Chapter 10: Tensor Analysis

Interpret the elements of the matrices in Chapter 3, Problems 11.18 to11.21, as components of stress tensors. In each case diagonalize the matrix and so find the principal axes of the stress (along which the stress is pure tension or compression). Describe the stress relative to these axes. (See Example 1.)

Show that in 2 dimensions (say the x , y plane), an inversion through the origin (that is $\mathit{x}\mathbf{\text{'}}\mathbf{=}\mathbf{-}\mathit{x}\mathbf{,}\mathbf{}\mathit{y}\mathbf{\text{'}}\mathbf{=}\mathbf{-}\mathit{y}$) is equivalent to a${\mathbf{180}}^{\mathbf{°}}$rotation of the plane about the axis. Hint:Compare Chapter 3, equation (7.13) with the negative unit matrix.

Any rotation of axes in three dimensions can be described by giving the nine direction cosines of the angle between the $\left(x,y,z\right)$axes and the $\left(x\text{'},y\text{'},z\text{'}\right)$axes. Show that the matrix A of these direction cosines in $\left(2.7\right)$or $\left(2.10\right)$is an orthogonal matrix. Hint: See Chapter 3, Section 9. Find $\mathit{A}{\mathit{A}}^{\mathbf{T}}$and use Problem 3.

Find the inertia tensor about the origin for a mass of uniform density =1, inside the part of the unit sphere where $\mathit{x}\mathbf{>}\mathbf{0}\mathbf{,}\mathit{y}\mathbf{>}\mathbf{0}\mathbf{,}$and find the principal moments of inertia and the principal axes. Note that this is similar to Example 5 but the mass is both above and below the $\left(x,y\right)$plane. Warning hint: This time don’t make the assumptions about symmetry that we did in Example 5.

Do Problem 3 in spherical coordinates; compare the results with Problem 8.3.

What are the physical components of the gradient in polar coordinates? [See (9.1)].The partial derivatives in (10.5) are the covariant components of$\mathbf{\nabla }\mathit{u}$. What relationdo you deduce between physical and covariant components? Answer the samequestions for spherical coordinates, and for an orthogonal coordinate system withscale factors${\mathit{h}}_{\mathbf{1}}\mathbf{,}{\mathit{h}}_{\mathbf{2}}\mathbf{,}{\mathit{h}}_{\mathbf{3}}$.

Generalize Problem 3 to see that the direct product of any two isotropic tensors (or a direct product contracted) is an isotropic tensor. For example show that${\mathit{ϵ}}_{\mathbf{i}\mathbf{j}\mathbf{k}}{\mathit{ϵ}}_{\mathbf{l}\mathbf{m}\mathbf{n}}$is an isotropic tensor (what is its rank?) and${\mathit{ϵ}}_{\mathbf{i}\mathbf{j}\mathbf{k}}{\mathit{ϵ}}_{\mathbf{l}\mathbf{m}\mathbf{n}}{\mathit{\delta }}_{\mathbf{j}\mathbf{n}}$is an isotropic tensor (what is its rank?).

In the text and problems so far, we have found the e vectors for Question: Using the results of Problem 1, express the vector in Problem 4in spherical coordinates.

Let be the tensor in . This is a -rank tensor and so has components. Most of the components are zero. Find the nonzero components and their values. Hint: See discussion after .

Show that ${\mathit{T}}_{\mathbf{i}\mathbf{j}\mathbf{k}\mathbf{l}\mathbf{m}}{\mathit{S}}_{\mathbf{l}\mathbf{m}}$is a tensor and find its rank (assuming that T and S are tensors of the rank indicated by the indices).