Problems
Problem 1.1
The components of vectors u, v, and w are given by \(u_{i}\), \(v_{i}\), \(w_{i}\). Verify that
$$\begin{aligned} \mathbf {u}\cdot \mathbf {v}&=u_{i}v_{i}, \\ \mathbf {u}\times \mathbf {v}&=\varepsilon _{ijk}\mathbf {e}_{i}u_{j}v_{k}, \\ \left( {\mathbf {u}\times \mathbf {v}}\right) \cdot \mathbf {w}&=\varepsilon _{ijk}u_{i}v_{j}w_{k}, \\ \left( {\mathbf {u}\times \mathbf {v}}\right) \cdot \mathbf {w}&=\mathbf {u} \cdot \left( {\mathbf {v}\times \mathbf {w}}\right) , \\ \left( {\mathbf {u}\times \mathbf {v}}\right) \times \mathbf {w}&=\left( { \mathbf {u}\cdot \mathbf {w}}\right) \mathbf {v}-\left( {\mathbf {v}\cdot \mathbf {w }}\right) \mathbf {u}, \\ \left( {\mathbf {u}\times \mathbf {v}}\right) ^{2}&=u^{2}v^{2}-\left( { \mathbf {u}\cdot \mathbf {v}}\right) ^{2}, \end{aligned}$$
where \(u^{2}=\left| \mathbf {u}\right| ^{2}\) and \(v^{2}=\left| \mathbf {v} \right| ^{2}.\)
Problem 1.2
Let A be a \(3\times 3\) matrix with entries \(A_{ij}\),
$$ \left[ \mathbf {A}\right] =\left[ { \begin{array}{*{20}c} {A_{11} } &{} {A_{12} } &{} {A_{13} } \\ {A_{21} } &{} {A_{22} } &{} {A_{23} } \\ {A_{31} } &{} {A_{32} } &{} {A_{33} } \\ \end{array}}\right] . $$
Verify that
$$\begin{aligned} \det \left[ \mathbf {A}\right]&=\varepsilon _{ijk}A_{1i}A_{2j}A_{3k}=\varepsilon _{ijk}A_{i1}A_{j2}A_{k3}, \\ \varepsilon _{lmn}\det \left[ \mathbf {A}\right]&=\varepsilon _{ijk}A_{il}A_{jm}A_{kn}=\varepsilon _{ijk}A_{li}A_{mj}A_{nk}, \\ \det \left[ \mathbf {A}\right]&=\frac{1}{6}\varepsilon _{ijk} \varepsilon _{lmn}A_{il}A_{jm}A_{kn}. \end{aligned}$$
Problem 1.3
Verify that
$$ \varepsilon _{ijk} \varepsilon _{imn} = \delta _{jm} \delta _{kn} - \delta _{jn} \delta _{km} . $$
Given that two \(3\times 3\) matrices of components
$$ \left[ \mathbf {A} \right] = \left[ { \begin{array}{*{20}c} {A_{11} } &{} {A_{12} } &{} {A_{13} } \\ {A_{21} } &{} {A_{22} } &{} {A_{23} } \\ {A_{31} } &{} {A_{32} } &{} {A_{33} } \\ \end{array}} \right] ,\;\;\left[ \mathbf {B} \right] = \left[ { \begin{array}{*{20}c} {B_{11} } &{} {B_{12} } &{} {B_{13} } \\ {B_{21} } &{} {B_{22} } &{} {B_{23} } \\ {B_{31} } &{} {B_{32} } &{} {B_{33} } \\ \end{array}} \right] $$
verify that if \(\left[ \mathbf {C} \right] = \left[ \mathbf {A} \right] \cdot \left[ \mathbf {B} \right] ,\) then the components of C are \(C_{ij} = A_{ik} B_{kj}.\) Thus if \(\left[ \mathbf {D} \right] = \left[ \mathbf {A} \right] ^{T} \left[ \mathbf {B} \right] ,\) then \(D_{ij} = A_{ki} B_{kj} .\)
Problem 1.4
Show that, if \(\left[ A_{ij}\right] \) is a frame rotation matrix,
$$\begin{aligned} \det \left[ {A_{ij}}\right]&=\left( {\mathbf {e^{\prime }}_{1}\times \mathbf { e^{\prime }}_{2}}\right) \cdot \mathbf {e^{\prime }}_{3}=1. \\ \left[ \mathbf {A}\right] ^{T}\left[ \mathbf {A}\right]&=\left[ \mathbf {A} \right] \left[ \mathbf {A}\right] ^{T}=\left[ \mathbf {I}\right] ,\;\;\left[ \mathbf {A}\right] ^{-1}=\left[ \mathbf {A}\right] ^{T},\;\;\det \left[ \mathbf { A}\right] =1. \end{aligned}$$
Problem 1.5
Verify that
$$ \varepsilon _{ijk}u_{i}v_{j}w_{k}=\det \left[ { \begin{array}{*{20}c} {u_1 } &{} {u_2 } &{} {u_3 } \\ {v_1 } &{} {v_2 } &{} {v_3 } \\ {w_1 } &{} {w_2 } &{} {w_3 } \\ \end{array}}\right] . $$
Consider a second-order tensor \(W_{ij}\) and a vector \(u_{i}=\varepsilon _{ijk}W_{jk}\). Show that if W is symmetric, u is zero, and if W is anti-symmetric the components of u are twice those of W in magnitude. This vector is said to be the axial vector of W.
Hence, show that the axial vector associated with the vorticity tensor of (1.14) is \(-\nabla \times \mathbf {u}\).
Problem 1.6
If \(\mathbf {D}\), \(\mathbf {S}\) and \(\mathbf {W}\) are second-order tensors, \( \mathbf {D}\) symmetric and \(\mathbf {W}\) anti-symmetric, show that
$$ \begin{array}{l} \mathbf {D}:\mathbf {S}=\mathbf {D}:\mathbf {S}^{T}=\mathbf {D}:\frac{1}{2}\left( {\mathbf {S}+\mathbf {S}^{T}}\right) , \\ \mathbf {W}:\mathbf {S}=-\mathbf {W}:\mathbf {S}^{T}=\mathbf {W}:\frac{1}{2} \left( {\mathbf {W}-\mathbf {W}^{T}}\right) , \\ \mathbf {D}:\mathbf {W}=0. \end{array} $$
Further, show that
$$ \begin{array}{l} \text {if }\mathbf {T}:\mathbf {S}=0\;\;\forall \mathbf {S}\text { then }\mathbf {T}=0, \\ \text {if }\mathbf {T}:\mathbf {S}=0\;\;\forall \text { symmetric }\mathbf {S}\text { then }\mathbf {T}\text { is anti-symmetric}, \\ \text {if }\mathbf {T}:\mathbf {S}=0\;\;\forall \text { anti-symmetric }\mathbf {S}\text { then }\mathbf {T}\text { is symmetric}\mathrm {.} \end{array} $$
Problem 1.7
Show that \(\mathbf {Q}\) is orthogonal if and only if \(\mathbf {H}=\mathbf {Q}- \mathbf {I}\) satisfies
$$ \mathbf {H}+\mathbf {H}^{T}+\mathbf {HH}^{T}=0,\;\;\;\;\mathbf {HH}^{T}=\mathbf {H }^{T}\mathbf {H}. $$
Problem 1.8
Show that, if \(\mathbf {S}\) is a second-order tensor, then \(I=\mathrm {tr}\ \mathbf {S}\), \(II=\mathrm {tr}\ \mathbf {S}^{2}\), \(III=\mathrm {\det \ }\mathbf {S }\) are indeed invariants. In addition, show that
$$ \det \left( {\mathbf {S}-\omega \mathbf {I}}\right) =-\omega ^{3}+I_{1}\omega ^{2}-I_{2}\omega +I_{3}. $$
If \(\omega \) is an eigenvalue of \(\mathbf {S}\) then \(\det \left( {\mathbf {S} -\omega \mathbf {I}}\right) =0.\) This is said to be the characteristic equation for \(\mathbf {S}\).
Problem 1.9
Apply the result above to find the square root of the Cauchy-Green tensor in a two-dimensional shear deformation
$$ \left[ \mathbf {C}\right] =\left[ { \begin{array}{*{20}c} {1 + \gamma ^2 } &{} \gamma \\ \gamma &{} 1 \\ \end{array}}\right] . $$
Investigate the corresponding formula for the square root of a symmetric positive definite tensor \(\mathbf {S}\) in three dimensions.
Problem 1.10
Write down all the components of the strain rate tensor and the vorticity tensor in a Cartesian frame.
Problem 1.11
Given that \(\mathbf {r}=x_{i}\mathbf {e}_{i}\) is the position vector, \(\mathbf { a}\) is a constant vector, and f(r) is a function of \(r=|\mathbf {r}|\), show that
$$ \nabla \cdot \mathbf {r}=3,\;\;\;\nabla \times \mathbf {r}=\mathbf {0} ,\;\;\;\nabla \left( {\mathbf {a}\cdot \mathbf {r}}\right) =\mathbf {a} ,\;\;\;\nabla f=\frac{1}{r}\frac{{df}}{{dr}}\mathbf {r}. $$
Problem 1.12
Show that the divergence of a second-order tensor \(\mathbf {S}\) in cylindrical coordinates is given by
$$\begin{aligned} \nabla \cdot \mathbf {S}&=\mathbf {e}_{r}\left( {\frac{{\partial S_{rr}}}{{ \partial r}}+\frac{{S_{rr}-S_{\theta \theta }}}{r}+\frac{1}{r}\frac{{ \partial S_{\theta r}}}{{\partial \theta }}+\frac{{\partial S_{zr}}}{{ \partial z}}}\right) \nonumber \\&+\mathbf {e}_{\theta }\left( {\frac{{\partial S_{r\theta }}}{{\partial r}}+ \frac{{2S_{r\theta }}}{r}+\frac{1}{r}\frac{{\partial S_{\theta \theta }}}{{ \partial \theta }}+\frac{{\partial S_{z\theta }}}{{\partial z}}+\frac{{ S_{\theta r}-S_{r\theta }}}{r}}\right) \nonumber \\&+\mathbf {e}_{z}\left( {\frac{{\partial S_{rz}}}{{\partial r}}+\frac{{S_{rz} }}{r}+\frac{1}{r}\frac{{\partial S_{\theta z}}}{{\partial \theta }}+\frac{{ \partial S_{zz}}}{{\partial z}}}\right) . \end{aligned}$$
(1.55)
Problem 1.13
Show that, in cylindrical coordinates, the Laplacian of a vector \(\mathbf {u}\) is given by
$$\begin{aligned} \nabla ^{2}\mathbf {u}&=\mathbf {e}_{r}\left[ {\frac{\partial }{{\partial r}} \left( {\frac{1}{r}\frac{\partial }{{\partial r}}\left( {ru_{r}}\right) } \right) +\frac{1}{{r^{2}}}\frac{{\partial ^{2}u_{r}}}{{\partial \theta ^{2}}} +\frac{{\partial ^{2}u_{r}}}{{\partial z^{2}}}-\frac{2}{{r^{2}}}\frac{{ \partial u_{\theta }}}{{\partial \theta }}}\right] \nonumber \\&+\mathbf {e}_{\theta }\left[ {\frac{\partial }{{\partial r}}\left( {\frac{1 }{r}\frac{\partial }{{\partial r}}\left( {ru_{\theta }}\right) }\right) + \frac{1}{{r^{2}}}\frac{{\partial ^{2}u_{\theta }}}{{\partial \theta ^{2}}}+ \frac{{\partial ^{2}u_{\theta }}}{{\partial z^{2}}}+\frac{2}{{r^{2}}}\frac{{ \partial u_{r}}}{{\partial \theta }}}\right] \nonumber \\&+\mathbf {e}_{z}\left[ {\frac{1}{r}\frac{\partial }{{\partial r}}\left( {r \frac{{\partial u_{z}}}{{\partial r}}}\right) +\frac{1}{{r^{2}}}\frac{{ \partial ^{2}u_{z}}}{{\partial \theta ^{2}}}+\frac{{\partial ^{2}u_{z}}}{{ \partial z^{2}}}}\right] . \end{aligned}$$
(1.56)
Problem 1.14
Show that, in cylindrical coordinates,
$$\begin{aligned} \mathbf {u}\cdot \nabla \mathbf {u}&=\mathbf {e}_{r}\left[ {u_{r}\frac{{ \partial u_{r}}}{{\partial r}}+\frac{{u_{\theta }}}{r}\frac{{\partial u_{r}}}{ {\partial \theta }}+u_{z}\frac{{\partial u_{r}}}{{\partial z}}-\frac{{ u_{\theta }u_{\theta }}}{r}}\right] \nonumber \\&+\mathbf {e}_{\theta }\left[ {u_{r}\frac{{\partial u_{\theta }}}{{\partial r}} +\frac{{u_{\theta }}}{r}\frac{{\partial u_{\theta }}}{{\partial \theta }}+u_{z} \frac{{\partial u_{\theta }}}{{\partial z}}+\frac{{u_{\theta }u_{r}}}{r}} \right] \nonumber \\&+\mathbf {e}_{z}\left[ {u_{r}\frac{{\partial u_{z}}}{{\partial r}}+\frac{{ u_{\theta }}}{r}\frac{{\partial u_{z}}}{{\partial \theta }}+u_{z}\frac{{ \partial u_{z}}}{{\partial z}}}\right] . \end{aligned}$$
(1.57)
Problem 1.15
The stress tensor in a material satisfies \(\nabla \cdot \mathbf {S}=\mathbf {0}\). Show that the volume-average stress in a region V occupied by the material is
$$\begin{aligned} \left\langle \mathbf {S}\right\rangle =\frac{1}{{2V}}\int _{S}{\left( {\mathbf { xt}+\mathbf {tx}}\right) \, dS}, \end{aligned}$$
(1.58)
where \(\mathbf {t}=\mathbf {n}\cdot \mathbf {S}\) is the surface traction. The quantity on the left side of (1.58) is called
the stresslet (Batchelor [4]).
Problem 1.16
Calculate the following integrals on the surface of the unit sphere
$$\begin{aligned} \left\langle {\mathbf {nn}} \right\rangle&= \frac{1}{S}\int _{S} {\mathbf {nn}dS} \end{aligned}$$
(1.59)
$$\begin{aligned} \left\langle {\mathbf {nnnn}} \right\rangle&= \frac{1}{S}\int _{S} {\mathbf { nnnn}dS} . \end{aligned}$$
(1.60)
These are the averages of various moments of a uniformly distributed unit vector on a sphere surface.
