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 Ex, Ey, Ez, the 3 Poisson's ratios nyz, nzx, nxy, and the 3 shear modulii Gyz, Gzx, Gxy.

The compliance matrix takes the form,

where .

Note that, in orthotropic materials, there is no interaction between the normal stresses sx, sy, sz and the shear strains eyz, ezx, exy

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.

 

 

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