Page 277 - Intro to Tensor Calculus

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The solution represents a longitudinal elastic wave propagating in the x−direction with speed α. The stress
wave associated with this displacement is determined from the constitutive equations. We find

∂u 1
σ xx =(λ + µ)e xx =(λ + µ) .
∂x
This produces the stress wave

(λ+µ)
0
− f (t − x/α), x ≤ αt
σ xx = α .
0, x > αt
Here there is a discontinuity in the stress wave front at x = αt.

Summary of Basic Equations of Elasticity

ij i
The equilibrium equations for a continuum have been shown to have the form σ + %b =0, where
,j
i
b are the body forces per unit mass and σ ij is the stress tensor. In addition to the above equations we
have the constitutive equations σ ij = λe kk δ ij +2µe ij which is a generalized Hooke’s law relating stress to
strain for a linear elastic isotropic material. The strain tensor is related to the displacement field u i by
1
the strain equations e ij = (u i,j + u j,i ) . These equations can be combined to obtain the Navier equations
2
µu i,jj +(λ + µ)u j,ji + %b i =0.
The above equations must be satisfied at all interior points of the material body. A boundary value
problem results when conditions on the displacement of the boundary are specified. That is, the Navier
equations must be solved subject to the prescribed displacement boundary conditions. If conditions on
the stress at the boundary are specified, then these prescribed stresses are called surface tractions and
ij
must satisfy the relations t i (n) = σ n j , where n i is a unit outward normal vector to the boundary. For
surface tractions, we need to use the compatibility equations combined with the constitutive equations and
equilibrium equations. This gives rise to the Beltrami-Michell equations of compatibility

1 ν
σ ij,kk + σ kk,ij + %(b i,j + b j,i )+ %b k,k =0.
1+ ν 1 − ν
Here we must solve for the stress components throughout the continuum where the above equations hold
subject to the surface traction boundary conditions. Note that if an elasticity problem is formed in terms of
the displacement functions, then the compatibility equations can be ignored.
For mixed boundary value problems we must solve a system of equations consisting of the equilibrium
equations, constitutive equations, and strain displacement equations. We must solve these equations subject
to conditions where the displacements u i are prescribed on some portion(s) of the boundary and stresses are
prescribed on the remaining portion(s) of the boundary. Mixed boundary value problems are more difficult
to solve.
For elastodynamic problems, the equilibrium equations are replaced by equations of motion. In this
case we need a set of initial conditions as well as boundary conditions before attempting to solve our basic
system of equations.