Stress-Strain relation (Hooke's Law)

The relationship between stress and strain in Hooke's Law is one of simple linear proportionality, with the constant of proportionality characterizing the elastic properties of the material.  For now, let this elastic coefficient be equal to one.  Then the x-y shear term, in Cartesian coordinates, is

In this form, it seems reasonable to say that the x-y component of the stress tensor produces a gradient in the y direction of the x displacement, and the y-x component produces a gradient in the x direction of the y displacement.  In the earlier blog, "Shearing Strain and the change in angle...," we saw that these gradients are each equal to one part of the total change in angle of two perpendicular planes at the point (x,y).  

The right hand side of the equation above is equal to twice the x-y component of the strain tensor, and since the x-y and y-x stress components are equal, the two's cancel, and we see another reason why the factor 1/2 comes into the strain tensor.

When you look at the drawing in the Shearing Strain blog mentioned above, it makes sense (to me, anyway) that the x-y and y-x stress components would each be responsible for the respective displacement gradients, since they act on different planes.  But can this be true?  The strain tensor is defined to contain both terms.  

We know that the stress tensor has to be symmetric (conservation of angular momentum), but we do not know that the two displacement gradients in the x-y strain tensor are equal.  If they were, it seems like it wouldn't be necessary to define the strain as the sum of the two gradients, i.e. to force the symmetry, so to speak.  So we are left with something of a mystery here.  

Somehow the stress tensor, representing the force on one plane, causes a strain affecting the displacement in that plane and the displacement in a perpendicular plane.  Maybe this will become more clear when we work out an example of shear.