# Material Symmetry - Isotropy - Lesson 11

The study of constitutive relations is continued here for solids. The question of material symmetry or isotropy will be discussed here. Isotropy means invariance with respect to direction. Isotropic tensor and its effect on any vector are discussed. First, the isotropy of hyperelastic solids is discussed.

### Response to a Question

In this lesson the response is provided to the following question:

• Can we not always correct the rotations by measuring the deformations from the ‘original’ reference configuration?

The discussion on material isotropy continues. It is seen that the rotation of the body does not matter or change the isotropy of the material. It is shown that strain energy density function is invariant to the reference configuration or it can be said that it is independent of the orientation of microstructures. Another way to say that the material is isotropic is to say that material is rotationally invariant.

In this lesson, it will be seen what more it means that material is isotropic. The steps followed will be most common as for material frame invariance. Polar decomposition is used here. Furthermore, the conditions for material invariance are discussed. Material frame invariance does not ensure material isotropy. Phi for single and composite crystals are shown.

In this lesson, it will be seen that all the strain energy functions that have been discussed in previous lessons are themselves isotropic. The functions are represented in form of examples. The first function is St. Venant Kirchhoff, the second is the compressible neo-Hookean model and the third one is the quadratic logarithmic model. This lesson concludes the basic study of constitutive relations.