Section 2.2
Solid Mechanics Part II
Kelly
23
2.2 One-dimensional Elastodynamics
In rigid body dynamics, it is assumed that when a force is applied to one point of an
object, every other point in the object is set in motion simultaneously.
On the other hand,
in static elasticity, it is assumed that the object is at rest and is in equilibrium under the
action of the applied forces; the material may well have undergone considerable changes
in deformation when first struck, but one is only concerned with the final static
equilibrium state of the object.
Elastostatics and rigid body dynamics are sufficiently accurate for many problems but
when one is considering the effects of forces which are applied
rapidly
, or for very short
periods of time, the effects must be considered in terms of the propagation of stress
waves.
2.2.1
The Wave Equation
Consider now the dynamic problem.
In this case one considers the equation of motion:
a
b
dx
d
ρ
σ
=
+
Equation of Motion
(2.2.1a)
dx
du
=
ε
Strain-Displacement Relation
(2.2.1b)
ε
σ
E
=
Constitutive Equation
(2.2.1c)
where
a
is the acceleration.
Expressing the acceleration in terms of the displacement, one
then obtains the dynamic version of Navier’s equation,
2
2
2
2
t
u
b
x
u
E
∂
∂
=
+
∂
∂
ρ
1-D Navier’s Equation
(2.2.2)
In most situations, the body forces will be negligible, and so consider the partial
differential equation
2
2
2
2
2
1
t
u
c
x
u
∂
∂
=
∂
∂
1-D Wave Equation
(2.2.3)
where
ρ
E
c
=
(2.2.4)
Equation 2.2.3 is the standard one-dimensional
wave equation
with wave speed
c
; note
from 2.2.4 that
c
has dimensions of velocity.
The solution to 2.2.3 (see below) shows that a
stress wave
travels at speed
c
through the
material from the point of disturbance, e.g. applied load.
When the stress wave reaches a

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