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首页> 外文期刊>Zeitschrift fur Angewandte Mathematik und Mechanik >A micromorphic approach to stress gradient elasticity theory with an assessment of the boundary conditions and size effects
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A micromorphic approach to stress gradient elasticity theory with an assessment of the boundary conditions and size effects

机译：对梯度弹性理论的微观方法，评估边界条件和尺寸效应

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Basing on thermodynamics principles, a stress gradient elasticity theory is presented whereby the material is modeled as a centro-symmetric micromorphic anisotropic material. A principle of minimum complementary energy and a stationarity principle of Hellinger-Reissner type are provided, by which the boundary conditions are uniquely determined in a form consistent with continuum mechanics. For 3-D solids, these conditions include, beside the three ordinary boundary conditions for the assigned boundary data (displacements and/or tractions), six extra boundary conditions by which a microstructure/continuum compatibility condition is enforced, generally through the vanishing of the normal derivative of the stress, ?_nσ = O, on the whole boundary surface. A typical boundary-value problem for isotropic materials with a single length scale parameter under static loads is discussed, showing that it is governed by the same Navier PDEs of classical elasticity, along with a set of Helmholtz PDEs as constitutive equations. Size effects are shown to arise from two distinct sources, that is, the spatial fluctuating of the body force and the double curl of the Hookean stress field. It is proved that a formulation of stress gradient elasticity theory based on the pure notion of nonsimple material (like the ones existing in the literature) leads to anomalies in the ordinary and extra boundary conditions. A comparison of the present theory with that by Eringen is also discussed. Two examples of stress gradient structural models are analytically worked out, namely, a thick-walled cylinder and a Kirchhoff-Love circular plate. For comparison, the solutions for strain gradient material are also presented. Graphic illustrations show that in both examples stiffening size effects like with the ancillary strain gradient models are found.
机译：基于热力学原理，提出了一种应力梯度弹性理论，其中该材料被建模为中心对称的微观各向异性材料。提供了最小互补能量和Hellinger-Reissner型的实质性原理的原理，通过该互补型的边界条件唯一地确定与连续式力学一致的形式。对于3-D固体，这些条件包括在分配边界数据（位移和/或牵引力）的三个常规边界条件旁边，六种额外的边界条件，通过该六个额外的边界条件，通常通过消失应力的正常导数，在整个边界表面上的应力，_nσ= o。讨论了在静载荷下具有单个长度参数的各向同性材料的典型边值问题，示出它由经典弹性的相同Navier PDE管理，以及一组Helmholtz PDE作为本构方程。显示尺寸效果从两个不同的来源出现，即体力的空间波动和卷绕应力场的双卷曲。事实证明，基于非纤维材料的纯粹概念的应力梯度弹性理论的制定（如文献中存在的那样）导致普通和额外的边界条件中的异常。还讨论了本理论与eringen的比较。应力梯度结构模型的两个例子进行了分析地解决，即厚壁圆筒和Kirchhoff-Love圆形板。为了比较，还提出了应变梯度材料的溶液。图形插图表明，在两个示例中，在两个示例中，都发现了与辅助应变梯度模型的加强尺寸效果。

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