Nonlinear Continuum Mechanics of Solids

This section defines tensors as invariant quantities and introduces tensorial operations in absolute notation. Emphasis is given to the definition of some special tensors playing an important role in continuum mechanics. In addition, some useful results such as the definition of the gradient and the divergence of tensors are presented. This section aims to present the mathematical background for an easy understanding of the following sections.

This section introduces various deformation, stretch and strain tensors to describe the deformation behaviour of a solid during an arbitrary motion. Their definitions are essentially based on the deformation gradient and the stretch variables introduced by the polar decomposition theorem. Emphasis is given to the eigenvalue problems of stretches presenting a suitable background for the definition of various strain measures within a unified concept. Attention is then dedicated to pull-back and push-forward operations which are of major importance for the construction of the LIE-derivatives. Finally the rate of the deformation tensor and the spin tensor are introduced and their geometrical interpretations are given.

After a detailed discussion of the Cauchy stress vector and stress tensor this section introduces various stress tensors, shows systematically their connections and finally relates them to the deformation measures of chapter 2 as energy conjugate quantities. Particular attention is given to the physical interpretation of the Cauchy stress tensor presenting the starting point of the derivations.

In this section the notion of material time derivative is introduced which is then used to define the velocity and the acceleration vector. Finally the material time derivatives of some geometrical variables such as those of volume, surface and line elements are given in spatial formulation.

This chapter is devoted to a systematical presentation of constitutive equations applicable to large strain analysis of arbitrary elastic bodies. The discussion starts with a brief survey of the principles which are relevant for the formulation of constitutive equations. Then, the definition of objective tensors is given serving in the sequel to assess the material objectivity of constitutive laws. Constitutive equations are presented for Cauchy material and for hyperelasticity. Special attention is given to isotropic elasticity to enlighten particularly the application of the material symmetry principle. In this context relevant isotropic material models such as Ogden, Mooney-Rivlin and Neo-Hookh models are presented. St. Venant-Kirchhoff model is then derived through a linearization technique. The last cited model in turn is used to derive by a similar procedure the Hookean law as the simplest model for elastic materials. Finally, useful connections between the Hookean material law and some nonlinear material models are established again through linearization.