A Causality Setting for Elasticity Theory

Abstract

We give a causality approach to nonlinear elasticity theory within a pure mechanics setting. Based on the notion of a fixed frame in and Euclidean invariance considerations, we show how a broadly general and novel evolutionary hypothesis concerning ‘cause’ and ‘effect’ must reduce to the classical statement of the balance of energy, and we obtain all of the classical balance laws of continuum mechanics. The concept of mass and its balance is derived within this theory. The mass density naturally emerges from the theory without preconception as an inertial scalar field for the body which is associated with the speed of its material particles and a measure of their kinetic behavior. Aside from the causality hypothesis and its invariance, the fundamental notions of body, motion, force and internal power are primitive.

Appendix

Here, we summarize several special properties of F-differentiable functionals, introduced in Sect. 4.1 and used throughout Sects. 4 and 5.

Lemma A.1

(Chain-rule) Let$\operatorname{\chi}_{t}$be a tame motion according to Definition 3.1on ℬ for all$t\in {\mathcal{I}}$. Let$\mathsf{R}_{\mathcal{P}}[\cdot , \cdot ]$be a F-differentiable functional for ℬ according to property (1) of Definition 4.1. Then, the following chain-rule holds:

$$ \frac{\mathrm{d}}{\mathrm{d}t}\mathsf{R}_{\mathcal{P}}\bigl[|\dot{\operatorname{\chi}}_{t}|^{2}, \nabla \operatorname{\chi}_{t}\bigr] = \delta _{1}\mathsf{R}_{\mathcal{P}}\bigl[| \dot{\operatorname{\chi}}_{t}|^{2}, \nabla \operatorname{\chi}_{t}|\, 2\dot{\operatorname{\chi}}_{t}\cdot \ddot{\operatorname{\chi}}_{t}\bigr] + \delta _{2}\mathsf{R}_{\mathcal{P}}\bigl[| \dot{\operatorname{\chi}}_{t}|^{2}, \nabla \operatorname{\chi}_{t}|\, \dot{ \overline{\nabla \operatorname{\chi}_{t}}}\,\bigr] $$

(A.1)

for all$t\in {\mathcal{I}}$.

Proof

Note that for a tame motion $\operatorname{\chi}_{t}$, it follows from Definition 3.1 that $\varphi _{t}:= |\dot{\operatorname{\chi}} _{t}|^{2} \in C^{+}({\mathcal{B}})\times C^{1}({\mathcal{I}})$ and $\boldsymbol{A}_{t}:= \nabla \operatorname{\chi}_{t} \in C^{\mathrm{Lin}^{+}}({\mathcal{B}}) \times C^{1}({\mathcal{I}})$ and that $\dot{\varphi }_{t}$ and $\dot{\boldsymbol{A}}_{t}$ are in $C({\mathcal{B}})$ for all $t\in {\mathcal{I}}$. Then, since $\mathsf{R}_{\mathcal{P}}[\cdot , \cdot ]$ is F-differentiable, we know that for all $\tau , t\in {\mathcal{I}}$

$$\begin{aligned} \mathsf{R}_{{\mathcal{P}}}[\varphi _{\tau }, \boldsymbol{A} _{\tau }] &= \mathsf{R}_{{\mathcal{P}}}[\varphi _{t} + \varphi _{\tau }- \varphi _{t}, \boldsymbol{A}_{t} + \boldsymbol{A}_{\tau }- \boldsymbol{A}_{t}] \\ & = \mathsf{R}_{{\mathcal{P}}}[\varphi _{t}, \boldsymbol{A}_{t}] + \delta _{1}\mathsf{R}_{{\mathcal{P}}}[\varphi _{t}, \boldsymbol{A}_{t}| \;\varphi _{\tau }- \varphi _{t}] + o\bigl(\|\varphi _{\tau }- \varphi _{t}\|\bigr) \\ & \quad {} + \delta _{2}\mathsf{R}_{{\mathcal{P}}}[\varphi _{t}, \boldsymbol{A} _{t}|\;\boldsymbol{A}_{\tau }- \boldsymbol{A}_{t}] + o\bigl(\| \boldsymbol{A}_{\tau }- \boldsymbol{A}_{t}\|\bigr). \end{aligned}$$

Thus, by computing

$$ \frac{\mathrm{d}}{\mathrm{d}t}\mathsf{R}_{{\mathcal{P}}}[\varphi _{t}, \boldsymbol{A} _{t}] = \lim_{\tau \rightarrow t}\frac{\mathsf{R}_{{\mathcal{P}}}[ \varphi _{t}, \boldsymbol{A}_{t}] - \mathsf{R}_{{\mathcal{P}}}[ \varphi _{\tau }, \boldsymbol{A}_{\tau }]}{t - \tau }, $$

we readily find

$$ \frac{\mathrm{d}}{\mathrm{d}t}\mathsf{R}_{{\mathcal{P}}}[\varphi _{t}, \boldsymbol{A} _{t}] = \delta _{1} \mathsf{R}_{\mathcal{P}}[\varphi _{t}, \boldsymbol{A}_{t}| \;\dot{\varphi }_{t}] + \delta _{2}\mathsf{R}_{ \mathcal{P}}[ \varphi _{t}, \boldsymbol{A}_{t}|\;\dot{\boldsymbol{A}} _{t}], $$

(A.2)

and by using the definitions of $\varphi _{t}$ and $\boldsymbol{A}_{t}$, we arrive at (A.1).  ■

Lemma A.2

(Mean-value theorem)

• (1) Let$\{\varphi ^{(1)}, \varphi ^{(2)}\} \in C^{+}({\mathcal{B}})$and$\boldsymbol{A}\in C^{\mathrm{Lin} ^{+}}({\mathcal{B}})$. Let$\mathsf{R}_{\mathcal{P}}[\cdot , \boldsymbol{A}]$be a F-differentiable functional for ℬ according to property (1) of Definition 4.1. Then, there exists$\bar{\varphi }\in C ^{+}({\mathcal{B}})$such that

  $$ \mathsf{R}_{{\mathcal{P}}}\bigl[\varphi ^{(1)}, \boldsymbol{A}\bigr] - \mathsf{R}_{{\mathcal{P}}}\bigl[\varphi ^{(2)}, \boldsymbol{A}\bigr] = \delta _{1}\mathsf{R}_{\mathcal{P}}\bigl[\bar{ \varphi }, \boldsymbol{A}|\, \varphi ^{(1)} - \varphi ^{(2)} \bigr]. $$

  (A.3)

• (2) Let$\{\boldsymbol{A}^{(1)}, \boldsymbol{A}^{(2)}\} \in C ^{\mathrm{Lin}^{+}}({\mathcal{B}})$be such that$\boldsymbol{A} ^{(2)} + t(\boldsymbol{A}^{(1)} - \boldsymbol{A}^{(2)}) \in C^{ \mathrm{Lin}^{+}}({\mathcal{B}})$for all$t\in [0, 1]$, and let$\varphi \in C^{+}({\mathcal{B}})$. Let$\mathsf{R}_{ \mathcal{P}}[\varphi , \cdot ]$be a F-differentiable functional for ℬ according to property (1) of Definition 4.1. Then, there exists$\bar{\boldsymbol{A}} \in C^{\mathrm{Lin}^{+}}({\mathcal{B}})$such that

  $$ \mathsf{R}_{{\mathcal{P}}}\bigl[\varphi , \boldsymbol{A}^{(1)} \bigr] - \mathsf{R}_{{\mathcal{P}}}\bigl[\varphi , \boldsymbol{A}^{(2)} \bigr] = \delta _{2}\mathsf{R}_{\mathcal{P}}\bigl[\varphi , \bar{ \boldsymbol{A}}|\, \boldsymbol{A}^{(1)} - \boldsymbol{A}^{(2)} \bigr]. $$

  (A.4)

Proof

To prove part (1), first note that $\varphi _{t}:= \varphi ^{(2)} + t(\varphi ^{(1)} - \varphi ^{(2)}) \in C^{+}({\mathcal{B}})$ for all $t \in [0,1]$. Now, define

$$ \omega (t):= \mathsf{R}_{{\mathcal{P}}}[\varphi _{t}, \boldsymbol{A}] \quad \forall t\in [0,1]. $$

Then, using the F-differentiability of $\mathsf{R}_{{\mathcal{P}}}[ \cdot , \boldsymbol{A}]$ in (4.3a), and (A.2) in the proof of Lemma A.1, we reach

$$ \dot{\omega }(t) = \delta _{1}\mathsf{R}_{{\mathcal{P}}}\bigl[\varphi _{t}, \boldsymbol{A}|\, \varphi ^{(1)} - \varphi ^{(2)}\bigr], $$

and the elementary mean value theorem for real-valued functions implies that

$$ \omega (1) - \omega (0) = \dot{\omega }(\bar{t}) $$

for some $\bar{t}\in [0,1]$, to complete the proof of (A.3).

To prove part (2), we first note that $\boldsymbol{A}_{t}:= \boldsymbol{A}^{(2)} + t(\boldsymbol{A}^{(1)} - \boldsymbol{A}^{(2)}) \in C^{\mathrm{Lin}^{+}}({\mathcal{B}})$ for all $t\in [0, 1]$. Then, defining

$$ \omega (t):= \mathsf{R}_{{\mathcal{P}}}[\varphi , \boldsymbol{A}_{t}] \quad \forall t\in [0,1], $$

and using the F-differentiability of $\mathsf{R}_{{\mathcal{P}}}[ \varphi , \cdot ]$ in (4.3b) we may follow an argument similar to that used above to reach (A.4).  ■

Remark A.1

Note that if $\{\varphi ^{(1)}, \varphi ^{(2)}\} \in C^{+}({\mathcal{B}}) \cap C^{1}({\mathcal{B}})$ in part (1) of Lemma A.2, then a similar argument shows that there exists a $\bar{\varphi } \in C^{+}({\mathcal{B}})\cap C^{1}({\mathcal{B}})$ for which (A.3) holds.

Note, also, that if $\boldsymbol{A}^{(1)}$ and $\boldsymbol{A}^{(2)}$ in part (2) of Lemma A.2 are gradients of vector-valued functions, then $\boldsymbol{A}_{t}$ is a gradient. This means that $\bar{\boldsymbol{A}}$ in (A.4), being equal to $\boldsymbol{A}_{\bar{t}}$ for some $\bar{t} \in [0, 1]$, also is a gradient. □

Lemma A.3

Let$\varphi \in C^{+}({\mathcal{B}})\cap C^{1}({\mathcal{B}})$and$\boldsymbol{A}\in C^{\mathrm{Lin}^{+}}({\mathcal{B}})$. Suppose that$\mathsf{R}_{\mathcal{P}}[\cdot , \cdot ]$is a F-differentiable functional for ℬ according to Definition4.1, and let

$$ \delta _{1}\mathsf{R}_{\mathcal{P}}[\varphi , \boldsymbol{A}|\, \psi ] = \delta _{1}\mathsf{R}_{\mathcal{P}}[0, \boldsymbol{A}| \; \psi ] $$

(A.5)

for all such$\varphi$and$\boldsymbol{A}$, and for all$\psi \in C^{1}({\mathcal{B}})$. Then,

$$ \mathsf{R}_{\mathcal{P}}[\varphi , \boldsymbol{A}] - \mathsf{R}_{\mathcal{P}}[0, \boldsymbol{A}] = \mathsf{L}_{ \mathcal{P}}[\boldsymbol{A}|\;\varphi ], $$

whereis continuous and F-differentiable, and$\mathsf{L}_{\mathcal{P}}[\boldsymbol{A}|\;\cdot ]$is linear for every$\boldsymbol{A}\in C^{\mathrm{Lin}^{+}}({\mathcal{B}})$.

Proof

In part (1) of Lemma A.2, set

$$ \varphi ^{(1)} = \varphi , \qquad \varphi ^{(2)} = 0. $$

Then, because of Remark A.1 following Lemma A.2, we know that there exists a $\bar{\varphi } \in C^{+}({\mathcal{B}})\cap C^{1}({\mathcal{B}})$ such that

$$ \mathsf{R}_{\mathcal{P}}[\varphi , \boldsymbol{A}] - \mathsf{R}_{ \mathcal{P}}[0, \boldsymbol{A}] = \delta _{1}\mathsf{R}_{\mathcal{P}}[\bar{ \varphi }, \boldsymbol{A}|\, \varphi ]. $$

This, together with the hypothesis (A.5), allows the conclusion

$$ \mathsf{R}_{\mathcal{P}}[\varphi , \boldsymbol{A}] - \mathsf{R}_{ \mathcal{P}}[0, \boldsymbol{A}] = \delta _{1}\mathsf{R}_{\mathcal{P}}[0, \boldsymbol{A}|\, \varphi ], $$

and by defining

$$ \mathsf{L}_{\mathcal{P}}[\boldsymbol{A}|\;\cdot ]:= \delta _{1} \mathsf{R}_{\mathcal{P}}[0, \boldsymbol{A}|\, \cdot ] $$

for every $\boldsymbol{A}\in C^{\mathrm{Lin}^{+}}({\mathcal{B}})$, and recalling the property (1) of F-differentiability in Definition 4.1, we complete this proof.  ■

Lemma A.4

Let$\varphi \in C^{+}({\mathcal{B}})$and let$\operatorname{\chi}$and$\boldsymbol{v}$be smooth fields over ℬ with$\nabla \operatorname{\chi}\in \mathrm{Lin}^{+}$. Supposeis a continuous and F-differentiable functional for ℬ according to Definition4.1, so that for every$\boldsymbol{A}\in C ^{\mathrm{Lin}^{+}}({\mathcal{B}})$there exists$\delta \mathsf{L}_{ \mathcal{P}}[\cdot |\, \cdot |\, \cdot ]$, linear in its second and third places, such that

$$ \mathsf{L}_{\mathcal{P}}[\boldsymbol{A}+ \boldsymbol{H}|\, \cdot ] = \mathsf{L}_{\mathcal{P}}[\boldsymbol{A}|\, \cdot ] + \delta \mathsf{L}_{\mathcal{P}}\bigl[\boldsymbol{A}|\, \cdot |\, \boldsymbol{H}\bigr] + o \bigl(\| \boldsymbol{H}\| \bigr) $$

(A.6)

for all$\boldsymbol{H}\in C^{\mathrm{Lin}}({\mathcal{B}})$with$\boldsymbol{A}+ \boldsymbol{H}\in C^{\mathrm{Lin}^{+}}({\mathcal{B}})$. Finally, suppose

$$ \delta \mathsf{L}_{\mathcal{P}}[\nabla \operatorname{\chi}|\, \varphi |\, \nabla \boldsymbol{v}] = 0 $$

(A.7)

for all such$\operatorname{\chi}$and$\boldsymbol{v}$. Then, there exists a linear functionalsuch that

$$ \mathsf{L}_{\mathcal{P}}[\nabla \operatorname{\chi}|\, \varphi ] = \mathsf{K}_{\mathcal{P}}[\varphi ]. $$

Proof

In part (2) of Lemma A.2, let us first identify $\mathsf{R}_{\mathcal{P}}[\varphi , \boldsymbol{A}]$ with $\mathsf{L}_{\mathcal{P}}[\boldsymbol{A}|\, \varphi ]$ and set

$$ \boldsymbol{A}^{(1)} = \nabla \operatorname{\chi}+ \nabla \boldsymbol{v},\qquad \boldsymbol{A}^{(2)} = \nabla \operatorname{\chi}. $$

Now, choose $\operatorname{\chi}(\boldsymbol{X}) = \boldsymbol{X}$ and $\boldsymbol{v}(\boldsymbol{X}) = (\lambda - 1)\boldsymbol{X}$, with $\lambda > 0$. In this case, we have $\boldsymbol{A}^{(1)} = \lambda \boldsymbol{\mathit{1}}$ and $\boldsymbol{A}^{(2)} = \boldsymbol{\mathit{1}}$, which implies that $\boldsymbol{A}_{t} = (1 + t(\lambda - 1) )\boldsymbol{\mathit{1}}$. Thus, $\boldsymbol{A}_{t}\in \mathrm{Lin}^{+}$ is the gradient of a vector-valued function for all $t\in [0,1]$, and we may use (A.4) (see also Remark A.1) to conclude that there exists $\bar{\boldsymbol{A}}\in C^{\mathrm{Lin}^{+}}({\mathcal{B}})$ which is a gradient and such that

$$ \mathsf{L}_{\mathcal{P}}[\lambda \boldsymbol{\mathit{1}}|\, \varphi ] - \mathsf{L}_{\mathcal{P}}[\boldsymbol{\mathit{1}}|\, \varphi ] = \delta \mathsf{L}_{\mathcal{P}}\bigl[\bar{\boldsymbol{A}}|\, \varphi |\, ( \lambda - 1) \boldsymbol{\mathit{1}}\bigr]. $$

But, because of (A.7), the right hand side here vanishes, which implies

$$ \mathsf{L}_{\mathcal{P}}[\lambda \boldsymbol{\mathit{1}}|\, \varphi ] = \mathsf{L}_{\mathcal{P}}[\boldsymbol{\mathit{1}}|\, \varphi ]. $$

As a second choice, suppose $\operatorname{\chi}$ is left unspecified with $\nabla \operatorname{\chi}\in {\mathrm{Lin}^{+}}$, and $\boldsymbol{v}( \boldsymbol{X}) = {\lambda \boldsymbol{X}- \operatorname{\chi}(\boldsymbol{X})}$, with $\lambda > 0$. In this case, we have $\boldsymbol{A}^{(1)} = \lambda \boldsymbol{\mathit{1}}$ and $\boldsymbol{A}^{(2)} = \nabla \operatorname{\chi}$, which implies that $\boldsymbol{A}_{t} = (1 - t)\nabla \operatorname{\chi}+ t \lambda \boldsymbol{\mathit{1}}$. Thus, for sufficiently small $\lambda > 0$, we see that $\boldsymbol{A}_{t}\in \mathrm{Lin}^{+}$, and is the gradient of a vector-valued function for all $t\in [0,1]$. Again, we may use (A.4) (see also Remark A.1) to conclude that there exists $\bar{\boldsymbol{A}}\in C^{\mathrm{Lin} ^{+}}({\mathcal{B}})$, the gradient of a vector-valued function, such that

$$ \mathsf{L}_{\mathcal{P}}[\lambda \boldsymbol{\mathit{1}}|\, \varphi ] - \mathsf{L}_{\mathcal{P}}[\nabla \operatorname{\chi}|\, \varphi ] = \delta \mathsf{L}_{\mathcal{P}}\bigl[\bar{\boldsymbol{A}}|\, \varphi |\, \lambda \boldsymbol{ \mathit{1}} - \nabla \operatorname{\chi}\bigr]. $$

But, again, the right hand side here vanishes because of (A.7), and we see that

$$ \mathsf{L}_{\mathcal{P}}[\lambda \boldsymbol{\mathit{1}}|\, \varphi ] = \mathsf{L}_{\mathcal{P}}[\nabla \operatorname{\chi}|\, \varphi ]. $$

From the above two choices, we take $\lambda > 0$, sufficiently small, and conclude that

$$ \mathsf{L}_{\mathcal{P}}[\nabla \operatorname{\chi}|\, \varphi ] = \mathsf{L}_{ \mathcal{P}}[ \boldsymbol{\mathit{1}}|\, \varphi ] =: \mathsf{K}_{ \mathcal{P}}[\varphi ], $$

to complete this proof.  ■