The main objective of this work is to estimate the compliance contribution tensor of the concave pore inhomogeneity surrounded by a transversely isotropic matrix. In this light, we make use of a recently developed adapted boundary conditions based Finite Elements Method to incorporate the matrix anisotropy and the correction of the bias induced by the bounded character of the mesh domain, which allows to accelerate the computation convergence without sacrificing its accuracy. The correction of the boundary conditions is given as functions of the Green tensor and its gradient as dependent on the anisotropic elasticity of the matrix material, which are rigorously calculated by means of the Fourier transform based integral method in particular for regularizing the singularities on the symmetric axis of the transverse isotropy. Simultaneously by complying with the numerical homogenization technique, the compliance contribution tensor is computed for different forms of pores (e.g. superspheroidal and superspherical ones, etc) embedded in an transversely isotropic matrix. The proposed numerical method is shown to be efficient and accurate after several appropriate assessment and validation by comparing its predictions, in some particular cases, with analytical results and some available numerical ones. Finally, the effect of the pore concavity on the compliance contribution tensor is quantitatively illustrated.

这项工作的主要目的是估计由横向各向同性矩阵包围的凹孔非均质性的顺应性贡献张量。有鉴于此，我们利用了最近开发的适应性边界条件基于有限元法的有限元方法，将矩阵各向异性和网格域的有界特征所引起的偏差的校正合并在一起，从而可以在不牺牲精度的情况下加快计算收敛速度。边界条件的校正是Green张量及其梯度的函数，取决于基体材料的各向异性弹性，它们是通过基于傅立叶变换的积分方法严格计算出来的，特别是为了使对称性上的奇异点正规化横观各向同性的轴。同时通过遵循数值均化技术，针对嵌入横向各向同性矩阵中的不同形式的孔（例如，超球形和超球形等）计算顺应性贡献张量。通过将其预测（在某些特定情况下）与分析结果和一些可用的数值方法进行比较，可以证明经过多次适当的评估和验证之后，所提出的数值方法是有效且准确的。最后，定量说明了孔洞凹度对柔度贡献张量的影响。