CONSERVATION LAW AND APPLICATION OF J-INTEGRAL IN MULTI-MATERIALS

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Unsymmetrical bending problems for the annular and circular thin plates under various supporting conditions (I)

In this paper we consider the nonlinear and unsymmetrical bendings of annular and circular thin plates under various supports. The uniformly valid asymptotic solutions have been derived by means of the perturbation method presented in [7].     

OnJ-integrals in the plane fracture of composite materials

By using a complex function method in this paper, the complex form of J-integral of mixed mode crack tip for unidirectional plate of linear-elastic orthotropic composites is obtained first by substituting crack tip stresses and displacements into general formula of J-integral. And then, the path-independence of this J-integral is proved. Finally, the computing formula of this J-integral is derived. As special examples, the complex forms, path-independence and computing formulae of J-integrals of mode Ⅰ and mode Ⅱ crack tips for unidirectional plate of linear-elastic orthotropic composites are given.     

Further investigation of interaction between interface macrocrack and parallel microcracks in bimaterial anisotropic solids

In this paper, with the aid of superimposing technique and the Pseudo Traction Method (PTM), the interaction problem between an interface macrocrack and parallel microcracks in the process zone in bimaterial anisotropic solids is reduced to a system of integral equations. After the integral equations are solved numerically, a conservation law among three kinds ofJ-integrals is obtained which are induced from the interface macrocrack tip, the microcrack and the remote field, respectively. This conservation law reveals that the microcrack shielding effect in such materials could be considered as the redistribution of the remoteJ-integral. The project supported by the National Natural Science Foundation of China, and the Doctorate Foundation of Xi'an Jiaotong University     

On Dual Conservation Laws in Linear Elasticity: Stress Function Formalism

Dual conservation laws of linear planar elasticity theory have been systematically studied based on stress function formalism. By employing generalized symmetry transformation or the Lie—Bäcklund transformation, a class of new dual conservation laws in planar elasticity have been discovered based on the Noether theorem and its Bessel—Hagen generalization. The physical implications of these dual conservation laws are discussed briefly.     

A crack perpendicular to and terminating at a bimaterial interface

Using dislocation simulation approach, the basic equation for a finite crack perpendicular to and terminating at a bimaterial interface is formulated. A novel expansion method is proposed for solving the problem. The complete solution to the problem, including the explicit formulae for theT stresses ahead of the crack tip and the stress intensity factors are presented. The stress field characteristics are analysed in detail. It is found that normal stresses {ie27-1} and {ie27-2} ahead of the crack tip, are characterised byQ fields if the crack is within a stiff material and the parameters |p _T | and |q _T | are very small, whereQ is a generalised stress intensity factor for a crack normal to and terminating at the interface. If the crack is within a weak material, the normal stresses {ie27-3} and {ie27-4} are dominated by theQ field plusT stress. This work was supported by the Swedish Research Council for Engineering Sciences.     

Experimental study and BEM analysis of fatigue crack growth in a cracked heterogeneous body

A model for analysing a soft-hard heterogeneous body with a crack in the hard region is presented in this paper. The result of fatigue experiments shows that mechanical heterogeneity affects the rate of propagation of fatigue crack. Meanwhile the results computed by BEM for cracked heterogeneous bodies under cycling loading indicate that the smaller the distance between the crack and the interface of hard and soft regions is, the smaller the amplitude of crack opening displacement, COD and ofJ-integral as well at the same step during the fatigue crack growth will be. The effect of heterogeneity on the rate of fatigue crack propagation is shown by the variation of J. The smaller the distance of the crack to the interface is, the smaller the rate of fatigue crack growth will be.     

Crack tip field andJ-integral with strain gradient effect

The mode I plane strain crack tip field with strain gradient effects is presented in this paper based on a simplified strain gradient theory within the framework proposed by Acharya and Bassani. The theory retains the essential structure of the incremental version of the conventionalJ ₂ deformation theory. No higher-order stress is introduced and no extra boundary value conditions beyond the conventional ones are required. The strain gradient effects are considered in the constitutive relation only through the instantaneous tangent modulus. The strain gradient measures are included into the tangent modulus as internal parameters. Therefore the boundary value problem is the same as that in the conventional theory. Two typical crack problems are studied: (a) the crack tip field under the small scale yielding condition induced by a linear elastic mode-IK-field and (b) the complete field for a compact tension specimen. The calculated results clearly show that the stress level near the crack tip with strain gradient effects is considerable higher than that in the classical theory. The singularity of the strain field near the crack tip is nearly equal to the square-root singularity and the singularity of the stress field is slightly greater than it. Consequently, theJ-integral is no longer path independent and increases monotonically as the radius of the calculated circular contour decreases. The project supported by the National Natural Science Foundation of China (19704100 and 10202023) and the Natural Science Foundation of Chinese Academy of Sciences (KJ951-1-20)     

多层材料结构的界面裂纹尖端复应力强度因子

本文建立了一种多层材料复合结构的界面裂纹问题分析模型。当两种材料之间插入第三种薄层弹性材料,裂纹位于第三种材料与第一或第二种弹性材料的界面上,且插页材料3~#的厚度相对于裂纹尺寸或平面内其他尺寸很小时,可以得到该问题裂纹尖端的复应力强度因子通式。本文用有限元法对结果进行了数值验证,并进行了有关问题的讨论。     

Path-independentJ-integral and its dual form in elastic-plastic solids with finite displacements

In this paper, based on energy variational principles of elastic-plastic solids, the path-independentJ-integral and its dual form in elastic-plastic solids with finite displacements are presented. Whose testification is given there after.     

Analysis of stress intensity factor in orthotropic bi-material mixed interface crack

Adopting the complex function approach, the paper studies the stress intensity factor in orthotropic bi-material interface cracks under mixed loads. With consideration of the boundary conditions, a new stress function is introduced to transform the problem of bi-material interface crack into a boundary value problem of partial differential equations. Two sets of non-homogeneous linear equations with 16 unknowns are constructed. By solving the equations, the expressions for the real bi-material elastic constant εt and the real stress singularity exponents λt are obtained with the bi-material engineering parameters satisfying certain conditions. By the uniqueness theorem of limit,undetermined coefficients are determined, and thus the bi-material stress intensity factor in mixed cracks is obtained. The bi-material stress intensity factor characterizes features of mixed cracks. When orthotropic bi-materials are of the same material, the degenerate solution to the stress intensity factor in mixed bi-material interface cracks is in complete agreement with the present classic conclusion. The relationship between the bi-material stress intensity factor and the ratio of bi-material shear modulus and the relationship between the bi-material stress intensity factor and the ratio of bi-material Young's modulus are given in the numerical analysis.     

有限尺寸下双材料界面附近裂纹位置对裂纹尖端的影响

随着复合材料的应用和发展,不同材料组成的界面结构越来越受到人们的重视.界面层两侧材料的性能相异会引起材料界面端奇异性,同时界面和界面附近存在裂纹会引起裂尖处的应力奇异性.因此双材料界面附近的力学分析是比较复杂的.论文建立双材料直角界面模型,在材料界面附近预设初始裂纹,计算了有限材料尺寸对界面应力场及其附近裂纹应力强度因子的影响.运用弹性力学中的Goursat公式求得直角界面端在有限尺寸下的应力场以及其应力强度系数.通过叠加原理和格林函数法进一步得到在直角界面端附近的裂纹尖端应力强度因子.计算结果表明,在适当范围内改变材料内裂纹与界面之间的距离,界面附近裂纹尖端的应力强度因子随着裂纹与界面距离的增加而减少,并且逐渐趋于稳定.分析结果可以为预测双材料结构复合材料界面失效位置提供参考.     

周期界面裂纹反平面问题的动态应力强度因子

在研究动载荷作用下复合材料层板结构的安全与可靠性问题以及在抗震设计中关于地层裂缝的运动等问题中,都与界面裂纹有关。本文研究了分布于两个半空间之间的周期界面裂纹在反平面剪切波作用下裂纹尖端应力强度因子的动态特性。文中利用有限 Pourier变换,将在一个周期带内的边值问题转化成求解一个带周期性奇异核的积分方程,再借助于Chebyshev 多项式求得问题的级数解,最后分析了应力场在裂纹尖端的奇异性,得到了裂纹尖端动态应力强度因子的计算公式,并通过数值计算给出了应力强度因子随入射波频率变化的特性曲线。     

A study ofJ-integral of the orthotropic composite material

The relation between J-integral near model I crack tip in the orthotropic plate and displacement derivative is derived in this paper. Meanwhile, the relation between stress intensity factor K _I and displacement is also given in this paper. With sticking film moire interferometry method, the three-point bending beam is tested, thus the values of J-integral and K _I can be obtained from the displacement field, and then the truth of relation formula between J-integral and K _I in the orthotropic composite materials is experimentally verified.     

界面下主微裂纹间干涉的研究

涉及两相正交各向异性体界面下裂纹间干涉问题的研究.多裂纹问题被分解为只含单裂纹的子问题,利用位错理论和裂面应力自由条件,列出一组可数值求解位错密度函数的奇异积分方程,从而求得应力强度因子.     

圆形界面裂纹反平面问题的基本奇异解

本文研究反平面集中力作用下,不同弹性材料的圆形界面上有多条裂纹的问题。运用复变函数的解析延拓技术与奇性主部分析方法,首次获得该问题的一般解答,求出了几种典型情况的封闭解;算出了应力强度因子,并由此导出一系列特殊结果,其中几个与已有文献完全吻合。     

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 stress concentration near a rigid line inclusion in a prestressed, elastic material. Part II.: Implications on shear band nucleation, growth and energy release rate

The full-field and asymptotic solutions derived in Part I of this article (for a lamellar rigid inclusion, embedded in a uniformly prestressed, incompressible and orthotropic elastic sheet, subject to a far-field deformation increment) are employed to analyse shear band formation, as promoted by the near-tip stress singularity. Since these solutions involve the prestress as a parameter, stress and deformation fields can be investigated near the boundary of ellipticity loss (but still within the elliptic range). In the vicinity of this boundary, the incremental stress and displacement fields evidence localized deformations with patterns organized into shear bands, evidencing inclinations corresponding to those predicted at ellipticity loss. These localized deformation patterns are shown to explain experimental results on highly deformed soft materials containing thin, stiff inclusions. Finally, the incremental energy release rate and incremental J-integral are derived, related to a reduction (or growth) of the stiffener. It is shown that this is always positive (or negative), but tends to zero approaching the Ellipticity boundary, which implies that reduction of the lamellar inclusion dies out and, simultaneously, shear bands develop.     

Constitutive behaviour of mixed mode loaded adhesive layer

Mixed mode testing of adhesive layer is performed with the Mixed mode double Cantilever Beam specimen. During the experiments, the specimens are loaded by transversal and/or shear forces; seven different mode mixities are tested. The J-integral is used to evaluate the energy dissipation in the failure process zone. The constitutive behaviour of the adhesive layer is obtained by a so called inverse method and fitting an existing mixed mode cohesive model, which uses a coupled formulation to describe a mode dependent constitutive behaviour. The cohesive parameters are determined by optimizing the parameters of the cohesive model to the experimental data. A comparison is made with the results of two fitting procedures. It is concluded that the constitutive properties are coupled, i.e. the peel and shear stress depend on both the peel and shear deformations. Moreover, the experiments show that the critical deformation in the peel direction is virtually independent of the mode mixity.     

双槽圆形截面管角裂纹应力强度因子

弯曲载荷作用下,双槽圆形截面管的角裂纹具有两个不同的奇异应力场和相应的应力强度因子,针对该异型薄壁管裂纹问题,提出了一种简单实用的应力强度因子求解方法。即利用守恒律,通过选取适当的三维积分路径,并结合初等力学的应力位移计算方法,显化了应力强度因子对J_2积分的贡献,建立了一个求解应力强度因子的方程。由于该方程不足以求解两个应力强度因子,利用材料力学平截面保持平面的变形假设,建立了应力强度因子之间的补充方程。将J_2积分与补充方程联立求解,既可得到弯曲载荷作用下双槽圆形截面管角裂纹的应力强度因子。对于其他异型薄壁管裂纹问题,该方法同样适用,计算过程简单。