Chapter 4

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) \(x_{1}=\exp \left(X_{1}\right), \quad x_{2}=-X_{3}, \quad x_{3}=X_{2}\).

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 \(\boldsymbol{F} v\).

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.

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 \(\boldsymbol{C} ?\)

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 \(\boldsymbol{U}\).

Let \(B=\left\\{\boldsymbol{X} \in \mathbb{E}^{3} \mid 00\) 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?

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) \(\varphi(\boldsymbol{X})=(\boldsymbol{I}+\epsilon \boldsymbol{A}) \boldsymbol{X}+\boldsymbol{c} .\)

Let \(B=\left\\{\boldsymbol{X} \in \mathbb{E}^{3} \mid 00\) 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)?

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 \(\boldsymbol{\psi}(\boldsymbol{\varphi}(\boldsymbol{X}, t), t)=\boldsymbol{X}\).

Let \(B=\left\\{\boldsymbol{X} \in \mathbb{E}^{3} \mid 00\) 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 \(t \in\left[0, t_{c}\right]\).

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)?

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 \(\omega\).

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 $$ \boldsymbol{L}(\boldsymbol{x}, t)=\frac{\partial}{\partial s} \widehat{\boldsymbol{U}}(\boldsymbol{x}, 0), \quad \boldsymbol{W}(\boldsymbol{x}, t)=\frac{\partial}{\partial s} \widehat{\boldsymbol{R}}(\boldsymbol{x}, 0). $$

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 $$ \boldsymbol{v}(\boldsymbol{x}, t)=\boldsymbol{\Omega}(t)(\boldsymbol{x}-\boldsymbol{c}(t))+\dot{\boldsymbol{c}}(t). $$

Let \(B=\left\\{\boldsymbol{X} \in \mathbb{E}^{3} \mid 00\) is a constant. Show that \(\varphi(\boldsymbol{X}, t)\) is volumepreserving.