Chapter 1

Given the vectors \(\boldsymbol{a}=1 \boldsymbol{i}+2 \boldsymbol{j}+3 \boldsymbol{k}, \boldsymbol{b}=1 \boldsymbol{i}+3 \boldsymbol{j}-2 \boldsymbol{k}\) and \(c=-2 i-1 j+0 k\), calculate: (a) \(a \cdot b\) (b) \(a \times b\) (c) \(a \cdot b \times c\) (d) \(\boldsymbol{a} \times(\boldsymbol{b} \times \boldsymbol{c})\), (e) \((\boldsymbol{a} \times \boldsymbol{b}) \times \boldsymbol{c}\).

Given a plane \(\Pi\) with normal \(\boldsymbol{n}=1 i-2 j+1 \boldsymbol{k}\) and the vector \(\boldsymbol{v}=3 \boldsymbol{i}+4 \boldsymbol{j}-2 \boldsymbol{k}\), calculate: (a) the projection of \(\boldsymbol{v}\) onto \(\Pi\), (b) the reflection of \(v\) with respect to \(\Pi\).

Calculate \(\delta_{i j} \delta_{i j}\) using the rules of index notation and the definition of the Kronecker delta.

Suppose a vector \(v\) satisfies the linear equation $$ \alpha \boldsymbol{v}+\boldsymbol{v} \times \boldsymbol{a}=\boldsymbol{b} $$ where \(\alpha \neq 0\) is a given scalar, and \(a\) and \(b\) are given vectors. Use the dot and cross product operations to solve the above equation for \(v .\) In particular, show that the unique solution is given by $$ v=\frac{\alpha^{2} b-\alpha(b \times a)+(b \cdot a) a}{\alpha\left(\alpha^{2}+|a|^{2}\right)} $$

Given two vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\), and a second-order tensor \(\boldsymbol{S}\), prove: (a) \(\boldsymbol{S}(\boldsymbol{a} \otimes \boldsymbol{b})=(\boldsymbol{S a}) \otimes \boldsymbol{b}\), (b) \((\boldsymbol{a} \otimes \boldsymbol{b}) \boldsymbol{S}=\boldsymbol{a} \otimes\left(\boldsymbol{S}^{T} \boldsymbol{b}\right)\) (c) \((\boldsymbol{a} \otimes \boldsymbol{b})^{T}=(\boldsymbol{b} \otimes \boldsymbol{a})\) Hint: Recall that two second-order tensors \(\boldsymbol{A}\) and \(\boldsymbol{B}\) are equal if and only if \(\boldsymbol{A} \boldsymbol{v}=\boldsymbol{B} \boldsymbol{v}\) for all \(\boldsymbol{v} \in \mathcal{V}\).

Consider any three vectors \(\boldsymbol{a}, \boldsymbol{b}\) and \(\boldsymbol{c}\) which are linearly independent, that is, \((a \times \boldsymbol{b}) \cdot \boldsymbol{c} \neq 0\). Show that: (a) \(\boldsymbol{a} \times \boldsymbol{b}, \boldsymbol{b} \times \boldsymbol{c}\) and \(\boldsymbol{c} \times \boldsymbol{a}\) are also linearly independent, (b) \((\boldsymbol{a} \times \boldsymbol{b}) \otimes \boldsymbol{c}+(\boldsymbol{b} \times \boldsymbol{c}) \otimes \boldsymbol{a}+(\boldsymbol{c} \times \boldsymbol{a}) \otimes \boldsymbol{b}=(\boldsymbol{a} \times \boldsymbol{b} \cdot \boldsymbol{c}) \boldsymbol{I}\).

A second-order tensor \(\boldsymbol{P}\) is a perpendicular projection if \(\boldsymbol{P}\) is symmetric and \(\boldsymbol{P}^{2}=\boldsymbol{P}\). Given two arbitrary unit vectors \(n \neq m\), determine which of the following are perpendicular projections: (a) \(P=I\), (b) \(\boldsymbol{P}=\boldsymbol{n} \otimes \boldsymbol{m}\), (c) \(\boldsymbol{P}=\boldsymbol{n} \otimes \boldsymbol{n}\), (d) \(P=I-n \otimes n\), (e) \(\boldsymbol{P}=\boldsymbol{n} \otimes \boldsymbol{m}+\boldsymbol{m} \otimes \boldsymbol{n}\).

Let \(Q\) be a second-order tensor and let \(\boldsymbol{I}\) be the identity tensor. Show that \(\boldsymbol{Q}\) is orthogonal if \(\boldsymbol{H}=\boldsymbol{Q}-\boldsymbol{I}\) satisfies $$ \boldsymbol{H}+\boldsymbol{H}^{T}+\boldsymbol{H} \boldsymbol{H}^{T}=\boldsymbol{O} $$

Show that the transpose of a second-order tensor \(S\) is uniquely defined and that \(\left[\boldsymbol{S}^{T}\right]=[\boldsymbol{S}]^{T}\)

Prove that a second-order tensor \(S\) cannot be both positivedefinite and skew- symmetric.

Let \(\boldsymbol{A}\) denote the change of basis tensor from a frame \(\left\\{e_{i}\right\\}\) to a frame \(\left\\{e_{i}^{\prime}\right\\}\) with representation \([\boldsymbol{A}]\) in \(\left\\{\boldsymbol{e}_{i}\right\\} .\) Let \(\boldsymbol{S}\) be a secondorder tensor with representation \([S]\) and \([S]^{\prime}\) in \(\left\\{e_{i}\right\\}\) and \(\left\\{e_{i}^{\prime}\right\\}\), respectively. Show that $$ [\boldsymbol{S}]^{\prime}=[\boldsymbol{A}]^{T}[\boldsymbol{S}][\boldsymbol{A}] $$

For an arbitrary second-order tensor \(\boldsymbol{A}=A_{i j} \boldsymbol{e}_{i} \otimes \boldsymbol{e}_{j}\) show that $$ \operatorname{det} \boldsymbol{A}=\frac{1}{6} \epsilon_{i j k} \epsilon_{p q r} A_{i p} A_{j q} A_{k r} $$ and deduce that \(\operatorname{det} \boldsymbol{A}=\operatorname{det} \boldsymbol{A}^{T}\).

For any two second-order tensors \(\boldsymbol{A}\) and \(\boldsymbol{B}\) show that $$ \operatorname{det}(\boldsymbol{A B})=(\operatorname{det} \boldsymbol{A})(\operatorname{det} \boldsymbol{B}) $$ Moreover, if \(\boldsymbol{A}^{-1}\) exists, show that $$ \operatorname{det} \boldsymbol{A}^{-1}=1 / \operatorname{det} \boldsymbol{A} $$

For any pair of vectors \(u\) and \(\boldsymbol{v}\) and any invertible second- order tensor \(\boldsymbol{F}\) show that $$ (\boldsymbol{F} \boldsymbol{u}) \times(\boldsymbol{F} \boldsymbol{v})=(\operatorname{det} \boldsymbol{F}) \boldsymbol{F}^{-T}(\boldsymbol{u} \times \boldsymbol{v}) $$

Show that: (a) \(|\operatorname{det} \boldsymbol{Q}|=1\) for any orthogonal tensor \(\boldsymbol{Q}\), (b) \(\operatorname{det} Q=1\) for any rotation tensor \(Q\).

Let \(Q\) be a rotation tensor and let \(u, v\) be arbitrary vectors. Show that: \((\mathrm{a})(\boldsymbol{Q} \boldsymbol{u}) \cdot(\boldsymbol{Q} v)=\boldsymbol{u} \cdot \boldsymbol{v}\) (b) \(|Q v|=|v|\), (c) \((\boldsymbol{Q} u) \times(\boldsymbol{Q} v)=\boldsymbol{Q}(\boldsymbol{u} \times \boldsymbol{v})\) Remark: The results in (a) and (b) together imply that the length of a vector and the angle between any two vectors are unchanged by a rotation. The result in (c) implies that rotations commute with the cross product operation; in particular, when two vectors are subject to a common rotation, the normal to their plane is subject to the same rotation.

Let \(Q \neq I\) be a rotation tensor. (a) Show that \(\lambda=1\) is always an eigenvalue of \(Q\). Hint: Use the characteristic polynomial and properties of determinants. (b) Show that there is only one independent eigenvector \(\boldsymbol{e}\) such that \(Q e=e\). Hint: Use part (c) of Exercise 21 to show that if there were more than one such independent eigenvector, then there must be three, which would imply \(Q=I\). (c) Let \(n\) be any unit vector orthogonal to \(e\). Show that \(Q n\) is also a unit vector orthogonal to \(\boldsymbol{e}\) and that the angle \(\theta \in[0, \pi]\) between \(n\) and \(Q n\) satisfies the relation $$ 1+2 \cos \theta=\operatorname{tr} Q $$ Hint: Express \(\boldsymbol{Q}\) in the frame \(\\{\boldsymbol{e}, \boldsymbol{n}, \boldsymbol{e} \times \boldsymbol{n}\\}\). Remark: The vector \(e\) in part (b) is called the rotation axis

Show that the principal invariants of a symmetric second-order tensor \(\boldsymbol{S}\) are given by $$ \begin{aligned} &I_{1}(\boldsymbol{S})=\lambda_{1}+\lambda_{2}+\lambda_{3} \\ &I_{2}(\boldsymbol{S})=\lambda_{1} \lambda_{2}+\lambda_{1} \lambda_{3}+\lambda_{2} \lambda_{3} \\ &I_{3}(\boldsymbol{S})=\lambda_{1} \lambda_{2} \lambda_{3} \end{aligned} $$ where \(\lambda_{i}\) are the eigenvalues of \(S\). Hint: Choose a simple frame in which to represent \(S\).

Let \(\boldsymbol{S}\) be a second-order tensor and let \(I_{2}(\boldsymbol{S})\) be its second principal invariant. Show that \(I_{2}(\boldsymbol{S})\) has the same numerical value regardless of the coordinate frame in which it is computed.

Let \(\boldsymbol{A}, \boldsymbol{B}\) and \(\boldsymbol{C}\) be second-order tensors. Show that $$ \boldsymbol{A}: \boldsymbol{B C}=\boldsymbol{A C}^{T}: \boldsymbol{B}=\boldsymbol{B}^{T} \boldsymbol{A}: \boldsymbol{C} $$

Suppose two symmetric second-order tensors \(S\) and \(E\) satisfy \(S=\mathbf{C} \boldsymbol{E}\), where \(\mathbf{C}\) is a fourth-order tensor with components \(C_{i j r s}=\lambda \delta_{i j} \delta_{r s}+\mu\left(\delta_{i r} \delta_{j s}+\delta_{i s} \delta_{j r}\right)\) and \(\lambda\) and \(\mu\) are scalar constants. Show that: (a) \(\boldsymbol{S}=\lambda(\operatorname{tr} \boldsymbol{E}) \boldsymbol{I}+2 \mu \boldsymbol{E}\) (b) \(\boldsymbol{E}=\frac{1}{2 \mu} \boldsymbol{S}-\frac{\lambda}{2 \mu(3 \lambda+2 \mu)}(\operatorname{tr} \boldsymbol{S}) \boldsymbol{I}\).