Constitutive equations for linear elasticity

In this chapter the constitutive equations for linear elasticity in which the free energy is a quadratic function, and the stress a linear function, of the small strain tensor are introduced. In general, the fourth-order linear elasticity tensor which relates the stress to the strain has twenty-one independent elastic constants. Most solids, however, exhibit some symmetry, the effect of which is to reduce the number of elastic constants. Forms of the linear elasticity tensor for some anisotropic materials, as well as the widely-used forms for an isotropic material, which has only two independent elastic constants, are discussed. The engineering elastic moduli known as the shear modulus and the bulk modulus, as well as the Young’s modulus and the Poisson’s ratio for isotropic linear elastic materials are introduced. Considerations of temperature changes necessitate the introduction of another tensorial material property called the thermal expansion tensor, which for an isotropic material reduces to a scalar coefficient of thermal expansion.