Solid Mechanics: Hooke's Law 
Transverse Isotropic Definition 
A special class of orthotropic
materials are those that have the same properties in one plane (e.g.
the xy plane) and different properties in the
direction normal to this plane (e.g. the zaxis). Such
materials are called transverse
isotropic, and they are described by 5 independent
elastic constants, instead of 9 for fully orthotropic.
Examples of transversely isotropic materials include some piezoelectric materials (e.g. PZT4, barium titanate) and fiberreinforced composites where all fibers are in parallel. 
Hooke's Law in Compliance Form 
By convention, the 5 elastic constants in
transverse isotropic constitutive equations are the Young's modulus
and poisson ratio in the xy symmetry plane,
E_{p} and n_{p}, the Young's modulus and
poisson ratio in the zdirection, E_{pz} and
n_{pz}, and the shear
modulus in the zdirection G_{zp}.
The compliance matrix takes the form,
where . 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 transverse isotropic 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. 
Copyright © efunda.com
