CONTINUUM MECHANICS for ENGINEERS

A most important feature of the field Eqs 6.4-1 through Eq 6.4-3 is that they
are linear in the unknowns. Consequently, if ( 1)
, ( 1)
, and ( 1)
σ are a solution
ij ε ij ui 1 * 1 * for body forces bi and surface tractions ti , whereas , , and are
a solution for body forces and surface tractions , then
ˆn ( )
( 2)
( 2)
( 2)
σ ij ε ij ui 2 *
2 *
bi
ti
ˆn ( )
( )
( )
( )
σij = + , εij = + , and ui = +
σ 1
ij σ 2
ij ε 1
ij ε 2
1
ij
ui 1 * 2 *
offer a solution for the situation where bi = bi + bi and t
( ˆn ) 1 * = + .
i
ti
This is a statement of the principle of superposition, which is extremely useful
for the development of solutions in linear elasticity.
For those problems in which the boundary conditions are given in terms
of displacements by Eq 6.4-4, it is convenient for us to eliminate the stress
and strain unknowns from the field equations so as to state the problem
solely in terms of the unknown displacement components. Thus, by substituting
Eq 6.4-2 into Hooke’s law (Eq 6.4-3) and that result into the equilibrium
equations (Eq 6.4-1), we obtain the three second-order partial
differential equations
ˆn ( ) 2 *
ti
ˆn ( )
µu i,jj + (λ + µ)u j,ji + ρb i = 0 (6.4-7)
which are known as the Navier equations. If a solution can be determined for
these equations that also satisfies the boundary condition Eq 6.4-4, that result
may be substituted into Eq 6.4-2 to generate the strains and those in turn
substituted into Eq 6.4-3a to obtain the stresses.
When the boundary conditions are given in terms of surface tractions
(Eq 6.4-5), the equations of compatibility for infinitesimal strains (Eq 4.7-32)
may be combined with Hooke’s law (Eq 6.4-3b) and the equilibrium equations
to arrive, after a certain number of algebraic manipulations, at the
equations
1
v
σ + σ + ρ( b + b )+ δ ρb
1+ v
1−v
ij,kk kk,ij i,j j,i ij k,k
= 0 (6.4-8)
which are known as the Beltrami-Michell stress equations of compatibility. In
combination with the equilibrium equations, these equations comprise a
system for the solution of the stress components, but it is not an especially
easy system to solve. As was the case with the infinitesimal strain equation
of compatibility, the body must be simply connected.
In elastodynamics, the equilibrium equations must be replaced by the
equations of motion (Eq 5.4-4) in the system of basic field equations. Therefore,
all field quantities are now considered functions of time as well as of
the coordinates, so that a solution for the displacement field, for example,
( )
( )
u ( 2)
i