General Continuum Mechanics

Nonlinearity occurs in solid mechanics and fluid mechanics. Nonlinear elasticity describes an elastic material behavior with large deformations, known as geometric nonlinearity (Section 7.1). This is distinguished from material nonlinearity as occurs in plasticity (Section 7.4). For fluids, the nonlinear behavior refers to higher-order velocity gradients in the viscous stresses. This is the subject of non-Newtonian fluid mechanics (Section 7.2). Such nonlinearity is in addition to the nonlinearity arising from convection and thermoviscous dissipation discussed in Section 5.4. A similar behavior in solids leads to viscoelasticity with time dependency of the strain tensor (Section 7.3). Finally we consider the combined effects of thermoelasticity, viscoelasticity, and plasticity, known as thermoviscoelastoplasticity (Section 7.5). The curvilinear coordinates introduced in Chap. 6 will be utilized in order to deal with curvatures that may arise in large deformations.

Nonlinear Elasticity

Stresses with large strains

Definitions of stresses with large strains on deformed surfaces are intimately associated with the deformed coordinates chosen. This has been a controversial subject in literature. Some of the methods for defining stresses on deformed surfaces are uncompromising with computational aspects. It is the purpose of this section to review historical developments and demonstrate some practical approaches for handling stresses with large strains (Green and Shield, 1950; Green and Adkins, 1960; Green and Naghdi, 1965; Ogden, 1984).