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 c11, c12, c21, and c22:
G is the shear modulus G = E/(2(1 + ν)), ν is the Poisson's ratio, and µ = 2Gν/(1 – ν)). The von Mises effective stress is computed as . 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 = 2Gν/(1 – 2ν)). The von Mises effective stress is computed as .
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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