Transversely Isotropic Elasticity

Transversely isotropic elasticity is a special subclass of orthotropy characterized by a plan of isotropy at every point in the material.

See Also
In Other Guides
Linear Elastic Behavior

Assuming the 1–2 plane to be the plane of isotropy at every point, transverse isotropy requires that E 1 = E 2 = E p , ν 31 = ν 32 = ν t p , ν 13 = ν 23 = ν p t , and G 13 = G 23 = G t , where p and t stand for “in-plane” and “transverse,” respectively. Thus, while ν t p has the physical interpretation of the Poisson's ratio that characterizes the strain in the plane of isotropy resulting from stress normal to it, ν p t characterizes the transverse strain in the direction normal to the plane of isotropy resulting from stress in the plane of isotropy. In general, the quantities ν t p and ν p t are not equal and are related by ν t p / E t = ν p t / E p . The stress-strain laws reduce to

{ ε 11 ε 22 ε 33 γ 12 γ 13 γ 23 } = [ 1 / E p − ν p / E p − ν t p / E t 0 0 0 − ν p / E p 1 / E p − ν t p / E t 0 0 0 − ν p t / E p − ν p t / E p 1 / E t 0 0 0 0 0 0 1 / G p 0 0 0 0 0 0 1 / G t 0 0 0 0 0 0 1 / G t ] { σ 11 σ 22 σ 33 σ 12 σ 13 σ 23 } = D e l { ε 11 ε 22 ε 33 γ 12 γ 13 γ 23 } .
where G p = E p / 2 ( 1 + ν p ) and the total number of independent constants is only five.

The restrictions on the elastic constants due to material stability are

E p , E t , G p , G t ⁢ > 0
| ν p | < 1
| ν p t | < ( E p / E t ) 1 / 2
| ν t p | < ( E t / E p ) 1 / 2
det ⁡ ( D e l ) = 1 − ν p 2 − 2 ν t p ν p t − 2 ν p ν t p ν p t > 0.

Parameters

Input Data Description
Parallel Young's modulus Young's modulus in the parallel direction, E p .
Normal Young's modulus Young's modulus in the normal direction, E t .
Parallel Poisson's ratio Poisson's ratio in the parallel direction, ν p .
Normal Poisson's ratio Poisson's ratio in the normal direction, ν t p .
Parallel shear modulus Shear modulus in the parallel direction, G p .