Introduction to Elasticity/Equilibrium example 3

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Example 3[edit | edit source]


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 | edit source]

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,