Continuum Mechanics in a Space of Any Dimension : I. Fundamental Laws and Constitutive Equations

Abstract

The basic concepts of the continuum mechanics in a space of any dimension are presented. The deformation measures are defined, and the three fundamental laws and the constitutive equations are defined according to an analogy of continuum mechanics in a three-dimensional space. For the isotropic elastic material in n - dimensional space, the stress is represented by a polynomial of degree (n - 1) of the left Cauchy-Green tensor, and its coefficients are scalar functions of the n invariants of the tensor. For the Stokes fluid, the stress is represented by a polynomial of degree (n - 1) of the stretching, and its coefficients are scalar functions of the n invariants and the mass density. For the continuum in a one-dimensional space, all of the quantities reduce to scalars. Then the identity relations demanded by the principle of frame-indifference become trivial relations, the difference between the isotropy and the anisotropy disappears, and the distinction between the fluid and the non-fluid material also disappears. If the material is incompressible, it reduces to a rigid material and the stress becomes completely undeterminate.