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Structural Mechanics

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:

·(cu)=k

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:

c11=(2G+μ00G),c12=(0μG0),c21=(0Gμ0),c22=(G002G+μ)

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 σ12+σ22σ1σ2 (plane stress)

  • von Mises effective stress (σ12+σ22)(ν2ν+1)+σ1σ2(2ν22ν1) (plane strain)

For details, see Plane Stress and Plane Strain.

Apps

PDE Solve partial differential equations in 2-D regions

Topics

Programmatic Workflow

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PDE App Workflow

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Concepts

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Plane stress and plane strain equations

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