Solving Elasticity Problems

[Lojzek]

I am trying to understand how to solve a certain type of elasticity problems.

Let's say that we have an isotropic material (2 elastic constants are known) in zero gravity with a shape of a half space
(limited by the boundary plane and infinite on one side of the plane).
Boundary conditions are: a known (but position dependent) force density in the boundary plane and zero stress tensor infinitely far from that plane.

1. Can we find the solution of this problem by demanding zero divergence of stress tensor and match between stress tensor and boundary conditions (where stress tensor is expressed from strain tensor with Hooke's law)?

2. Is it possible to obtain an analitic solution in case when the boundary condition is a delta function
(the boundary plane is loaded by a finite point force)? What is the solution for this case?

3. Can we construct the solution for a general load on the boundary plane by summing/integrating the solutions
for delta function load? Is this the correct approach or do we need another method?

4. Does anyone know a good link about this type of problems?

 

[Mapes]

The case of a point load on the surface of a semi-infinite medium is known as Boussinesq's problem (sometimes the case of a tangential load is called Cerruti's problem). Analytical solutions exist, and yes, the solution for general loads can be found by combining or integrating the point load solutions. More info can be found on the web and via Google Books; also, you can find good coverage in Johnson's Contact Mechanics.

 

[Lojzek]

Thanks. I found the solution of Boussinesq problem in the book Fundamentals of surface mechanics.