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:
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:
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 same equations describe both plane stress and plane strain and use the following parameters:
Shear modulus G = E/(2(1 + ν))
Poisson's ratio ν
µ = 2Gν/(1 – ν)) (plane stress)
µ = 2Gν/(1 – 2ν)) (plane strain)
von Mises effective stress (plane stress)
von Mises effective stress (plane strain)
For details, see Plane Stress and Plane Strain.
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