ABSTRACT
A general description of plane stress at any point of a solid is shown in Figure 13.1, where all the stress components in the z direction (σz, tyz, and txz) are zero.
The three-dimensional generalized Hooke’s law for an isotropic material is [1,2]
e σ u σ σx x y zE
= - +1 [ ( )]
(13.1)
e σ u σ σy y x zE
= - +1 [ ( )] (13.2)
e σ u σ σz z x yE
= - +1 [ ( )] (13.3)
g
t g
t g txy
G G G = = =, , (13.4)
where E is the elastic modulus or Young’s modulus, u is the Poisson’s ratio, and G is the shear modulus, which is defined by
G E=
+2 1( )u
For plane stress, Equations (13.1) through (13.4) become
e σ u σx x yE
= -1 [ ( )] (13.5)
e σ u σy y xE
= -1 [ ( )]
(13.6)
e u σ σz x yE
= - +1 [ ( )] (13.7)
g
t u t xy
G E = =
+2 1( ) (13.8)
Equation (13.7) indicates that without z-direction restraints, the strain ez may not be zero, even though the stress in the z direction, σz, is zero. Equations (13.5) through (13.8) can be written in matrix form as shown by Equation (13.9).
