In the previous lecture we have seen the constitutive equations for various types of (that is, nature of) materials. There are 81 independent elastic constants for generally anisotropic material and two for an isotropic material. Let us summarize the reduction of elastic constants from generally anisotropic to isotropic material.

1. For a generally anisotropic material there are 81 independent elastic constants.
   
   
2. With additional stress symmetry the number of independent elastic constants reduces to 54.
   
   
3. Further, with strain symmetry this number reduces to 36.
   
   
4. A hyperelastic material with stress and strain symmetry has 21 independent elastic constants. The material with 21 independent elastic constants is also called as anisotropic or aelotropic material.
   
   
5. Further reduction with one plane of material symmetry gives 13 independent elastic constants. These materials are known as monoclinic materials.
   
   
6. Additional orthogonal plane of symmetry reduces the number of independent elastic constants to 9. These materials are known as orthotropic materials. Further, if a material has two orthogonal planes of symmetry then it is also symmetric about third mutually perpendicular plane. A unidirectional lamina is orthotropic in nature.
   
   
7. For a transversely isotropic material there are 5 independent elastic constants. Plane 2-3 is transversely isotropic for the lamina shown in Figure 3.7.
   
   
8. For an isotropic material there are only 2 independent elastic constants.

Principal Material Directions:

The interest of this course is unidirectional lamina or laminae and laminate made from stacking of these unidirectional laminae. Hence, we will introduce the principal material directions for a unidirectional fibrous lamina. These are denoted by 1-2-3 directions. The direction 1 is along the fibre. The directions 2 and 3 are perpendicular to the direction 1 and mutually perpendicular to each other. The direction 3 is along the thickness of lamina. The principal directions for a unidirectional lamina are shown in Figure 3.7.

Engineering Constants:

The elastic constants which form the stiffness matrix are not directly measured from laboratory tests on a material. One can measure engineering constants like Young’s modulus, shear modulus and Poisson’s ratio from laboratory tests. The relationship between engineering constants and elastic constants of stiffness matrix is also not straight forward. This relationship can be developed with the help of relationship between engineering constants and compliance matrix coefficients.
In order to establish the relationship between engineering constants and the compliance coefficients, we consider an orthotropic material in the principal material directions. If this orthotropic material is subjected to a 3D state of stress, the resulting strains can be expressed in terms of these stress components and engineering constants as follows: