 |
|||||||||
|
|||||||||
|
Chapter 4
Solutions to simple boundary and initial value problems
4.4 Simple dynamic solutions for linear elastic materials
In this section we summarize and derive the solutions to various elementary problems in dynamic linear elasticity.
4.4.1: Surface subjected to time varying normal pressure
An
isotropic, linear elastic half space with shear modulus
Solution: The displacement and stress fields in the solid (as a function of time and position) are
where
Evidently,
a stress pulse equal in magnitude to the surface pressure propagates
vertically through the half-space with speed
Notice
that the velocity of the solid is constant in the region
Derivation: The solution can be derived as follows. The governing equations are
Now: 1. Symmetry considerations indicate that the displacement field must have the form
Substituting this equation into the
strain-displacement equations shows that the only nonzero component of strain
is 2. The stress-strain law then shows that
In addition, the shear stresses are all zero
(because the shear strains are zero), and while 3. The only nonzero linear momentum balance equation is therefore
Substituting for stress from (2) yields
where
4. This is a 1-D wave equation with general solution
where f and g are two functions that must be chosen to satisfy boundary and initial conditions. 5. The initial conditions are
where the prime denotes differentiation with respect
to its argument. Solving these
equations (differentiate the first equation and then solve for
where A is some constant. 6. Observe that 7. Next, use
the boundary condition
where B is a constant of integration. 8. Finally, B can
be determined by setting t=0 in the
result of (7) and recalling from step (5) that
as stated.
4.4.2: Surface subjected to time varying shear traction
An
isotropic, linear elastic half space with shear modulus
It is straightforward to show that in this case
where
4.4.3: 1-D Bar subjected to end loading
This solution is a cheat, because it doesn’t satisfy the full 3D equations of elasticity, but it turns out to be quite accurate.
A
long thin rod occupying the region
We
cheat by modeling this as a 1-D problem.
We assume that
where
It is straightforward to solve the equation to see that
4.4.4 Plane waves in an infinite solid
A plane wave that travels in direction p at speed c has a displacement field of the form
where p is a unit vector. Again, to visualize this motion, consider the special case
In
this solution, the wave has a planar front, with normal vector p. The wave travels in direction p at
speed c. Ahead of the front, the solid is at
rest. Behind it, the solid has
velocity a. For
We seek plane wave solutions of the Cauchy-Navier equation of motion
Substituting a plane wave solution for u we see that
where
is a symmetric, positive definite tensor known as the `Acoustic Tensor.’ Plane wave solutions to the Cauchy-Navier equation must therefore satisfy
This requires
Evidently
for any wave propagation direction, there are three wave speeds, and three
corresponding displacement directions, which follow from the eigenvalues and
eigenvectors of
where
so that
By inspection, there are two eigenvectors that satisfy this equation 1.
2.
The
two wave speeds are evidently those we found in our 1-D calculation
earlier. So there are two types of
plane wave in an isotropic solid. The
S-wave travels at speed
4.4.5: Summary of Wave Speeds in isotropic elastic solids.
It is worth summarizing the three wave speeds calculated in the preceding sections. Recall that It
is straightforward to show that, for all positive definite materials (those
with positive definite strain energy density
There are also special kinds of waves (called Rayleigh and Stoneley waves) that travel near the surface of a solid, or near the interface between two dissimilar solids, respectively. These waves have their own speeds. Rayleigh waves are discussed in more detail in Section 5.5.3.
4.4.6: Reflection of waves traveling normal to a free surface
Suppose that a longitudinal wave with stress state
is
incident on a free surface at
To
visualize the wave, imagine that it is a front, such as would be generated by
applying a constant uniform pressure at
At
time
(we need a + in the argument because the wave travels to the left and has negative velocity). For the stress to vanish at the free surface, we must have
so,
and the full solution consists of both incident and reflected waves
As a specific example, consider a plane, constant-stress wave that is incident on a free surface. The histories of stress and velocity in the solid are illustrated in the figures above. In this case: 1. Behind the incident stress wave, the stress is
constant, with magnitude 2. At time 3. Behind the reflected wave, the solid is stress free,
and, the solid has constant velocity
4.4.7: Reflection and Transmission of waves normal to an interface
The
material on the left has mass density
is
incident on a bi-material interface at
As before, waves will be reflected at the bi-material interface. This time, however, some of the energy will be reflected, while some will be transmitted into the adjacent solid. Guided by the solution to the preceding problem, we assume that the stress associated with the reflected and transmitted waves have the form
The functions g and h must be chosen to satisfy stress and displacement continuity at the interface. Stress continuity requires that
To satisfy displacement continuity, we make the acceleration continuous
which may be integrated to give
where C is a constant of integration. Setting t=0 shows that C must vanish, since f=g=h=0 at t=0. The two conditions (1) and (2) may now be solved for g and h to see that Reflected
wave Transmitted
wave
Where the coefficients of reflection and transmission are given by
Results for a shear wave
approaching the interface follow immediately from the preceding calculation,
by simply setting
4.4.8: Simple example involving plane wave propagation: the plate impact experiment
A plate impact experiment is used to measure the plastic properties of materials at high rates of strain. In typical experiment, a large, elastic flyer plate is fired (e.g. by a gas-gun) at a stationary target plate. The specimen is a thin film of material, which is usually deposited on the surface of the flyer plate. When the flyer plate impacts the target, plane pressure and shear waves begin to propagate through both plates. The experiment is designed so that the target and flier plates remain elastic, while the thin film specimen deforms plastically. A laser interferometer is used to monitor the velocity of the back face of the target plate: these measurements enable the history of stress and strain in the film to be reconstructed.
A full analysis of the plate impact experiment will not be
attempted here
The resulting stress and motion in the plate is most
conveniently displayed on “(x-t)
diagrams” as shown in the figures on the next page. The graphs can be used to deduce the velocity
and stress in both flyer and target at any position x and time t in both
plates. The solution consists of
triangular regions (of time and position) of constant velocity and stress,
separated by lines with slope equal to the longitudinal wave speed
The figures illustrate the following sequence of events: 1. Just after impact, plane pressure
waves propagate in opposite directions through the flyer and target. Behind the traveling wave fronts, both
plates have velocity
2. At time 3. At time 4. At time 5. At time 6. Thereafter, the target continues to
propagate with constant velocity
Derivation: The solution can be constructed using the simple 1-D solutions given in 4.4.1 and 4.4.6. For example, to find the stress and velocity associated with the waves generated by the initial impact: 1. At the moment of impact, both flyer and target are subjected to a sudden pressure. Wave motion in both solids can be analyzed using the solution given in 4.4.1. 2. Let 3. Let 4. From Section 4.4.1 we know that the velocity change and stress are related by
5. The target and flyer must have the
same velocity at the impact surface.
Therefore 6. The horizontal stress must be equal
in both solids at the impact surface.
Therefore 7. The four equations in steps 4-6 can
be solved to yield The changes in stress and velocity that occur at each reflection can then be deduced using the results at the end of Section 4.4.6. Alternatively the (x-t) diagrams can be constructed directly, by first drawing all the characteristic lines, and then deducing the velocity and stress in each sector of the diagram by noting that (i) the change in stress and velocity across each line must be constant; (ii) the overall momentum of the solid must be conserved, and (iii) the total energy of the solid must be conserved.
|
|||||||||
|
(c) A.F. Bower, 2008 |
The strain-displacement relation 
