Solid Mechanics: Hooke's Law
Hooke's Law for Plane Stress

Hooke's Law for Plane Stress
For the simplification of plane stress, where the stresses in the z direction are considered to be negligible, , the stress-strain compliance relationship for an isotropic material becomes,

The three zero'd stress entries in the stress vector indicate that we can ignore their associated columns in the compliance matrix (i.e. columns 3, 4, and 5). If we also ignore the rows associated with the strain components with z-subscripts, the compliance matrix reduces to a simple 3x3 matrix,

The stiffness matrix for plane stress is found by inverting the plane stress compliance matrix, and is given by,

Note that the stiffness matrix for plane stress is NOT found by removing columns and rows from the general isotropic stiffness matrix.

Plane Stress Hooke's Law via Engineering Strain
Some reference books incorporate the shear modulus G and the engineering shear strain gxy, related to the shear strain exy via,

The stress-strain compliance matrix using G and gxy are,

The stiffness matrix is,

The shear modulus G is related to E and n via,

 

 

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