERR), which represents the energy available for separating the crack surfaces during the crack propagation and excludes the loading-mode-dependent plastic dissipation. Therefore the surface-forming ERR based fracture criterion has wider applicability, including elastic–plastic crack propagation problems. Several formulae are derived for calculating the surface-forming ERR. From the most concise formula, it is interesting to note that the surface-forming ERR can be computed using only the stress and deformation of the current moment, and the definition of the energy density or work density is avoided. When an infinitesimal contour is chosen, the expression can be further simplified. For any fracture behaviors, the surface-forming ERR is proven to be path-independent, and the path-independence of its constituent term, so-called J_s-integral, is also investigated. The physical meanings and applicability of the proposed surface-forming ERR, traditional ERR, J_s-integral and J-integral are compared and discussed. Besides, we give an interpretation of Rice paradox by comparing the cohesive fracture model and the surface-forming ERR based fracture criterion.     

Electro-mechanical crack systems: bound theorems,dual finite elements and error estimation

Because of the complexity of piezoelectric crack problems, it is hard to obtain closed-form solutions, and numerical methods are largely resorted to. Hence, the upper/lower bound estimation of piezoelectric fracture parameters is of theoretical and practical importance. in this paper, the path-independent integral I, which is the dual of the J-integral, for electro-mechanical coupling crack systems, is presented. The related bound theorems are established for J and I. Piezoelectric dual finite elements are presented for the numerical implementation of the bound analysis. Moreover, an error estimator is presented for the assessment of numerical accuracy of the piezoelectric fracture parameters.     

Stress intensity factor analysis of a three-dimensional interfacial corner between anisotropic bimaterials under thermal stress

A numerical method using a path-independent H-integral based on the conservation integral was developed to analyze the singular stress field of a three-dimensional interfacial corner between anisotropic bimaterials under thermal stress. In the present method, the shape of the corner front is smooth. According to the theory of linear elasticity, asymptotic stress near the tip of a sharp interfacial corner is generally singular as a result of a mismatch of the materials’ elastic constants. The eigenvalues and the eigenfunctions are obtained using the Williams eigenfunction method, which depends on the anisotropic materials’ properties and the geometry of an interfacial corner. The order of the singularity related to the eigenvalue is real, complex or power-logarithmic. The amplitudes of the singular stress terms can be calculated using the H-integral. The stress and displacement around an interfacial corner for the H-integral are obtained using finite element analysis. In this study, a proposed definition of the stress intensity factors of an interfacial corner, which includes those of an interfacial crack and a homogeneous crack, is used to evaluate the singular stress fields. Asymptotic solutions of stress and displacement around an interfacial corner front are uniquely obtained using these stress intensity factors. To prove the accuracy of the present method, several different kinds of examples are shown such as interfacial corners or cracks in three-dimensional structures.     

压电材料中的微裂纹屏蔽问题分析

分析当主裂纹与一个微裂纹在远场I型力（KI）和远场电位移（Ke)作用下的相互干涉问题,得出了在微裂纹的位置角和方向角周时独立变化时,微裂纹对主裂纹的屏蔽作用的全局使命主裂纹扩展,通过电算还发现Ortiz在各向同性材料和各向异性材料中得出的“微裂纹群对主裂纹最大屏蔽效应产生在微裂纹方向与最大主应力垂直的方向”在压电材料中不再成立,进而提出除Hutchinson指出微裂纹屏蔽效应两个来源（即：材料有效刚度的降低和残余应力的释放）外的另一个来源,微裂纹对主裂砂电场的扰动,在对主微裂纹J积分分析时发现J2积分与J1积分具有同等重要的地位。