Eshelby's inhomogeneity problem is solved within the second form of Mindlin's first strain gradient elasticity theory for the prediction of the effective elastic properties of composites. Considering Green's function technique, an integral equation is established for an ellipsoidal inhomogeneity embedded in a homogeneous elastic medium and subjected to non-uniform boundary conditions. Within isotropic elasticity, the mean strain inside a spherical inhomogeneity is detailed to provide analytical results. In addition to the elastic properties of the inhomogeneity and the matrix, the strain localization depends on five gradient elastic constants, introduced by the first strain gradient elasticity theory. The effective bulk and shear moduli of a two-phase composite are predicted through Mori-Tanaka's scheme. The strain localization and the effective elastic moduli are then expressed within some simplified gradient elasticity theories. To test the relevance of the developed model, its predictions are compared with those of some investigations and the effective elastic properties are analyzed for a metal matrix composite. Finally, some comparisons with experimental data are performed to estimate the characteristic length scale parameters and gradient elastic constants of local phases.

中文翻译：

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Mindlin 第一应变梯度弹性理论中的 Eshelby 不均匀性模型及其在复合材料中的应用


Eshelby 的不均匀性问题在 Mindlin 的第一应变梯度弹性理论的第二种形式中得到解决，用于预测复合材料的有效弹性特性。考虑格林函数技术，为嵌入均匀弹性介质中并受到非均匀边界条件的椭球不均匀性建立了一个积分方程。在各向同性弹性中，详细描述了球形不均匀性内的平均应变，以提供分析结果。除了不均匀性和基体的弹性特性外，应变局域化还取决于第一个应变梯度弹性理论引入的五个梯度弹性常数。两相复合材料的有效体积模量和剪切模量是通过 Mori-Tanaka 的方案预测的。然后，应变局部化和有效弹性模量在一些简化的梯度弹性理论中表示。为了检验所开发模型的相关性，将其预测与一些研究的预测进行了比较，并分析了金属基复合材料的有效弹性特性。最后，与实验数据进行了一些比较，以估计局部相的特征长度尺度参数和梯度弹性常数。