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Journal of Elasticity
Erschienen in: Journal of Elasticity 1/2018

05.02.2018

On the Uniqueness of Energy Minimizers in Finite Elasticity

verfasst von: Jeyabal Sivaloganathan, Scott J. Spector

Erschienen in: Journal of Elasticity | Ausgabe 1/2018

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Abstract

The uniqueness of absolute minimizers of the energy of a compressible, hyperelastic body subject to a variety of dead-load boundary conditions in two and three dimensions is herein considered. Hypotheses under which a given solution of the corresponding equilibrium equations is the unique absolute minimizer of the energy are obtained. The hypotheses involve uniform polyconvexity and pointwise bounds on derivatives of the stored-energy density when evaluated on the given equilibrium solution. In particular, an elementary proof of the uniqueness result of Fritz John (Commun. Pure Appl. Math. 25:617–634, 1972) is obtained for uniformly polyconvex stored-energy densities.
Vorheriger Artikel The Bending-Gradient Theory for Thick Plates: Existence and Uniqueness Results
Nächster Artikel Partial Constraint Singularities in Elastic Rods

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Fußnoten
1
Here \({\mathbf {N}}=(\mathbf{A},\mathbf{B},t)\) for some 3 by 3 matrices \(\mathbf{A}\) and \(\mathbf{B}\) and some scalar \(t>0\).
 
2
Our hypotheses on the stored-energy differs from that in [34]. See Remark 5.4.
 
3
For interesting examples of nonuniqueness for both compressible and incompressible materials see, e.g., [3, §9], [14, §5.8], [1, 8, 29‐31, 44, 45], and the references therein.
 
4
This assumption allows for a piecewise \(C^{1}\) boundary, for example, a rectangle.
 
5
In particular, the stored-energy density may therefore be piecewise continuous.
 
6
If \({\mathscr {S}}=\varnothing\), then \({\mathbf {u}}\in C^{2}({\mathscr {B}};{\mathbb {R}}^{n})\cap{\mathscr {A}}\) suffices.
 
7
In general, one can only prove that a minimizer is a weak solution of alternative forms of the equilibrium equations. See [5, Theorem 2.4] and the references therein. However, Lemma 2.9 shows that additional hypotheses may imply that a minimizer is in fact a weak equilibrium solution.
 
8
See, e.g., Del Piero and Rizzoni [19] and the references therein for results concerning weak relative minimizers in Elasticity. See Grabovsky and Mengesha [25, 26] for results concerning the relationship between such minimizers and strong relative minimizers, although not for Elasticity.
 
9
This terminology for (3.1) has previously been used in [52].
 
10
For \({\mathbf {K}}\in\operatorname{Lin}\) and \({\mathbf {M}}=({\mathbf {F}},{\mathbf {A}})\in{{\mathscr {E}}_{3}}\) we write \({\mathbf {K}}{\mathbf {M}}:=({\mathbf {K}}{\mathbf {F}},{\mathbf {K}}{\mathbf {A}})\).
 
11
The equality on the boundary is to be taken in the sense of trace.
 
12
We say that \({\mathbf {A}}\) is a point of convexity of the differentiable, real-valued function \({\mathbf {X}}\mapsto\phi({\mathbf {X}})\) whenever the graph of \(\phi\) is (globally) above its tangent plane at \({\mathbf {A}}\), i.e., \(\phi({\mathbf {X}})\ge\phi({\mathbf {A}}) + \nabla\phi({\mathbf {A}})\cdot({\mathbf {X}}-{\mathbf {A}})\) for all \({\mathbf {X}}\).
 
13
Zhang [63] instead assumes that \(W({\mathbf {F}})=a|{\mathbf {F}}|^{p}+b|\operatorname{cof}{\mathbf {F}}|^{q}+\varPhi({\mathbf {F}},\operatorname{cof}{\mathbf {F}},\det{\mathbf {F}})\) with \(a>0\), \(b>0\), \(p\ge2\), \(q\ge p/(p-1)\), and \(\varPhi\) convex.
 
14
We write \(\mathrm{Sym}_{n}\) for those \({\mathbf {H}}\in\mathrm{Lin}_{n}\) that satisfy \({\mathbf {H}}={\mathbf {H}}^{\mathrm {T}}\); \(\mathrm{Psym}_{n}\) denotes those \({\mathbf {H}}\in\mathrm{Sym}_{n}\) that are strictly positive definite.
 
15
The proof in [34, Eqns. (8)–(11)] is also not compatible with \(W({\mathbf {F}})=+\infty\) when \(\det{\mathbf {F}}=0\).
 
16
More generally, it is not difficult to show that the results in [34] are valid for any \(W\in C^{3}(\mathrm{Lin}_{n};{\mathbb {R}})\) with \({\mathbf {S}}({\mathbf {I}})=\mathbf{0}\) and \({\mathbb {C}}({\mathbf {I}})\) strongly elliptic, where \({\mathbb {C}}({\mathbf {F}}):=\partial{\mathbf {S}}/\partial{\mathbf {F}}\) here denotes the elasticity tensor. In particular, for a \(W\) that satisfies (5.13), \(\mu>0\) and \(\mu+\lambda>0\) suffice. See, e.g., [14, Theorem 4.10.2] for examples of polyconvex energies that are consistent with (5.13).
 
17
Blatz and Ko [11] showed that the experimental data of Bridgman [12], for certain solid rubbers, is compatible with (6.2)1 with \(h(t)= t^{-13.3} \); see p. 238 and equation (50) (with \(f=1\)) in [11].
 
18
A slight change in the boundary conditions at \(y=\pm L\) allows one to prove that buckled solutions exist. See [50] and also [39, Chap. 10].
 
19
The symbol ⨍ here denotes the integral of the indicated function divided by the area of the region of integration.
 
20
At an arbitrary minimizer this is not possible due to the Jacobian becoming negative with an additive variation. However, at a radial deformation this difficulty can be overcome. See, e.g., [4, §7.3], [51, pp. 133–135], or [56, Theorem 2.6.19] for details.
 
21
Modulo a rotation, \({\mathbf {I}}\) is in fact the unique absolute minimizer of \({\mathbf {F}}\mapsto W_{\pm}({\mathbf {F}})\).
 
22
For example, \(h'(t)=-{\omega}_{\mathrm {o}} t^{-m}\), \(m>0\), cf. footnote 17.
 
23
Recall that \(\operatorname{Psym}_{n}\) denotes the set of strictly positive-definite, symmetric \({\mathbf {C}}\in\operatorname{Lin}_{n}\).
 
24
The uniqueness of the absolute minimizer also requires a result of Ciarlet and Mardare [15, Theorem 2.1].
 
25
However, see [16] where the implicit function theorem is used to obtain existence for small data.
 
26
See, e.g., [14, p. 51]. Alternatively, one can derive (B.1) from the characteristic polynomial.
 
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Metadaten
Titel
On the Uniqueness of Energy Minimizers in Finite Elasticity
verfasst von
Jeyabal Sivaloganathan
Scott J. Spector
Publikationsdatum
05.02.2018
Verlag
Springer Netherlands
Erschienen in
Journal of Elasticity / Ausgabe 1/2018
Print ISSN: 0374-3535
Elektronische ISSN: 1573-2681
DOI
https://doi.org/10.1007/s10659-018-9671-8

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