SAM-Consult - Hooke's Law for Orthotropic Materials

Solid Mechanics: Hooke's Law
Hooke's Law for Orthotropic Materials

Orthotropic Definition

Some engineering materials, including certain piezoelectric materials (e.g. Rochelle salt) and 2-ply fiber-reinforced composites, are orthotropic.

By definition, an orthotropic material has at least 2 orthogonal planes of symmetry, where material properties are independent of direction within each plane. Such materials require 9 independent variables (i.e. elastic constants) in their constitutive matrices.

In contrast, a material without any planes of symmetry is fully anisotropic and requires 21 elastic constants, whereas a material with an infinite number of symmetry planes (i.e. every plane is a plane of symmetry) is isotropic, and requires only 2 elastic constants.

Hooke's Law in Compliance Form

By convention, the 9 elastic constants in orthotropic constitutive equations are comprised of 3 Young's modulii E_x, E_y, E_z, the 3 Poisson's ratios n_yz, n_zx, n_xy, and the 3 shear modulii G_yz, G_zx, G_xy.

The compliance matrix takes the form,

where .

Note that, in orthotropic materials, there is no interaction between the normal stresses s_x, s_y, s_z and the shear strains e_yz, e_zx, e_xy

The factor 2 multiplying the shear modulii in the compliance matrix results from the difference between shear strain and engineering shear strain, where , etc.

Hooke's Law in Stiffness Form

The stiffness matrix for orthotropic materials, found from the inverse of the compliance matrix, is given by,

where,

The fact that the stiffness matrix is symmetric requires that the following statements hold,

The factor of 2 multiplying the shear modulii in the stiffness matrix results from the difference between shear strain and engineering shear strain, where , etc.