In this appendix, we present a particular family of motions and temperatures, given by the following functions of space and time:

$\begin{array}{ll}\mathbf{x}\hfill & =\mathbf{\chi }\left(\mathbf{X},t\right)={\mathbf{X}}_0+\left[{\mathbf{F}}_0+\left(t-t_0\right)\mathbf{A}\right]\left(\mathbf{X}-{\mathbf{X}}_0\right),\hfill \\ \text{Θ}\hfill & =\widehat{\text{Θ}}\left(\mathbf{X},t\right)={\text{Θ}}_0+\left(t-t_0\right)a+\left[{\mathbf{g}}_0+\left(t-t_0\right)\mathbf{a}\right]·{\mathbf{F}}_0\left(\mathbf{X}-{\mathbf{X}}_0\right),\hfill \end{array}$

where t₀, Θ₀, and a are arbitrary constant scalars, X₀, g₀, and a are arbitrary constant vectors, and F₀ and A are arbitrary constant tensors. Loosely speaking, (D.1) gives a family of motions and temperatures with an infinite number of members. From (D.1), with the aid of results (2.99) and (3.22), we can explicitly calculate

$\begin{array}{ll}\mathbf{F}\hfill & =\widehat{\mathbf{F}}\left(\mathbf{X},t\right)=\frac{\partial \mathbf{\chi }\left(\mathbf{X},t\right)}{\partial \mathbf{X}}={\mathbf{F}}_0+\left(t-t_0\right)\mathbf{A},\hfill \\ \stackrel{.}{\mathbf{F}}\hfill & =\frac{\partial \widehat{\mathbf{F}}\left(\mathbf{X},t\right)}{\partial t}=\mathbf{A},\hfill \\ \mathbf{g}\hfill & =\widehat{\mathbf{g}}\left(\mathbf{X},t\right)={\mathbf{F}}^{-\text{T}}\frac{\partial \widehat{\text{Θ}}\left(\mathbf{X},t\right)}{\partial \mathbf{X}}={\left[{\mathbf{F}}_0+\left(t-t_0\right)\mathbf{A}\right]}^{-\text{T}}{\mathbf{F}}_0^T\left[{\mathbf{g}}_0+\left(t-t_0\right)\mathbf{a}\right],\hfill \\ \stackrel{.}{\mathbf{g}}\hfill & =\frac{\partial \widehat{\mathbf{g}}\left(\mathbf{X},t\right)}{\partial t}={\mathbf{A}}^{-\text{T}}{\mathbf{F}}_0^{\text{T}}\left[{\mathbf{g}}_0+\left(t-t_0\right)\mathbf{a}\right]+{\left[{\mathbf{F}}_0+\left(t-t_0\right)\mathbf{A}\right]}^{-\text{T}}{\mathbf{F}}_0^{\text{T}}\mathbf{a},\hfill \\ \stackrel{.}{\text{Θ}}\hfill & =\frac{\partial \widehat{\text{Θ}}\left(\mathbf{X},t\right)}{\partial t}=a+\mathbf{a}·{\mathbf{F}}_0\left(\mathbf{X}-{\mathbf{X}}_0\right).\hfill \end{array}$

At the particular place X = X₀ and time t = t₀, the above functions take the values

$\mathbf{x}=\mathbf{\chi }\left({\mathbf{X}}_0,t_0\right)={\mathbf{X}}_0,\text{Θ}=\widehat{\text{Θ}}\left({\mathbf{X}}_0,t_0\right)={\text{Θ}}_0,\mathbf{F}={\mathbf{F}}_0,$

and

$\stackrel{.}{\mathbf{F}}=\mathbf{A},\mathbf{g}={\mathbf{g}}_0,\stackrel{.}{\mathbf{g}}={\mathbf{A}}^{-\text{T}}{\mathbf{F}}_0^{\text{T}}{\mathbf{g}}_0+\mathbf{a},\stackrel{.}{\text{Θ}}=a.$

It follows that

$\mathbf{L}=\stackrel{.}{\mathbf{F}}{\mathbf{F}}^{-1}=\mathbf{A}{\mathbf{F}}_0^{-1},\mathbf{D}=\frac{1}{2}\left(\mathbf{L}+{\mathbf{L}}^{\text{T}}\right)=\frac{1}{2}\left(\mathbf{A}{\mathbf{F}}_0^{-1}+{\mathbf{F}}_0^{-\text{T}}{\mathbf{A}}^{\text{T}}\right)$

at place x = X = X₀ and time t = t₀.

Recall that t₀, Θ₀, a, X₀, g₀, a, F₀, and A are arbitrary quantities. We have therefore explicitly exhibited a family of thermomechanical processes (D.1) in which the temperature Θ, deformation gradient F, rate of deformation gradient $\stackrel{.}{\mathbf{F}}$ (or, alternatively, velocity gradient L or rate of deformation D), temperature gradient g, rate of temperature gradient $\stackrel{.}{\mathbf{g}}$, and rate of temperature $\stackrel{.}{\text{Θ}}$ can be chosen independently at any place and time.