The drawing on the left is intended to represent a 3-dimensional solid body, acted upon by force PR on the right, and PL on the left. Within the body is a surface S.
Imagine that we have a fixed coordinate system (x,y,z) such that the positive x axis is normal to the surface S. We use the label SR for S when it represents a part of the surface of the volume VR, and SL for the part of the surface of the volume VL. (VR and VL together give the total volume).
It seems clear that each component of PR and PL in the x, y and z directions must be equal in magnitude, but with opposite sign. Otherwise, since the net force on a body is equal to the rate of change of its momentum with time, the body would accelerate in the direction of the greater force.
In elasticity theory, it is usually assumed that the material body is somehow held fixed in space, i.e. with balanced applied forces.
Now let's consider the surface S within the body: Here S, SL and SR all occupy the same region in space, but SR has normal vectors pointing into the region VL, and SL's normal vectors point into VR. At each point on S, the normal vectors for SR and SL have components in x, y and z which are equal in magnitude and opposite in direction.
Let's start with SL. We choose a point (x,y,z) on SL, and take the normal vector there to point in the positive x direction (into VR). Then the normal vector has only one non-zero component, as shown in the Cauchy formula on the left.
As an aside, note that we have used the symmetry of the stress matrix, which allows us to equate the off-diagonal components as shown.
Using matrix multiplication, we find that each component of the stress vector contains only one component of the stress tensor, as shown below.
The stress vector is determined by only three components of the stress tensor - the normal stress in the 11 or xx direction, and two shear stresses in the yx and zx directions.
Since the normal vector is of unit length, i.e. n = (1,0,0), that is n1 = 1, the equations on the left show that the x component of the stress vector (that is, of the force acting on the surface SL) is given by the xx component of the stress tensor.
Similarly the y component of the force t is given by the yx component of the stress tensor, and the z component of force is given by the zx component of stress.
Put another way, the yx component of the stress tensor is the y component of the force acting on the x plane. By"x plane," we mean the plane tangent to the surface SL at the point (x,y,z) whose normal vector points in the x direction. It's actually the yz plane, but calling it the x plane makes it easier to identify with the yx component of the stress tensor.
Similarly, the zx component of the stress tensor is the z component of the force acting on the x plane.
We now have a simple example of a particular stress vector acting on a surface within a body. The next step is to consider the stress vector acting on the same plane tangent to the same surface, but now imagining that surface to to belong to the remaining part of the body.
Referring again to the drawing at the top, we use the same coordinate system we had earlier, and consider the surface SR, which is part of the surface of the volume VR. We again choose the same point (x,y,z) that we chose before, with the same plane tangent to SL (and therefore to SR), only this time the normal vector has components (-1,0,0) in our coordinate system.
All we have to do to get stress vector components acting at that point is to replace the x component of n with -1. We get the equations shown below, and find that the components have the same magnitude as before, only they are in the opposite (-x) direction. Since this is true for any point (x,y,z) on the surface drawn anywhere in the body, it must be that the the stress vectors are balanced. Note that the stress components depend only on the location of (x,y,z) and not on the orientation of the surface through this point.
Therefore the stress vectors are balanced. If we choose a different surface through the same point, the components of n will change (the coordinate system is the same), but the stress vectors will still balance.
Truesdell pointed out that this result is actually a consequence of the conservation of momentum: The sum of all the external forces is zero (that PL and PR are equal and opposite). Otherwise, if there is a net force, the body will accelerate, and will no longer be in equilibrium.
Instead, the body will deform as a function of time. The result will be that the surface S will change over time, as will the normal vector: n=n(t).
Furthermore, if the applied forces PL and PR are unequal, then the integrals of t over the surfaces SL and SR will be unequal, since each of those integrals must balance the applied forces on their respective regions VL and VR. From this it is clear that the stress vectors cannot balance, even at a given instant of time. From this we conclude that when the body accelerates, there must be a discontinuity in the stress tensor across the surface S. That is, the value of the stress components will be different if we approach S from the right (VR) or the left (VL) side. This is the subject of the theory of shock waves.
We won't pause now (maybe later) to study shock waves, but suffice it to say that we would expect the acceleration necessary to cause a jump condition within the body of a material would have to be quite large - say of the magnitude that occurs when a massive object from space accelerates towards the earth.
In the next blog, we will return to our main subject - reversible, elastic deformations of homogeneous materials - and introduce some of the concepts needed to actually solve problems.
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