Relation Among Elastic Constants
Relation Among Elastic Constants:
Relation between E, G and u : Let us establish a relation among the elastic constants E,G and u. Consider a cube of material of side ‘a' subjected to the action of the shear and complementary shear stresses as shown in the figure and producing the strained shape as shown in the figure below.
Assuming that the strains are small and the angle A C B may be taken as 450.
Therefore strain on the diagonal OA
= Change in length / original length
Since angle between OA and OB is very small hence OA @ OB therefore BC, is the change in the length of the diagonal OA
Now this shear stress system is equivalent or can be replaced by a system of direct stresses at 450 as shown below. One set will be compressive, the other tensile, and both will be equal in value to the applied shear strain.
Thus, for the direct state of stress system which applies along the diagonals: