Abstract

The problem of the mechanics of cracks in an orthotropic plate is considered. A general solution form is proposed as a generalized Papkovich–Neuber representation for the plane problem of the theory of elasticity of an orthotropic body. This representation allows us to write the general solution in displacements through two vector potentials satisfying the generalized harmonic equations. The dependences between the vector potentials of Papkovich–Neuber and the stress function are presented. It is shown that there is a complex-valued solution form through four analytic functions of complex variables associated with coefficients that are the roots of the characteristic equation corresponding to the generalized biharmonic equation. It is proved the statement that the operator of the problem is written only through conjugate analytic functions of complex variables, and the general solution is written through arbitrary linear combinations of four functions of complex variables. A general form of solutions with a given singularities is presented, including representations for both cracks in Mode I and cracks in Mode II. The conditions are analyzed that make it possible to obtain generalized non-singular solutions for cracks of Mode I and II. Finally, we establish the conditions for the regularization of singular solutions through the solutions of the generalized Helmholtz equations that correspond to a particular version of the gradient theory of elasticity.

中文翻译：

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正交各向异性弹性理论奇异解的正则化

摘要

考虑了正交各向异性板中裂纹的力学问题。对于正交异性体弹性理论的平面问题，提出了一种通用的求解形式，作为广义的Papkovich-Neuber表示。这种表示使我们可以通过两个满足广义谐波方程的矢量势来写位移的一般解。提出了Papkovich–Neuber矢量势与应力函数之间的依赖关系。通过对与系数相关联的复变数的四个解析函数，得出复值解形式，这些系数是与广义双调和方程相对应的特征方程的根。证明了该问题的算子仅通过复变量的共轭解析函数来写，而一般解是通过复变量的四个函数的任意线性组合来写的。给出了具有给定奇点的解决方案的一般形式，包括模式I裂纹和模式II裂纹的表示。分析的条件使得有可能获得I型和II型裂纹的广义非奇异解。最后，我们通过对应于弹性梯度理论特定版本的广义Helmholtz方程的解，为奇异解的正则化建立了条件。通用解是通过复杂变量的四个函数的任意线性组合来写的。给出了具有给定奇点的解的一般形式，包括模式I裂纹和模式II裂纹的表示。分析的条件使得有可能获得I型和II型裂纹的广义非奇异解。最后，我们通过对应于弹性梯度理论特定版本的广义Helmholtz方程的解，为奇异解的正则化建立了条件。通用解是通过复杂变量的四个函数的任意线性组合来写的。给出了具有给定奇点的解决方案的一般形式，包括模式I裂纹和模式II裂纹的表示。分析的条件使得有可能获得I型和II型裂纹的广义非奇异解。最后，我们通过对应于弹性梯度理论特定版本的广义Helmholtz方程的解，为奇异解的正则化建立了条件。分析的条件使得有可能获得I型和II型裂纹的广义非奇异解。最后，我们通过对应于弹性梯度理论特定版本的广义Helmholtz方程的解，为奇异解的正则化建立了条件。分析的条件使得有可能获得I型和II型裂纹的广义非奇异解。最后，我们通过对应于弹性梯度理论特定版本的广义Helmholtz方程的解，为奇异解的正则化建立了条件。