Solid Mechanics: Hooke's Law
Hooke's Law for Transversely Isotropic Materials

Transverse Isotropic Definition
A special class of orthotropic materials are those that have the same properties in one plane (e.g. the x-y plane) and different properties in the direction normal to this plane (e.g. the z-axis). 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. PZT-4, barium titanate) and fiber-reinforced 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 x-y symmetry plane, Ep and np, the Young's modulus and poisson ratio in the z-direction, Epz and npz, and the shear modulus in the z-direction Gzp.

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.

 

 

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