Question

Show by the quotient rule (Section 3 ) that ${\mathbf{C}}_{\mathbf{ijkm}}$in $\mathbf{(}\mathbf{7}\mathbf{.}\mathbf{5}\mathbf{)}$is a ${\mathbf{4}}^{\mathbf{th}}$-rank tensor.

Short answer

The statement has been proven.

Step-by-step solution

Given Information

The tensor $C_{ijkm}$ .

Definition of a strain tensor.

Solids are strained under applied forces, resulting in a change in volume and shape is called strain tensor.

Verify the statement.

The tensor $C_{ijkm}$.

Let S is the strain tensor.

Correspondence between point mechanics and mechanics of an elastic body.

$$\begin{gathered} r\leftrightarrow s \\ F=-\nabla U\leftrightarrow P \\ F=ma\leftrightarrow P_{ij} \\ F=C_{ijkl}{S}_{kl} \end{gathered}$$

In the above statement$P_{ij}=C_{ijkl}{S}_{kl}$

C is a fourth rank tensor.

Hence, the statement has been proven.