Question

For Hooke’s Law, F = kx, the spring constant, k, describes the force required to deform a spring. For a three-dimensional object, we can generalize Hooke’s Law to describe the stress required to strain a material: where stress can be written as a [9x1] tensor, strain can also be written as a [9x1] tensor, and E is an “elasticity” tensor, which is analogous to our spring constant. How many elements, n, in the elasticity tensor are required to satisfy the rules of matrix multiplication? Give a reason why fewer unique parameters than n populate the elasticity tensor. An example of the stress tensor is given below:

Answer 1

Answcy, * how many Elements, n, in the elasticity densor are required to satisfy the rules os mateix multiplicalion ?. - The stress tensor is axi tensor, - The Strain tensor is also a 9x1 tensor. The elasticity tensor E should be a 9x9 Hensor, which implies that it will have a total of 81 Elementy, to satisfy the matize multiplication . The actual number os unique parameters in the Elasticity Hensor is not 81 and is way dessey than that Due to symmetries in the stress and strain tensors. Ethere is symmetry is the stress tensor).