EN224: Linear Elasticity

Division of Engineering

6.3 Field Equations Implied by the Fundamental 2D System

Before developing techniques to solve our field equations, we need some preliminary results. We will explore the 2D versions of strain and stress compatibility.

Summary of field equations for plane elasticity

where

Strain Equations of Compatibility

Let and define

Then

Proof: Substitute…

Conversely, let be a symmetric two-tensor field on a simply connected region satisfying

Then a displacement field exists satisfying

Proof:

Rewrite the compatibility equation as

Now define

Observe that

Hence

The symmetries here again imply that functions and exist such that

Hence

Stress Equations of Compatibility

The stress equations of compatibility actually turn out to be more useful to us in 2D (you will recall that in 3D, they were virtually useless!)

Let , and let

Then

Proof:

From the preceding section, we have

Invert the stress-strain relation:

Substitute in the strain equation of compatibility to see that

Now use equilibrium

Finally, eliminate the term involving shear stress and rearrange to obtain

Conversely, let

Then there exists a displacement field such that

Proof: Reverse the argument given above to obtain

Then use the result for the strain equation of compatibility.