Tensile stresses in an elastic body accelerated by an attractive body force

Jaydeep,

The problem you posed is an interesting one.  Some classical solutions in this regard can br found in Love's book on Elasticity Theory. You can also find these solutions, in particular, the solutions of a gravitating sphere as well as a uniformly rotating sphere in A. I. Lurie's book (Theory of Elasticity, Springer, 2005), Chapter V, Sections 3.12 and 3.13.

Of course, if you are talking about accelerating sphere, you will need to resort to the equations of dynamic elasticity. Lame's representation for the displacement vector in terms of a scalar potential and a vector potential allows the equations of dynamic elasticity with body forces to be written down as 2 simple inhomogeneous wave equations (1 scalar wave equation and 1 vector wave equation). They can be solved for the potential functions with your particular body force. You then determine the dispacements and stresses in the sphere, in particular, determine the stresses on the surface of the sphere, and you will see that they are non-zero. But, the original sphere is stress free on the surface. So, you will need to remove these surface stresses. Now, this leads to another classical problem: Determine the elastic field in a sphere with prescribed dynamic tractions on its surface. You will need to solve the equations of dynamic elasticity without body forces for the sphere (in spherical coordinates).This problem can be solved by some standard methods of dynamic elasticity. You will find these methods for instance, in Eringen's book on Elastodynamics. The stresses and displacements found this way should be added to those from the body-force state to give the complete solution to the problem. 

I have just silhoutted a scheme that I believe should work. You can contact me at Mujibur.Rahman1@gmail.com, if you need any further detail.

Hope this helps!

Mujibur Rahman