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



Strain Equations of Compatibility

Let and define




Proof: Substitute…

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

Then a displacement field exists satisfying


Rewrite the compatibility equation as

Now define

Observe that


The symmetries here again imply that functions and exist such that



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



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.