Example 3 edit


If a material is incompressible (  = 0.5), a state of hydrostatic stress ( ) produces no strain. The corresponding stress-strain relation can be written as


where   is an unknown hydrostatic pressure which will generally vary with position. Also, the condition of incompressibility requires that the dilatation



Show that the stress components and the hydrostatic pressure   must satisfy the equations


where   is the body force.

Solution edit

We have,   Also,




Since  , the above relation gives  . Therefore,


The strain-stress relations are


Differentiating the strains so that they correspond to the compatibilityrelation is two-dimensions, we have


In terms of the compatibility equation,


From the two-dimensional equilibrium equations,


Therefore, differentiating w.r.t   and   respectively,






Substituting back into the compatibility equation,