Solve PDEs that model plane stress and strain in solid
mechanics

In structural mechanics, the equations that relate stress and
strain arise from the balance of forces in the material medium. Plane
stress is a condition that prevails in a flat plate in the *x*-*y* plane,
loaded only in its own plane and without *z*-direction
restraint. The displacement equations for plane stress are:

where *u* is the displacement in the *x*-direction, **k** is a vector of volume forces, and **c** is a rank four tensor that can be written
as four 2-by-2 matrices *c*_{11}, *c*_{12}, *c*_{21},
and *c*_{22}:

*G* is the shear modulus *G* = *E*/(2(1 + *ν*)), *ν* is
the Poisson's ratio, and *µ* = 2*G**ν*/(1 – *ν*)).
The von Mises effective stress is computed as $$\sqrt{{\sigma}_{1}^{2}+{\sigma}_{2}^{2}-{\sigma}_{1}{\sigma}_{2}}$$.
For details, see Structural Mechanics — Plane Stress.

Plane strain is a deformation state where there are no displacements
in the *z*-direction, and the displacements in the *x*-
and *y*-directions are functions of *x* and *y* but
not *z*. The stress-strain relation is only slightly
different from the plane stress case, and the same set of material
parameters is used. Thus, for the plane strain equations, the *µ* parameter
in the *c* tensor is defined as *µ* = 2*G**ν*/(1 – 2*ν*)).
The von Mises effective stress is computed as $$\sqrt{\left({\sigma}_{1}^{2}+{\sigma}_{2}^{2}\right)\left({\nu}^{2}-\nu +1\right)+{\sigma}_{1}{\sigma}_{2}\left(2{\nu}^{2}-2\nu -1\right)}$$.

Plane strain problems are less common than plane stress problems.
An example is a slice of an underground tunnel that lies along the *z*-axis.
It deforms in essentially plane strain conditions. For details, see Structural Mechanics — Plane Strain.

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