Abstract
In this chapter, we explain the necessity of introducing what we commonly call constitutive structures and indicate in an isothermal setting a mechanical dissipation inequality as a source of a priori constitutive restrictions. We focus attention essentially on the elastic behavior, and when there are large strains, we show the physical incompatibility between objectivity of the elastic energy and the convexity of the energy itself with respect to the deformation gradient. Then we discuss the elastic behavior in the small-strain regime. We present the notions of material isomorphism and material symmetries, indicating how they allow us to distinguish between simple solids and fluids. Among material symmetries, we discuss isotropy at length. We then include some digressions on viscous materials, showing how the introduction of the incompressibility internal constraint in this case leads to the Navier–Stokes equations.
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Notes
- 1.
The existence of the free energy can be determined in an abstract setting, without reference to a specific material class, by resorting to the notions of state, process, and action. We do not need to specify otherwise the state, except to declare that it is an element of a Hausdorff topological space. Paths in the state space represent state transformations. They are determined by processes, operators acting over the state space and representing ways in which the body “perceives” the external environment. Finally, actions are functionals assigning to every state transformation a number. Actions depend on the initial state of the path and the process. We commonly require that actions be continuous with respect to the states and additive with respect to the prolongations of paths by means of “subsequent” state transformations (the external power that we have defined previously is an action, for example). In this setting, the free energy is a concept associated with an action or a class of actions. It is a state function such that the difference of its values over an arbitrary pair of states is bounded by the infimum of the action considered, provided that the two states can be connected by a path. We do not go into details. We just mention these aspects for the sake of completeness and to open a window onto a wider landscape. For further details, the reader can refer to Miroslav Šilhavý’s treatise referenced at the end of this book as a suggestion for further reading. Here, we just assume the possibility of defining the free energy as an integral over the volume of a differentiable density.
- 2.
Notice that the choice \(P =\tilde{ P}(x,C)\) corresponds to \(S =\tilde{ S}(x,C)\).
- 3.
We exploit the identification of \(\mathbb{R}^{3}\) with its dual.
- 4.
The definition is different from that proposed in 1959 (and refined in 1972) by Noll. In fact, he calls a material element at x solid when its symmetry group is included in the full orthogonal group and not only its special subgroup of rotations. In other words, in the definition of solids, Noll includes reflections. Then he calls a material element having the full unimodular group as symmetry group fluid. In proposing such a definition, however, Noll does not impose the orientation-preserving nonlinear constraint \(\det F> 0\). In this way, he can select as changes in observers those involving the full orthogonal group. In other words, with y and y′ the actual placements of the same material element evaluated by two observers, \(\mathcal{O}\) and \(\mathcal{O}'\) respectively, which differ from each other by a time-parameterized family of isometries, we have always \(y' = y + w(t) + Q(t)(y - y_{0})\), with \(w(t) \in \mathbb{R}^{3}\), but now Q(t) ∈ O(3) instead of Q(t) belonging just to SO(3). In this setting, even the notion of objectivity changes, because it involves the full orthogonal group, i.e., reflections in addition to rotations.
- 5.
The definition is justified here by the exclusion of reflections due to the orientation-preserving constraint. In the absence of it, we usually call the tensor-valued functions satisfying (4.26) for all elements of O(3) isotropic.
- 6.
Recall that a second-rank tensor is called spherical when it is of the form αI, with α a scalar and I the unit second-rank tensor.
- 7.
The pressure appears also in different contexts. An example is nonviscous compressible fluids.
- 8.
Here μ has nothing to do with d μ, which denotes the volume measure.
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Mariano, P.M., Galano, L. (2015). Constitutive Structures: Basic Aspects. In: Fundamentals of the Mechanics of Solids. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-3133-0_4
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DOI: https://doi.org/10.1007/978-1-4939-3133-0_4
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