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

Hooke's Law for Plane Strain
For the case of plane strain, where the strains in the z direction are considered to be negligible, , the stress-strain stiffness relationship for an isotropic material becomes,

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

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

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

Plane Strain Hooke's Law via Engineering Strain
The stress-strain stiffness matrix expressed using the shear modulus G and the engineering shear strain is,

The compliance matrix is,

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

 

 

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