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.