A First Course in Continuum Mechanics

Oscar Gonzalez, Andrew M. Stuart

Chapter 4 Kinematics - all with Video Answers

Educators

Chapter Questions

Problem 1

Let $B=\left\{\boldsymbol{X} \in \mathbb{E}^{3}|| X_{i} \mid<1\right\}$ be the reference configuration of a body as shown below, and consider the deformation $\boldsymbol{x}=\boldsymbol{\varphi}(\boldsymbol{X})$ defined in components by $x_{1}=X_{1}, x_{2}=X_{3}+4, x_{3}=-X_{2}$.
(a) Sketch the deformed configuration $B^{\prime}=\varphi(B)$ relative to the same coordinate frame and indicate the points in $B^{\prime}$ which correspond to the vertices $a, b, c$ in $B$.
(b) Find the components of the displacement field $\boldsymbol{u}$.

Mahendra K

Numerade Educator

01:03

Problem 2

Let $B=\left\{\boldsymbol{X} \in \mathbb{E}^{3}|| X_{i} \mid<1\right\}$ and $B^{\prime}=\left\{\boldsymbol{x} \in \mathbb{E}^{3}|| x_{1} \mid<\right.$ 1, $\left.\left|x_{3}\right|<1,\left|x_{2}-6\right|<3\right\}$ be the reference and deformed configurations of a body as shown below, and consider a deformation $\boldsymbol{x}=\boldsymbol{\varphi}(\boldsymbol{X})$ of the form $x_{1}=p X_{2}+q, x_{2}=r X_{1}+s, x_{3}=X_{3}$.
By comparing the faces of $B^{\prime}$ and $B$ find constants $p, q, r, s$ such that $B^{\prime}=\varphi(B)$.

John Nicolle

Numerade Educator

08:28

Problem 3

Let $B=\left\{\boldsymbol{X} \in \mathbb{E}^{3}|| X_{i} \mid<1\right\}$ be the reference configuration of a body. For each deformation $\boldsymbol{x}=\boldsymbol{\varphi}(\boldsymbol{X})$ given below (i) sketch the deformed configuration $B^{\prime}$, (ii) find the components of $\nabla \varphi,($ iii) determine if $\varphi$ is one-to-one, and (iv) determine if $\boldsymbol{\varphi}$ is admissible:
(a) $x_{1}=1+X_{1}, \quad x_{2}=\frac{1}{2} X_{2}+5, \quad x_{3}=2 X_{3}$,
(b) $x_{1}=X_{1}, \quad x_{2}=3, \quad x_{3}=X_{3}$,
(c) $x_{1}=X_{1}, \quad x_{2}=X_{2}, \quad x_{3}=4-X_{3}$.

Frank Lin

Numerade Educator

11:37

Problem 4

Consider a deformation $\boldsymbol{x}=\boldsymbol{\varphi}(\boldsymbol{X})$ defined in components by
$$
x_{1}=X_{1}^{2}, \quad x_{2}=X_{3}^{2}, \quad x_{3}=X_{2} X_{3}
$$
(a) Find the components of $\nabla \varphi$ and determine $\operatorname{det} \nabla \varphi$.
(b) Is $\varphi$ admissible for arbitrary $B ?$

Amit Srivastava

Numerade Educator

00:48

Problem 5

For each deformation $\boldsymbol{x}=\boldsymbol{\varphi}(\boldsymbol{X})$ given below find the components of the deformation gradient $\boldsymbol{F}$ and determine if $\varphi$ is homogeneous or non-homogeneous:
(a) $x_{1}=X_{1}, \quad x_{2}=X_{2} X_{3}, \quad x_{3}=X_{3}-1$,
(b) $x_{1}=2 X_{2}-1, \quad x_{2}=X_{3}, \quad x_{3}=3+5 X_{1}$
(c) $x_{1}=\exp \left(X_{1}\right), \quad x_{2}=-X_{3}, \quad x_{3}=X_{2}$.

Jack Chen

Numerade Educator

02:02

Problem 6

Let $\varphi$ be a homogeneous deformation with deformation gradient $\boldsymbol{F}$. For any two points $\boldsymbol{X}$ and $\boldsymbol{Y}$ show that
$$
\varphi(\boldsymbol{X})=\boldsymbol{\varphi}(\boldsymbol{Y})+\boldsymbol{F}(\boldsymbol{X}-\boldsymbol{Y}).
$$

David Nguyen

Numerade Educator

01:04

Problem 7

Let $\boldsymbol{\varphi}: B \rightarrow B^{\prime}$ be a homogeneous deformation with deformation gradient $\boldsymbol{F}$, and let $\boldsymbol{X}(\sigma)=\boldsymbol{X}_{0}+\sigma \boldsymbol{v}$ be a line segment through the point $\boldsymbol{X}_{0}$ in $B$ with direction $\boldsymbol{v}$. Show that $\boldsymbol{\varphi}(\boldsymbol{X}(\sigma))$ is a line segment through the point $\varphi\left(\boldsymbol{X}_{0}\right)$ in $B^{\prime}$ with direction $\boldsymbol{F} v$.

Carson Merrill

Numerade Educator

02:16

Problem 8

One way to visualize a deformation is to show how curves etched or marked on the surface of a body appear before and after the deformation. Suppose a reference configuration $B$ is marked with lines as follows:
Which of the deformations visualized below appear to be homogeneous? Which appear to be non-homogeneous?
(a)
(b)
(c)

Surendra Kumar

Numerade Educator

01:04

Problem 9

Prove Result 4.1.

Carson Merrill

Numerade Educator

02:56

Problem 10

Prove Result $4.2$.

Nick Johnson

Numerade Educator

02:56

Problem 11

Prove Result 4.3.

Nick Johnson

Numerade Educator

01:07

Problem 12

Let $\boldsymbol{F}=\boldsymbol{R} \boldsymbol{U}=\boldsymbol{V} \boldsymbol{R}$ be the right and left polar decompositions of a deformation gradient $\boldsymbol{F} .$ Show that:
(a) $\boldsymbol{U}$ and $\boldsymbol{V}$ have the same eigenvalues,
(b) if $\left\{\boldsymbol{u}_{i}\right\}$ is a frame of eigenvectors of $\boldsymbol{U}$, then $\left\{\boldsymbol{R} \boldsymbol{u}_{i}\right\}$ is a frame of eigenvectors of $\boldsymbol{V}$. Thus, in general, $\boldsymbol{U}$ and $\boldsymbol{V}$ have different eigenvectors.

Victor Salazar

Numerade Educator

06:32

Problem 13

Consider a deformation $\varphi$ defined in components by
$$
[\boldsymbol{\varphi}(\boldsymbol{X})]=\left\{\begin{array}{c}
p X_{1}+a \\
q X_{2}+b \\
r X_{3}+c
\end{array}\right\},
$$
where $p, q, r, a, b, c$ are constants.
(a) Find conditions on $p, q, r$ for $\varphi$ to be an admissible deformation of $B=\mathbb{E}^{3}$.
(b) Given an arbitrary point $\boldsymbol{Y}$ find the components of $\boldsymbol{d}, \boldsymbol{s}$ and $\boldsymbol{r}$ such that $\boldsymbol{\varphi}=\boldsymbol{r} \circ \boldsymbol{s} \circ \boldsymbol{d}$, where $\boldsymbol{d}$ is a translation, $\boldsymbol{s}$ is a stretch from $\boldsymbol{Y}$ and $\boldsymbol{r}$ is rotation about $\boldsymbol{Y}$.

Khoobchandra Agrawal

Numerade Educator

03:49

Problem 14

Let $\boldsymbol{Y}$ be an arbitrary point and let $\boldsymbol{Q}$ be an arbitrary rotation tensor and consider the deformation
$$
\boldsymbol{\varphi}(\boldsymbol{X})=\boldsymbol{Y}+\boldsymbol{Q}(\boldsymbol{X}-\boldsymbol{Y})
$$
In particular, $\varphi$ is a rotation about $\boldsymbol{Y}$. Find the deformation gradient $\boldsymbol{F}$ and the Cauchy-Green strain tensor $\boldsymbol{C} .$ Does $\boldsymbol{F}$ depend on $\boldsymbol{Q} ?$ What about $\boldsymbol{C} ?$

Nick Johnson

Numerade Educator

02:02

Problem 15

Suppose the deformation gradient at a point $\boldsymbol{X}_{0}$ in a body has components
$$
[\boldsymbol{F}]=\left(\begin{array}{lll}
1 & 0 & 0 \\
0 & 2 & 1 \\
0 & 1 & 2
\end{array}\right).
$$
Find the components of the Cauchy-Green strain tensor $\boldsymbol{C}$ and the right stretch tensor $\boldsymbol{U}$.

Elliott Walker

Numerade Educator

04:22

Problem 16

Let $B=\left\{\boldsymbol{X} \in \mathbb{E}^{3} \mid 0<X_{i}<1\right\}$ and consider a deformation $\boldsymbol{x}=\boldsymbol{\varphi}(\boldsymbol{X})$ of the form $x_{1}=p X_{1}, x_{2}=q X_{2}, x_{3}=r X_{3}$, where $p, q, r$ are constants. Notice that, because $\varphi$ is homogeneous, the Cauchy-Green strain tensor $\boldsymbol{C}$ will be constant.
(a) Find the components of $\boldsymbol{C}$.
(b) Find the stretch $\lambda$ in (i) the direction parallel to the edge $\overline{a b}$ and (ii) the direction parallel to the diagonal $\overline{a c}$.

Malika Singh

Numerade Educator

02:16

Problem 17

Let $B=\left\{\boldsymbol{X} \in \mathbb{E}^{3} \mid 0<X_{i}<1\right\}$ and consider a deformation $\boldsymbol{x}=\boldsymbol{\varphi}(\boldsymbol{X})$ of the form $x_{1}=X_{1}+\alpha X_{2}, x_{2}=X_{2}, x_{3}=X_{3}$, where $\alpha>0$ is a constant. Such a deformation is called a simple shear in the $\boldsymbol{e}_{1}, \boldsymbol{e}_{2}$-plane.
(a) Find the components of the Cauchy-Green strain tensor $\boldsymbol{C}$.
(b) Find the shear $\gamma$ for the direction pair $\left(\boldsymbol{e}_{1}, \boldsymbol{e}_{2}\right)$ and for the pair $\left(\boldsymbol{e}_{1}, \boldsymbol{e}_{3}\right)$. What happens to these shears in the limits $\alpha \rightarrow 0$ and $\alpha \rightarrow \infty$ ?
(c) Find the extreme values of the stretch $\lambda$ and the directions in which these occur. Do the directions of extreme stretch correspond to any of the diagonal directions $\pm \frac{1}{\sqrt{2}}\left(\boldsymbol{e}_{1}+\boldsymbol{e}_{2}\right)$ or $\pm \frac{1}{\sqrt{2}}\left(\boldsymbol{e}_{1}-\boldsymbol{e}_{2}\right)$ in the $\boldsymbol{e}_{1}, \boldsymbol{e}_{2}$-plane?

Surendra Kumar

Numerade Educator

00:57

Problem 18

Show that the shear $\gamma(\boldsymbol{e}, \boldsymbol{d})$ between any two right principal directions $e$ and $\boldsymbol{d}$ is zero.

Mayukh Banik

Numerade Educator

09:08

Problem 19

Prove Result $4.5$.

Donald Albin

Numerade Educator

00:01

Problem 20

Let $\boldsymbol{c}$ be an arbitrary vector, $\boldsymbol{A}$ an arbitrary tensor and $\epsilon$ an arbitrary scalar. Supposing the components of $\boldsymbol{c}$ and $\boldsymbol{A}$ are of order unity and $0 \leq \epsilon \ll 1$, determine which of the following deformations are small:
(a) $\boldsymbol{\varphi}(\boldsymbol{X})=\boldsymbol{X}+\boldsymbol{c}$
(b) $\varphi(\boldsymbol{X})=\boldsymbol{A} \boldsymbol{X}+\boldsymbol{c}$,
(c) $\varphi(\boldsymbol{X})=\epsilon \boldsymbol{A} \boldsymbol{X}+\boldsymbol{c}$,
(d) $\varphi(\boldsymbol{X})=(\boldsymbol{I}+\epsilon \boldsymbol{A}) \boldsymbol{X}+\boldsymbol{c} .$

Jacob Fry

Numerade Educator

01:02

Problem 21

Let $B=\left\{\boldsymbol{X} \in \mathbb{E}^{3} \mid 0<X_{i}<L\right\}$ and consider the simple shear deformation $\boldsymbol{x}=\boldsymbol{\varphi}(\boldsymbol{X})$ given by $x_{1}=X_{1}+(\alpha / L) X_{2}, x_{2}=X_{2}$ $x_{3}=X_{3}$, where $\alpha>0$ is a constant.
(a) Show that $\varphi$ is small when $\alpha \ll L$ and find the components of the infinitesimal strain tensor $\boldsymbol{E}$
(b) Use the series expansion $\sqrt{1+\epsilon^{2}}=1+\frac{\epsilon^{2}}{2}-\frac{\epsilon^{4}}{8}+\cdots$ to show that
$$
\frac{\overline{a b^{\prime}}-\overline{a b}}{\overline{a b}}=\frac{(\alpha / L)^{2}}{2}-\frac{(\alpha / L)^{4}}{8}+\cdots
$$
(Here $\overline{a b}$ denotes length.) Is this result consistent with the interpretation of $E_{22}$ and its value found in part (a)?
(c) Use the series expansion arctan $(\epsilon)=\epsilon-\frac{\epsilon^{3}}{3}+\frac{\epsilon^{5}}{5}-\cdots$ to show that
$$
\angle b a c-\angle b^{\prime} a c=(\alpha / L)-\frac{(\alpha / L)^{3}}{3}+\frac{(\alpha / L)^{5}}{5}-\cdots
$$
(Here $\angle b a c$ denotes angle.) Is this result consistent with the interpretation of $E_{12}$ and its value found in part (a)?

Raj Bala

Numerade Educator

06:32

Problem 22

Consider a rigid deformation of the form
$$
\boldsymbol{\varphi}(\boldsymbol{X})=\boldsymbol{R} \boldsymbol{X}+\boldsymbol{c},
$$
where $\boldsymbol{R}$ is a small rotation in the sense that $\boldsymbol{R}=\boldsymbol{I}+\mathcal{O}(\epsilon)$ for some $0 \leq \epsilon \ll 1 .$ In particular, suppose
$$
\boldsymbol{R}=\exp (\epsilon \boldsymbol{W})=\boldsymbol{I}+\epsilon \boldsymbol{W}+\frac{\epsilon^{2}}{2} \boldsymbol{W}^{2}+\cdots,
$$
where $\boldsymbol{W}$ is an arbitrary skew-symmetric tensor with components of order unity (see Result 1.7).
(a) Find the displacement field $\boldsymbol{u}$ and show that $\boldsymbol{\varphi}$ is a small deformation for any reference configuration $B$.
(b) Show that, if terms of order $\mathcal{O}\left(\epsilon^{2}\right)$ are neglected, then $\varphi$ has the form of an infinitesimally rigid deformation.

Khoobchandra Agrawal

Numerade Educator

02:34

Problem 23

Let $B=\left\{\boldsymbol{X} \in \mathbb{E}^{3} \mid 0<X_{i}<1\right\}$ and consider the motion $\varphi$ defined in components by
$$
[\boldsymbol{\varphi}(\boldsymbol{X}, t)]=\left\{\begin{array}{l}
\exp \left(t X_{1}\right)+X_{1}-1 \\
\exp \left(t X_{2}\right)+X_{2}-1 \\
\exp \left(t X_{3}\right)+X_{3}-1
\end{array}\right\}.
$$
Determine the position of the boundary points $\left[\boldsymbol{X}_{0}\right]=(0,0,0)^{T}$ and $\left[\boldsymbol{X}_{1}\right]=(1,1,1)^{T}$ as a function of $t .$

Xiaomeng Zhang

Numerade Educator

01:04

Problem 24

Let $B=\mathbb{E}^{3}$ and consider the motion $\boldsymbol{x}=\boldsymbol{\varphi}(\boldsymbol{X}, t)$ defined by
$$
x_{1}=e^{t} X_{1}+X_{3}, \quad x_{2}=X_{2}, \quad x_{3}=X_{3}-t X_{1}
$$
(a) Show that the inverse motion $\boldsymbol{X}=\boldsymbol{\psi}(\boldsymbol{x}, t)$ is given by
$$
X_{1}=\frac{x_{1}-x_{3}}{t+e^{t}}, \quad X_{2}=x_{2}, \quad X_{3}=\frac{t x_{1}+e^{t} x_{3}}{t+e^{t}}
$$
(b) Verify that $\boldsymbol{\varphi}(\boldsymbol{\psi}(\boldsymbol{x}, t), t)=\boldsymbol{x}$ and $\boldsymbol{\psi}(\boldsymbol{\varphi}(\boldsymbol{X}, t), t)=\boldsymbol{X}$.

Aman Gupta

Numerade Educator

02:12

Problem 25

Consider the motion in Exercise 24 and let $\Omega(\boldsymbol{X}, t)$ and $\Gamma(\boldsymbol{x}, t)$ be material and spatial fields defined by
$$
\Omega(\boldsymbol{X}, t)=X_{1}+t, \quad \Gamma(\boldsymbol{x}, t)=x_{1}+t
$$
(a) Find the spatial description $\Omega_{s}$ of $\Omega$.
(b) Find the material description $\Gamma_{m}$ of $\Gamma$.

Eric Mockensturm

Numerade Educator

08:01

Problem 26

Consider the motion from Exercise 24 and consider the spatial field $\Gamma(\boldsymbol{x}, t)=x_{1}+t$
(a) Find the material time derivative $\dot{\Gamma}(\boldsymbol{x}, t)$.
(b) Find the partial time derivative $\frac{\partial}{\partial t} \Gamma(\boldsymbol{x}, t)$ and verify that $\dot{\Gamma}(\boldsymbol{x}, t) \neq \frac{\partial}{\partial t} \Gamma(\boldsymbol{x}, t)$.

Mahnoor Amin

Numerade Educator

02:53

Problem 27

Let $\boldsymbol{x}=\boldsymbol{\varphi}(\boldsymbol{X}, t)$ be the motion considered in Exercise 24 .
(a) Find the components of the material velocity $\boldsymbol{V}(\boldsymbol{X}, t)$ and the spatial velocity $\boldsymbol{v}(\boldsymbol{x}, t)$
(b) Find the components of the material acceleration $\boldsymbol{A}(\boldsymbol{X}, t)$ and the spatial acceleration $\boldsymbol{a}(\boldsymbol{x}, t)$
(c) Verify that $\boldsymbol{a}(\boldsymbol{x}, t) \neq \frac{\partial}{\partial t} \boldsymbol{v}(\boldsymbol{x}, t)$.

Eric Mockensturm

Numerade Educator

03:18

Problem 28

Let $B=\left\{\boldsymbol{X} \in \mathbb{E}^{3} \mid 0<X_{i}<1\right\}$ and consider the motion $\boldsymbol{x}=\boldsymbol{\varphi}(\boldsymbol{X}, t)$ defined by
$$
x_{1}=t X_{2}^{2}+X_{1}, \quad x_{2}=t X_{1}+X_{2}, \quad x_{3}=t X_{3}+X_{3}
$$
(a) Find the components of the deformation gradient and find the largest number $t_{c}>0$ for which $\operatorname{det} \boldsymbol{F}(\boldsymbol{X}, t)>0$ for all $\boldsymbol{X} \in B$ and $t \in\left[0, t_{c}\right]$
(b) Find the components of the inverse motion $\boldsymbol{X}=\boldsymbol{\psi}(\boldsymbol{x}, t)$ for $t \in\left[0, t_{c}\right]$.
(c) Find the components of the material velocity field $\boldsymbol{V}(\boldsymbol{X}, t)$ and the spatial velocity field $\boldsymbol{v}(\boldsymbol{x}, t)$ for $t \in\left[0, t_{c}\right]$.

Narayan Hari

Numerade Educator

02:47

Problem 29

Let $B=\mathbb{E}^{3}$ and consider the motion $\boldsymbol{x}=\boldsymbol{\varphi}(\boldsymbol{X}, t)$ defined by
$$
x_{1}=(1+t) X_{1}, \quad x_{2}=X_{2}+t X_{3}, \quad x_{3}=X_{3}-t X_{2}
$$
Moreover, consider the spatial field $\phi(\boldsymbol{x}, t)=t x_{1}+x_{2}$
(a) Show that $\operatorname{det} \boldsymbol{F}(\boldsymbol{X}, t)>0$ for all $t \geq 0$ and find the components of the inverse motion $\boldsymbol{X}=\boldsymbol{\psi}(\boldsymbol{x}, t)$ for all $t \geq 0$
(b) Find the components of the spatial velocity field $\boldsymbol{v}(\boldsymbol{x}, t)$.
(c) Find the material time derivative of $\phi$ using the definition $\dot{\phi}=\left[\dot{\phi}_{m}\right]_{s}$
(d) Find the material time derivative of $\phi$ using Result 4.7. Do you obtain the same result as in part (c)?

Benjamin Arndell

Numerade Educator

03:18

Problem 30

Consider the deformation $\boldsymbol{x}=\boldsymbol{\varphi}(\boldsymbol{X}, t)$ given by
$$
\begin{aligned}
&x_{1}=\cos (\omega t) X_{1}+\sin (\omega t) X_{2} \\
&x_{2}=-\sin (\omega t) X_{1}+\cos (\omega t) X_{2} \\
&x_{3}=(1+\alpha t) X_{3}
\end{aligned}
$$
Notice that this deformation corresponds to rotation (with rate \omega) in the $\boldsymbol{e}_{1}, \boldsymbol{e}_{2}$-plane together with extension (with rate $\alpha$ ) along the $\boldsymbol{e}_{3}$-axis.
(a) Find the components of the inverse motion $\boldsymbol{X}=\boldsymbol{\psi}(\boldsymbol{x}, t)$.
(b) Find the components of the spatial velocity field $\boldsymbol{v}(\boldsymbol{x}, t)$.
(c) Find the components of the rate of strain and spin tensors $\boldsymbol{L}(\boldsymbol{x}, t)$ and $\boldsymbol{W}(\boldsymbol{x}, t)$. Verify that $\boldsymbol{L}$ is determined by $\alpha$, whereas $\boldsymbol{W}$ is determined by $\omega$.

Narayan Hari

Numerade Educator

03:20

Problem 31

Consider a motion $\varphi: B \times[0, \infty) \rightarrow \mathbb{E}^{3}$. For any fixed $t \geq 0$ let $\boldsymbol{v}$ be the spatial velocity field in the current configuration $B_{t}$ and let $\boldsymbol{\psi}$ be the inverse motion. For any $s>0$ let $\widehat{\boldsymbol{\varphi}}_{s}: B_{t} \rightarrow B_{t+s}$ be the motion which coincides with $\varphi_{t+s}: B \rightarrow B_{t+s}$ in the sense that
$$
\widehat{\boldsymbol{\varphi}}(\boldsymbol{x}, s)=\left.\boldsymbol{\varphi}(\boldsymbol{X}, t+s)\right|_{\boldsymbol{X}=\boldsymbol{\psi}(\boldsymbol{x}, t)}, \quad \forall \boldsymbol{x} \in B_{t}
$$
(a) Show that $\widehat{\varphi}(\boldsymbol{x}, 0)=\boldsymbol{x}$ for all $\boldsymbol{x} \in B_{t}$.
(b) Show that $\frac{\partial}{\partial s} \widehat{\varphi}(\boldsymbol{x}, 0)=\boldsymbol{v}(\boldsymbol{x}, t)$ for all $\boldsymbol{x} \in B_{t}$.
(c) Let $\widehat{\boldsymbol{F}}(\boldsymbol{x}, s)=\nabla^{x} \widehat{\boldsymbol{\varphi}}(\boldsymbol{x}, s)$ be the deformation gradient associated with $\widehat{\varphi}$. Show that $\widehat{\boldsymbol{F}}(\boldsymbol{x}, 0)=\boldsymbol{I}$ and $\frac{\partial}{\partial s} \widehat{\boldsymbol{F}}(\boldsymbol{x}, 0)=\nabla^{x} \boldsymbol{v}(\boldsymbol{x}, t)$
(d) Let $\widehat{\boldsymbol{E}}=\operatorname{sym}\left(\nabla^{x} \widehat{\boldsymbol{u}}\right)=\operatorname{sym}(\widehat{\boldsymbol{F}}-\boldsymbol{I})$ be the infinitesimal strain tensor associated with $\widehat{\varphi}$. Show that
$$
\boldsymbol{L}(\boldsymbol{x}, t)=\frac{\partial}{\partial s} \widehat{\boldsymbol{E}}(\boldsymbol{x}, 0)
$$
(e) Consider the right polar decomposition $\widehat{\boldsymbol{F}}=\widehat{\boldsymbol{R}} \widehat{\boldsymbol{U}}$, where $\widehat{\boldsymbol{U}}^{2}=\widehat{\boldsymbol{F}}^{T} \widehat{\boldsymbol{F}} .$ Show that
$$
\boldsymbol{L}(\boldsymbol{x}, t)=\frac{\partial}{\partial s} \widehat{\boldsymbol{U}}(\boldsymbol{x}, 0), \quad \boldsymbol{W}(\boldsymbol{x}, t)=\frac{\partial}{\partial s} \widehat{\boldsymbol{R}}(\boldsymbol{x}, 0).
$$

Ahmad Reda

Numerade Educator

01:17

Problem 32

Consider a motion $\varphi: B \times[0, \infty) \rightarrow \mathbb{E}^{3}$ with spatial velocity field $\boldsymbol{v}$, spatial acceleration field $\boldsymbol{a}$, spatial vorticity field $\boldsymbol{w}$ and spatial spin field $\boldsymbol{W}$
(a) Show that $2 \boldsymbol{W} \boldsymbol{c}=\boldsymbol{w} \times \boldsymbol{c}$ for any arbitrary vector $\boldsymbol{c}$. (Thus $\boldsymbol{w}$ is the axial vector of $2 \boldsymbol{W}$.)
(b) Show that the acceleration field satisfies
$$
\boldsymbol{a}=\frac{\partial \boldsymbol{v}}{\partial t}+\boldsymbol{w} \times \boldsymbol{v}+\nabla^{x}\left(\frac{1}{2}|\boldsymbol{v}|^{2}\right).
$$

Subhadeepta Sahoo

Numerade Educator

01:30

Problem 33

Consider an arbitrary rigid motion $\varphi: B \times[0, \infty) \rightarrow \mathbb{E}^{3}$ of the form
$$
\boldsymbol{\varphi}(\boldsymbol{X}, t)=\boldsymbol{R}(t) \boldsymbol{X}+\boldsymbol{c}(t)
$$
where $\boldsymbol{R}(t)$ is a rotation tensor and $\boldsymbol{c}(t)$ is a vector.
(a) Find the inverse motion $\boldsymbol{\psi}(\boldsymbol{x}, t)$.
(b) Let $\boldsymbol{\Omega}(t)=\dot{\boldsymbol{R}}(t) \boldsymbol{R}(t)^{T} .$ Show that $\boldsymbol{\Omega}(t)$ is skew-symmetric.
(c) Show that the spatial velocity field can be written in the form
$$
\boldsymbol{v}(\boldsymbol{x}, t)=\boldsymbol{\Omega}(t)(\boldsymbol{x}-\boldsymbol{c}(t))+\dot{\boldsymbol{c}}(t).
$$

Narayan Hari

Numerade Educator

02:48

Problem 34

Prove Result $4.10$.

Harmender Singh Yadav

Numerade Educator

03:26

Problem 35

Show that
$$
\int_{\partial \Omega_{t}} \boldsymbol{w}(\boldsymbol{x}, t) \otimes \boldsymbol{n}(\boldsymbol{x}) d A_{\boldsymbol{x}}=\int_{\partial \Omega} \boldsymbol{w}_{m}(\boldsymbol{X}, t) \otimes \boldsymbol{G}(\boldsymbol{X}, t) \boldsymbol{N}(\boldsymbol{X}) d A_{\boldsymbol{X}}
$$
where the tensor $\boldsymbol{G}$ and unit normals $\boldsymbol{n}$ and $\boldsymbol{N}$ are as defined in Result 4.11.

Khoobchandra Agrawal

Numerade Educator

02:06

Problem 36

Prove Result $4.12$.

Narayan Hari

Numerade Educator

07:28

Problem 37

Let $B=\left\{\boldsymbol{X} \in \mathbb{E}^{3} \mid 0<X_{i}<1\right\}$ and consider the simple shear motion $\boldsymbol{x}=\boldsymbol{\varphi}(\boldsymbol{X}, t)$ defined by
$$
x_{1}=X_{1}+\alpha t X_{2}, \quad x_{2}=X_{2}, \quad x_{3}=X_{3}
$$
where $\alpha>0$ is a constant. Show that $\varphi(\boldsymbol{X}, t)$ is volumepreserving.

Dr. Rajveer Singh

Numerade Educator