The strain tensor, and the stress-strain relationship

In order to proceed from the simple example of tension in a rod to a more general formulation of stress and strain, we need to consider the work done in deforming an elastic material.

We start with the definition: work is equal to force times distance - for example, think of rolling a ball uphill.  You multiply the required force by the distance traveled to obtain the work done.  Note that the work is recoverable,  since you can let the ball roll back downhill.  In pushing the ball uphill, you are storing potential energy, which can be recovered as kinetic energy by letting it roll back down.  The stored energy will be completely recovered only if none of it is needed to overcome friction.  More generally, the process is reversible if none of the stored energy is converted into heat.

Similarly, the work required to deform an elastic body is the force acting at each point in the body times the displacement of that point from its position when no force was applied.  The deformation is reversible if the body returns to its original state when the force is removed.  Note that this force is the limit of the force per unit volume, as the volume shrinks to a point, and not the force per unit area.

This force was introduced in an earlier blog, "The stress principle of Euler and Cauchy," with the first equation shown below.  The stress vector was given in terms of the stress tensor in the blog "Cauchy's Stress Theorem,"  as shown in the second equation below.  This latter formula can be written in tensor notation as given in the third equation, with the convention that repeated subscripts are summed.  So when you see a repeated subscript, imagine a summation sign, with the sum going from k=1 to k=3.  In this case, the equation means that the ith component of the stress vector depends on all three components of the surface normal vector, and also on three components of the stress tensor.

Finally, substituting the right hand side of the third equation in place of the stress vector into the first equation will gives us the relationship between the force density vector and the stress tensor shown in the fourth equation.  Recall that the area S is the entire area enclosing the volume V, and the equation holds for any volume chosen within the material body.  Note that the integrand on the right is given by the ith component of a vector, which has (at most) three terms, given by the summation over the subscript k.  

The meaning of this last equation is the same as the first equation:  It states that the forces acting on the interior of a part of the material body may be entirely determined from the forces acting on the enclosing surface.

But wait a minute!  In the earlier blogs on the balance of forces, and on the stretched piece of rubber, we learned that the total force acting on any portion of a body in equilibrium is zero.  Since this is true for any volume, it follows that f =0 at each point in the material.  You can imagine a point in the body being pulled in various directions, and if the net force were not zero, the body would deform in the neighborhood of that point.  

The two sides of the last equation above are non-zero only if the material is in the process of being deformed. Just as with the ball on the hill, you're only working while you're rolling it, at least as far as the ball is concerned.  (The energy your body requires to hold it there is another story).  

So there is a term missing from the equation.  To find it, we need to consider the conservation of momentum, which we'll do in the next blog.