基于非协调元特征法的奇异性问题分析

提出了一个基于位移的、分析平面尖劈尖端奇性应力场和位移场问题的非协调FE特征分析法．该方法与过去原有求解裂纹尖端近似场的有限元特征分析方法导出公式的出发点不同，并且采用的单元形式为非协调元，尖劈尖端邻域内的位移场假定没有采用奇异变换技术，运用该方法处理了若干尖劈和接头的算例，所有的计算结果表明，该方法较原有方法使用的单元少而且精度高，具有应用广泛性。     

Asymptotic analysis of an impermeable crack in an electrostrictive material subjected to electric loading

The simple asymptotic problem of an impermeable crack in an electrostrictive ceramic under electric loading is analyzed. Closed form solutions of elastic fields are obtained by using the complex function theory. It is found that the K_I-dominant region is very small compared to the electric saturation zone. A fracture parameter for an electrostrictive material subjected to electric loading is discussed. In order to investigate the influence of the transverse electric displacement on fracture behavior under the small-scale conditions, we also consider the modified boundary layer problem of a crack in an electrostrictive material. Analytic solutions of electric displacement fields for the asymptotic problem are obtained based on the nonlinear dielectric theory from a modified boundary layer analysis. The shape of the electric displacement saturation zone is shown to depend on the transverse electric displacement. Stress intensity factors induced by the electrostrictive strains are evaluated using the nonlinear solution of the electric displacements. It is found that the transverse electric displacement affects strongly the variation of the mode mixity.     

弹塑性双材料界面裂纹尖端场的渐近分析和位移匹配

本文对两种硬化指数的弹塑性材料界面裂纹尖端场进行了分析。通过对渐近场的计算，讨论了尖端场位移匹配问题和一阶静水压力场的存在，使应力解更加完备。     

三维切口/裂纹结构的扩展边界元法分析

在线弹性理论中,三维 V 形切口/裂纹结构尖端区域存在多重应力奇异性,常规数值方法不易求解. 本文提出和建立了三维扩展边界元法 (XBEM),用于分析三维线弹性 V 形切口/裂纹结构完整的位移和应力场. 先将三维线弹性 V 形切口/裂纹结构分为尖端小扇形柱和挖去小扇形柱后的外围结构. 尖端小扇形柱内的位移函数采用自尖端径向距离 $r$ 的渐近级数展开式表达,其中尖端区域的应力奇异指数、位移和应力特征角函数通过插值矩阵法获得. 而级数展开式各项的幅值系数作为基本未知量. 挖去扇形域后的外围结构采用常规边界元法分析. 两者方程联立求解可获得三维 V 形切口/裂纹结构完整的位移和应力场,包括切口/裂纹尖端区域精细的应力场. 扩展边界元法具有半解析法特征,适用于一般三维 V 形切口/裂纹结构完整位移场和应力场的分析,其解可精细描述从尖端区域到整体结构区域的完整应力场. 作者研制了三维扩展边界元法程序,文中给出了两个算例,通过计算结果分析,表明了扩展边界元法求解三维 V 形切口/裂纹结构完整应力场的准确性和有效性.      

Interaction integrals for fracture analysis of functionally graded piezoelectric materials

This paper presents domain form of the interaction integrals based on three independent formulations for computation of the stress intensity factors and electric displacement intensity factor for cracks in functionally graded piezoelectric materials. Conservation integrals of J-type are derived based on the governing equations for piezoelectric media and the crack tip asymptotic fields of homogeneous piezoelectric medium as auxiliary fields. Each of the formulation differs in the way auxiliary fields are imposed in the evaluation of interaction integral and each of them results in a consistent form of the interaction integral in the sense that extra terms naturally appears in their derivation to compensate for the difference in the chosen crack tip asymptotic fields of homogeneous and functionally graded piezoelectric medium. The additional terms play an important role of ensuring domain independence of the presented interaction integrals. Comparison of the numerically evaluated intensity factors through the three consistent formulations with those obtained using displacement extrapolation method is presented by means of two examples.     

ANALYSIS OF 3-D NOTCHED/CRACKED STRUCTURES BY USING EXTENDED BOUNDARY ELEMENT METHOD 1)

在线弹性理论中,三维 V 形切口/裂纹结构尖端区域存在多重应力奇异性,常规数值方法不易求解. 本文提出和建立了三维扩展边界元法 (XBEM),用于分析三维线弹性 V 形切口/裂纹结构完整的位移和应力场. 先将三维线弹性 V 形切口/裂纹结构分为尖端小扇形柱和挖去小扇形柱后的外围结构. 尖端小扇形柱内的位移函数采用自尖端径向距离 $r$ 的渐近级数展开式表达,其中尖端区域的应力奇异指数、位移和应力特征角函数通过插值矩阵法获得. 而级数展开式各项的幅值系数作为基本未知量. 挖去扇形域后的外围结构采用常规边界元法分析. 两者方程联立求解可获得三维 V 形切口/裂纹结构完整的位移和应力场,包括切口/裂纹尖端区域精细的应力场. 扩展边界元法具有半解析法特征,适用于一般三维 V 形切口/裂纹结构完整位移场和应力场的分析,其解可精细描述从尖端区域到整体结构区域的完整应力场. 作者研制了三维扩展边界元法程序,文中给出了两个算例,通过计算结果分析,表明了扩展边界元法求解三维 V 形切口/裂纹结构完整应力场的准确性和有效性.     

Modified boundary layer analysis of an electrode in an electrostrictive material

A thin electrode embedded in an electrostrictive material under electric loading is investigated. In order to obtain an asymptotic form of electric fields and elastic fields near the electrode edge, we consider a modified boundary layer problem of an electrode in an electrostrictive material under the small scale saturation condition. The exact electric solution for the electrode is obtained by using the complex function theory. It is found that the shape of the electric displacement saturation zone is sensitive to the transverse electric displacement. A perturbation solution of stress fields induced by incompatible electrostrictive strains for the small value of the transverse electric displacement is obtained. The influence of transverse electric displacement on a microcrack initiation from the electrode edge is also discussed.     

Higher-order analysis of near-tip fields around an interfacial crack between two dissimilar power law hardening materials

By means of an asymptotic expansion method of a regular series, an exact higher-order analysis has been carried out for the near-tip fields of an interfacial crack between two different elastic-plastic materials. The condition of plane strain is invoked. Two group of solutions have been obtained for the crack surface conditions: (1) traction free and (2) frictionless contact, respectively. It is found that along the interface ahead of crack tip the stress fields are co-order continuous while the displacement fields are cross-order continuous. The zone of dominance of the asymptotic solutions has been estimated. The project supported by the National Natural Science Foundation of China     

Numerical analysis of an electrode embedded between dissimilar electrostrictive materials

The asymptotic problem of an electrode that is embedded between dissimilar electrostrictive materials and subjected to electric loading is numerically analyzed by using the finite element method. Electrostatic analysis for the asymptotic problem is conducted under the small-scale saturation condition on the basis of the mathematical equivalence between anti-plane shearing and electrostatics. The distribution of the electric displacement fields is obtained. It is shown that the shapes of the saturation zones are affected by the ratios of the permittivities and the saturated electric displacements between the dissimilar electrostrictive materials. Stress fields that are generated for matching incompatible strains, which are induced by non-uniform electric displacement fields, are numerically calculated for various combinations of the material properties. Also, stress intensity factors for arbitrary small cracks that are initiated from the edge of an electrode are evaluated. Behaviors of cracking that may take place at the edge of an electrode are discussed.     

求解含钝裂纹体应力场的扩展单元方法

本文基于钝裂纹端部位移场的渐近解和等参元构造方法,开发了一种新的适合钝裂纹端部应力场计算的扩展单元法,为了消除不同单元间的位移不协调又在扩展单元的基础上提出过渡单元.和常规的等参元相比,扩展单元除了以节点位移为待求未知量外,它们额外增加了Ⅰ型和Ⅱ型广义应力强度因子作为未知量.根据这个理论我们编制了有限元的程序并计算了算例,算例表明,在网格较大的情况下,与常规等参元计算方案相比,扩展单元和过渡单元法更好地接近理论值,它具有计算精度高、减少缺陷附近的单元数量和计算时间等优点.     

Stress intensity factors for an interface crack with impermeable surfaces between dissimilar electrostrictive materials

The asymptotic problem of a semi-infinite interface crack between dissimilar electrostrictive materials that are subjected to electric loading is numerically analyzed by using the finite element method. Numerical results of electric displacement fields are obtained on the basis of the mathematical equivalence of the mode III problem and an electrostatic problem. The shape and the size of saturation zones are explored as a function of the ratio of the saturated electric displacements of dissimilar electrostrictive materials. In contrast with conventional wisdom, the ratio of the permittivities is shown to exert a negligibly small influence on electric displacement fields. For various combinations of the material properties of dissimilar electrostrictive materials, stress fields and stress intensity factors are systemically calculated by using the numerical results of electric displacement fields. The effects of the electric, elastic, and electrostrictive properties on stress intensity factors are demonstrated.     

幂硬化弹塑性-刚性界面Ⅲ型定常扩展裂纹尖端的渐近场

对幂硬化弹塑性材料－刚性材料界面上裂纹以定常方式扩展的Ⅲ型问题进行弹塑性渐近分析，给出裂纹尖端的应力，应变和位移场解。通过数值计算，考察了不同Mach数以及裂纹尖端混合参数对场解的构造以及应力，应变分布的影响，为给出合理的断裂准则提供理论依据。     

插值矩阵法分析双材料平面V形切口奇异阶

对二维V形切口问题提出奇异阶分析的一个新方法.首先,以V形切口尖端附近位移场沿其径向渐近展开为基础,将其线弹性理论控制方程转换成切口尖端附近关于周向变量的常微分方程组特征值问题,然后将数值求解两点边值问题的插值矩阵法进一步拓展为求解一般常微分方程组特征值问题,插值矩阵法是在离散节点上采用微分方程中待求函数的最高阶导数作为基本未知量.由此,V形切口的应力奇性阶问题通过插值矩阵法获得,同时相应的切口附近位移场和应力场特征向量一并求出.     

AN INVESTIGATION ON ASYMPTOTIC NEAR-TIP FIELDS OF DYNAMIC CRACK IN AN ELASTIC-PERFECTLY PLASTIC COMPRESSIBLE MATERIAL

The stress and deformation fields near the tip of a mode-I dynamic crack steadilypropagating in an elastic-perfectly plastic compressible material are considered under plane strain condi-tions. Within the framework of infinitesimal displacement gradient theory, the material is character-ized by the Von Mises yield criterion and the associated J_2 flow theory of plasticity. Through rigorousmathematical analysis, this paper eliminates the possibilities of elastic unloading and continuousasymptotic fields with singular deformation, and then constructs a fully continuous and boundedasymptotic stress and strain field. It is found that in this solution there exists a parameter (?)_0 whichcannot be determined by asymptotic analysis but may characterize the effect of the far field. Lastly thevariations of continuous stresses, velocities and strains around the crack tip are given numerically fordifferent values of (?)_0.     

ASYMPTOTIC DYNAMIC SOLUTION TO THE MODE Ⅰ PROPAGATING CRACK-TIP FIELD IN ELASTIC-VISCOPLASTIC MATERIAL

A new elastic-viscoplastic mode was proposed to analyze the stress and strain fields surrounding the tip of a propagating mode Ⅰ cracks. A proper displacement pattern was suggested and asymptotic equations were derived, and numerical solutions were illustrated. The analysis and calculation show that the crack-tip field is of logarithmic singularity for smaller viscosity, however no solution exists for large viscosity. By a careful analysis and comparison, it is found that the present results retain all merits of those given by Gao Yu-chen, while removing existing problems.     

ASYMPTOTIC DYNAMIC SOLUTION TO THE MODE PROPAGATING CRACK-TIP FIELD IN ELASTIC- VISCOPLASTIC MATERIAL

A new elastic-viscoplastic mode was proposed to analyze the stress and strain fields surrounding the tip of a propagating mode Ⅰ cracks. A proper displacement pattern was suggested and asymptotic equations were derived, and numerical solutions were illustrated. The analysis and calculation show that the crack-tip field is of logarithmic singularity for smaller viscosity, however no solution exists for large viscosity. By a careful analysis and comparison, it is found that the present results retain all merits of those given by Gao Yu-chen, while removing existing problems.     

Effects of characteristic material lengths on mode III crack propagation in couple stress elastic–plastic materials

The asymptotic fields near the tip of a crack steadily propagating in a ductile material under Mode III loading conditions are investigated by adopting an incremental version of the indeterminate theory of couple stress plasticity displaying linear and isotropic strain hardening. The adopted constitutive model is able to account for the microstructure of the material by incorporating two distinct material characteristic lengths. It can also capture the strong size effects arising at small scales, which results from the underlying microstructures. According to the asymptotic crack tip fields for a stationary crack provided by the indeterminate theory of couple stress elasticity, the effects of microstructure mainly consist in a switch in the sign of tractions and displacement and in a substantial increase in the singularity of tractions ahead of the crack-tip, with respect to the classical solution of LEFM and EPFM. The increase in the stress singularity also occurs for small values of the strain hardening coefficient and is essentially due to the skew-symmetric stress field, since the symmetric stress field turns out to be non-singular. Moreover, the obtained results show that the ratio η introduced by Koiter has a limited effect on the strength of the stress singularity. However, it displays a strong influence on the angular distribution of the asymptotic crack tip fields.