Abstract
The balance laws discussed in the previous chapter represent general principles that all deformable bodies must satisfy. They do not distinguish between fluids and solids, and are equally applicable to all bodies. It also turns out that the number of equations found so far is insufficient for determining the deformations and stresses in an arbitrary deformable body.
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Notes
- 1.
Generally, \(d\mathscr {W}\) is not the total differential of any function. This is true only when the forces are conservative, i.e. \(\varvec{F}_{\text {tot}}=-\varvec{\nabla }U\) for some \(U=U(\varvec{r})\). In this case \(d\mathscr {W}=-dU\), and the right-hand side of this last equation is a true total differential.
- 2.
This result can be proved more rigorously: if \(\varvec{A}\), \(\varvec{B}\in {{\mathbf {\mathtt{{Sym}}}}}\) such that \(\varvec{A}\,{:}\,\varvec{C}=0\) and \(\varvec{B}\,{:}\,\varvec{C}=0\) for all \(\varvec{C}\in {{\mathbf {\mathtt{{Sym}}}}}\),  then \(\varvec{A}=\alpha \varvec{B}\), for some \(\alpha \in \mathbb {R}\).
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Coman, C.D. (2020). Constitutive Relationships. In: Continuum Mechanics and Linear Elasticity. Solid Mechanics and Its Applications, vol 238. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-1771-5_4
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