In this article, we present a computationally efficient homogenization technique for linear coupled diffusion–mechanics problems. It considers a linear chemo-mechanical material model at the fine scale, and relies on a full separation of scales between the time scales governing diffusion and mechanical phenomena, and a relaxed separation of scales for diffusion between the matrix and the inclusion. When the characteristic time scales associated with mass diffusion are large compared to those linked to the deformation, the mechanical problem can be considered to be quasi-static, and a full separation of scales can be assumed, whereas the diffusion problem remains transient. Using equivalence of the sum of virtual powers of internal and transient forces between the microscale and the macroscale, a homogenization framework is derived for the mass diffusion, while for the mechanical case, considering its quasi-static nature, the classical equivalence of the virtual work of internal forces is used instead. Model reduction is then applied at the microscale. Assuming a relaxed separation of scales for diffusion phenomena, the microscopic fields are split into steady-state and transient parts, for which distinct reduced bases are extracted, using static condensation for the steady-state part and the solution of an eigenvalue problem for the transient part. The model reduction at the microscale results in emergent macroscopic enriched field variables, evolution of which is described with a set of ordinary differential equations which are inexpensive to solve. The net result is a coupled diffusion–mechanics enriched continuum at the macroscale. Numerical examples are conducted for the cathode–electrolyte system characteristic of a lithium ion battery. The proposed reduced order homogenization method is shown to be able to capture the coupled behavior of this system, whereby high computational gains are obtained relative to a full computational homogenization method.

中文翻译：

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耦合力学的多尺度瞬态扩散的丰富连续体

在本文中，我们提出了一种计算有效的均质化技术，用于解决线性耦合扩散力学问题。它考虑了精细尺度上的线性化学机械材料模型，并依赖于控制扩散和机械现象的时间尺度之间尺度的完全分离，以及矩阵和内含物之间扩散尺度的宽松分离。当与质量扩散相关的特征时间尺度比与变形相关的特征尺度大时，机械问题可以被认为是准静态的，并且可以假定尺度完全分离，而扩散问题仍然是瞬态的。使用微尺度和宏观尺度之间的内力和瞬态力的虚拟力之和的等价性，对于质量扩散，导出了一个均质化框架，而对于机械情况，考虑到其准静态性质，取而代之的是使用内力虚拟功的经典当量。然后在微尺度上应用模型简化。假设对扩散现象的尺度进行了轻松的分离，则微观区域被分为稳态部分和瞬态部分，对于稳态部分，使用静态凝聚和瞬态特征值问题的解决方案，可以提取出明显的还原碱基。部分。在微观尺度上的模型简化导致涌现出的宏观富集场变量，其演化由一组普通的微分方程式描述，而这些方程通常难以解决。最终结果是在宏观尺度上耦合了扩散-力学的连续体。针对锂离子电池的阴极-电解质系统特性进行了数值示例。所提出的降阶均质化方法显示出能够捕获该系统的耦合行为，从而相对于完整的计算均质化方法可以获得较高的计算增益。