The recent advances of solid mechanics of polymer networks are that they can be well-modelled by a physically-based size-dependent constitutive relation via a simplified strain gradient elasticity theory. However, boundary value problems of plate models composed of polymer networks have not been reported, which limit wide applications of the models in the engineering science. In this paper, we systematically established a variationally consistent boundary value problems of Mindlin plate models for polymer networks leading to the framework of a simplified strain gradient elasticity. This study considers the strain energy produced by the strain gradient in the thickness direction and proposes a well-posed boundary value problem for a Mindlin plate with arbitrary boundaries, discussing possible boundary conditions, especially higher-order nonconventional ones. The senses of stress resultants and double stresses acting on the face of a volume element are firstly explained. Surprisingly, it is found that unexpected corner condition related to normal derivatives of shear force, bending moment, and twisting moment exists for plates with irregular boundaries—contradicting conventional mechanics notions of plates. For illustrative purpose, static bending analyses of a simply supported rectangular plate subjected to a uniformly distributed loading and a concentrated loading are provided. The effective Young's modulus predicted by this approach agrees well with reported result in the open literature. This work may be helpful in developing efficient numerical methods and offers new insights into the existence of corner condition in Mindlin plates within the context of a simplified strain gradient elasticity theory.

中文翻译：

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关于由聚合物网络制成的 Mindlin 板的尺寸依赖性力学


聚合物网络固体力学的最新进展是，它们可以通过简化的应变梯度弹性理论，通过基于物理的尺寸依赖本构关系很好地建模。然而，由聚合物网络组成的板模型的边界值问题尚未被报道，这限制了模型在工程科学中的广泛应用。在本文中，我们系统地建立了聚合物网络的 Mindlin 板模型的变分一致边值问题，从而建立了简化的应变梯度弹性框架。本研究考虑了应变梯度在厚度方向上产生的应变能，并提出了具有任意边界的 Mindlin 板的适定边界值问题，讨论了可能的边界条件，尤其是高阶非常规边界条件。首先解释了作用在体积单元表面的应力合力和双应力的感觉。令人惊讶的是，研究发现，对于边界不规则的板，存在与剪切力、弯矩和扭力矩的法向导数相关的意外角条件，这与板的传统力学概念相矛盾。为了便于说明，本文提供了简支矩形板在均匀分布载荷和集中载荷作用下的静态弯曲分析。这种方法预测的有效杨氏模量与公开文献中报告的结果非常吻合。这项工作可能有助于开发有效的数值方法，并在简化的应变梯度弹性理论的背景下为 Mindlin 板中存在角条件提供新的见解。