This text is intended to be a broad introduction to isotropic elasticity, with a focus on providing complete derivations for strain, stress, and their constitutive relationships for both “finite” hyperelasticity and “infinitesimal” linear elasticity. The complications that arise due to rigid body rotations were discussed, and the importance of “work-conjugate” tensor pairs was emphasized. A particular “rate” approach to the handling of rigid body rotations was discussed, which included a particular treatment for the “frame of reference” problem with respect to our hyperelasticity model as well as for our linear infinitesimal elasticity model.

Specific hyperelastic material models were introduced, along with a description of specific methodologies that are currently used for fitting to experimental data. Lastly, the constitutive equations for linear infinitesimal elasticity were developed, including the derivation of Young’s Modulus and Poisson Ratio, which are material properties that may be familiar to the reader. We concluded this chapter by considering all of the important equations in solid mechanics: the equations of equilibrium, the equations relating stress and strain, and the equations relating strain to displacement. These, or similar, equations, are used in any FEA code on the market. With the completion of this text, we now have the foundation that is necessary in order to investigate many challenging engineering and materials problems. We also have the tools to study more specific topics in solid mechanics, such as anisotropy, viscoelasticity, and metal plasticity. Further research in the area of solid mechanics is also needed, particularly in the treatment of brittle-type damage, for those so inclined.