Introduction to Elasticity/Equilibrium example 3

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Example 3[edit | edit source]

Given:

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:

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,

Therefore,

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,

Adding,

Hence,

Substituting back into the compatibility equation,

Hence,