For a solid surface or interface that is subjected to transverse loading, the influence of its flexural resistibility to bending deformation becomes significant. A spherical inhomogeneity or void embedded in an infinite elastic medium under the application of nonhydrostatic loads represents a typical example. In this work, we consider the most fundamental loading of a far-field unidirectional tension. Analytical displacements and stresses are developed by the coupling of a Steigmann–Ogden surface mechanical model, the simple method of Boussinesq displacement potentials, the semi-inverse method of elasticity, and Legendre series representations of spherical harmonics. The problem is then solved by converting the equilibrium equations of displacement into a linear system with respect to the Legendre series coefficients. The developed solutions are general in the sense that they may reduce to their classical or Gurtin–Murdoch counterparts as special cases. Analytical expressions reveal that the derived solution depends on four dimensionless ratios from among surface material parameters, shear moduli ratio, and inhomogeneity or void radius. In particular, instead of depending on both flexural parameters in the moment–curvature relation, one fixed combination is sufficient to represent the surface flexural rigidity. This is in contrast with the influence of the in-plane elastic stiffness, in which both surface Lamé parameters matter. Parametric studies further demonstrate that, for metallic inhomogeneities or voids with radii between 10 nm and 100 nm, the effects of surface flexural rigidity on stress distributions and stress concentrations are significant.

中文翻译：

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基于 Steigmann-Ogden 表面模型的远场单向加载下球形纳米不均匀性的解析解

对于承受横向载荷的固体表面或界面，其抗弯能力对弯曲变形的影响变得显着。在非静水载荷作用下，嵌入无限弹性介质中的球形不均匀性或空隙就是一个典型的例子。在这项工作中，我们考虑了远场单向张力的最基本载荷。解析位移和应力是通过耦合 Steigmann-Ogden 表面力学模型、Boussinesq 位移势的简单方法、弹性的半逆方法和球谐函数的 Legendre 级数表示来开发的。然后通过将位移的平衡方程转换为关于勒让德级数系数的线性系统来解决该问题。所开发的解决方案在某种意义上是通用的，因为它们可能会简化为它们的经典或 Gurtin-Murdoch 对应方案作为特殊情况。解析表达式表明，导出的解取决于表面材料参数、剪切模量比和不均匀性或空隙半径中的四个无量纲比。特别是，不是依赖于弯矩-曲率关系中的两个弯曲参数，一个固定的组合就足以表示表面弯曲刚度。这与平面内弹性刚度的影响形成对比，其中两个表面拉梅参数都很重要。参数研究进一步表明，对于半径在 10 nm 和 100 nm 之间的金属不均匀性或空隙，表面抗弯刚度对应力分布和应力集中的影响是显着的。