图宾-明德林梯度弹性理论中的有限元法混合公式化

IF 0.7 4区 材料科学 Q4 MATERIALS SCIENCE, CHARACTERIZATION & TESTING
O. Yu. Chirkov, L. Nazarenko, H. Altenbach
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引用次数: 0

摘要

针对 Toupin-Mindlin 梯度弹性理论问题的有限元方法混合表述是合理的。该理论允许考虑材料微观结构尺寸所产生的尺度效应,尤其是在具有经典弹性限制的问题中。研究了边界问题的变分公式,其中应变、应力及其梯度与位移一起作为等效参数进入变分方程。这些方程的主要特点是只涉及位移的一阶偏导数,而经典问题公式中的微分方程涉及位移的四阶导数,而位移的拉格朗日变分方程则包含其双重微分。基于混合有限元法的求解大大简化了近似函数的选择,因为不需要使用有限元来确保元边界处第一位移导数的连续性。这种以位移、应变、应力及其梯度的单独近似为基础的计算方法适用于解决弹性理论的边界问题,其中包括应变梯度。对于混合方法变分方程,条件的制定是为了确保梯度弹性理论问题解的唯一性和混合近似的稳定性。这个条件是用正交投影算子定义的,它在应变分布的经典近似和混合近似之间建立了一一对应关系。在混合近似稳定性的最弱要求下,为位移和应变的变分方程的实际应用提出了一个非常适合的公式。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Mixed Formulation of Finite Element Method Within Toupin–Mindlin Gradient Elasticity Theory

The mixed formulation of the finite element method for the problems of the Toupin–Mindlin gradient theory of elasticity is justified. This theory permits accounting for scale effects stemming from the material microstructure dimensions, particularly in problems with the limitations of classical elasticity. The variational formulation of the boundary problem is examined where strains, stresses, and their gradients enter into the variational equations along with displacements as equivalent arguments. The key feature of these equations is that they involve only the first-order partial derivatives of displacements, in contrast to the differential equations of the classical problem formulation that involve the derivatives of displacements to the fourth order inclusive, and the Lagrange variational equation in displacements, which incorporates their double differentiation. Their solution based on the mixed finite element method greatly simplifies the choice of approximation functions since there is no need to use finite elements that ensure the continuity of the first displacement derivatives at the element boundaries. This formulation based on a separate approximation of displacements, strains, stresses, and their gradients, is applied to solving the boundary problems of elasticity theory, where the strain gradient is included. For the mixed-method variational equations, the condition is formulated so as to ensure the uniqueness of the solution and stability of the mixed approximation for gradient elasticity theory problems. This condition is defined with the orthogonal projection operator that establishes a one-to-one correspondence between the classical and mixed approximation of strain distributions. A well-suited formulation for the practical application of variational equations for displacements and strains is proposed with the weakest requirements for mixed approximation stability.

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来源期刊
Strength of Materials
Strength of Materials MATERIALS SCIENCE, CHARACTERIZATION & TESTING-
CiteScore
1.20
自引率
14.30%
发文量
89
审稿时长
6-12 weeks
期刊介绍: Strength of Materials focuses on the strength of materials and structural components subjected to different types of force and thermal loadings, the limiting strength criteria of structures, and the theory of strength of structures. Consideration is given to actual operating conditions, problems of crack resistance and theories of failure, the theory of oscillations of real mechanical systems, and calculations of the stress-strain state of structural components.
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