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A First Course in Continuum Mechanics

Oscar Gonzalez, Andrew M. Stuart

Chapter 1

Tensor Algebra - all with Video Answers

Educators

WM

Chapter Questions

03:50

Problem 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) $(\boldsymbol{a} \times \boldsymbol{b}) \times \boldsymbol{c}$.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
06:26

Problem 2

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 $\Pi$.

Gaurav Kalra
Gaurav Kalra
Numerade Educator
02:34

Problem 3

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

WM
William Mead
Numerade Educator
16:53

Problem 4

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
$$
v=\frac{\alpha^{2} b-\alpha(b \times a)+(b \cdot a) a}{\alpha\left(\alpha^{2}+|a|^{2}\right)}
$$

Anthony Ramos
Anthony Ramos
Numerade Educator
02:57

Problem 5

Let $\boldsymbol{a}, \boldsymbol{b}$ and $\boldsymbol{c}$ be vectors. Find scalars $\lambda$ and $\mu$ such that
$$
(\boldsymbol{a} \times \boldsymbol{b}) \times \boldsymbol{c}=\lambda \boldsymbol{b}-\mu \boldsymbol{a}
$$

CD
Chandana Deeksha
Numerade Educator
01:56

Problem 6

Let $\boldsymbol{v}$ be an arbitrary vector and let $\boldsymbol{n}$ be an arbitrary unit vector. Show that:
(a) $v=(v \cdot \boldsymbol{n}) \boldsymbol{n}-(\boldsymbol{v} \times \boldsymbol{n}) \times n$
(b) $v=(v \cdot n) n+(v \otimes n) n-(n \otimes v) n$.
Remark: The identity in (a) shows that $\boldsymbol{v}$ can always be decomposed into parts parallel and perpendicular to $n .$

Anthony Ramos
Anthony Ramos
Numerade Educator
09:16

Problem 7

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 $\boldsymbol{v} \in \mathcal{V}$.

Jacob Fry
Jacob Fry
Numerade Educator
03:22

Problem 8

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) $(\boldsymbol{a} \times \boldsymbol{b}) \otimes \boldsymbol{c}+(\boldsymbol{b} \times \boldsymbol{c}) \otimes \boldsymbol{a}+(\boldsymbol{c} \times \boldsymbol{a}) \otimes \boldsymbol{b}=(\boldsymbol{a} \times \boldsymbol{b} \cdot \boldsymbol{c}) \boldsymbol{I}$.

WM
William Mead
Numerade Educator
04:38

Problem 9

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) $\boldsymbol{P}=\boldsymbol{n} \otimes \boldsymbol{m}+\boldsymbol{m} \otimes \boldsymbol{n}$.

Muhammad Nawaz
Muhammad Nawaz
Numerade Educator
01:23

Problem 10

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
$$
\boldsymbol{H}+\boldsymbol{H}^{T}+\boldsymbol{H} \boldsymbol{H}^{T}=\boldsymbol{O}
$$

Raj Bala
Raj Bala
Numerade Educator
01:23

Problem 11

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

Raj Bala
Raj Bala
Numerade Educator
01:23

Problem 12

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

Raj Bala
Raj Bala
Numerade Educator
01:51

Problem 13

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
$$
[\boldsymbol{S}]^{\prime}=[\boldsymbol{A}]^{T}[\boldsymbol{S}][\boldsymbol{A}]
$$

Victor Salazar
Victor Salazar
Numerade Educator
02:23

Problem 14

Consider a vector $\boldsymbol{a}$ with representation $[\boldsymbol{a}]=(1,1,1)^{T}$ in a coordinate frame $\left\{e_{i}\right\} .$ If $S$ is an anti-clockwise rotation through an angle of $\pi / 4$ about $\boldsymbol{a}$, find $[\boldsymbol{S}]$.

Sanat Mukherjee
Sanat Mukherjee
Numerade Educator
01:01

Problem 15

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 $\operatorname{det} \boldsymbol{A}=\operatorname{det} \boldsymbol{A}^{T}$.

Raj Bala
Raj Bala
Numerade Educator
01:18

Problem 16

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
$$
\operatorname{det} \boldsymbol{A}^{-1}=1 / \operatorname{det} \boldsymbol{A}
$$

Jacob Denson
Jacob Denson
Numerade Educator
04:16

Problem 17

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

Foster Wisusik
Foster Wisusik
Numerade Educator
02:00

Problem 18

Prove Result $1.1 .$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
03:45

Problem 19

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

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
02:07

Problem 20

Prove Result $1.6$.

Ankur S
Ankur S
Numerade Educator
01:40

Problem 21

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.

Fuzail Shakir
Fuzail Shakir
Numerade Educator
03:45

Problem 22

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

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
01:23

Problem 23

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 $S$.

Raj Bala
Raj Bala
Numerade Educator
01:23

Problem 24

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.

Raj Bala
Raj Bala
Numerade Educator
03:26

Problem 25

Prove Result $1.10$ for symmetric $\boldsymbol{S}$.

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
02:07

Problem 26

Prove Result $1.11$.

Ankur S
Ankur S
Numerade Educator
01:33

Problem 27

For any vectors $\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}$ and $\boldsymbol{d}$ show
$$
(\boldsymbol{a} \otimes \boldsymbol{b}):(\boldsymbol{c} \otimes \boldsymbol{d})=(\boldsymbol{a} \cdot \boldsymbol{c})(\boldsymbol{b} \cdot \boldsymbol{d})
$$

Hunza Gilgit
Hunza Gilgit
Numerade Educator
06:38

Problem 28

For any given coordinate frame $\left\{e_{i}\right\}$ show that the dyadic basis $\left\{\boldsymbol{e}_{i} \otimes \boldsymbol{e}_{j}\right\}$ for $\mathcal{V}^{2}$ is orthonormal in the inner product (1.17)

Chris Trentman
Chris Trentman
Numerade Educator
01:24

Problem 29

Prove Result 1.13.

Nick Johnson
Nick Johnson
Numerade Educator
01:24

Problem 30

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

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
20:00

Problem 31

Let $a$ and $b$ be vectors and let $\boldsymbol{A}$ be a second-order tensor. Define $\mathbf{C}$ to be the fourth-order tensor given by $\mathbf{C}(\boldsymbol{S})=\boldsymbol{A} \boldsymbol{S}$. Show that, if $a$ is an eigenvector of $A$ with eigenvalue $\alpha$, then
$$
(\mathbf{C}(\boldsymbol{a} \otimes \boldsymbol{b})) \boldsymbol{v}=\alpha(\boldsymbol{b} \cdot \boldsymbol{v}) \boldsymbol{a}
$$

Chris Trentman
Chris Trentman
Numerade Educator
View

Problem 32

Find the components $C_{i j k l}$ of the fourth-order tensor defined by
$$
\mathbf{C}(\boldsymbol{S})=\frac{1}{2}\left(\boldsymbol{S}-\boldsymbol{S}^{T}\right)
$$

Victor Salazar
Victor Salazar
Numerade Educator
02:07

Problem 33

Prove Result $1.14$.

Ankur S
Ankur S
Numerade Educator
01:03

Problem 34

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) $\boldsymbol{E}=\frac{1}{2 \mu} \boldsymbol{S}-\frac{\lambda}{2 \mu(3 \lambda+2 \mu)}(\operatorname{tr} \boldsymbol{S}) \boldsymbol{I}$.

Raj Bala
Raj Bala
Numerade Educator