Page 17 - Introduction to Continuum Mechanics

2 Introduction

and (2) constitutive equations defining idealized materials. The general principles are axioms
considered to be self-evident from our experience with the physical world, such as conservation
of mass, balance of linear momentum, of moment of momentum, of energy, and the entropy
inequality law. Mathematically, there are two equivalent forms of the general principles: (1)
the integral form, formulated for a finite volume of material in the continuum, and (2) the field
equations for differential volume of material (particle) at every point of the field of interest.
Field equations are often derived from the integral form. They can also be derived directly
from the free body of a differential volume. The latter approach seems to suit beginners. In
this text both approaches are presented, with the integral form given toward the end of the
text. Field equations are important wherever the variations of the variables in the field are
either of interest by itself or are needed to get the desired information. On the other hand, the
integral forms of conservation laws lend themselves readily to certain approximate solutions.

The second major part of the theory of continuum mechanics concerns the "constitutive
equations" which are used to define idealized material. Idealized materials represent certain
aspects of the mechanical behavior of the natural materials. For example, for many materials
under restricted conditions, the deformation caused by the application of loads disappears with
the removal of the loads. This aspect of the material behavior is represented by the constitutive
equation of an elastic body. Under even more restricted conditions, the state of stress at a point
depends linearly on the changes of lengths and mutual angle suffered by elements at the point
measured from the state where the external and internal forces vanish. The above expression
defines the linearly elastic solid. Another example is supplied by the classical definition of
viscosity which is based on the assumption that the state of stress depends linearly on the
instantaneous rates of change of length and mutual angle. Such a constitutive equation defines
the linearly viscous fluid. The mechanical behavior of real materials varies not only from
material to material but also with different loading conditions for a given material. This leads
to the formulation of many constitutive equations defining the many different aspects of
material behavior. In this text, we shall present four idealized models and study the behavior
they represent by means of some solutions of simple boundary-value problems. The idealized
materials chosen are (1) the linear isotropic and anisotropic elastic solid (2) the incompressible
nonlinear isotropic elastic solid (3) the linearly viscous fluid including the inviscid fluid, and
(4) the Non-Newtonian incompressible fluid.

One important requirement which must be satisfied by all quantities used in the formulation
of a physical law is that they be coordinate-invariant. In the following chapter, we discuss such
quantities.