. Now we
imagine distorting this solid. To describe the distortion, we write down the
change in position of each bit of the solid sample.
For fluids, we studied a representative volume flowing with the liquid, moving past us in the laboratory frame of reference. For our treatment of solids, the representative volume will always stay "close to home." We will emphasize cases in which the acceleration of the solid sample is zero. We will be concerned with small displacements from the "home" position, and the forces necessary to cause these displacements.
We ask how forces deform the shape of a stationary nonrigid object. To study these deformations, we will arrange the forces applied to the object so that the total force is zero. We will think of the applied forces as a "perturbation," and the change in shape of the solid as the "response" to that perturbation.
We call the change in shape of the solid the strain. Our aim is to find a simple representation of common strains that can occur in solids.
We will restrict our study to a comparison of the initial (undistorted) shape and the final (distorted) shape. We generally will not analyze the period of transition, just after the forces are applied, when the object is moving towards its new shape. The only exception will be a calculation of the speed of sound, for which local motion is accelerated.
This perturbation/response description is called a phenomenological description because we do not derive it from first principles, such as Newton's laws. The relation between perturbation and response is different for each different material, and may depend on which way the sample is oriented. Each material is characterized by "elastic constants" which relate applied forces to changes in shape.
A similar approach was used studying the viscous force in fluids. We will be satisfied to discover these elastic constants experimentally, and save them in tables of data. The calculation of elastic constants from interatomic forces is left to a more advanced course.
One bit of evidence for microscopic structure in solids is the observed anisotropy of the elastic constants: In a block of wood, for example, the resistance to stretching is different for forces applied parallel to the wood grain, from the resistance with the forces perpendicular to the wood grain. Until our ideas are firmly established, we will assume that our samples are isotropic. Blocks of plastic or brass or foam rubber are examples of materials that are isotropic enough for our purposes.
DISPLACEMENT FIELDS
When our object is perturbed by applied forces, we say that it is "stressed."
When an object is stressed, its shape distorts. We imagine an unstressed solid
sample, with a Cartesian coordinate system for reference. We identify a point
in the solid by its position vector in our coordinate system,
. Now we
imagine distorting this solid. To describe the distortion, we write down the
change in position of each bit of the solid sample.
Since the displacement of a bit of sample is not necessarily small, we
cannot simply refer to it as a differential,
. We define a
displacement vector for a bit of sample,
. If the
sample is distorted, then different bits of sample have different displacements.
One bit may be pressed closer to another, or pulled farther apart, or slid
sideways.
We continue to identify each bit by its undistorted "home"
position vector,
, and now also
identify each bit's displacement vector in the same way. We have formed a displacement
field, 
.
Two most familiar displacement fields are discussed briefly below:
|
| Rigid body translational displacement field. Each displacement |
| vector is identical. |
For the uniform displacement field shown in the figure, every point in the object is displaced by the same vector. The object moves as a whole, undistorted. The object is said to move "rigidly." Introductory physics dealt extensively with rigid body motion. We will not discuss it further here.
|
| Rigid body rotational displacement field. Compare to velocity field |
| for fluid rotational flow. |
For the uniform displacement field shown in the figure, the body moves rigidly again, but this time it rotates. Compare this field to the velocity field for the rotating bucket.
It will turn out that we must take some care to eliminate rotations from our displacement fields.
|
| Compression of a cube by uniform pressure. The compressed cube |
| is shaded. |
Perhaps the simplest perturbation to begin a survey of displacement fields is the uniform pressure that was discussed for fluids. We may imagine placing our sample at the bottom of a tall column of water. We "turn off" gravity, so we can ignore any variation of pressure from the bottom of the sample to the top. In that case, the pressure on all faces of our sample is uniform.
The pressure causes a simple distortion in the sample: The volume decreases, while the shape remains unchanged. In the figure, a cube is compressed to a smaller cube. (We are assuming isotropic material here.) We could represent this response by simply stating the change in volume.
A better representation describes the way that any segment of the cube changes: We calculate the change of volume per unit volume.
This is consistent with the field approach taken in the sections on fluids.
For a particular sample of volume
, we measure
the change in volume,
, and
calculate the dilatation:
Note that the volume in the figure decreases, so that
and the
dilatation are negative. Yet, in the fluids notation, the pressure is said to
be positive. For solids, we change notation:
We now adopt the convention that a compressing pressure is a negative pressure.
|
| Expansion of a cube by a uniform positive pressure; two dimensional |
| section of the displacement field. |
The figure shows a two dimensional representation of the displacement field associated with the dilatation strain. In the figure, the coordinate system has been chosen to be fixed at the center of the cube. A three dimensional view would show arrows converging towards the center. It is clear that this displacement field has a (negative) divergence.
Compression of a cube by uniform pressure, two dimensional section of the displacement field.
With the origin at the center of the cube, and the x,y, and z axes along the
cube edge directions, the displacement vector,
, for a
particular bit of the material in the cube is in the direction opposite to the
position vector,
, of that bit.
Moreover, it can be shown that the displacement vector is proportional to
. We can
write the displacement field as
and the divergence of that field is
|
| Expansion of a cube by a uniform positive pressure. Shaded areas |
| have moved. |
As with fluid velocity fields, the divergence gives us a quick
characterization of the displacement field. Let us compare this to the
dilation. The figure shows a cube, to which a positive uniform pressure has
been applied, so that the volume increases. (Positive uniform pressure is
harder to apply, but easier to sketch.) If the side of the cube has length L,
and the displacement field is now positive:
, then each
the length of each side has increased by a distance
.
Expansion of a cube by uniform pressure. Shaded areas have moved.
Note that the coordinate system has been shifted so that the origin is now
at the back, lower, left corner of the cube. This was done to make the figure
easier to draw and interpret. The top shaded surface has moved up a distance
; the front
shaded surface has moved towards us a distance
, and the
right shaded surfaces has moved a distance
to the right.
The picture omits the motion of the edges of the cube. Thus it underestimates the volume change by an amount
The picture estimates the increase in volume as
Since the volume increase is proportional to
, and the
error above is proportional to
, we are
justified in ignoring the error, as long as
.
With this estimate of the volume change, the dilatation is
,
the same as the divergence of the displacement field, which is now positive also.
We find the same answer for
if we
calculate the new expanded volume as
and (assuming
) keep only
the two largest terms:
We have found that the strain of a solid subjected to uniform pressure can be written as
where e is the symbol that we give for the dilatation.
From now on, we will follow tradition, and express strains in terms of ratios of distances, or as combinations of derivatives of the displacement field. We will not use the vector derivative operators developed for fluids.
Return to index.
Return to survey.
Go forward to tensile strain.