Interface crack problems for mode II of double dissimilar orthotropic composite materials

The fracture problems near the similar orthotropic composite materials are interface crack tip for mode Ⅱ of double disstudied. The mechanical models of interface crack for mode Ⅱ are given. By translating the governing equations into the generalized hi-harmonic equations, the stress functions containing two stress singularity exponents are derived with the help of a complex function method. Based on the boundary conditions, a system of non-homogeneous linear equations is found. Two real stress singularity exponents are determined be solving this system under appropriate conditions about bimaterial engineering parameters. According to the uniqueness theorem of limit, both the formulae of stress intensity factors and theoretical solutions of stress field near the interface crack tip are derived. When the two orthotropic materials are the same, the stress singularity exponents, stress intensity factors and stresses for mode II crack of the orthotropic single material are obtained.     

Interface crack problems for mode Ⅱ of double dissimilar orthotropic composite materials

The fracture problems near the interface crack tip for mode Ⅱ of double dissimilar orthotropic composite materials are studied. The mechanical models of interface crack for mode Ⅱ are given. By translating the governing equations into the generalized bi-harmonic equations,the stress functions containing two stress singularity exponents are derived with the help of a complex function method. Based on the boundary conditions,a system of non-homogeneous linear equations is found. Two real stress singularity exponents are determined be solving this system under appropriate conditions about himaterial engineering parameters. According to the uniqueness theorem of limit,both the formulae of stress intensity factors and theoretical solutions of stress field near the interface crack tip are derived. When the two orthotropic materials are the same,the stress singularity exponents,stress intensity factors and stresses for mode Ⅱ crack of the orthotropic single material are obtained.     

SH波在正交各向异性功能梯度无限长条中心裂缝处的散射

研究了正交各向异性功能梯度材料无限长条中心裂缝对SH波的散射问题,为方便起见,材料两个方向的剪切模量和密度假定为指数模型.通过Fourier积分变换,将问题转化为对偶积分方程的求解.然后,用Cop-son方法求解对偶积分方程,定义了标准动应力强度因子,通过数值算例,讨论了在SH波作用下,裂缝尖端的标准动应力强度因子与入射波的频率、材料参数之间的关系.     

SINGULAR SOLUTIONS OF ANISOTROPIC PLATE WITH AN ELLIPTICAL HOLE OR A CRACK

In the present paper, closed form singular solutions for an infinite anisotropic plate with an elliptic hole or crack are derived based on the Stroh-type formalism for the general anisotropic plate. With the solutions, the hoop stresses and hoop moments around the elliptic hole as well as the stress intensity factors at the crack tip under concentrated in-plane stresses and bending moments are obtained. The singular solutions can be used for approximate analysis of an anisotropic plate weakened by a hole or a crack under concentrated forces and moments.They can also be used as fundamental solutions of boundary integral equations in BEM analysis for anisotropic plates with holes or cracks under general force and boundary conditions.     

Fracture of a rectangular piezoelectromagnetic body

The singular stress, electric fields and magnetic fields in a rectangular piezoelectromagnetic body containing a center Griffith crack under longitudinal shear are obtained. Fourier transforms and Fourier sine series are used to reduce the mixed boundary value problems of the crack, which is assumed to be impermeable, to dual integral equations. The solution of the dual integral equations is then expressed in terms of Fredholm integral equations of the second kind. Expressions for stresses, electric displacements and magnetic inductions in the vicinity of the crack tip are derived. Also obtained are the field intensity factors and the energy release rates. Numerical results obtained show that the geometry of the rectangular body have significant influence on the field intensity factors and the energy release rates.     

A numerical study of crack tip constraint in ductile single crystals

In this work, the effect of crack tip constraint on near-tip stress and deformation fields in a ductile FCC single crystal is studied under mode I, plane strain conditions. To this end, modified boundary layer simulations within crystal plasticity framework are performed, neglecting elastic anisotropy. The first and second terms of the isotropic elastic crack tip field, which are governed by the stress intensity factor K and T-stress, are prescribed as remote boundary conditions and solutions pertaining to different levels of T-stress are generated. It is found that the near-tip deformation field, especially, the development of kink or slip shear bands, is sensitive to the constraint level. The stress distribution and the size and shape of the plastic zone near the crack tip are also strongly influenced by the level of T-stress, with progressive loss of crack tip constraint occurring as T-stress becomes more negative. A family of near-tip fields is obtained which are characterized by two terms (such as K and T or J and a constraint parameter Q) as in isotropic plastic solids.     

Pre-kinking of a moving crack in a magnetoelectroelastic material under in-plane loading

A constant moving crack in a magnetoelectroelastic material under in-plane mechanical, electric and magnetic loading is studied for impermeable crack surface boundary conditions. Fourier transform is employed to reduce the mixed boundary value problem of the crack to dual integral equations, which are solved exactly. Steady-state asymptotic fields near the crack tip are obtained in closed form and the corresponding field intensity factors are expressed explicitly. The crack speed influences the singular field distribution around the crack tip and the effects of electric and magnetic loading on the crack tip fields are discussed. The crack kinking phenomena is investigated using the maximum hoop stress intensity factor criterion. The magnitude of the maximum hoop stress intensity factor tends to increase as the crack speed increases.     

Cracked elastic layer under compressive mechanical loads

We consider boundary value problem in which an elastic layer containing a finite length crack is under compressive loading. The crack is parallel to the layer surfaces and the contact between crack surfaces are either frictionless or with adhesive friction or Coulomb friction.Based on fourier integral transformation techniques the solution of the formulated problems is reduced to the solution of a singular integral equation, then, using Chebyshev’s orthogonal polynomials, to an infinite system of linear algebraic equations. The regularity of these equations is established. The expressions for stress and displacement components in the elastic layer are presented. Based on the developed analytical algorithm, extensive numerical investigations have been conducted.The results of these investigations are illustrated graphically, exposing some novel qualitative and quantitative knowledge about the stress field in the cracked layer and their dependence on geometric and applied loading parameters. It can be seen from this study that the crack tip stress field has a mode II type singularity.     

Mode III fracture analysis by Trefftz boundary element method

This paper presents a hybrid Trefftz (HT) boundary element method (BEM) by using two indirect techniques for mode III fracture problems. Two Trefftz complete functions of Laplace equation for normal elements and a special purpose Trefftz function for crack elements are proposed in deriving the Galerkin and the collocation techniques of HT BEM. Then two auxiliary functions are introduced to improve the accuracy of the displacement field near the crack tips, and stress intensity factor (SIF) is evaluated by local crack elements as well. Furthermore, numerical examples are given, including comparisons of the present results with the analytical solution and the other numerical methods, to demonstrate the efficiency for different boundary conditions and to illustrate the convergence influenced by several parameters. It shows that HT BEM by using the Galerkin and the collocation techniques is effective for mode III fracture problems. The project supported by the National Natural Science Foundation of China(10472082). The English text was polished by Keren Wang.     

APPLICATION OF TREFFTZ BEM TO ANTI-PLANE PIEZOELECTRIC PROBLEM

Anti-plane electroelastic problems are studied by the Trefftz boundary elementmethod (BEM) in this paper. The Trefftz BEM is based on a weighted residual formulation andindirect boundary approach. In particular the point-collocation and Galerkin techniques, in whichthe basic unknowns are the retained expansion coefficients in the system of complete equations,are considered, Furthermore, special Trefftz functions and auxiliary functions which satisfy ex-actly the specified boundary conditions along the slit boundaries are also used to derive a specialpurpose element with local defects. The path-independent integral is evaluated at the tip of acrack to determine the energy release rate for a mode Ⅲ fracture problem. In final, the accuracyand efficiency of the Trefftz boundary element method are illustrated by an example and thecomparison is made with other methods.     

Anti-plane analysis on a finite crack in a one-dimensional hexagonal quasicrystal strip

The anti-plane fracture problem for a finite crack in a one-dimensional hexagonal quasicrystal strip is analyzed. By using Fourier transforms, the mixed boundary value problems are reduced to the dual integral equations. The solution of the dual integral equations is then expressed by the complete elliptic integrals of the first and the third kinds. The expressions for stress, strains, displacements and field intensity factors of the phonon and phason fields near the crack tip are obtained exactly. The path-independent integral derived by a conservation law equals the energy release rate, which can be used as a fracture criterion for a mode III fracture problem.     

Analytic solutions to crack tip plastic zone under various loading conditions

Based on stress field equations and Hill yield criterion, the crack tip plastic zone is determined for orthotropic materials and isotropic materials under small-scale yielding condition. An analytical solution to calculating the crack tip plastic zone in plane stress states is presented. The shape and size of the plastic zone are analyzed under different loading conditions. The obtained results show that the crack tip plastic zones present “butterfly-like” shapes, and the elastic–plastic boundary is smooth. The size of the plastic zone for orthotropic composites is less at the crack tip for various loading conditions, compared with the case of isotropic materials. Crack inclination angle and loading conditions affect greatly the size and shape of crack tip plastic zone. The mode I crack has a crucial effect on the plastic zone for mixed mode case in plane stress state. The plastic zone for pure mode I crack and pure mode II crack have a symmetrical distribution to the initial crack plane.     

A directional crack path criterion for crack growth in ductile materials subjected to shear and compressive loading under plane strain conditions

A directional crack growth criterion in a compressed elastic perfectly plastic material is considered. The conditions at the crack-tip are evaluated for a straight stationary crack with a small incipient kink. Remote load is a combined hydrostatic pressure and pure shear applied via a boundary layer. Crack surfaces in contact are assumed to develop homogenous Coulomb friction.The crack opening displacement of an extended kink is examined in a finite element analysis to judge the risk of opening mode failure. It has been found that the direction that maximizes the crack opening displacement of an extended kink tip coincides very well with a prediction of the crack growth direction obtained by using a criterion for continued crack growth direction discussed by the authors elsewhere [Int. J. Fract. 108 (2001) 351].Moreover, the by the model predicted incipient crack growth directions are qualitatively comparable with reported crack paths obtained in ductile materials in a limited number of experiments performed under a combined load of in-plane shear and compression.     

Scattering of SH-wave by cracks originating at an elliptic hole and dynamic stress intensity factor

The method of complex function and the method of Green‘s function are used to investigate the problem of SH-wave scattering by radial cracks of any limited length along the radius originating at the boundary of an elliptical hole, and the solution of dynamicstress intensity factor at the crack tip was given. A Green‘s function was constructed for the problem, which is a basic solution of displacement field for an elastic half space containing a half elliptical gap impacted by anti-plane harmonic linear source force at any point of its horizontal boundary. With division of a crack technique, a series of integral equations can be established on the conditions of continuity and the solution of dynamic stress intensity factor can be obtained. The influence of an elliptical hole on the dynamic stress intensity factor at the crack tip was discussed.     

Crack-tip field on mode Ⅱ interface crack of double dissimilar orthotropic composite materials

Two systems of non-homogeneous linear equations with 8 unknowns are obtained.This is done by introducing two stress functions containing 16 undetermined coefficients and two real stress singularity exponents with the help of boundary conditions.By solving the above systems of non-homogeneous linear equations,the two real stress singularity exponents can be determined when the double material parameters meet certain conditions.The expression of the stress function and all coefficients are obtained based on the uniqueness theorem of limit.By substituting these parameters into the corresponding mechanics equations,theoretical solutions to the stress intensity factor,the stress field and the displacement field near the crack tip of each material can be obtained when both discriminants of the characteristic equations are less than zero.Stress and displacement near the crack tip show mixed crack characteristics without stress oscillation and crack surface overlapping.As an example,when the two orthotropic materials are the same,the stress singularity exponent,the stress intensity factor,and expressions for the stress and the displacement fields of the orthotropic single materials can be derived.     

Fracture Mechanics of Corrosion Cracking in Concrete by Acoustic Emission

Nucleation of microcracks can be detected and analyzed by acoustic emission (AE), by which crack kinematics of locations, types and rientations are quantitatively estimated. The procedure was applied to clarify mechanisms of corrosion cracking. Based on fracture mechanics, numerical analysis was conducted by the boundary element method (BEM). Relations between the stress intensity factors and crack types were investigated by BEM solutions. In experiments, four types of crack patterns were nucleated by employing expansive agent. Following the surface crack, the diagonal crack and/or the horizontal crack propagated. The internal crack extended after the surface crack was terminated. Depending on the crack types, contributions of mode-I and mode-II were varied. According to AE results, four crack patterns observed differently consisted of tensile, mixed-mode and shear cracks. It is demonstrated that mechanisms of corrosion cracking in concrete are dominantly of mode-I failure along with a minor contribution of mixed-mode and mode-II.     

Dynamic stress intensity factor KⅢ and dynamic crack propagation characteristics of anisotropic materials

Based on the mechanics of anisotropic materials,the dynamic propagation problem of a mode Ⅲ crack in an infinite anisotropic body is investigated.Stress,strain and displacement around the crack tip are expressed as an analytical complex function,which can be represented in power series.Constant coefficients of series are determined by boundary conditions.Expressions of dynamic stress intensity factors for a mode Ⅲ crack are obtained.Components of dynamic stress,dynamic strain and dynamic displacement around the crack tip are derived.Crack propagation characteristics are represented by the mechanical properties of the anisotropic materials,i.e.,crack propagation velocity M and the parameter α.The faster the crack velocity is,the greater the maximums of stress components and dynamic displacement components around the crack tip are.In particular,the parameter α affects stress and dynamic displacement around the crack tip.     

On anti-plane shear behavior of a Griffith permeable crack in piezoelectric materials by use of the non-local theory

In this paper, the non-local theory of elasticity is applied to obtain the behavior of a Griffith crack in the piezoelectric materials under anti-plane shear loading for permeable crack surface conditions. By means of the Fourier transform the problem can be solved with the help of a pair of dual integral equations with the unknown variable being the jump of the displacement across the crack surfaces. These equations are solved by the Schmidt method. Numerical examples are provided. Unlike the classical elasticity solutions, it is found that no stress and electric displacement singularity is present at the crack tip. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum stress hypothesis. The finite hoop stress at the crack tip depends on the crack length and the lattice parameter of the materials, respectively. The project supported by the National Natural Science Foundation of China (50232030 and 10172030)     

On the half-infinite crack problem in thermo-electro-elasticity

This paper concerns itself with the fundamental solutions of the thermo-electro-elastic field in an infinite medium, weakened by a half-infinite plane crack with two identical point thermal loads applied on the crack surfaces. The corresponding mixed boundary value problem is solved by virtue of the potential theory method conjugated with the general solutions. The boundary governing equations are solved by using the results available in literature. Exact and complete three-dimensional (3D) fundamental solutions are presented in terms of elementary functions. The singularity at the crack tip is analyzed explicitly. The obtained solutions will be of high significance to the related BEM analysis.     

基于数字散斑相关方法测定Ⅰ型裂纹应力强度因子

提出了一种通过数字散斑相关方法测定金属材料Ⅰ型裂纹尖端位置和应力强度因子的实验方法.实验采用疲劳试验机对含Ⅰ型缺口的Cr12MoV钢试件预制裂纹,通过数字散斑相关方法测试试件在三点弯曲加载条件下裂纹的扩展过程及裂尖区域的位移场.将位移场数据代入裂尖位移场方程组,采用牛顿-拉普森方法求解含未知参量的裂尖非线性位移场方程组,计算裂尖位置和应力强度因子.实验结果表明,采用该方法可以准确地测定金属材料Ⅰ型裂纹应力强度因子、裂尖位置及裂纹扩展长度,解决了以往研究中因不能准确测定裂纹尖端位置,而无法准确计算Ⅰ型裂纹裂尖断裂参数的难题,揭示了金属材料裂纹扩展过程中应力强度因子演化特征.