The interface crack problem for bonded piezoelectric and orthotropic layers under antiplane shear loading

The primary objective of this paper is to study the influence of the electroelastic interactions on the stress intensity factor in bonded layers of piezoelectric and orthotropic materials containing a crack along the interface under antiplane shear. Attention is given to a two-layer hybrid laminate formed by adding a layer of piezoelectric ceramic to a unidirectional graphite/epoxy composite or an aluminum layer. Electric displacement or electric field is prescribed on the surfaces of the piezoelectric layer. The problem is formulated in terms of a singular integral equation which is solved by using a relatively simple and efficient technique. A number of examples are given for various material combinations. The results show that the effect of the electroelastic interactions on the stress intensity factor and the energy release rate can be highly significant. This revised version was published online in July 2006 with corrections to the Cover Date.     

Electroelastic analysis of an interface anti-plane shear crack in a layered piezoelectric plate

The problem of an anti-plane interface crack in a layered piezoelectric plate composed of two bonded dissimilar piezoelectric ceramic layers subjected to applied voltage is considered. It is assumed that the crack is either impermeable or permeable. An integral transform technique is employed to reduce the problem considered to dual integral equations, then to a Fredholm integral equation by introducing an auxiliary function. Field intensity factors and energy release rate are obtained in explicit form in terms of the auxiliary function. In particular, by solving analytically a resulting singular integral equation, they are determined explicitly in terms of given electromechanical loadings for the case of two bonded layers of equal thickness. Some numerical results are presented graphically to show the influence of the geometric parameters on the field intensity factors and the energy release rate.     

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.     

Numerical analysis of dynamic T stress of moving interfacial crack

A method to extract dynamic T stress term of moving interfacial crack is proposed. Anisotropic bimaterial which has subsonic crack propagation is considered, and interaction energy method is applied. Stress fields by the constant T stress and stress fields by the point force moving with the crack are obtained by using the series expansion method and Stroh formalism. J based interaction energy (J^I) between the constant T stress and the point force is calculated by Yeh formulation and the relation between interaction energy and T stress is obtained. Energy release rate and T stress of a moving interfacial crack are calculated numerically. Elastodynamic finite element code is developed to investigate fracture parameters for the propagating crack. Four nodes linear elastodynamic element is used and Newmark formulae are applied to integrate displacement and velocity. Node release method is adapted to simulate crack propagation along the interface. The energy release rate is calculated in the area moving with crack. T stress term is calculated from the interaction energy with a stress field formed by the moving point force. Five examples are solved to show the validity and time history of energy release rate and T stress. The energy release rate calculated from numerical analysis agrees well with an analytic solution and experimental results. The T stress of homogeneous specimen under the steady state condition shows a slightly different value compared with the stationary result. It is observed that the T stress of polymethyl methacrylate–steel specimen shows continuous change and the T stress of aluminum-polymethyl methacrylate specimen shows discontinuous jump when the initial crack initiates. From the result of the variation of T stress, the effect of T stress on the stability of crack propagation is observed.     

Calculation of transient strain energy release rates under impact loading based on the virtual crack closure technique

This paper describes an interface element to calculate the strain energy release rates based on the virtual crack closure technique (VCCT) in conjunction with finite element analysis (FEA). A very stiff spring is placed between the node pair at the crack tip to calculate the nodal forces. Dummy nodes are introduced to extract information for displacement openings behind the crack tip and the virtual crack jump ahead of the crack tip. This interface element leads to a direct calculation of the strain energy release rate (both components G_I and G_II) within a finite element analysis without extra post-processing. Several examples of stationary cracks under impact loading were examined. Dynamic stress intensity factors were converted from the calculated transient strain energy release rate for comparison with the available solutions by the others from numerical and experimental methods. The accuracy of the element is validated by the excellent agreement with these solutions. No convergence difficulty has been encountered for all the cases studied. Neither special singular elements nor the collapsed element technique is used at the crack tip. Therefore, the fracture interface element for VCCT is shown to be simple, efficient and robust in analyzing crack response to the dynamic loading. This element has been implemented into commercial FEA software ABAQUS^® with the user defined element (UEL) and should be very useful in performing fracture analysis at a structural level by engineers using ABAQUS^®.     

Dynamic stress intensity factor due to concentrated loads on a propagating semi-infinite crack in orthotropic materials

The elastodynamic response of an infinite orthotropic material with a semi-infinite crack propagating at constant speed under the action of concentrated loads on the crack faces is examined. Solution for the stress intensity factor history around the crack tip is found for the loading modes I and II. Laplace and Fourier transforms along with the Wiener-Hopf technique are employed to solve the equations of motion. The asymptotic expression for the stress near the crack tip is analyzed which lead to a closed-form solution of the dynamic stress intensity factor. It is found that the stress intensity factor for the propagating crack is proportional to the stress intensity factor for a stationary crack by a factor similar to the universal function k(v) from the isotropic case. Results are presented for orthotropic materials as well as for the isotropic case.     

Griffith crack moving along the interface of two dissimilar piezoelectric materials

The problem of an anti-plane Griffith crack moving along the interface of dissimilar piezoelectric materials is solved by using the integral transform technique. It is shown from the result that the intensity factors of anti-plane stress and electric displacement are dependent on the speed of the Griffith crack as well as the material coefficients. When the two piezoelectric materials are identical, the present result will reduce to the result for the problem of an anti-plane moving Griffith crack in homogeneous piezoelectric materials. This revised version was published online in August 2006 with corrections to the Cover Date.     

An augmented cohesive zone element for arbitrary crack coalescence and bifurcation in heterogeneous materials

We demonstrate that traditional cohesive zone (CZ) elements cannot be accurate when used in conjunction with solid elements with arbitrary intra‐element cracking capability, because they cannot capture the load transfer between cohesive interfaces and the solid elements when crack bifurcation or coalescence occurs. An augmented cohesive zone (ACZ) element based on the augmented finite element method formulation is therefore proposed. The new element allows for arbitrary separation of the cohesive element in accordance with the crack configuration of the abutting solid elements, thus correctly maintaining the non‐linear coupling between merging or bifurcating cracks. Numerical accuracy and effectiveness of the proposed ACZ element are demonstrated through several examples. Copyright © 2011 John Wiley & Sons, Ltd.     

The mode I crack problem for layered piezoelectric plates

The plane strain singular stress problem for piezoelectric composite plates having a central crack is considered. For the case of the crack which is normal to and ends at the interface between the piezoelectric plate and the elastic layer, the order of stress singularity around the tip of the crack is obtained. The Fourier transform technique is used to formulate the problem in terms of a singular integral equation. The singular integral equation is solved by using the Gaus–Jacobi integration formula. Numerical calculations are carried out, and the main results presented are the variation of the stress intensity factor as functions of the geometric parameters, the piezoelectric material properties and the electrical boundary conditions of the layered composites.     

Kinking of an interface crack in an orthotropic bimaterial

We investigated the asymptotic problem of a kinked interface crack in an orthotropic bimaterial under in‐plane loading conditions. The stress intensity factors at the tip of the kinked interface crack are described in terms of the stress intensity factors of the interface crack prior to the kink combined with a dimensionless matrix function. Using a modified Stroh formalism and an orthotropy rescaling technique, the matrix function was obtained from the solutions of the corresponding problem in transformed bimaterial. The effects of orthotropic and bimaterial parameters on the matrix function were examined. A reduction in the number of dependent material parameters on the matrix function was made using the modified Stroh formalism. Moreover, the explicit dependence of one orthotropic parameter on the matrix function was determined using an orthotropic rescaling technique. The effects of the other material parameters on the matrix function were numerically examined. The energy release rate was obtained for a kinked interface crack in an orthotropic bimaterial.     

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。      

压电复合材料中III型唇形裂纹问题的解析解

采用复变函数方法和保角映射技术,研究了压电复合材料中含唇形裂纹的无限大体远场受反平面机械载荷和面内电载荷作用下的反平面问题,利用复变函数中的留数定理和Cauchy积分公式,分别获得了电不可通和电可通两种边界条件下裂纹尖端场强度因子和机械应变能释放率的解析表达式。当唇形裂纹的高度趋于零时,可得到无限大压电复合材料中Griffith裂纹的解析解。若不考虑电场作用,所得解退化为经典材料的已知结果。数值算例显示了裂纹的几何尺寸和机电载荷对机械应变能释放率的影响规律。结果表明: 唇形裂纹高度的增加会阻碍裂纹的扩展;机械载荷总是促进裂纹的扩展;电载荷对裂纹扩展的影响与裂纹面电边界条件有关。     

Off-axis fatigue crack growth and the associated energy release rate in composite laminates

Stable matrix crack growth behaviour under mechanical fatigue loading has been studied in a quasi-isotropic (0/90/-45/+45)_s GFRP laminate. Detailed experimental observations were made on the accumulation of cracks and on the growth of individual cracks in +45° as well as 90° plies. A generalised plain strain finite element model of the damaged laminate has been constructed. This model has been used to relate the energy release rate of growing cracks to the crack growth rate via a Paris relation.     

基于虚拟裂纹闭合技术的应变能释放率分析

基于虚拟裂纹闭合技术(VCCT),建立了复合材料层合板层间裂纹尖端的应变能释放率(SERR)三维有限元计算模型。该模型考虑了裂纹尖端大转动和离散单元形状变化对应变能释放率计算的影响,修正了裂纹尖端应变能释放率的计算方法。利用该模型计算了裂纹长度为15 mm和35 mm时纯Ⅰ型和纯Ⅱ型的应变能释放率,纯Ⅰ型应变能释放率分别为 207 J/m2和 253 J/m2;纯Ⅱ型应变能释放率分别为 758 J / m 2和 1040 J / m2;计算值与试验值吻合得很好。同时,该模型计算了混合型不同比值 R=(G_Ⅱ/G_Ⅰ+G_Ⅱ)的长裂纹层合板层间断裂过程的应变能释放率,其中Ⅰ型和Ⅱ型应变能释放率计算值与试验平均值的最大误差为 11.4%,最小误差为 0.4%。该模型能有效计算裂纹尖端的应变能释放率。     

Interaction effect crack–interfacial crack using finite element method

The interaction effect of an interfacial crack–microcrack modifies considerably the fracture behaviour of S45C/Si₃N₄ bimaterial. This work aims at studying the interaction effect of a crack located in one of the materials constituting the assembly near the interface, and that between an interfacial crack and a microcrack parallel to the interface by using the finite element method. The effect of transverse and longitudinal interaction distances between the interfacial crack and the microcrack are highlighted. The stress intensity factor of the interacting cracks and the bimaterial mechanical properties influence on the conditions of deviation and propagation of crack by interface and intercrack are examined.     

XFEM analysis of the effects of voids,inclusions and other cracks on the dynamic stress intensity factor of a major crack

This paper is devoted to the extraction of the dynamic stress intensity factor (DSIF) for structures containing multiple discontinuities (cracks, voids and inclusions) by developing the extended finite element method (XFEM). In this method, four types of enrichment functions are used in the framework of the partition of unity to model interface discontinuity within the classical finite element method. In this procedure, elements that include a crack segment, the boundary of a void or the boundary of an inclusion are not required to conform to discontinuous edges. The DSIF is evaluated by the interaction integral. After the effectiveness of the implemented XFEM program is verified, the effects of voids, inclusions and other cracks on the DSIF of a stationary major crack are investigated by using XFEM. The results show that the dynamic effects have an influence on the path independence of the interaction integral, and these voids, inclusions and other cracks have a significant effect on the DSIF of the major crack.     

A new look at energy release rate in fracture mechanics

The energy balance for fracture in elastic/perfectly plastic solids is examined using the finite element method. An extension-release procedure that gives numerically converged solutions is employed in the numerical simulation of crack extensions in elastic/plastic solids. Increments of work and energy during crack extension are calculated for various loading conditions. Several conclusions are obtained. First, the elastic separation work of creating new crack surfaces is shown to be negligible, indicating that the Griffith-type energy release does not exist. Second, as the yield stress increases, the plastic dissipation work rate associated with crack extension converges to the energy release rate in the limiting elastic solid. The latter result can be adopted to interpret the classical energy release rate in elastic solids as plastic dissipation work rate taken in the limit as the yield stress approaches infinity during crack extension. Lastly, it is shown that the energy release rate obtained according to Irwin's plastic zone adjustment approach is equal to the plastic dissipation work rate for the original crack, provided the plastic zone size is less than 10% of the original crack size.     

Higher order asymptotic solutions of V-notch tip fields for damaged nonlinear materials under antiplane shear loading

The higher order solutions of stress and deformation fields near the tip of a sharp V-notch in a power-law hardening material with continuous damage formation are analytically investigated under antiplane shear loading condition. The interaction between a macroscopic sharp notch and distributed microscopic damage is considered by describing the effect of damage in terms of a damage variable in the framework of damage mechanics. A deformation plasticity theory coupled with damage and a damage evolution law are formulated. A hodograph transformation is employed to determine the solution of damaged nonlinear notch problem in the stress plane. Then, inversion of the stress plane solution to the physical plane is performed. Consequently, higher order terms in the asymptotic solutions of the notch tip fields are obtained. Analytical expressions of the dominant and second order singularity exponents and associated angular distribution functions of notch tip stress and strain are presented. Effects of damage and strain hardening exponents and notch angle on the singular behavior of the notch tip quantities are discussed detailly. It is found that damage can lead to a weaker singularity of the dominant term of stress on one hand, but to stronger singularities of the second order term of stress and the dominant and second order terms of strain compared to that for undamaged case on the other. Also, both hardening exponent and notch angle have important effects on the notch tip quantities. Moreover, reduction of the notch tip solutions to a damaged nonlinear crack problem is carried out, and higher order solutions of the crack tip fields are obtained. Effects of damage and hardening exponents on the dominant and second order terms in the crack tip solutions are detailly discussed. Discussions on some other special cases are also presented, which shows that if damage exponent equals to zero, then the present solutions can be easily reduced to the solutions for undamaged cases. This revised version was published online in July 2006 with corrections to the Cover Date.     

The energy release rate of mode II fractures in layered snow samples

Before a dry snow slab avalanche is released, a shear failure along a weak layer or an interface has to take place. This shear failure disconnects the overlaying slab from the weak layer. A better understanding of this fracture mechanical process, which is a key process in slab avalanche release, is essential for more accurate snow slope stability models. The purpose of this work was to design and to test an experimental set-up for a mode II fracture test with layered snow samples and to find a method to evaluate the interfacial fracture toughness or alternatively the energy release rate in mode II. Beam-shaped specimens were cut out of the layered snow cover, so that they consisted of two homogeneous snow layers separated by a well defined interface. In the cold laboratory 27 specimens were tested using a simple cantilever beam test. The test method proved to be applicable in the laboratory, although the handling of layered samples was delicate. An energy release rate for snow in mode II was calculated numerically with a finite element (FE) model and analytically using an approach for a deeply cracked cantilever beam. An analytical bilayer approach was not suitable. The critical energy release rate G _c was found to be 0.04 ± 0.02 J m⁻². It was primarily a material property of the weak layer and did not depend on the elastic properties of the two adjacent snow layers. The mixed mode interfacial fracture toughness for a shear fracture along a weak layer estimated from the critical energy release rate was substantially lower than the mode I fracture toughness found for snow of similar density.     

On the dynamic energy release rate in functionally graded materials

By using the effective shear modulus and mass density, the influence of functional gradient on dynamic energy release rate is discussed under the condition of constant velocity of crack propagation.