Introduction to Elasticity/Compatibility

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Compatibility[edit | edit source]

For an arbitrary strain field , the strain-displacement relation is the partial differential equation that can be solved to obtain the displacement . The solution to this equation must exist and must be unique.

Uniqueness (Kirchhoff's Theorem)[edit | edit source]

If two displacement fields and correspond to the same strain field, then , where is a rigid displacement field.

Existence (Compatibility Theorem)[edit | edit source]

The strain field corresponding to a continuous displacement field satisfies the compatibility equation

In index notation

The converse also holds if the body is simply connected. The compatibility equations can also be written as

The compatibility condition also implies the following relationship between the infinitesimal strain tensor and the axial vector corresponding to the infinitesimal rotation tensor:

Sample homework problems[edit | edit source]

Problem 1[edit | edit source]

Show that the compatibility relation for plane stress is satisfied by unrestrained thermal expansion (, ), where is the coefficient of thermal expansion and is the temperature, provided that the temperature is a two-dimensional harmonic function, i.e.,

Hence deduce that, subject to certain restrictions which you should explicitly specify, no thermal stresses will be induced in a thin body with a steady-state, two-dimensional temperature distribution and no surface tractions.


Solution[edit | edit source]

The plane stress compatibility equation is

Plugging in the expressions for strain,

or,

The above equation is the steady-state heat conduction equation without any internal sources.

If there are no surface tractions, the state satisfies the BCs. Since the steady-state heat conduction equation is also the compatibility equation, compatibility is automatically satisfied by the above stress state. Therefore, no thermal stresses are induced in this situation. However, extra conditions need to be applied if the body is multiply-connected.