The incompatibility of linearized piecewise smooth strain field, arising out of volumetric and surface densities of topological defects and metric anomalies, is investigated. First, general forms of compatibility equations are derived for a piecewise smooth strain field, defined over a simply connected domain, with either a perfectly bonded or an imperfectly bonded interface. Several special cases are considered and discussed in the context of existing results in the literature. Next, defects, representing dislocations and disclinations, and metric anomalies, representing extra matter, interstitials, thermal, and growth strains, etc., are introduced in a unified framework which allows for incorporation of their bulk and surface densities, as well as for surface densities of defect dipoles. Finally, strain incompatibility relations are derived both on the singular interface, and away from it, with sources in terms of defect and metric anomaly densities. With appropriate choice of constitutive equations, the incompatibility relations can be used to determine the state of internal stress within a body in response to the given prescription of defects and metric anomalies.

中文翻译：

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拓扑缺陷和度量异常作为分段平滑应变场不兼容的来源

研究了由拓扑缺陷的体积密度和表面密度以及度量异常引起的线性化分段平滑应变场的不相容性。首先，为分段平滑应变场导出兼容性方程的一般形式，该应变场定义在简单连接域上，具有完美结合或不完美结合的界面。在文献中现有结果的背景下考虑和讨论了几个特殊情况。接下来，缺陷（代表位错和向错）和度量异常（代表额外物质、间隙、热和生长应变等）被引入到一个统一的框架中，该框架允许合并它们的体积和表面密度，以及表面缺陷偶极子的密度。最后，应变不相容性关系是在奇异界面上和远离奇异界面导出的，来源是缺陷和度量异常密度。With appropriate choice of constitutive equations, the incompatibility relations can be used to determine the state of internal stress within a body in response to the given prescription of defects and metric anomalies.