功能梯度材料的黏弹性断裂问题

功能梯度材料(FGM)是一种不同于传统复合材料的新型工程复合材料 [1], 国内外关于FGM的断裂力学方面的研究发展非常迅速. 关于FGM静态裂纹问题,学者们研究了不同类型裂纹尖端场的应力强度因子 [2-5], 探讨了有限长裂纹在不用载荷作用下的传播等问题. 而关于动态裂纹问题,也已经取得很大成就 [6-9]. FGM一个很重要的应用是高温结构材料,在强大的热环境中,很多材料都呈现出黏弹性. 因此,研究FGM的黏弹性断裂力学非常具有实际价值.对此,众多研究 [10-14]提出不同的分析模型,并在不同受载条件,通过理论计算,分析了黏弹性裂纹尖端场的力学 行为.本文考查了功能梯度材料板条中界面裂纹垂直于梯度方向时的黏弹性断裂问题,首先利用有限元法求解线弹性功能梯度材料板条的裂纹尖端场,然后根据黏弹性的对应性原理,求解出黏弹性功能梯度材料板条裂纹问题的应力场强度因子.      

断裂力学判据的评述

从Inglis 和Griffith 的著名论文到Irwin 和Rice 等的奠基性贡献,对断裂力学中的线弹性断裂力学的K判据,界面断裂力学的G判据,和弹塑性断裂力学的J 判据作了扼要的综述. 介绍了在界面断裂力学G判据的基础上提出的界面断裂力学的K判据,以说明断裂力学的判据存在改进的可能性. 在综述中归纳出断裂力学判据中目前还没有较好解决的几个问题. 在总结以往断裂力学研究经验的基础上,指出裂纹端应力奇异性的源是对断裂力学判据存在的问题作进一步研究的切入点. 探讨了裂纹端应变间断的奇点是裂纹端应力奇异性的源的问题,从而对裂纹端应力强度因子的物理意义进行了讨论. 最后,阐述了进行可靠的裂纹端应力场的弹塑性分析是改进弹塑性断裂力学判据的关键,而进行可靠的裂纹端应力场的弹塑性分析的前提是要通过裂纹端应力奇异性的源的研究来获得作用在裂纹端的造成裂纹端应变间断的有限值应力.      

Torsional impact response of a penny-shaped crack in a functional graded strip

The torsional impact response of a penny-shaped crack in a nonhomogeneous strip is considered. The shear modulus is assumed to be functionally graded such that the mathematics is tractable. Laplace and Hankel transforms were used to reduce the problem to solving a Fredholm integral equation. The crack tip stress field is obtained by considering the asymptotic behavior of Bessel function. Explicit expressions of both the dynamic stress intensity factor and the energy density factor were derived. And it is shown that, as crack driving force, they are equivalent for the present crack problem. Investigated are the effects of material nonhomogeneity and strip‘s highness on the dynamic fracture behavior.Numerical results reveal that the peak of the dynamic stress intensity factor can be suppressed by increasing the nonhomogeneity parameter of the shear modulus, and that the dynamic behavior varies little with the adjusting of the strip‘ s highness.     

Pseudo plane stress plastic field for mode I crack growth

The pseudo plane stress field for a mode I crack growth is analyzed for both perfectly plastic and power law hardening plastic materials. When finite strain is taken into account, it is found that for perfectly plastic materials, the plastic domain is a narrow strip ahead of the crack tip. For power law hardening plastic material, the plastic domain contains a strip and a region ahead of the strip. The fracture criterion is discussed. The energy dissipated in the plastic strip is found to be proportional to the square of the thickness. Singular solutions to the field are ruled out by analysis.     

The collinear crack problem for an orthotropic functionally graded coating–substrate structure

A theoretical treatment of antiplane crack problem of two collinear cracks on the two sides of and perpendicular to the interface between a functionally graded orthotropic strip bonded to an orthotropic homogeneous substrate is put forward. Various internal cracks and crack terminating at the interface and crack crossing the interface configurations are investigated, respectively. The problem is formulated in terms of a singular integral equation with the crack face displacement as the unknown variable. 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 the orthotropy and nonhomogeneous parameters and crack interactions on the mode III stress intensity factors are investigated.     

Analysis of dynamic growth of a tensile crack in an elastic-plastic material

The influence of inertia on the stress and deformation fields near the tip of a crack growing in an elastic-plastic material is studied. The material is characterized by the von Mises yield criterion and J₂ flow theory of plasticity. The crack grows steadily under plane strain conditions in the tensile opening mode. Features of the stress and deformation state at points near the moving crack tip are described for elastic-perfectly plastic response and for several crack propagation speeds. It is found that inertia has a significant effect on the elastic-plastic response of material particles near the crack tip, and that elastic unloading may occur behind the crack tip for higher speeds. The relationship between the applied crack driving force, represented by a remote stress intensity factor, and the crack tip speed is examined on the basis of a critical crack tip opening angle growth criterion. The calculated result is compared with dynamic fracture toughness versus crack speed data for a 4340 steel.     

功能梯度压电板条中电绝缘型运动裂纹的电弹性场

基于三维弹性理论和压电理论，对材料系数按指数函数规律分布的功能梯度压电板条中的反平面运动裂纹问题进行了求解。利用Fourier积分变换方法将电绝缘型运动裂纹问题化为对偶积分方程，并进一步归结为易于求解的第二类Fredholm积分方程。通过渐近分析，获得了裂纹尖端应力、应变、电位移和电场的解析解，给出了裂纹尖端场各个变量的角分布函数，并求得了裂纹尖端场的强度因子，分析了压电材料物性梯度参数、几何尺寸及裂纹运动速度对它们的影响。结果表明，对于电绝缘型裂纹，功能梯度压电板条中运动裂纹尖端附近的各个场变量都具有-1／2阶的奇异性；当裂纹运动速度增大时，裂纹扩展的方向会偏离裂纹面。     

Impact failure prediction of Mode III crack in orthotropic functionally graded strip

Mode III impact of a crack in an orthotropic functionally graded strip is investigated. The shear moduli in two directions of the material are assumed to vary proportionately with gradient. Laplace transform and Fourier cosine transform are used to reduce the problem to solving a Fredholm integral equation. The crack tip stress field is obtained by considering the asymptotic behavior of Bessel function. Energy density factor criterion is applied to obtain the maximum of minimum energy density and direction of crack initiation. Numerical results are given graphically. The effects of orthotropy, nonhomogeneity and height of the strip on the energy density factor are discussed.     

The scattering of harmonic elastic anti-plane shear waves by a finite crack in infinitely long strip using the non-local theory

In this paper, the scattering of harmonic anti-plane shear waves by a finite crack in infinitely long strip is studied using the non-local theory. The Fourier transform is applied and a mixed boundary value problem is formulated. Then a set of dual integral equations is solved using the Schmidt method instead of the first or the second integral equation method. A one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress occurring at the crack tips. Contraty to the classical elasticity solution, it is found that no stress singularity is present at the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the width of the strip and the lattice parameter. Supported by the Post Doctoral Science Foundation of Heilongjiang Province, the Natural Science Foundation of Heilongjiang Province and the National Foundation for Excellent Young Investigators.     

A general solution method for an anti-plane shear crack dynamically accelerating along a bimaterial interface

We develop a general solution method for a dynamically accelerating crack under anti-plane shear conditions along the interface between two different homogeneous isotropic elastic materials. The crack is initially at rest, and after loading is applied the crack-tip speed which may accelerate up to the shear wave speed of the more compliant material. The analysis includes an exact, closed-form expression for the stress intensity factor for an arbitrary time-dependent crack-face traction, as well as expressions for computing the crack-face displacements and the stress in front of the crack. We also present some numerical examples for fixed loads and for loads moving with the crack tip, using a stress intensity factor fracture criterion, in order to examine the predicted effect of material mismatch on interfacial fracture.     

Anti-plane shear of an arbitrary oriented crack in a functionally graded strip bonded with two dissimilar half-planes

An internal crack located within a functionally graded material (FGM) strip bonded with two dissimilar half-planes and under an anti-plane load is considered. The crack is oriented in an arbitrary direction. The material properties of strip are assumed to vary exponentially in the thickness direction and two half-planes are assumed to be isotropic. Governing differential equations are derived and to reduce the difficulty of the problem dealing with solution of a system of singular integral equations Fourier integral transform is employed. Semi closed form solution for the stress distribution in the medium is obtained and mode III stress intensity factor (SIF), at the crack tip is calculated and its validity was verified. Finally, the effects of nonhomogeneous material parameter and crack orientation on the stress intensity factor are studied.     

功能梯度材料裂纹尖端动态应力场

研究受反平面剪切作用的功能梯度材料动态裂纹问题,通过积分变换-对偶积分方程方法推出了裂纹尖端动态应力场,时间域内的动态应力强度因子由Laplace数值反演获得,研究结果表明功能梯度材料的梯度越大,相应的裂纹问题的动态应力强度因子值越低。     

Investigation of the Dynamic Behavior of a Griffith Crack in a Piezoelectric Material Strip Subjected to the Harmonic Elastic Anti-plane Shear Waves by Use of the Non-local Theory

In this paper, the dynamic behavior of a Griffith crack in a piezoelectric material strip subjected to the harmonic anti-plane shear waves is investigated by use of the non-local theory for impermeable crack surface conditions. To overcome the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress and the electric displacement near at the crack tip. By means of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations. These equations are solved using the Schmidt method. Contrary to the classical solution, it is found that no stress and electric displacement singularity is present near the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress near the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the thickness of the strip, the circular frequency of incident wave and the lattice parameter.     

压电材料裂纹顶端条状电饱和区模型的力学分析

在线性压电本构方程框架下,对裂纹顶端条状电饱和区模型进行了严格的数学分析．完整地考虑了各向异性力电耦合效应．建立了电饱和区尺寸与外加电场的依赖关系．证实了当裂纹垂直极化轴时,压电材料的断裂应力随着外加正电场的增加而减小,随着外加负电场的增加而增加．当裂纹平行于极化轴时,与极化轴平行的外加电场对断裂应力无影响     

Griffith crack moving in a piezoelectric strip

Summary  The dynamic problem of an impermeable crack of constant length 2a propagating along a piezoelectric ceramic strip is considered under the action of uniform anti-plane shear stress and uniform electric field. The integral transform technique is employed to reduce the mixed-boundary-value problem to a singular integral equation. For the case of a crack moving in the mid-plane, explicit analytic expressions for the electroelastic field and the field intensity factors are obtained, while for an eccentric crack moving along a piezoelectric strip, numerical results are determined via the Lobatto–Chebyshev collocation method for solving a resulting singular integral equation. The results reveal that the electric-displacement intensity factor is independent of the crack velocity, while other field intensity factors depend on the crack velocity when referred to the moving coordinate system. If the crack velocity vanishes, the present results reduce to those for a stationary crack in a piezoelectric strip. In contrast to the results for a stationary crack, applied stress gives rise to a singular electric field and applied electric field results in a singular stress for a moving crack in a piezoelectric strip. Received 14 August 2001; accepted for publication 24 September 2002 The author is indebted to the AAM Reviewers for their helpful suggestions for improving this paper. The work was supported by the National Natural Science Foundation of China under Grant 70272043.     

Anti-plane stress intensity, energy release and energy density at crack tips in a functionally graded strip with linearly varying properties

The fracture problem of a crack in a functionally graded strip with its properties varying in a linear form along the strip thickness under an anti-plane load is considered. The embedded anti-plane crack is located in the middle of strip half way through the thickness. The third mode stress intensity factor is derived using two different methods. In the first method, by employing Fourier integral transforms, the governing equation is converted to a singular integral equation, which is subsequently solved numerically by the collocation method based on Chebyshev polynomials. Then, the problem is solved by means of finite element method in which quadrilateral 8-node singular elements around each crack tip are used. After inspecting the validity of the solution technique, effects of crack geometry and non-homogeneous material parameter on the stress intensity, energy release and energy density are studied and the results of analytical and FEM solutions are compared.     

Crack tip asymptotic character of anti-plane stress and strain rate for linear fractional constitutive relations

Linear-fractional strain rate and stress relations are used to simulate materials undergoing steady state creep. The crack tip asymptotic character of the stress and strain rate field is obtained in exact and approximate form. In the limit as the radial distance emanating from the crack tip approaches zero, the stress field corresponds to that for an ideal plastic material while the exact and approximate solutions tend to coincide. Discussed is the nonhomogeneous singular character of the strain rate field that possess different orders of singularities in a circular region around the crack tip.     

A MIXED MODE FRACTURE CRITERION BASED ON THE MAXIMUM TANGENTIAL STRESS IN BRITTLE INCLUSION

A closed-form solution for predicting the tangential stress of an inclusion located in mixed mode Ⅰ and Ⅱ crack tip field was developed based on the Eshelby equivalent inclusion theory. Then a mixed mode fracture criterion, including the fracture direction and the critical load, was established based on the maximum tangential stress in the inclusion for brittle inclusioninduced fracture materials. The proposed fracture criterion is a function of the inclusion fracture stress, its size and volume fraction, as well as the elastic constants of the inclusion and the matrix material. The present criterion will reduce to the conventional one as the inclusion having the same elastic behavior as the matrix material. The proposed solutions are in good agreement with detailed finite element analysis and measurement.     

Effect of orthotropy on crack propagation

The tendency of moving cracks to spread along the preferred directions of material anisotropy is treated. Depending on the velocity of crack propagation, the change of material properties in orthogonal planes is shown to affect the bifurcation characteristics. The problem is reduced to a system of dual integral equations that can be solved in a standard fashion. Of particular interest is the dynamic stress field near the tip of a moving crack in an orthotropic material. Although the 1√r stress singularity is preserved with r being the radial distance measured from the crack tip, the angular variations of the stresses are dependent on crack speed and material anisotropy. The possibility of crack bifurcation is examined by application of the strain energy density criterion for several composite systems. Crack branching is found to be enhanced by material anisotropy, a phenomenon that is not uncommon in composite materials.     

A small-scale magnetic-yielding model for an infinite magnetostrictive plane with a crack-like flaw

The effect of an external magnetic field on the fracture toughness of magnetostrictive materials has been investigated by determining the local stress fields around the tip of a very slender elliptical flaw embedded in an infinite magnetostrictive plane subjected to magnetic loading, based on the assumption of linear magnetization. In this paper, the above-mentioned analytical approach is extended to develop a small-scale magnetic-yielding model. The magnetic saturation zone is constructed and the distributions of magnetic field and magnetization are obtained around the tip of a slender elliptical crack. Based on the complex potential theory, the stress field is obtained in the vicinity of the tip of the slender elliptical crack by implementing the continuity conditions of displacement and resultant force at the interface between the magnetic saturation and magnetoelastic zones. The stress fields near the tip of the slender elliptical crack are obtained for two kinds of soft ferromagnetic materials each with a small induction magnetostrictive modulus. The theoretical results obtained show that the stresses in the neighborhood of a crack-tip are finite even when the elliptical crack reduces to a sharp crack, and are much smaller than the yield stress or the nominal fracture stress of the material. This suggests that, generally, the magnetic field has no obvious effects on the apparent fracture toughness of soft ferromagnetic materials, which is in agreement with the existing experimental results published in the existing literature. In addition, the theoretical analysis illustrates that no crack is magnetically impermeable, and the corresponding boundary conditions are inappropriate for fracture analysis of soft ferromagnetic materials.