Chapter 39: Elastic Materials – Deformation & Motion

The chapter provides a rigorous mathematical foundation for analyzing the localized distortions, or strain, within generalized elastic materials, emphasizing the necessity of employing a tensor of strain to accurately capture the six independent components of stretch, compression, and shear in three dimensions. Initial concepts introduce homogeneous strain and the definition of normal strain (e_xx), which represents fractional expansion along an axis, and the more complex shear strains (e_xy), which describe angle changes, all components being derived from the displacement vector's derivatives. Building upon this, the tensor of elasticity is introduced to quantify the potential energy density (W) stored in the strained material, establishing the generalized, three-dimensional form of Hooke's Law where energy is proportional to the squares of the strain components. This energy relationship is defined by numerous elastic constants (C_ijkl), which decrease in number for materials exhibiting high symmetry, simplifying to three independent constants for cubic crystals and just two—the Lamé constants (mu and lambda)—for isotropic materials. The dynamics of motions in an elastic body are examined by applying Newton’s laws to a volume element, balancing internal forces (stresses) and external forces, which results in wave equations showing that disturbances propagate as distinct longitudinal sound waves and transverse shear waves in isotropic media. The visualization technique of photoelasticity, which uses polarized light to observe internal stresses in strained models, is also noted. Finally, the text addresses nonelastic behavior, showing how stress-strain curves deviate from linearity at high stresses, leading to material failure, fracture, or complex inertial recovery phenomena, and details the method for calculating the elastic constants from first principles by modeling the atomic lattice (e.g., an ionic crystal) as a network of springs and computing the total change in spring potential energy under strain.