Transverse Isotropy:

Introduction:

In this lecture, we are going to see some more simplifications of constitutive equation and develop the relation for isotropic materials.

First we will see the development of transverse isotropy and then we will reduce from it to isotropy.

First Approach: Invariance Approach

This is obtained from an orthotropic material. Here, we develop the constitutive relation for a material with transverse isotropy in x₂-x₃ plane (this is used in lamina/laminae/laminate modeling). This is obtained with the following form of the change of axes.

(3.30)

Now, we have

From this, the strains in transformed coordinate system are given as:

(3.31)

Here, it is to be noted that the shear strains are the tensorial shear strain terms.

For any angle α,

(3.32)

and therefore, W must reduce to the form

(3.33)

Now, let us write the left hand side of the above equation using the matrix as given in Equation (3.26) and engineering shear strains. In the following we do some rearrangement as

Similarly, we can write the right hand side of previous equation using rotated strain components. Now, for W to be invariant it must be of the form as in Equation (3.33).

1. If we observe the terms containing and   in the first bracket, then we conclude that   is unchanged.

2. Now compare the terms in the second bracket. If we have  then the first of Equation (3.32) is satisfied.
3. Now compare the third bracket. If we have , then the third of Equation (3.32) is satisfied.
4. Now for the fourth bracket we do the following manipulations. Let us assume that and is unchanged. Then we write the terms in fourth bracket as
   

To have W to be invariant we need to have so that the third of Equation (3.32) is satisfied.

Thus, for transversely isotropic material (in plane x₂-x₃) the stiffness matrix becomes

(3.34)