Page 243 - Intro to Tensor Calculus

237

• Conservation of linear momentum sometimes called the Cauchy equation of motion.

j
j
σ ij ,i + %b = %f , j =1, 2, 3.
• Conservation of angular momentum

σ ij = σ ji
• Strain tensor for linear elasticity
1
e ij = (u i,j + u j,i ).
2
If we assume that the continuum is in equilibrium, and there is no motion, then the velocity and
acceleration terms above will be zero. The continuity equation then implies that the density is a constant.
The conservation of angular momentum equation requires that the stress tensor be symmetric and we need
find only six stresses. The remaining equations reduce to a set of nine equations in the fifteen unknowns:

3 displacements u 1 ,u 2,u 3
6 strains e 11 ,e 12 ,e 13 ,e 22 ,e 23 ,e 33

6 stresses σ 11 ,σ 12 ,σ 13 ,σ 22 ,σ 23 ,σ 33

Consequently, we still need additional information if we desire to determine these unknowns.
Note that the above equations do not involve any equations describing the material properties of the
continuum. We would expect solid materials to act differently from liquid material when subjected to external
forces. An equation or equations which describe the material properties are called constitutive equations.
In the following sections we will investigate constitutive equations for solids and liquids. We will restrict
our study to linear elastic materials over a range where there is a linear relationship between the stress and
strain. We will not consider plastic or viscoelastic materials. Viscoelastic materials have the property that
the stress is not only a function of strain but also a function of the rates of change of the stresses and strains
and consequently properties of these materials are time dependent.