材料弹性常数关系推导在材料力学教学中的应用研究

Research on the Application of Deriving the Relationship between Elastic Constants of Materials in the Teaching of Mechanics of Materials

摘要: 弹性模量、剪切模量和泊松比等弹性常数是材料力学和弹性力学中重要的知识点，也是后续力学课程学习的基础。各向同性线弹性材料只有两个独立的弹性常数，因此其弹性模量、剪切模量和泊松比之间具有直接关联关系。在力学基础教学中充分利用这一关系的不同推导方法，可将相关力学知识串联融合，加深学生的理解，同时也可拓展教师授课的深度和广度。鉴于此，本文分别梳理了在单向拉伸应力状态、纯剪切应力状态和一般应力状态下各向同性材料弹性常数之间关系的多种推导方法，并揭示了该关系式实质上是各向同性线弹性体弹性完全对称性的反映，以丰富教学素材，培养学生举一反三的创新实践能力。

Abstract: Elastic constants such as elastic modulus, shear modulus, and Poisson’s ratio are important knowledge points in the mechanics of materials and elasticity, and are also the foundation for subsequent mechanics courses. Isotropic linear elastic materials have only two independent elastic constants, so there is a relation between the elastic modulus, shear modulus, and Poisson’s ratio. By fully utilizing the different derivation methods of this relationship in basic mechanics teaching, relevant mechanical knowledge can be connected and integrated. This can deepen students’ understanding, and also expand the depth and breadth of teachers’ teaching. In view of this, this article summarizes various derivation methods for the relationship between the elastic constants of isotropic materials under different conditions such as uniaxial tensile stress state, pure shear stress state, and general stress state, and reveals that the relation between the elastic modulus, shear modulus, and Poisson’s ratio is essentially a reflection of the full symmetry of the isotropic linear elasticity, which can enrich teaching materials and cultivate students’ innovative practical ability.