Abstract
We provide an approximation result for the pure traction problem of linearized elasticity in terms of local minimizers of finite elasticity, under the constraint of vanishing average curl for admissible deformation maps. When suitable rotations are included in the constraint, the limit is shown to be the linear elastic equilibrium associated to rotated loads.
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1 Introduction
We consider the pure traction problem of finite elasticity and its relation with the linear elastic problem. If \(\Omega \subset {\mathbb{R}}^{3}\) is the reference configuration of an elastic body and \(\mathbf {y}:\Omega \to \mathbb{R}^{3}\) is the deformation field, we introduce the global energy
where \(\mathbf {i}\) denotes the identity map on \(\Omega \) and \(\mathcal {W}:\Omega \times \mathbb{R}^{3\times 3}\to [0,+\infty ]\) is the strain energy density. Here, ℒ is the load functional whose typical form is
where \(\mathbf{f}\colon \Omega \to {\mathbb{R}}^{3}\) is a body force field, \(\mathbf{g}\colon \partial \Omega \to {\mathbb{R}}^{3}\) is a surface force field, \(\mathbf{v}=\mathbf {y}-\mathbf {i}\) is the displacement field and \(\mathit{dS}\) is the surface measure. For every \({\mathbf{x}}\in \Omega \), the function \(\mathcal {W}({\mathbf{x}},\cdot )\) is assumed to be frame indifferent, uniquely minimized at rotations with value 0, \(C^{2}\)-smooth and quadratically growing out of rotations; in addition, \(\mathcal {W}\) shall satisfy the natural condition
Letting \(h>0\) be an adimensional parameter, we shall consider the natural rescaling \(h^{-2}\mathcal {G}(\cdot ;h\mathcal {L})\). We also introduce the associated linear elastic energy
where \(\mathbb{E}\mathbf{u}:=\mathrm{sym}(\nabla \mathbf{u})\) is the infinitesimal strain tensor and \(\mathbf {I}\) is the identity matrix. This standard expression is formally obtained by linearization around the identity: by introducing the rescaled displacement \(\mathbf {u}=h^{-1}\mathbf {v}\), by writing the deformation field as \(\mathbf {i}+h\mathbf {u}\) and by considering the rescaled energies \(h^{-2}\mathcal {G}(\mathbf {i}+h\mathbf {u};h\mathcal {L})\), functional \(\mathcal {F}_{0}(\mathbf {u};\mathcal {L})\) is the pointwise limit obtained by performing a Taylor expansion for small \(h\).
Recently, several contributions [15, 18, 19, 21, 23] have analyzed the variational limit of the rescaled functionals \(h^{-2}\mathcal {G}(\cdot ;h\mathcal {L})\) as \(h\to 0\). For this purpose, it is necessary to assume that ℒ is equilibrated (i.e., with null resultant and null momentum) and that
where \(\mathit{SO}(3)\) denotes the special orthogonal group, thus preventing the rescaled energy being driven to \(-\infty \) by rigid motions as \(h\) becomes small. In contrast to the case of the Dirichlet problem [1, 9, 15, 22, 25], global minimizers (or quasi-minimizers) of \(h^{-2}\mathcal {G}(\cdot ;h\mathcal {L})\) over \(H^{1}(\Omega ,\mathbb{R}^{3})\) do not necessarily converge to minimizers of the associated linearized elastic energy \(\mathcal {F}_{0}(\cdot ;\mathcal {L})\), as might be expected from classical continuum mechanics literature, so it seems appropriate to shift the attention to the asymptotic behavior of suitably constrained minimizers in order to recover a minimizer of \(\mathcal {F}_{0}(\cdot ;\mathcal {L})\) in the limit. Indeed, the results in [15, 21, 23] show that, without further assumptions on the external loads, such as the absence of axes of equilibrium, there holds
for some suitable \(\mathbf {R}_{*}\in \mathit{SO}(3)\), and possibly
where \(\mathbf {R}_{*}\mathcal {L}\) is the notation for the rotated load functional, defined by \(\mathbf {R}_{*}\mathcal {L}(\cdot )= \mathcal {L}(\mathbf {R}_{*}^{T}\, \cdot )\).
Before introducing constrained approximations, let us mention that an alternative standard way for rigorously obtaining linearized elasticity is to consider the equilibrium problems associated to small loads. Equilibrium configurations of the (rescaled) traction problem for the equilibrated loads \((\mathbf {f},\mathbf {g})\) are given by the deformations \(\mathbf{y}\colon \Omega \to {\mathbb{R}}^{3}\) which solve
where \(\mathbf{n}\) denotes the exterior unit normal vector to \(\Omega \). Under suitable conditions on \(\mathbf{f}\) and \(\mathbf{g}\), classical results based on the implicit function theorem, described for instance in [7, 28, 29], allow to prove that there exists a solution \(\mathbf {y}_{h}\) to the above problem in a small neighborhood of the identity and that \(\mathbf{u}_{h}:=h^{-1}({\mathbf{y}_{h}-\mathbf{i}})\to \mathbf{u}_{0} \) in a suitable Sobolev space over \(\Omega \), where \(\mathbf{u}_{0}\) solves the linearized problem
that is, \(\mathbf{u}_{0}\) is a minimizer of the linearized elastic energy \(\mathcal {F}_{0}(\cdot ;\mathcal {L})\). A drawback of this scheme is that it works only if the external loads satisfy the following integrability condition
along with further nondegeneracy constraints (the simplest one being again the absence of axes of equilibrium) that we shall discuss in Sect. 3 along with a precise statement and some literature review.
In this paper, we would like to avoid the additional assumptions on external loads that are required both by the above classical approach and by the approximation with global minimizers of the rescaled functionals \(h^{-2}\mathcal {G}(\cdot ;h\mathcal {L})\). The approach we propose is to include a constraint on admissible deformation maps, without affecting the limit problem. The first step towards this goal is a suitable definition of local minimizer of functionals (1.1), which takes into account that the expected limit functional \(\mathcal {F}_{0}(\cdot ;\mathcal {L})\) can be restricted, without loss of generality, to those \(\mathbf{u}\in H^{1}(\Omega ,\mathbb{R}^{3})\) such that
provided that ℒ is equilibrated (as for the notation, we shall often omit \(d{\mathbf{x}}\) in volume integrals). Indeed, if \(\mathbf{u}\in H^{1}(\Omega ,{\mathbb{R}}^{3})\) and \({\mathbf{w}}:=|\Omega |^{-1}\int _{\Omega }\mathrm{curl}\,\mathbf{u}\), it is immediate to check that \(\mathcal {F}_{0}(\mathbf{u}-\tfrac{1}{2} {\mathbf{w}}\wedge {\mathbf{x}};\mathcal {L})= \mathcal {F}_{0}(\mathbf{u};\mathcal {L})\), where ∧ denotes the cross product. In the infinitesimal theory, the above constraint has a clear mechanical meaning: the infinitesimal rotation tensor \(\mathrm{skew}(\nabla \mathbf{u})\), whose axial vector is \(\tfrac{1}{2}\,\mathrm{curl}\,\mathbf{u}\), is vanishing in average. In this perspective we shall define constrained local minimizers \(\mathbf {y}\) of \(\mathcal {G}(\cdot ;\mathcal {L})\), the constraint being
by requiring that \(\mathcal {G}(\mathbf {y};\mathcal {L})\le \mathcal {G}(\mathbf {y}+ \varepsilon \boldsymbol{\psi};\mathcal {L})\) for every \(\boldsymbol{\psi}\) such that \(\int _{\Omega }\mathop{{\mathrm{curl}}}\nolimits \boldsymbol{\psi}= \mathbf {0}\) and for any small enough \(\varepsilon \) (we refer to Sect. 2 for the rigorous definition). In finite strain theory, the meaning of (1.7) is obtained from the polar decomposition. Letting \(\mathbf {H}:=|\Omega |^{-1}\int _{\Omega}\nabla \mathbf {y}\) be the average deformation gradient, and assuming that \(\mathbf {H}\) is positive definite, the polar decomposition of \(\mathbf {H}\) provides a unique \(\mathbf {R}\in \mathit{SO}(3)\) such that \(\mathbf {H}=\mathbf {R}\sqrt{\mathbf {H}^{T}\mathbf {H}} \). From the latter relation, and since \(\mathbf {H}\) is positive definite, we deduce that \(\mathbf {R}=\mathbf {I}\) if and only if \(\mathbf {H}^{T}=\mathbf {H}\), i.e., \(\int _{\Omega}\mathrm{skew}(\nabla \mathbf {y})=\mathbf {0}\). Therefore, (1.7) means that the rotation tensor associated with the average deformation gradient is the identity. Of course, (1.7) is equivalent to (1.6) if \(\mathbf {u}\) is the (rescaled) displacement associated to the deformation \(\mathbf {y}\), i.e., \(\mathbf {y}=\mathbf {i}+h\mathbf{u}\), and in this case \(|\Omega |^{-1}\int _{\Omega}\nabla \mathbf {y}\) is positive definite for small \(h\).
We notice that nontrivial rotations close to the identity are not admissible under the constraint (1.7), and more precisely a nontrivial rotation satisfies (1.7) if and only if the rotation angle is of 180 degrees (see Remark 2.3 below). In fact, since we deal with energetically stable configurations (local minimizers), let us mention that, in the stability theory of elastic equilibria, other constraints such as the zero moment condition by Beatty [3] (see also [10, 11]) have been introduced for preventing instabilities due to rigid rotations. The zero moment condition restricts admissible deformations to those \(\mathbf {y}\) that satisfy \(\int _{\Omega}\mathbf {f}\wedge \mathbf {y}\, d{\mathbf{x}} +\int _{\partial \Omega}\mathbf {g}\wedge \mathbf {y}\,\mathit{dS}= \mathbf {0}\). It is interesting to notice that in a simple problem of homogeneous normal boundary traction, the zero moment condition coincides with (1.7). Indeed, if \(\mathbf {f} \equiv \mathbf {0}\) and \(\mathbf {g}=\lambda \mathbf {n}\), where \(\lambda \in \mathbb{R}\) and \(\mathbf {n}\) is the exterior unit normal to \(\partial \Omega \), then \(\int _{\partial \Omega} \mathbf {g}\wedge \mathbf {y}\,\mathit{dS}= \lambda \int _{ \Omega}\mathrm{curl}\,\mathbf {y}\).
According to the above definition of constrained local minimizers, in our main result (see Theorem 2.6 in Sect. 2), under additional assumptions on \({\mathcal {W}}\), we show that if external loads are equilibrated then there exist constrained local minimizers \(\mathbf {y}_{h}\in H^{1}(\Omega ,\mathbb{R}^{3})\) for \(\mathcal {G}(\cdot ;h \mathcal {L})\) such that by letting \(\mathbf{u}_{h}:=h^{-1}(\mathbf {y}_{h}-\mathbf {i})\) there hold
as \(h\to 0\), where \(\mathbf{u}_{0}\) minimizes \(\mathcal {F}_{0}(\cdot ;\mathcal {L})\) over \(H^{1}(\Omega ,{\mathbb{R}}^{3})\).
Finally, by strengthening the hypotheses on external loads, namely by also assuming (1.3), an almost immediate consequence of the main result is that, given \(\mathbf {R}\in \mathit{SO}(3)\) belonging to the rotation kernel
for any small enough \(h\) there exists a constrained local minimizer \(\mathbf {y}_{h}\) for \(\mathcal {G}(\cdot ; h\mathcal {L})\) (this time the constraint for admissible deformations \(\mathbf {y}\) being \(\int _{\Omega }\mathop{{\mathrm{curl}}}\nolimits ( \mathbf {R}^{T}\mathbf {y})=\mathbf {0}\)), and
This, in a nutshell, is what might be considered the essence of the approximation of minimizers of linear elasticity with constrained local minimizers of finite elasticity: when condition (1.3) is satisfied then close to any \(\mathbf {R}\in \mathcal{S}^{0}_{\mathcal{L}}\) there exists a sequence of constrained local minimizers of functionals (1.1) such that the corresponding energies converge to the energy of the linearly elastic problem where ℒ is replaced by \(\mathbf {R}^{T}\mathcal {L}\). This further clarifies that global minimizers of (1.1) are not a good choice in order to approximate minimizers of the linear elastic energy \(\mathcal {F}_{0}(\cdot ;\mathcal {L})\).
Plan of the Paper
In Sect. 2 we introduce all the assumptions of the theory and rigorously state the main results, which are proved in Sect. 4. In Sect. 3 we give a brief comparison between our approach and the classical results about the asymptotic behavior of equilibrium states of traction problems via implicit function theorem. In Sect. 5 we revisit some examples of [21] by applying our convergence results of constrained local minimizers.
2 Main Results
In this section we introduce the basic notations and assumptions, then we state the main results. In the following, \(\Omega \) is a bounded open connected Lipschitz subset of \(\mathbb{R}^{3}\), representing the reference configuration of the body. \(\mathbb{R}^{3\times 3}\) is the set of real \(3\times 3\) matrices. \(\mathbb{R}^{3\times 3}_{\mathrm{sym}}\) (resp. \(\mathbb{R}^{3\times 3}_{\mathrm{skew}}\)) is the set of symmetric (resp. skew-symmetric) matrices. \(\mathbb{R}^{3\times 3}_{+}\) denotes the set of matrices with positive determinant.
Assumptions on the Elastic Energy Density
We let \(\mathcal {W} : \Omega \times \mathbb{R}^{3 \times 3} \to [0, +\infty ]\) be a Carathéodory function. We will consider the following assumptions.
Moreover, we shall consider the following regularity property: there exist an open neighborhood \(\mathcal {U}\) of \(\mathit{SO}(3)\) in \({\mathbb{R}}^{3\times 3}\), an increasing function \(\omega :\mathbb{R}_{+}\to \mathbb{R}\) satisfying \(\lim _{t\to 0^{+}}\omega (t)=0\) and a constant \(K>0\) such that for a.e. \({\mathbf{x}}\in \Omega \)
We introduce a growth condition from below: there exist \(C>0\) such that for a.e. \({\mathbf{x}}\in \Omega \)
where \(\mathrm{dist}(\mathbf {F}, \mathit{SO}(3)):=\inf \{|\mathbf {F}-\mathbf {R}|: \mathbf {R}\in \mathit{SO}(3)\}\) and \(|\mathbf {F}|^{2}:=\mathrm{Tr}(\mathbf {F}^{T}\mathbf {F})\). A second growth condition from below that we shall consider is the following: there exist \(C'>0\), \(\tilde{C}>0\), \(s\ge 2\), \(q\ge \tfrac{s}{s-1}\) and \(r>1\) such that for a.e. \({\mathbf{x}}\in \Omega \)
Finally, we introduce a polyconvexity condition: for a.e. \({\mathbf{x}}\in \Omega \)
Model Energy Densities
We present here two instances of energy densities \({\mathcal {W}}\) which satisfy the above assumptions and for which the main result of the present paper (see Theorem 2.6 below) applies. For simplicity, we consider the homogeneous case. Incompressible materials are usually modeled by isochoric-type energies \(\mathcal{W}_{\mathrm{iso}}\) defined on the set of matrices with unitary determinant \(\{\mathbf{F}\in {\mathbb{R}}^{3\times 3}\colon \det \mathbf{F}=1\}\) and one can pass to the corresponding compressible model by letting
where \(\mathcal{W}_{\mathrm{vol}}(\mathbf{F})=g(\det \mathbf{F})\) for some convex \(g\colon {\mathbb{R}}_{+}\to {\mathbb{R}}\) of class \(C^{2}\) in a neighborhood of 1 and such that
In addition, the function \(g\) is required to grow faster than linearly at infinity, i.e.
In view of the isochoric-volumetric decomposition (2.1), a model energy density is identified by the choice of \({\mathcal {W}}_{\mathrm{iso}}\), while \({\mathcal {W}}_{\mathrm{vol}}\) is left in a general form satisfying the above restrictions.
The first example is an energy of Yeoh type [30, 31], which is defined by choosing
with coefficients \(c_{k}> 0\). It is easy to check that with this choice the energy density satisfies all the assumptions from (\(\boldsymbol {\mathcal {W}1}\)) to (\(\boldsymbol {\mathcal {W}6}\)), provided that \(r\ge 2\) in (2.3). Indeed, the validity of (\(\boldsymbol {\mathcal {W}1}\)), (\(\boldsymbol {\mathcal {W}2}\)), (\(\boldsymbol {\mathcal {W}3}\)) is trivial. The validity of (\(\boldsymbol {\mathcal {W}6}\)) follows from the polyconvexity of the map \(\mathbb{R}^{3\times 3}_{+}\ni \mathbf {F}\mapsto \frac{|\mathbf {F}|^{2}}{(\det \mathbf {F})^{2/3}}\). The inequality in (\(\boldsymbol {\mathcal {W}4}\)) is satisfied if \(\mathrm{dist}(\mathbf {F},\mathit{SO}(3))\) is small enough, as shown in [1, Remark 2.8], and therefore (\(\boldsymbol {\mathcal {W}4}\)) directly follows from the following claim: there are positive constants \(a_{1}\), \(a_{2}\) such that \(\mathcal {W}(\mathbf {F})\ge a_{1}|\mathbf {F}|^{3}-a_{2}\) for every \(\mathbf {F}\in \mathbb{R}^{3\times 3}\). In order to prove such a claim, it is clear that we can reduce to consider the regime \(\mathrm{dist}(\mathbf {F},\mathit{SO}(3))\gg 1\) (i.e., \(|\mathbf {F}|\gg 1\)). By (2.3) there is \(t_{0}>1\) such that \(g(t)\ge C'' t^{r}\) for \(t\ge t_{0}\) and the claim is obvious for \(\det \mathbf {F}< t_{0}\) so that we may assume \(\det \mathbf {F}\ge t_{0}\). If \(|\mathbf {F}|\le (\det \mathbf {F})^{2/3}\) we have \(g(\det \mathbf {F})\ge C''(\det \mathbf {F})^{r}\ge C''|\mathbf {F}|^{3r/2}\) and the claim follows since \(r\ge 2\); else if \(|\mathbf {F}|\ge (\det \mathbf {F})^{2/3}\) then \(|\mathbf {F}|^{2}/(\det \mathbf {F})^{2/3}\) is large and Young inequality entails the existence of suitable positive constants \(c_{3}'\), \(c_{3}''\) such that
The claim is proved since \(r\ge 2\), and it implies (\(\boldsymbol {\mathcal {W}5}\)) by means of the elementary inequality \(|\mathrm{cof}\,\mathbf {F}|\le 2|\mathbf {F}|^{2}\).
A second example of energy density for hyperelastic materials is given by Ogden type energies, see for instance [7, Chap. 4], identified by the following choice
where \(n\), \(m\) are positive integers and \(c_{i},\gamma _{i},d_{j},\delta _{j}> 0\). While hypotheses (\(\boldsymbol {\mathcal {W}1}\)), (\(\boldsymbol {\mathcal {W}2}\)) and (\(\boldsymbol {\mathcal {W}3}\)) are again easily seen to be fulfilled by this model, some care must be taken concerning the remaining assumptions. In particular, one can prove that (\(\boldsymbol {\mathcal {W}5}\)) and (\(\boldsymbol {\mathcal {W}6}\)) hold true, after suitably restricting the values of the exponents \(\gamma _{i}\) and \(\delta _{j}\). More precisely, in order to obtain polyconvexity of the function \(\mathcal{W}\) (i.e. assumption (\(\boldsymbol {\mathcal {W}6}\))), one shall assume that
Moreover, if we let \(\gamma :=\max \{\gamma _{i}\colon i=1,\dots ,n\}\) and \(\delta :=\max \{\delta _{j}\colon j=1,\dots ,m\}\), we have that (\(\boldsymbol {\mathcal {W}5}\)) holds with \(r>1\) as in (2.3) together with
provided \(s\geq 2\) and \(q\geq s/(s-1)\). We point out that \(\delta \geq 3\) and \(r>1\) imply \(q>1\). For instance, (\(\boldsymbol {\mathcal {W}5}\)) is satisfied if \(\gamma \geq 5/2\) and \(r\ge 4\) The proof of (\(\boldsymbol {\mathcal {W}5}\)) and (\(\boldsymbol {\mathcal {W}6}\)) can be found in [4, Proposition 6, Proposition 7]. Finally, for the proof of (\(\boldsymbol {\mathcal {W}4}\)) one can see [1, Remark 2.8] and no further restrictions on \(\gamma _{i}\) and \(\delta _{j}\) are needed.
Energy Functionals
Let \(\mathcal {W}\) satisfy assumptions (\(\boldsymbol {\mathcal {W}1}\)), (\(\boldsymbol {\mathcal {W}2}\)), (\(\boldsymbol {\mathcal {W}3}\)) and (\(\boldsymbol {\mathcal {W}4}\)). We denote by \((H^{1}(\Omega ,\mathbb{R}^{3}))^{*}\) the dual of the Sobolev space \(H^{1}(\Omega ,\mathbb{R}^{3})\). Given \(\mathcal {L}\in (H^{1}(\Omega ,\mathbb{R}^{3}))^{*}\), we introduce the following energy functionals. We let \(\mathcal {G}(\cdot ;\mathcal {L}):H^{1}(\Omega ,\mathbb{R}^{3})\to (-\infty ,+\infty ]\) be defined by
and for \(\mathbf{u}\in H^{1}(\Omega ,\mathbb{R}^{3})\) we let
We define \(\mathcal {F}_{0}:H^{1}(\Omega ,\mathbb{R}^{3})\to (-\infty ,+\infty )\) as
If \(h> 0\) we also define the rescaled functionals
We refer to ℒ as the load functional. Thanks to the Sobolev embedding \(H^{1}(\Omega ,\mathbb{R}^{3})\hookrightarrow L^{6}(\Omega ,\mathbb{R}^{3})\), it can be always written as
for suitable \(\mathbf {f}_{*}\in L^{\frac{6}{5}}(\Omega ,\mathbb{R}^{3})\), \(\mathbf {G}_{*} \in L^{2}(\Omega ,\mathbb{R}^{3\times 3})\). We assume that ℒ is equilibrated, i.e.,
We denote with \(\|\mathcal {L}\|_{*}\) its norm in the space \((H^{1}(\Omega ,\mathbb{R}^{3}))^{*}\). By (2.8) and by Korn and Poincaré inequalities, there exists a constant \(K_{\Omega}>0\) such that
for every \(\mathbf{u}\in H^{1}(\Omega ,\mathbb{R}^{3})\).
Concerning the elastic energy, the following rigidity inequality is crucial, see [1, 12, 13]: there exists a constant \(\tilde{C}_{\Omega }\) such that for every \(\mathbf {y} \in H^{1}(\Omega ,\mathbb{R}^{3} )\) there is \(\mathbf {R}\in \mathit{SO}(3)\) (depending on \(\mathbf {y}\)) such that
For every \(\mathbf {y}\in H^{1}(\Omega ,\mathbb{R}^{3})\), we combine the latter with (\(\boldsymbol {\mathcal {W}4}\)) and obtain the existence of \(\mathbf {R} \in \mathit{SO}(3)\) (depending on \(\mathbf {y}\)) such that
where \(C_{\Omega }:=\tilde{C}_{\Omega }/C\) and \(C\) is the constant in (\(\boldsymbol {\mathcal {W}4}\)).
Main Results
We state the main result, showing that there exist suitably constrained local minimizers of the rescaled functionals (2.7) that converge to a global minimizer of the linear elastic energy (2.6) as \(h\to 0\). We need the precise notion of local minimizer under the constraint of vanishing average curl. In the next definition we make use of the following notation: for \(\mathbf {R}\in \mathit{SO}(3)\), we let
In particular, \(H^{1}_{\mathbf {I}}(\Omega ,\mathbb{R}^{3})\) is the linear subspace of \(H^{1}(\Omega ,\mathbb{R}^{3})\) made of vector fields with vanishing average curl.
Definition 2.1
constrained local minimizer
Let \(\mathcal {W}\) satisfy assumptions (\(\boldsymbol {\mathcal {W}1}\)), (\(\boldsymbol {\mathcal {W}2}\)), (\(\boldsymbol {\mathcal {W}3}\)) and (\(\boldsymbol {\mathcal {W}4}\)) and let \(\mathbf {R}\in \mathit{SO}(3)\). We say that \(\mathbf {y}\in H^{1}_{\mathbf {R}}(\Omega ,\mathbb{R}^{3})\) is a local minimizer for \(\mathcal {G}(\cdot ; \mathcal {L})\) over \(H^{1}_{\mathbf {R}}(\Omega ,\mathbb{R}^{3})\) if for every \(\boldsymbol{\psi}\in H^{1}_{\mathbf {R}}(\Omega ,\mathbb{R}^{3})\) there exists \(\varepsilon _{0}=\varepsilon _{0}(\mathbf {y},\boldsymbol{\psi})>0\) such that for every \(\varepsilon \in [0,\varepsilon _{0}]\) there holds \(\mathcal {G}(\mathbf {y};\mathcal {L})\le \mathcal {G}(\mathbf {y}+ \varepsilon \boldsymbol{\psi}; \mathcal {L})\). We say that \(\mathbf{u}\in H^{1}_{\mathbf {R}}(\Omega ,\mathbb{R}^{3})\) is a local minimizer for \(\mathcal {F}(\cdot ; \mathcal {L})\) over \(H^{1}_{\mathbf {R}}(\Omega ,\mathbb{R}^{3})\) if \(\mathbf {y}({\mathbf{x}})=\mathbf {R}{\mathbf{x}}+\mathbf{u}\) is a local minimizer for \(\mathcal {G}(\cdot ;\mathcal {L})\) over \(H^{1}_{\mathbf {R}}(\Omega ,\mathbb{R}^{3})\).
Remark 2.2
If \(\mathbf {y}({\mathbf{x}})=\mathbf {R}{\mathbf{x}}+\mathbf{u}\) we have \(\mathrm{curl}( \mathbf {R}^{T}\mathbf {y})= \mathrm{curl}(\mathbf {R}^{T}\mathbf{u})\). In the distinguished case \(\mathbf {R}=\mathbf {I}\) we have \(\mathrm{curl}\, \mathbf {y}=\mathrm{curl}\,\mathbf{u}\) and in particular \(\mathbf {i}+\mathbf {u}\in H^{1}_{\mathbf {I}}(\Omega ,\mathbb{R}^{3})\) if and only if \(\mathbf{u}\in H^{1}_{\mathbf {I}}(\Omega ,\mathbb{R}^{3})\).
Remark 2.3
If \(\mathbf {y}({\mathbf{x}})= {\mathbf {R}}_{*}{\mathbf{x}}\) for some \(\mathbf {R}_{*}\in \mathit{SO}(3)\), the condition \(\int _{\Omega }\mathrm{curl}\,\mathbf {y}=\mathbf {0}\) reads \(\mathrm{skew}(\mathbf {R}_{*})=\mathbf {0}\) and it is satisfied if and only if either \(\mathbf {R}_{*}=\mathbf {I}\) or the rotation is of 180 degrees. In fact, by the classical Euler-Rodrigues formula yielding the representation
for some suitable \(\theta \in \mathbb{R}\) and \(\mathbf {W}\in \mathbb{R}^{3\times 3}_{\mathrm{skew}}\) with \(|\mathbf {W}|^{2}=|\mathbf {W}^{2}|^{2}=2\), we see that the condition \(\mathrm{skew}(\mathbf {R}_{*})=\mathbf {0}\) is satisfied if and only if the rotation angle is either \(\theta =0\) or \(\theta =\pi \) (and if \(\theta =\pi \) we have \(|\mathbf {R}_{*}-\mathbf {I} |=2\sqrt {2}\)).
Remark 2.4
Given \(h>0\), \(\mathcal {F}_{h}\) is defined by (2.7). By rescaling \(\varepsilon _{0}\) we notice that \(\mathbf{u}\in H^{1}_{\mathbf {I}}(\Omega ,\mathbb{R}^{3})\) is a local minimizer for \(\mathcal {F}_{h}( \cdot ;\mathcal {L})\) over \(H^{1}_{\mathbf {I}}(\Omega ,\mathbb{R}^{3})\) if and only if \({\mathbf{v}}:=h\mathbf{u}\) is a local minimizer for \(\mathcal {F}(\cdot ;h\mathcal {L})\) over \(H^{1}_{\mathbf {I}}(\Omega ,\mathbb{R}^{3})\).
Remark 2.5
By applying the classical results of [2], if \(\mathcal {W}\) satisfies (\(\boldsymbol {\mathcal {W}1}\)), (\(\boldsymbol {\mathcal {W}2}\)), (\(\boldsymbol {\mathcal {W}3}\)), (\(\boldsymbol {\mathcal {W}4}\)), (\(\boldsymbol {\mathcal {W}5}\)), (\(\boldsymbol {\mathcal {W}6}\)) and if \(\mathcal {L}\in (H^{1}(\Omega ,{\mathbb{R}}^{3}))^{*}\) satisfies (2.8), then a global minimizer of \(\mathcal {F}(\cdot ;\mathcal {L})\) over \(H^{1}(\Omega ,{\mathbb{R}}^{3})\) does exist. Nevertheless, it has been recently shown in [21] that following a sequence of global minimizers for the rescaled functionals \(\mathcal {F}_{h}(\cdot ;\mathcal {L})\), the limit energy as \(h\to 0\) is not necessarily equal to \(\min _{H^{1}(\Omega ,\mathbb{R}^{3})}\mathcal {F}_{0}(\cdot ;\mathcal {L})\). Indeed, in a pure traction problem, this limit can be strictly lower than the minimal value of \(\mathcal {F}_{0}(\cdot ;\mathcal {L})\) over \(H^{1}(\Omega ,\mathbb{R}^{3})\). This motivates the analysis of constrained local minimizers for obtaining the linear elastic energy as limit of rescaled finite elasticity energies.
We are in a position to state our main result.
Theorem 2.6
Suppose that (\(\boldsymbol {\mathcal {W}1}\)), (\(\boldsymbol {\mathcal {W}2}\)), (\(\boldsymbol {\mathcal {W}3}\)), (\(\boldsymbol {\mathcal {W}4}\)), (\(\boldsymbol {\mathcal {W}5}\)), (\(\boldsymbol {\mathcal {W}6}\)) hold true. Let \(\mathcal {L} \in (H^{1}(\Omega ,\mathbb{R}^{3}))^{*}\) satisfy (2.8). Then there exist a vanishing sequence \((h_{j})_{j\in \mathbb{N}}\subset (0,1)\) and a sequence \((\mathbf{u}_{j})_{j\in \mathbb{N}}\subset H^{1}_{\mathbf {I}}(\Omega , \mathbb{R}^{3})\) such that \(\mathbf{u}_{j}\) is a local minimizer for \(\mathcal {F}_{h_{j}}(\cdot ;\mathcal {L})\) over \(H^{1}_{\mathbf {I}}(\Omega ,\mathbb{R}^{3}) \) for any \(j\in \mathbb{N}\) according to Definition 2.1, and moreover
where \(\mathbf{u}_{*}\) is a minimizer for \(\mathcal {F}_{0}(\cdot ;\mathcal {L})\) over \(H^{1}(\Omega ,\mathbb{R}^{3})\).
The above Theorem 2.6 expresses the fact that the minimizer of the linear elastic energy \(\mathcal {F}_{0}(\cdot ;\mathcal {L})\) (which is unique up to infinitesimal rigid displacements) gets approximated by a sequence of constrained local minimizers of rescaled finite elasticity functionals. We stress that the vanishing average curl constraint disappears in the limit problem, since the stored linear elastic energy and the load functional are invariant by infinitesimal rigid displacements, due to (2.8). Since only local minimizers are involved, the statement requires (2.8) and no further assumptions on the external loads. In particular, assumption (1.3), appearing in [21] for the analysis of the asymptotic behavior of global minimizers (or quasi-minimizers) of rescaled finite elasticity functionals, is not required. Similarly, assumptions about axes of equilibrium of external loads (see Sect. 3) are not required as well.
However, if besides (2.8) we additionally assume (1.3), we can draw some interesting consequences of Theorem 2.6. Under assumption (1.3), it is not difficult to see that the rotation kernel of ℒ, defined by (1.8), is a subgroup of \(\mathit{SO}(3)\), see [22, Remark 2.2]. \(\mathcal{S}^{0}_{\mathcal{L}}\) contains the identity matrix and it is possibly reduced to it. Moreover, if \(\mathbf {R}\in \mathcal{S}^{0}_{\mathcal{L}}\), then for every \(\mathbf {S}\in \mathit{SO}(3)\) we get
that is, \(\mathbf {R}^{T}\mathcal {L}\) still satisfies (1.3) (hence it also satisfies (2.8), see [22, Remark 2.1]) for every \(\mathbf {R}\in \mathcal{S}^{0}_{\mathcal{L}}\). Therefore the map \(\beta : \mathcal{S}^{0}_{\mathcal{L}}\to \mathbb{R}\)
is well defined and continuous. We notice that the energy identity satisfied by solutions to the minimization problem (2.11) implies that \(\beta (\mathbf {R})=-\tfrac{1}{2}\mathcal {B}(\mathbf {R}^{T}\mathcal {L}, \mathbf {R}^{T}\mathcal {L})\), where \(\mathcal {B}(\cdot ,\cdot )\) is the Betti form associated to couples of equilibrated loads, defined by
Here, \(\mathbf{u}_{i}\) are solutions to the linear elastic problem with external loads \(\mathcal {L}_{i}\), \(i=1,2\). The degeneracy properties of the Betti form are relevant in the analysis of the nonlinear elastic equations for pure traction problems, see [6]. A nondegeneracy condition on \(\beta \) also appears in the following results which are straightforward consequences of Theorem 2.6.
Corollary 2.7
Assume that (\(\boldsymbol {\mathcal {W}1}\)), (\(\boldsymbol {\mathcal {W}2}\)), (\(\boldsymbol {\mathcal {W}3}\)), (\(\boldsymbol {\mathcal {W}4}\)), (\(\boldsymbol {\mathcal {W}5}\)), (\(\boldsymbol {\mathcal {W}6}\)) hold true and that \(\mathcal {L} \in (H^{1}(\Omega ,\mathbb{R}^{3}))^{*}\) satisfies (2.8) and (1.3) Let \(\mathbf {R}\in \mathcal {S}^{0}_{\mathcal {L}}\). Then there exist a vanishing sequence \((h_{j})_{j\in \mathbb{N}}\subset (0,1)\) and a sequence \((\mathbf {y}_{j})_{j\in \mathbb{N}}\subset H^{1}_{\mathbf {R}}(\Omega , \mathbb{R}^{3})\) such that \(\mathbf {y}_{j}\) is a local minimizer for \(\mathcal {G}(\cdot ; h_{j}\mathcal {L})\) over \(H^{1}_{\mathbf {R}}(\Omega :\mathbb{R}^{3})\) for every \(j\in \mathbb{N}\) according to Definition 2.1and
Corollary 2.8
In the same assumptions of Corollary 2.7, suppose that \(\mathcal{S}^{0}_{\mathcal{L}}\setminus \{\mathbf {I}\}\neq \emptyset \) and that \(\beta \) from (2.11) is not a constant map. Then for every \(n\in \mathbb{N}\) there exist \(\{\mathbf {R}_{k}:\, k=1,2...n\}\subset \mathcal {S}^{0}_{\mathcal {L}}\) and \(h_{0}=h_{0}(n)> 0\) such that for every \(h\in (0,h_{0})\) the functional \(\mathcal {G}(\cdot \,; h\mathcal {L})\) has a local minimizer \(\mathbf {y}_{k}\) over \(H^{1}_{\mathbf {R}_{k}}(\Omega ,\mathbb{R}^{3})\) for every \(k=1,2...n\) (according to Definition 2.1) and \(\mathcal {G}(\mathbf {y}_{k}; h\mathcal {L})\neq \mathcal {G}(\mathbf {y}_{k'}; h\mathcal {L})\) if \(k\neq k'\).
The last corollary shows that the nonlinear elastic energy might have several constrained local minimizers (each corresponding to a different constraint) at different energy levels. The condition requiring that \(\beta \) is not a constant map is verified for instance in the example from [21, Theorem 2.7], which we will further discuss in Sect. 5.
3 The Classical Approach via Implicit Function Theorem
In this section we briefly review some known results concerning existence and asymptotic behavior of equilibrium configurations of hyperelastic bodies, that make use of a perturbative method which relies on the implicit function theorem. This is done for the sake of a comparison with the assumptions of our result in Theorem 2.6. Indeed, the classical results that we are going to describe hereafter require stronger summability and nondegeneracy assumptions on external loads, as well as more regularity of \(\mathcal {W}\) (in turn, they do not need polyconvexity).
Throughout this section, we assume \(\Omega \subset {\mathbb{R}}^{3}\) to be of class \(C^{2}\), and besides (\(\boldsymbol {\mathcal {W}1}\)) and (\(\boldsymbol {\mathcal {W}2}\)) we shall consider the following additional regularity hypothesis on the strain energy density
along with the coercivity condition
Given a couple of external (dead) loads \(\mathbf{f}\colon \Omega \to {\mathbb{R}}^{3}\) and \(\mathbf{g}\colon \partial \Omega \to {\mathbb{R}}^{3}\) acting on the elastic body \(\Omega \) as forces of body and surface type, respectively, equilibrium configurations are given by the deformations \(\mathbf{y}\colon \Omega \to {\mathbb{R}}^{3}\) which solve
In view of the above assumptions on \({\mathcal {W}}\), it is trivial to observe that, if there are no external forces, i.e. \(\mathbf {f}=\mathbf {g}=\mathbf {0}\), then the identity map \(\mathbf{i}\) solves (3.1), as well as the map \(\mathbf {y}({\mathbf{x}})=\mathbf {R}{\mathbf{x}}+\mathbf {c}\), for every \(\mathbf {R}\in \mathit{SO}(3)\) and every \(\mathbf {c}\in {\mathbb{R}}^{3}\). Therefore one may expect that problem (3.1) admits a solution when the couple \((\mathbf{f},\mathbf{g})\) consists of a small perturbation of \((\mathbf{0},\mathbf{0})\) and that the gradient of such solution is close, in some suitable sense, to the identity matrix (or at least to a rotation). This approach is based on a careful use of the implicit function theorem and it has been pursued for the first time in the works by Stoppelli [26, 27], where loads are of the form \((h\mathbf{f},h\mathbf{g})\), for some fixed \((\mathbf{f},\mathbf{g})\) and small parameter \(h>0\). This approach for traction problems of the type (3.1) has been further developed by several authors, see, e.g., [5, 6, 16, 17]. We also mention the comprehensive book [29] for a thorough description and [7, Sect. 6.7] for an exhaustive exposition of the literature concerning this topic (see also [8, 28]).
We now outline the results, starting by fixing the functional framework. We consider the Nemitsky operators associated with \(-\mathrm{div}\,(D{\mathcal {W}}({\mathbf{x}},\nabla \mathbf{y}({\mathbf{x}})))\) and \(D{\mathcal {W}}({\mathbf{x}},\nabla \mathbf{y}({\mathbf{x}}))\,\mathbf{n}\), that is
We remark that it is hereafter fundamental to have \(p>3\) in order to obtain that \(W^{1,p}(\Omega ,{\mathbb{R}}^{3})\) is a Banach algebra along with the continuous embedding
In particular, this ensures that the Nemitsky operator (3.2) is of class \(C^{1}\), see [29, Lemma 2.1, Chapter V]. This motivates assumptions (1.5) for external loads. Given a small parameter \(h>0\), we are then interested in existence and asymptotic behavior of solutions to the rescaled nonlinear problem (1.4), which formally coincide with critical points of the functional \(\mathcal {G}(\cdot ;h\mathcal {L})\) defined by (2.5), with the load functional given by (1.2). We easily notice that, if (1.4) possesses a solution, the forces must necessarily satisfy the following compatibility condition
In addition, after simple computations, one can observe that, if \(\mathbf{y}_{h}\in W^{2,p}(\Omega ,{\mathbb{R}}^{3})\) solves (1.4), then it must necessarily satisfy the zero moment condition
When (3.5) holds, we say that the loads \(\mathbf{f}\) and \(\mathbf{g}\) are equilibrated with respect to the deformation \(\mathbf{y}_{h}\). (3.4) can be imposed as an a priori condition on the loads while (3.5) may be only regarded as an a posteriori condition. Nevertheless, we assume the loads \(\mathbf{f}\) and \(\mathbf{g}\) to satisfy
Loosely speaking, a justification of this assumption lies in the fact that a solution to (1.4) (thus verifying (3.5)) is expected to be a small perturbation of the identity map \(\mathbf{i}\). We just say that loads are equilibrated if (3.4) and (3.6) hold. Moreover, heuristically speaking, another natural condition which, together with (3.6), goes in the direction of ruling out the chance that \(\nabla \mathbf {y}_{h}\) converges to a rotation \(\mathbf {R}\neq \mathbf {I}\) is the absence of axes of equilibrium for the couple \((\mathbf {f},\mathbf {g})\). We recall that \(\mathbf{a}\in {\mathbb{R}}^{3}\) is said to be an axis of equilibrium for \((\mathbf{f},\mathbf{g})\) if, for any rotation \(\mathbf{R}\in \mathit{SO}(3)\) around \(\mathbf{a}\), there holds
which is equivalent to
The presence of axes of equilibrium for a couple \((\mathbf{f},\mathbf {g})\) is linked with the astatic load matrix, defined as
In particular, if we call \(\lambda _{1},\lambda _{2},\lambda _{3}\in {\mathbb{R}}\) the eigenvalues of the matrix \(K_{\mathbf{f},\mathbf{g}}\) (which is symmetric in view of (3.6)), it is known that the absence of axes of equilibrium is equivalent to the following assumption:
We refer to [5, Sect. 3] for a more detailed characterization of this property. We also observe that the couple of assumptions (3.4) and (3.6) coincides with (2.8). Furthermore, we notice that if loads are equilibrated and satisfy (1.3), assumption (3.7) (i.e., absence of axes of equilibrium) can be rephrased by prescribing that the rotation kernel (1.8) of ℒ only contains the identity matrix (this can be seen for instance by combining [23, Proposition 6.2] and [5, Proposition 3.8]).
The following result is contained in [29, Corollary 6.9, Chapter V]. We point out that for such a result the assumptions on the lack of axes of equilibrium for the loads \((\mathbf{f},\mathbf{g})\) could be relaxed, see for instance Theorem 6.8 and Corollary 6.10 in Chapter V of [29] and the thorough analysis in [5, 6]. However, to the best of our knowledge, even the results with weaker assumptions do not allow the full generality for the equilibrated couple \((\mathbf{f},\mathbf{g})\): for instance, they do not cover the case in which every axis in \(\mathbb{R}^{3}\) is an axis of equilibrium for \((\mathbf {f},\mathbf {g})\) (which is equivalent to saying that the rotation kernel of ℒ is the whole of \(\mathit{SO}(3)\)), a case which we consider in our example in Sect. 5.
Theorem 3.1
\(\mbox{[29, Corollary 6.9, Chapter V]}\)
Suppose that \(\Omega \) is of class \(C^{2}\) and that (\(\boldsymbol {\mathcal {W}1}\)), (\(\boldsymbol {\mathcal {W}2}\)), (\(\boldsymbol {\mathcal {W}7}\)) and (\(\boldsymbol {\mathcal {W}8}\)) are satisfied. Let \(p>3\). Let \(\mathbf{f}\in L^{p}(\Omega ,{\mathbb{R}}^{3})\) and \(\mathbf{g}\in W^{1-1/p,p}(\partial \Omega ,{\mathbb{R}}^{3})\) satisfy (3.4) and (3.6) and let us assume that the eigenvalues \((\lambda _{i})_{i=1,\dots ,3}\) of the astatic load \(K_{\mathbf{f},\mathbf{g}}\) satisfy (3.7). Then there exists \(h_{0}>0\) such that for any \(h\in (0,h_{0})\) there exists a deformation \(\mathbf{y}_{h}\in W^{2,p}(\Omega ,{\mathbb{R}}^{3})\) which solves (1.4). Moreover the map
extended to the identity \(\mathbf{i}\) for \(h=0\), belongs to \(C^{1}([0,h_{0});W^{2,p}(\Omega ,{\mathbb{R}}^{3}))\).
It is now immediate to investigate the asymptotic behavior of the deformation \(\mathbf{y}_{h}\) as \(h\to 0\). A direct consequence of Theorem 3.1 is that
which, in view of (3.3), in turn implies that
Being the map \(\mathbf{Y}\) from (3.8) of class \(C^{1}\), we can perform a first order Taylor expansion at \(h=0\), which guarantees the existence of a function \(\mathbf{u}_{0}\in W^{2,p}(\Omega ,{\mathbb{R}}^{3})\) such that
Since
we can carry out a Taylor expansion of \(D\mathcal {W}({ \mathbf{x}},\cdot )\) near the identity, that is
for some \(\boldsymbol{\xi}_{h}({\mathbf{x}})\) belonging to the segment \([\mathbf{I},\mathbf{I}+h\nabla \mathbf{u}_{h}({\mathbf{x}})]\), and thus \(\boldsymbol{\xi}_{h}\) is uniformly converging to \(\mathbf{I}\) as \(h\to 0^{+}\). Now, in view of the equation (1.4) satisfied by \(\mathbf{y}_{h}=\mathbf {i}+h\mathbf{u}_{h}\) and of (3.9) we deduce that
for all \({\mathbf{v}}\in H^{1}(\Omega ,{\mathbb{R}}^{3})\). Then, thanks to the regularity of \(\mathcal{W}\) we can pass to the limit as \(h\to 0\) and obtain
for all \({\mathbf{v}}\in H^{1}(\Omega ,{\mathbb{R}}^{3})\), i.e. \(\mathbf{u}_{0}\) weakly solves
This means that \(\mathbf{u}_{0}\) is an equilibrium configuration for the linearized elastic problem with loads \((\mathbf{f},\mathbf{g})\) and as such it minimizes functional \(\mathcal {F}_{0}(\cdot ;\mathcal {L})\) over \(H^{1}(\Omega ,{\mathbb{R}}^{3})\).
4 Proof of the Main Results
Let us state a preliminary result about Korn inequalities. We state it for \(W^{1,p}(\Omega ,\mathbb{R}^{3})\) vector fields with generic \(p\in (1,+\infty )\). It is based on the standard Korn inequality, see [24], which yields the existence of a constant \(D_{p,\Omega }\) such that for every \(\mathbf{u}\in W^{1,p}(\Omega ,\mathbb{R}^{3\times 3})\) there holds
Lemma 4.1
Let \(p\in (1,+\infty )\). There are constants \(Z_{p,\Omega }\) and \(Q_{p,\Omega }\) (only depending on \(p\), \(\Omega \)) such that
and
Proof
Suppose first that \(\displaystyle \int _{\Omega }|\nabla \mathbf{u}|^{p-2}\mathrm{curl} \,\mathbf{u}=0\). Then for every \(\mathbf {W}\in {\mathbb{R}}^{3\times 3}_{\mathrm{skew}}\),
implying that the unique projection of \(\nabla \mathbf{u}\) on the vector space of constant skew-symmetric tensor fields over \(\Omega \) is \(\mathbf {0}\). Therefore, (4.1) gives (4.3).
In order to prove that (4.2) holds, assume by contradiction that there exists a sequence \((\mathbf{u}_{j})_{j\in \mathbb{N}}\subset W^{1,p}(\Omega ,{\mathbb{R}}^{3})\) such that \(\int _{\Omega }\mathrm{curl}\,\mathbf{u}_{j}=\mathbf {0}\) and \(\int _{\Omega }|\nabla \mathbf{u}_{j}|^{p}=1\) for every \(j\in \mathbb{N}\), and \(\int _{\Omega }|\mathbb{E}\mathbf{u}_{j}|^{p}\to 0\) as \(j\to \infty \). By (4.1) and since \(\int _{\Omega }\mathrm{skew}(\nabla \mathbf{u}_{j})=\mathbf {0}\), there exists a sequence \((\mathbf {W}_{j})_{j\in \mathbb{N}}\subset \mathbb{R}^{3\times 3}_{ \mathrm{skew}}\) such that
and since \(\int _{\Omega }|\mathbb{E}\mathbf{u}_{j}|^{p}\to 0\) we conclude that \(\mathbf {W}_{j}\to \mathbf {0}\) and that \(\int _{\Omega }|\nabla \mathbf{u}_{j}-\mathbf {W}_{j}|^{p}\to 0\) as \(j\to \infty \). But then we have
and the right hand side goes to 0 as \(j\to \infty \), contradicting the fact that \(\int _{\Omega }|\nabla \mathbf{u}_{j}|^{p}=1\) for every \(j\in \mathbb{N}\) and thus proving (4.2). □
Let us now state a simple result about the minimization of the linear elastic energy.
Lemma 4.2
Let \(M>0\). Suppose that (\(\boldsymbol {\mathcal {W}1}\)), (\(\boldsymbol {\mathcal {W}2}\)), (\(\boldsymbol {\mathcal {W}3}\)), (\(\boldsymbol {\mathcal {W}4}\)), hold and that ℒ satisfies (2.8) and \(\|\mathcal {L}\|_{*} \le {\mathit{CM} }/{K_{\Omega}}\), where \(K_{\Omega}\) is the constant in (2.9) and \(C\) is the constant in (\(\boldsymbol {\mathcal {W}4}\)). If \(\mathbf{u}_{*}\in H^{1}(\Omega ,\mathbb{R}^{3})\) minimizes \(\mathcal {F}_{0}(\cdot ;\mathcal {L})\) over \(H^{1}(\Omega ,{\mathbb{R}}^{3})\), then \(\int _{\Omega }|\mathbb{E}\mathbf{u}_{*}|^{2}\,d{\mathbf{x}}\le M^{2}\).
Proof
Let us consider a solution \(\mathbf{u}_{*}\in H^{1}(\Omega ,\mathbb{R}^{3})\) to the linear elastic problem
whose existence is ensured by the compatibility conditions (2.8) by standard arguments. Being a minimizer, \(\mathbf{u}_{*}\) satisfies the energy identity
From (4.5), by the basic ellipticity estimate \(\mathrm{sym}\mathbf {F}\,D^{2}\mathcal {W}({\mathbf{x}},\mathbf {I})\, \mathrm{sym}\mathbf {F}\ge C|\mathrm{sym}\mathbf {F}|^{2}\), where \(C\) is the constant in (\(\boldsymbol {\mathcal {W}4}\)), by taking into account (2.9) we get
that is,
Thanks to the assumption \(\|\mathcal {L}\|_{*} \le {\mathit{CM}}/{K_{\Omega}}\), we conclude. □
For every \(R > 0\) and for every \(\mathcal {L}\in (H^{1}(\Omega ,\mathbb{R}^{3}))^{*}\), in the sequel we shall consider the following constrained minimization problems of finite elasticity
Lemma 4.3
Suppose that (\(\boldsymbol {\mathcal {W}1}\)), (\(\boldsymbol {\mathcal {W}2}\)), (\(\boldsymbol {\mathcal {W}3}\)), (\(\boldsymbol {\mathcal {W}4}\)), (\(\boldsymbol {\mathcal {W}5}\)), (\(\boldsymbol {\mathcal {W}6}\)) hold and that \(\mathcal {L} \in (H^{1}(\Omega ,\mathbb{R}^{3}))^{*}\) satisfies (2.8). Let \(R>0\). Then there exists a solution to problem (4.6) and there exists a solution to problem (4.7).
Proof
Let \((\mathbf{u}_{n})_{n\in \mathbb{N}}\subset H^{1}(\Omega ,\mathbb{R}^{3})\) denote a minimizing sequence for problem (4.6). Since \(\mathcal {F}(\mathbf {0};\mathcal {L})=0\), it is not restrictive to assume that \(\mathcal {F}(\mathbf{u}_{n};\mathcal {L})\le 1\) for every \(n\in \mathbb{N}\). Moreover, by (2.10) there exists a sequence \((\mathbf {R}_{n})_{n\in \mathbb{N}}\subset \mathit{SO}(3)\) such that
From (2.9) and (4.8), since \(\int _{\Omega }|\mathbb{E}\mathbf{u}_{n}|^{2}\le R\), we deduce that the sequence \((\nabla \mathbf{u}_{n})_{n\in \mathbb{N}}\) is bounded in \(L^{2}(\Omega ,\mathbb{R}^{3\times 3})\). Thanks to Poincaré inequality and to (2.8), we deduce that up to subsequences \(\nabla \mathbf{u}_{n}\) weakly converge to \(\nabla \mathbf{u}\) for some \(\mathbf{u}\in H^{1}(\Omega ,\mathbb{R}^{3})\), that \(\mathcal {L}( \mathbf{u}_{n})\to \mathcal {L}( \mathbf{u})\), and that \(\mathbb{E}\mathbf{u}_{n}\) weakly converge to \(\mathbb{E}\mathbf{u}\) in \(L^{2}(\Omega ,\mathbb{R}^{3})\). The weak \(L^{2}(\Omega ,\mathbb{R}^{3\times 3})\) lower semicontinuity of the map \(\mathbf {F}\mapsto \int _{\Omega }|\mathbf {F}|^{2}\) yields \(\int _{\Omega }|\mathbb{E}\mathbf{u}|^{2}\le R\). On the other hand, with the notation \(\mathbf {y}_{n}({\mathbf{x}}):={\mathbf{x}}+h\mathbf{u}_{n}({\mathbf{x}})\), the assumption (\(\boldsymbol {\mathcal {W}5}\)) implies, thanks to the classical results by Ball [2, §6], that \(\mathrm{cof}\nabla \mathbf {y}_{n}\) (resp. \(\det \nabla \mathbf {y}_{n}\)) weakly converge in \(L^{q}(\Omega ,\mathbb{R}^{3\times 3})\) to \(\mathrm{cof}\nabla \mathbf {y}\) (resp. weakly converge in \(L^{r}(\Omega )\) to \(\det \nabla \mathbf {y}\)). By the polyconvexity assumption (\(\boldsymbol {\mathcal {W}6}\)) on the Carathéodory function \(\mathcal {W}\), and since \(\mathcal {L}( \mathbf{u}_{n})\to \mathcal {L}( \mathbf{u})\) as \(n\to \infty \), we deduce that
Since \((\mathbf{u}_{n})_{n}\) is by assumption a minimizing sequence, we conclude that \(\mathbf{u}\in H^{1}(\Omega ,\mathbb{R}^{3})\) is a solution to problem (4.6).
The same argument shows that there exists a solution to problem (4.7), since the weak \(L^{2}(\Omega ,\mathbb{R}^{3\times 3})\) convergence of \(\nabla \mathbf{u}_{n}\) to \(\nabla \mathbf{u}\) implies that \(\int _{\Omega }\mathrm{curl}\,\mathbf{u}_{n}\to \int _{\Omega } \mathrm{curl}\,\mathbf{u}\). □
Remark 4.4
It is worth noticing that a solution \(\mathbf{u}\) to problem (4.7) is not a priori guaranteed to be a constrained local minimizer in the sense of Definition 2.1. Indeed it may happen that \(\mathbf{u}\) satisfies \(\int _{\Omega }|\mathbb{E}\mathbf{u}|^{2}= R\), so that for instance by taking \(\boldsymbol{\psi}=\mathbf{u}\), for every \(\varepsilon >0\) the function \(\mathbf{u}+\varepsilon \boldsymbol{\psi}\) is not admissible for problem (4.7).
Remark 4.5
Let \(h>0\) and \(M>0\). Then by Lemma 4.3 there exists a solution \({\mathbf{v}}_{h}\) to the problem
and thus \(\mathbf{u}_{h}=:h^{-1}{\mathbf{v}}_{h}\) is a solution to
The next lemma provides convergence to constrained linear elasticity and generalizes results from [19, 20]. The proof is a \(\Gamma \)-convergence argument. With respect to the analysis in [19, 20], here we crucially take advantage of the constraints in the minimization problem, that allow to avoid further assumptions on external loads.
Lemma 4.6
Suppose that (\(\boldsymbol {\mathcal {W}1}\)), (\(\boldsymbol {\mathcal {W}2}\)), (\(\boldsymbol {\mathcal {W}3}\)), (\(\boldsymbol {\mathcal {W}4}\)), (\(\boldsymbol {\mathcal {W}5}\)), (\(\boldsymbol {\mathcal {W}6}\)) hold and that \(\mathcal {L}\in (H^{1}(\Omega ,{\mathbb{R}}^{3}))^{*}\) satisfies (2.8). Let \(M>0\) and let \((h_{j})_{j\in \mathbb{N}}\subset (0,1)\) be a vanishing sequence. For any \(j\in \mathbb{N}\), let \(\mathbf{u}_{j}:=\mathbf{u}_{h_{j}}\), where \(\mathbf{u}_{h_{j}}\) is a solution to problem (4.9) (with \(h_{j}\) in place of \(h\), see Remark 4.5). Then there exists \(\mathbf{u}_{*}\in H^{1}(\Omega ,{\mathbb{R}}^{3})\) such that \(\int _{\Omega }|\mathbb{E}\mathbf{u}_{*}|^{2}\le M^{2} \) and such that, up to subsequences, \(\mathbb{E}\mathbf{u}_{j}\rightharpoonup \mathbb{E}\mathbf{u}_{*}\) weakly in \(L^{2}(\Omega ,\mathbb{R}^{3\times 3})\) as \(j\to \infty \) and
Proof
Step 1 (lower bound). Suppose that \(\sup _{j\in \mathbb{N}}{\mathcal {F}}_{h_{j}}(\tilde{\mathbf{u}}_{j}; \mathcal {L})<+\infty \) and that \(\mathbb{E}\tilde{\mathbf{u}}_{j}\rightharpoonup \mathbb{E} \tilde{\mathbf{u}}\) weakly in \(L^{2}(\Omega ,\mathbb{R}^{3\times 3})\) for some sequence \((\tilde{\mathbf{u}}_{j})_{j\in \mathbb{N}}\subset H^{1}(\Omega , \mathbb{R}^{3})\) such that \(\int _{\Omega }\mathrm{curl}\,\tilde{\mathbf{u}}_{j}=\mathbf {0}\) for every \(j\in \mathbb{N}\) and some \(\tilde {\mathbf{u}}\in H^{1}(\Omega ,\mathbb{R}^{3})\). By the weak \(L^{2}(\Omega ,\mathbb{R}^{3\times 3})\) lower semicontinuity of the map \(\mathbf {F}\mapsto \int _{\Omega }|\mathbf {F}|^{2}\), we get
By applying [20, Lemma 5.1] we obtain the existence of \(\mathbf {W}\in {\mathbb{R}}^{3\times 3}_{\mathrm{skew}}\) such that \(\sqrt{h_{j}}\nabla \tilde{\mathbf{u}}_{j}\to \mathbf {W}\) in \(L^{2}(\Omega ,\mathbb{R}^{3\times 3})\) as \(j\to \infty \). However, since \(\int _{\Omega }\mathrm{curl}\,\tilde{\mathbf{u}}_{j}=\mathbf {0}\), by Lemma 4.1 we deduce that the sequence \((\nabla \tilde{\mathbf{u}}_{j})_{j\in \mathbb{N}}\) is bounded in \(L^{2}(\Omega ,{\mathbb{R}}^{3})\), thus forcing \(\mathbf {W}\equiv \mathbf {0}\). From [20, Lemma 5.2] we also deduce that \(\mathbf {1}_{B_{j}}\,\mathbb{E}\tilde{\mathbf{u}}_{j}\rightharpoonup \mathbb{E}\tilde {\mathbf{u}}\) weakly in \(L^{2}(\Omega ,{\mathbb{R}}^{3\times 3})\) as \(j\to \infty \), where \(B_{j}:=\{{\mathbf{x}}\in \Omega :|\sqrt{h_{j}}\nabla \tilde{\mathbf{u}}_{j}({ \mathbf{x}})|\le 1\}\). We stress that [20, Lemma 5.1, Lemma 5.2] are stated under the additional incompressibility constraint requiring \(\mathcal {W}(\mathbf {F})=+\infty \) if \(\det \mathbf {F}\neq 1\), but they are true (with the very same proof) even without such a constraint.
By the frame indifference assumption (\(\boldsymbol {\mathcal {W}1}\)), there exists a function \(\mathcal {V}\) such that for a.e. \({\mathbf{x}}\in \Omega \)
By (\(\boldsymbol {\mathcal {W}3}\)) we deduce that for a.e. \({\mathbf{x}}\in \Omega \) the function \(\mathcal {V}({\mathbf{x}},\cdot )\) is \(C^{2}\) smooth in a suitable neighborhood \(\tilde{\mathcal {U}}\) of the origin in \({\mathbb{R}}^{3\times 3}\), with a \({\mathbf{x}}\)-independent modulus of continuity \(\eta :\mathbb{R}_{+}\to \mathbb{R}\), which is increasing and vanishing at \(0^{+}\). By following the proof of [20, Lemma 5.3], we define
By (\(\boldsymbol {\mathcal {W}2}\)), (\(\boldsymbol {\mathcal {W}3}\)) and by a Taylor expansion, we get the existence of \(j_{0}\in \mathbb{N}\) such that for every \(j>j_{0}\) the estimate
holds for a.e. \({\mathbf{x}}\in B_{j}\), since \(h_{j}|\mathbb{D}\tilde{\mathbf{u}}_{j}({\mathbf{x}})|\le 2\sqrt{h_{j}}\) and thus \(h_{j} \mathbb{D}\tilde{\mathbf{u}}_{j}({\mathbf{x}})\in \tilde{\mathcal {U}}\) for a.e. \({\mathbf{x}}\in B_{j}\) if \(j\) is large enough. We deduce
for every \(j>j_{0}\), where the last inequality is due \(D^{2}\mathcal {W}({\mathbf{x}},\mathbf {I})=D^{2}\mathcal {V}({\mathbf{x}},\mathbf{0})\) and to the monotonicity of the modulus of continuity \(\eta \), and we also recall that \(\eta (2\sqrt{h_{j}})\to 0\) as \(j\to \infty \). By recalling that \(\sqrt{h_{j}}\nabla \tilde{\mathbf{u}}_{j}\) strongly converge to zero in \(L^{2}(\Omega ,{\mathbb{R}}^{3\times 3})\), hence a.e. in \(\Omega \) up to subsequences, and since \((\mathbf {1}_{B_{j}}h_{j}\nabla \tilde{\mathbf{u}}_{j}^{T}\nabla \tilde{\mathbf{u}}_{j})_{j\in \mathbb{N}}\) is a bounded sequence in \(L^{2}(\Omega ,\mathbb{R}^{3\times 3})\), we deduce that up to subsequences \(\mathbf {1}_{B_{j}}h_{j}\nabla \tilde{\mathbf{u}}_{j}^{T}\nabla \tilde{\mathbf{u}}_{j}\) weakly converge to zero in \(L^{2}(\Omega ,\mathbb{R}^{3\times 3})\) so that \(\mathbf {1}_{B_{j}}\mathbb{D}\tilde{\mathbf{u}}_{j}\) weakly converge in \(L^{2}(\Omega ,\mathbb{R}^{3\times 3})\) to \(\mathbb{E}\tilde{\mathbf{u}}\) as \(j\to \infty \). Since the map \(\mathbf {F}\mapsto \int _{\Omega }\mathbf {F}^{T}\, D^{2}\mathcal {W}({ \mathbf{x}},\mathbf {I})\,\mathbf {F}\,d{\mathbf{x}}\) is weakly \(L^{2}(\Omega ,{\mathbb{R}}^{3\times 3})\) lower semicontinuous, and since \(\mathcal {L}\in (H^{1}(\Omega ,{\mathbb{R}}^{3}))^{*}\) and (2.8) imply \(\mathcal {L}(\tilde{\mathbf{u}}_{j} )\to \mathcal {L}(\tilde{\mathbf{u}})\) thanks to Korn and Poincaré inequalities, we conclude that
Step 2 (upper bound). Let now \(\bar{\mathbf{u}}\in H^{1}(\Omega ,\mathbb{R}^{3})\) be such that \(\int _{\Omega } |\mathbb{E}\bar{\mathbf{u}}|^{2}\le M^{2} \). Let \(\delta _{j}:=h_{j}^{1/5}\) and \(\bar{\mathbf{u}}_{j}:=\bar{\mathbf{u}}\ast \rho _{j}\), where \(\rho _{j}({\mathbf{x}}):=\delta _{j}^{-3}\rho ({\mathbf{x}}/\delta _{j})\) and \(\rho :\mathbb{R}^{3}\to \mathbb{R}\) is the standard unit symmetric mollifier. We notice that \(\bar{\mathbf{u}}_{j}\to \bar{\mathbf{u}}\) in \(H^{1}(\Omega ,\mathbb{R}^{3})\) as \(j\to +\infty \) and that the elementary estimate \(\|\rho _{j}\|_{W^{1,\infty}(\mathbb{R}^{3})}\le 2\delta _{j}^{-4}\| \rho \|_{W^{1,\infty}(\mathbb{R}^{3})}\) holds. Therefore, by Young inequality we obtain
where \(\Omega '\) is a suitable open neighbor of \(\Omega \) (and \(\bar{\mathbf{u}}\) is understood as a not relabeled \(H^{1}(\Omega ',\mathbb{R}^{3})\) extension). We deduce \(h\|\bar{\mathbf{u}}_{j}\|_{W^{1,\infty}(\Omega ,\mathbb{R}^{3})}\to 0\) as \(j\to +\infty \). Therefore \(\mathbf {I}+h_{j}\nabla \bar{\mathbf{u}}_{j}\in \mathcal {U}\) for a.e. \({\mathbf{x}}\) in \(\Omega \) if \(j\) is large enough, where \(\mathcal {U}\) is the neighborhood of \(\mathit{SO}(3)\) that appears in assumption (\(\boldsymbol {\mathcal {W}3}\)), which then implies, by Taylor expansion,
The latter limit is zero since \(h_{j}\nabla \bar{\mathbf{u}}_{j}\to 0\) in \(L^{\infty}(\Omega ,\mathbb{R}^{3\times 3})\), as \(\omega \) is increasing with \(\lim _{t\to 0^{+}}\omega (t)= 0\), and since \(\bar{\mathbf{u}}_{j}\to \bar {\mathbf{u}}\) in \(H^{1}(\Omega ,\mathbb{R}^{3})\) as \(j\to +\infty \), which also implies \(\mathcal {F}_{0}(\bar{\mathbf{u}}_{j};\mathcal {L})\to \mathcal {F}_{0}( \bar{\mathbf{u}};\mathcal {L})\), so that by writing \(|\mathcal {F}_{h_{j}}(\bar{\mathbf{u}}_{j};\mathcal {L})- \mathcal {F}_{0}( \bar{\mathbf{u}};\mathcal {L})|\le |\mathcal {F}_{h_{j}}(\bar{\mathbf{u}}_{j}; \mathcal {L})- \mathcal {F}_{0}(\bar{\mathbf{u}}_{j};\mathcal {L})|+ | \mathcal {F}_{0}(\bar{\mathbf{u}}_{j};\mathcal {L})- \mathcal {F}_{0}( \bar{\mathbf{u}};\mathcal {L})|\) we deduce
Moreover, by using Young inequality again we deduce that for any \(j\in \mathbb{N}\)
Step 3 (convergence). Since we are assuming
for every \(j\in \mathbb{N}\), it is readily seen that \(\sup _{j\in \mathbb{N}}{\mathcal {F}}_{h_{j}}(\mathbf{u}_{j};\mathcal {L})<+ \infty \). Moreover, since \(\int _{\Omega }|\mathbb{E}\mathbf{u}_{j}|^{2}\le M^{2}\) for every \(j\in \mathbb{N}\), up to subsequences there exists \(\mathbf{u}_{*}\in H^{1}(\Omega ,\mathbb{R}^{3})\) such that \(\mathbb{E}\mathbf{u}_{j}\rightharpoonup \mathbb{E}\mathbf{u}_{*}\) weakly in \(L^{2}(\Omega ,\mathbb{R}^{3\times 3})\), see [19, Lemma 3.2]. By Step 1 we get \(\int _{\Omega }|\mathbb{E}\mathbf{u}_{*}|^{2}\le M^{2} \), see (4.11), and
To every \(\bar{\mathbf{u}}\in H^{1}(\Omega ,\mathbb{R}^{3})\) such that \(\int _{\Omega }|\mathbb{E}\bar{\mathbf{u}}|^{2}\le M^{2} \), we associate the sequence \((\bar{\mathbf{u}}_{j})_{j\in \mathbb{N}}\subset H^{1}(\Omega , \mathbb{R}^{3})\) constructed as in Step 2. Therefore, such a sequence converges to \(\bar {\mathbf{u}}\) strongly in \(H^{1}(\Omega ,\mathbb{R}^{3})\) with \(h_{j}\|\nabla \bar{\mathbf{u}}_{j}\|_{L^{\infty}(\Omega ,{\mathbb{R}}^{3 \times 3})}\to 0\) as \(j\to +\infty \), and it satisfies \(\int _{\Omega }|\mathbb{E}\bar{\mathbf{u}}_{j}|^{2}\le M^{2} \) for every \(j\in \mathbb{N}\), see (4.12). For every \(j\in \mathbb{N}\) we let \(\mathbf {w}_{j}:=|\Omega |^{-1}\int _{\Omega }\mathrm{curl}\, \bar{\mathbf{u}}_{j}\) so that
since \(\mathrm{curl}(\mathbf {a}\wedge {\mathbf{x}})=2\mathbf {a}\) for every \(\mathbf {a}\in {\mathbb{R}}^{3}\). Thus \(\mathbf {w}_{j}\to \mathbf {w}:=|\Omega |^{-1}\int _{\Omega } \mathrm{curl}\,\bar{\mathbf{u}}\) as \(j\to \infty \). We deduce that \(\bar{\mathbf{u}}_{j}-\tfrac{1}{2}\mathbf {w}_{j}\wedge {\mathbf{x}}\) strongly converge in \(H^{1}(\Omega ,{\mathbb{R}}^{3})\) to \(\bar{\mathbf{u}}-\tfrac{1}{2}\mathbf {w}\wedge {\mathbf{x}}\) as \(j\to +\infty \), that \(h_{j}\|\nabla (\bar{\mathbf{u}}_{j}-\tfrac{1}{2}\mathbf {w}_{j}\wedge { \mathbf{x}})\|_{L^{\infty}(\Omega ,{\mathbb{R}}^{3\times 3})}\to 0\) as \(j\to \infty \), and that for every \(j\in \mathbb{N}\) there holds
By (4.14), (4.13) and by Step 2 we get
where the last equality is due to the invariance of \(\mathcal {F}_{0}(\cdot ;\mathcal {L})\) by infinitesimal rigid displacements. By the arbitrariness of \(\bar{\mathbf{u}}\), we deduce that \(\mathbf{u}_{*}\) is a solution to the minimization problem in the right hand side of (4.10), and that indeed (4.10) holds. □
Next we provide the key lemma towards the proof of the main theorem.
Lemma 4.7
Assume that (\(\boldsymbol {\mathcal {W}1}\)), (\(\boldsymbol {\mathcal {W}2}\)), (\(\boldsymbol {\mathcal {W}3}\)), (\(\boldsymbol {\mathcal {W}4}\)), (\(\boldsymbol {\mathcal {W}5}\)), (\(\boldsymbol {\mathcal {W}6}\)) hold true, that \(\mathcal {L} \in (H^{1}(\Omega ,\mathbb{R}^{3}))^{*}\) satisfies (2.8) and that
where \(C_{\Omega }\) is the constant in (2.10), \(K_{\Omega}\) is the constant in (2.9) and \(C\) is the constant in (\(\boldsymbol {\mathcal {W}4}\)). Then there exists \(\bar{h}>0\) such that if \(h\in (0,\bar{h})\) and if
then \(\displaystyle \int _{\Omega }|\mathbb{E} {\mathbf{v}}_{h}|^{2}< h^{2}\).
Proof
Step 1. Let us consider a minimizer \(\mathbf{u}_{*}\in H^{1}(\Omega ,\mathbb{R}^{3})\) of the linear elastic problem (4.4), whose existence is ensured by the compatibility conditions (2.8) on ℒ. It satisfies the energy identity (4.5), and since by (4.15) we have \(\|\mathcal {L}\|_{*}\le \tfrac{C}{2K_{\Omega}}\), by applying Lemma 4.2 with \(M=1/2\) we deduce \(\int _{\Omega }|\mathbb{E}\mathbf{u}_{*}|^{2}\le 1/4\). Let \(h\in (0,1)\) and let \(\bar{\mathbf{u}}_{h}:=\mathbf{u}_{*}\ast \rho _{h}\), where \(\rho _{h}({\mathbf{x}})=\delta _{h}^{-3}\rho ({\mathbf{x}}/\delta _{h})\) and \(\delta _{h}=h^{1/5}\), so that the argument in Step 2 of the proof of Lemma 4.6 yields \(\bar{\mathbf{u}}_{h}\to \mathbf{u}_{*}\) in \(H^{1}(\Omega ,\mathbb{R}^{3})\) along with \(h\|\nabla \bar{\mathbf{u}}_{h}\|_{L^{\infty}(\Omega ,\mathbb{R}^{3 \times 3})}\to 0\) as \(h\to 0\), and moreover
We let \(\mathbf{u}_{h}^{*}:=\bar{\mathbf{u}}_{h}-\tfrac{1}{2}\mathbf {w}_{h} \wedge {\mathbf{x}}\), where \({\mathbf{w}}_{h}:=|\Omega |^{-1}\int _{\Omega }\mathrm{curl}\, \bar{\mathbf{u}}_{h}\), and \(\mathbf {w}:=|\Omega |^{-1}\int _{\Omega }\mathrm{curl}\,\mathbf{u}_{*}\), so that also along \(\mathbf{u}_{h}^{*}\) we have
since the linear elastic energy is unaffected by the addition of infinitesimal rigid displacements, and similarly we have \(\mathcal {L}(\mathbf{u}_{h}^{*})\to \mathcal {L}(\mathbf{u}_{*})\) as \(h \to 0\) thanks to (2.8). Moreover, since \(\mathbf{u}_{h}^{*}\to \mathbf{u}_{*}-\tfrac{1}{2}{\mathbf{w}}\wedge {\mathbf{x}}\) in \(H^{1}(\Omega ,\mathbb{R}^{3})\) as \(h\to 0\), and since \(\int _{\Omega }|\mathbb{E}\mathbf{u}_{*}|^{2}\le 1/4\), we see that for small enough \(h\) there holds \(\int _{\Omega }|h\mathbb{E}\mathbf{u}_{h}^{*}|^{2}=\int _{\Omega }|h \mathbb{E}\bar{\mathbf{u}}_{h}|^{2}\le h^{2}\), so that \(h\mathbf{u}^{*}_{h}\) is admissible for problem (4.16).
Step 2. For every \(h\in (0,1)\), let \({\mathbf{v}}_{h} \) as in (4.16) and \(\mathbf{u}_{h}:=h^{-1}{\mathbf{v}}_{h}\). By Step 1 we get
as \(h\to 0\). On the other hand, by (2.10) and thanks to the Euler-Rodrigues formula yielding the representation
for some suitable \(\theta _{h}\in \mathbb{R}\) and \(\mathbf {W}_{h}\in \mathbb{R}^{3\times 3}_{\mathrm{skew}}\) with \(|\mathbf {W}_{h}|^{2}=|\mathbf {W}_{h}^{2}|^{2}=2\), we have
By taking into account that \(\int _{\Omega }\mathop{{\mathrm{curl}}}\nolimits {\mathbf{v}}_{h}=\mathbf {0}\), i.e., \(\int _{\Omega }\mathrm{skew} (\nabla {\mathbf{v}}_{h})=\mathbf {0}\), an application of Jensen inequality yields
From (4.16), (4.17) and (4.19), by (2.9) and by the energy identity (4.5) we deduce
as \(h\to 0\). By setting
thanks to (4.20) we deduce that
as \(h\to 0\). In particular, since \(\int _{\Omega }|\mathbb{E}\mathbf{u}_{h}|^{2}\le 1\), we deduce that \(|\Omega ||\mathbf {A}_{h}|^{2}\le 2+2C_{\Omega }K_{\Omega}\|\mathcal {L} \|_{ *}+o(1)\) as \(h\to 0\). Then we get \(|\Omega | | \mathbf {B}_{h}|^{2} \le C_{\Omega }K_{\Omega}\|\mathcal {L}\|_{*}+o(1) \) as \(h\to 0\). Since both \(|\mathbf {A}_{h}|\) and \(|\mathbf {B}_{h}|\) are bounded as \(h\to 0\), and since \(|\mathbf {W}_{h}|^{2}=|\mathbf {W}_{h}^{2}|^{2}=2\), from (4.21) we deduce that \(\lim _{h\to 0}\theta _{h}=0\), therefore \(\sin \theta _{h}\sim \theta _{h}\), \(1-\cos \theta _{h}\sim \theta _{h}^{2}/2\), and thus from (4.21) again we get \(\mathbf {A}_{h}\to 0\) as \(h\to 0\).
Step 3. We end the proof by contradiction, assuming that there exists a vanishing sequence \((h_{n})_{n\in \mathbb{N}}\subset (0,1)\) such that
for all \(n\in \mathbb{N}\). Thanks to Lemma 4.6 (applied with \(M=1\)), along a not relabeled subsequence we have \(\mathbb{E}\mathbf{u}_{h_{n}}\rightharpoonup \mathbb{E}\mathbf{u}_{*}\) weakly in \(L^{2}(\Omega ,{\mathbb{R}}^{3\times 3})\) and thanks to (4.23) and to (4.22) we have
as \(n\to +\infty \). We pass to the limit as \(n\to +\infty \) and by recalling that \(\mathbf {A}_{h_{n}}\to 0\) we obtain
which is a contradiction with (4.15). □
Corollary 4.8
Suppose that the assumptions of Theorem 2.6are satisfied and let
where \(\gamma (\Omega ,C)\) is defined in Lemma 4.7. There exist \(h_{0}>0\) such that if \(0< h< h_{0}\) and \({\mathbf{v}}_{h}\) is a solution to
then \({\mathbf{v}}_{h}\) is a local minimizer for \(\mathcal {F} (\cdot ; h\mathcal {L})\) over \(H^{1}_{\mathbf {I}}(\Omega ,\mathbb{R}^{3})\) and \(\mathbf{u}_{h}:=h^{-1}{\mathbf{v}}_{h}\) is a local minimizer for \(\mathcal {F}_{h} (\cdot ; \mathcal {L})\) over \(H^{1}_{\mathbf {I}}(\Omega ,\mathbb{R}^{3})\).
Proof
The existence of a solution to problem (4.24) is due to Lemma 4.3 and Remark 4.5. Since \(\|M_{0}^{-1}\mathcal {L}\|_{*}< \gamma (\Omega , C)\), we may apply Lemma 4.7 with \(M_{0}^{-1}\mathcal {L}\) in place of ℒ, thus finding \(h_{0}>0\) such that \({\mathbf{v}}_{h}\) satisfies \(\int _{\Omega }|\mathbb{E}{\mathbf{v}}_{h}|^{2}< M_{0}^{2}h^{2}\) as soon as \(h\in (0,h_{0})\).
Let \(\boldsymbol {\psi}\in H^{1}(\Omega ,\mathbb{R}^{3})\) be such that \(\int _{\Omega }\mathrm{curl}\,\boldsymbol{\psi}=\mathbf {0}\). For any given \(h\in (0, h_{0})\), the continuity of the map \([0,1]\ni \varepsilon \mapsto \int _{\Omega }|\mathbb{E}{\mathbf{v}}_{h}+ \varepsilon \boldsymbol{\psi}|^{2}\) shows that there exists \(\varepsilon _{0}=\varepsilon _{0}({\mathbf{v}}_{h},\boldsymbol{\psi})\) such that for every \(\varepsilon <\varepsilon _{0}\) there holds \(\int _{\Omega }|\mathbb{E}{\mathbf{v}}_{h}+\varepsilon \boldsymbol{\psi}|^{2}< M_{0}^{2}h^{2}\). Thus for every \(\varepsilon <\varepsilon _{0}\) we have that \({\mathbf{v}}_{h}+\varepsilon \boldsymbol{\psi}\) is admissible for problem (4.24) and so \(\mathcal {F}({\mathbf{v}}_{h}; h\mathcal {L})\le \mathcal {F}({\mathbf{v}}_{h}+ \varepsilon \boldsymbol{\psi}; h\mathcal {L})\) as claimed. The statement about \(\mathbf{u}_{h}\) follows from Remark 2.4. □
Proof of Theorem 2.6
Let \(M_{0}\) and \(h_{0}\) be as in Corollary 4.8 and let \((h_{j})_{j\in \mathbb{N}}\subset (0,h_{0})\) be a vanishing sequence. For every \(j\in \mathbb{N}\), let \({\mathbf{v}}_{j}\) be a solution to problem (4.24) (with \(h_{j}\) in place of \(h\)), whose existence is ensured by Remark 4.5. By Corollary 4.8, for every \(j\in \mathbb{N}\) we have that \({\mathbf{v}}_{j}\) is a local minimizer for \(\mathcal {F} (\cdot ; h_{j}\mathcal {L})\) over \(H^{1}_{\mathbf {I}}(\Omega ,{\mathbb{R}}^{3\times 3})\), and \(\mathbf{u}_{j}:=h_{j}^{-1}{\mathbf{v}}_{j}\) is a local minimizer for \(\mathcal {F}_{h_{j}}(\cdot ;\mathcal {L})\) over \(H^{1}_{\mathbf {I}}(\Omega ,{\mathbb{R}}^{3\times 3})\). Moreover, by Lemma 4.6 we get, up to subsequences, \(\mathbb{E}\mathbf{u}_{j}={h_{j}}^{-1}\mathbb{E}{\mathbf{v}}_{j} \rightharpoonup \mathbb{E}\mathbf{u}_{*}\) weakly in \(L^{2}(\Omega ,{\mathbb{R}}^{3\times 3})\) as \(j\to \infty \) and
where \(\mathbf{u}_{*}\in H^{1}(\Omega ,\mathbb{{\mathbb {R}}}^{3})\) is a solution to
Since the choice of \(M_{0}\) from Corollary 4.8 is such that \(M_{0}^{-1}\|\mathcal {L}\|_{*}< \gamma (\Omega , C) \), we obtain \(\|\mathcal {L}\|_{*}\le \tfrac{\mathit{CM}_{0}}{K_{\Omega}}\), thus by Lemma 4.2 we deduce that \(\mathbf{u}_{*}\) minimizes \(\mathcal {F}_{0}(\cdot ;\mathcal {L})\) over the whole \(H^{1}(\Omega ,\mathbb{R}^{3})\). □
Proof of Corollary 2.7
Since \(\mathbf {R}\in \mathcal {S}^{0}_{\mathcal {L}}\), then \(\mathbf {R}^{T}\mathcal {L}\) satisfies (2.8) as remarked in Sect. 2. Hence, by Theorem 2.6 there exist a sequence \((h_{j})_{j\in \mathbb{N}}\subset (0,1)\) and local minimizers \({\mathbf{v}}_{j}\) for \(\mathcal {F}(\cdot \,; h_{j}\mathbf {R}^{T}\mathcal {L})\) over \(H^{1}_{\mathbf {I}}(\Omega ,\mathbb{R}^{3})\) such that
where \(\beta \) is defined by (2.11). By setting \(\mathbf {y}_{j}({\mathbf{x}}):= \mathbf {R} {\mathbf{x}}+h_{j}\mathbf {R}{\mathbf{v}}_{j}\) we get \(\int _{\Omega }\mathop{{\mathrm{curl}}}\nolimits ( \mathbf {R}^{T}\mathbf {y}_{j})=\mathbf{0}\). By taking into account frame indifference and again that \(\mathbf {R}\in \mathcal {S}^{0}_{\mathcal {L}}\) we get
Therefore \(\mathbf {y}_{j}\) is a local minimizer for \(\mathcal {G}(\cdot \,; h_{j}\mathcal {L})\) over \(H^{1}_{\mathbf {R}}(\Omega ,\mathbb{R}^{3})\) and
as claimed. □
Proof of Corollary 2.8
Since \(\mathcal {S}^{0}_{\mathcal {L}}\setminus \{\mathbf {I}\}\neq \emptyset \) and since \(\beta \) is not a constant map then \(\beta (\mathcal {S}^{0}_{\mathcal {L}})=[a,b]\) for some couple of reals \(a\), \(b\), \(a< b\), and so for every \(n\in \mathbb{N}\) there exists \(\{\mathbf {R}_{k}:\, k=1,2...n\}\subset \mathcal {S}^{0}_{\mathcal {L}}\) such that \(\beta (\mathbf {R}_{k})\neq\beta (\mathbf {R}_{k}')\) if \(k\neq k'\). By Corollary 2.7 for every vanishing sequence \((h_{j})_{j\in \mathbb{N}}\subset (0,1)\) and for every \(k=1,2....n\) there exist a subsequence \((h_{j}^{(k)})_{j\in \mathbb{N}}\) and local minimizers \(\mathbf {y}_{j}^{(k)}\) for \(\mathcal {G}(\cdot \,; h_{j}^{(k)}\mathcal {L})\) over \(H^{1}_{\mathbf {R}_{k}}(\Omega ,\mathbb{R}^{3})\) such that
We let
It is readily seen that for every \(k=1,2....n\) there exists \(h_{0}^{(k)}> 0\) such that, for every \(0< h< h_{0}^{(k)}\), the functional \(\mathcal {G}(\cdot \,; h\mathcal {L})\) has a local minimizer \(\mathbf {y}^{(k)}\) over \(H^{1}_{\mathbf {R}_{k}}(\Omega ,\mathbb{R}^{3})\) satisfying
The result follows by choosing \(h_{0}:=\min \{h_{0}^{(k)}: k=1,2...n\}\). □
5 Constrained Local Minimizers at Different Energy Levels: An Example
In this Section we give an explicit example in which the situation described in Corollary 2.7 and Corollary 2.8 occurs. To this aim we shall consider the Yeoh type energy density defined by (2.1) and (2.4), with \(\mathcal {W}_{\mathrm{vol}}(\mathbf {F})=g(\det \mathbf {F})\), where \(g:\mathbb{R}_{+}\to \mathbb{R}\) is the convex \(C^{2}\) function (satisfying (2.2) and (2.3) with \(r=2\)) obtained by setting
for some \(c>0\). This is a usual choice, see [14].
By taking into account that for every \(\mathbf {B}\in \mathbb{R}^{3\times 3}_{\mathrm{sym}}\) there holds
as \(\varepsilon \to 0\), and since
we get
and we choose from now on \(c_{1}=2\), \(c=\tfrac{4}{3}\) so that
Let now
\(B:=\{(x,y)\in \mathbb{R}^{2}: x^{2}+y^{2}<1\}\), let \(\phi :B\to {\mathbb{R}}\) be the radial function whose radial profile (still denoted by \(\phi \)) is
and let us consider a load functional of the form
It is readily seen that \(\phi (1)=\phi '(1)=0\) and that \(\mathbf {f}\in L^{p}(\Omega ,{\mathbb{R}}^{3})\) if and only if \(p< 2\) (so that assumption (1.5) is not satisfied) and thus \(\mathcal {L}\in (H^{1}(\Omega ,{\mathbb{R}}^{3}))^{*}\). It is also easy to see that ℒ is equilibrated. We claim that \(\mathcal {L}((\mathbf {R}-\mathbf {I}){\mathbf{x}})=0\) for every \(\mathbf {R}\in \mathit{SO}(3)\), that is, condition (1.3) is satisfied and \(\mathcal {S}^{0}_{\mathcal {L}}\equiv \mathit{SO}(3)\) (which implies that every direction in \(\mathbb{R}^{3}\) is an axis of equilibrium of ℒ, because the astatic load is zero). Indeed if \(\mathbf {R}\in \mathit{SO}(3)\) then by the Euler-Rodrigues formula there exist \(\theta \in [0,2\pi )\) and \(\mathbf {a}\in \mathbb{R}^{3}\), \(|\mathbf {a}|=1\), such that for every \({\mathbf{x}}\in \Omega \)
hence by setting \(c(\mathbf {a},\theta )=(1-\cos \theta )(\pi (1-a_{3}^{2})-1)\) we have
as claimed.
Let us now consider the following rotation of \(\pi /2\) around the \(z\) axis
We claim that \(\beta ({\mathbf {R}}_{*})< \beta (\mathbf {I})\), that is, the map \(\beta \) defined in (2.11) takes two different values as required in Corollary 2.8. To this aim we need the following
Lemma 5.1
Let \(\Omega \) as in (5.1), \(\phi \) as in (5.2) and ℒ as in (5.3). Then
Proof
Let \(B\) the unit ball in \(\mathbb{R}^{2}\) and
It is easily seen that \(\mathcal {K}\) is weakly closed in \(H^{1}(\Omega ,\mathbb{R}^{3})\) so there are minimizers of \({\mathcal {F}}_{0}(\cdot \,;{\mathbf {R}}_{*}^{T}\mathcal {L})\) over \(\mathcal {K}\) (recalling that \(\mathbf {R}_{*}^{T}\mathcal {L}\) satisfies (2.8) since the \(z\) axis is an axis of equilibrium for ℒ). Therefore
If \(u\) is a minimizer of the right hand side of (5.4), \(0< \varepsilon < 1\), \(B_{\varepsilon }:=\{(x,y)\in \mathbb{R}^{2}:x^{2}+y^{2}<\varepsilon \}\) and \(\zeta \in H^{2}_{0}(B\setminus B_{\varepsilon })\), by taking into account that \(\Delta \phi \in L^{2}(B\setminus B_{\varepsilon })\), a first variation argument for the minimization problem in the right hand side of (5.4) yields
and after integration by parts we obtain the Euler-Lagrange equation
for every \(0< \varepsilon < 1\), where \(\Delta ^{2}\) denotes the planar biharmonic operator. On the other hand if \(u\) is a minimizer then by using \(u\) as test function we easily get
In view of (5.4), and since \({\mathcal {F}}_{0}(\mathbf {0}\,;{\mathbf {R}}_{*}^{T}\mathcal {L})=0\), in order to conclude it is enough to show that \(\min _{{\mathbf{v}}\in \mathcal {K}}{\mathcal {F}}_{0}({\mathbf{v}}\,;{\mathbf {R}}_{*}^{T} \mathcal {L})\neq 0\). Assume by contradiction that \(\min _{{\mathbf{v}}\in \mathcal {K}}{\mathcal {F}}_{0}({\mathbf{v}}\,;{\mathbf {R}}_{*}^{T} \mathcal {L})= 0\): then by (5.5) we get
that is \(u_{xy}=u_{yy}-u_{xx}=0\) a.e. in \(B\), hence \(\Delta ^{2} u=0\) a.e. in \(B\) and from the above Euler-Lagrange equation we deduce \(\Delta \phi \equiv 0\) in \(B\setminus B_{\varepsilon }\), a contradiction since \(\Delta \phi = 3-\frac{2}{r}\) in \(B\setminus B_{\varepsilon }\). □
We can now conclude by proving the claim, i.e., by proving that \(\beta ({\mathbf {R}}_{*})< \beta (\mathbf {I})\). Let \({\mathbf{v}}=(v_{1},v_{2},v_{3})\in H^{1}(\Omega ,\mathbb{R}^{3})\), \(\tilde {{\mathbf{v}}}:=(v_{1},v_{2})\) and let \(\tilde {\mathbf{u}}\in H^{1}(B,\mathbb{R}^{2})\) be defined by \(\tilde{\mathbf{u}}(x_{1},x_{2}):=\int _{0}^{1} \tilde{{\mathbf{v}}}(x_{1},x_{2},x_{3}) \,dx_{3}\). By Jensen inequality it is readily seen that
where \(\widetilde{\mathbb{E}}(\cdot )\) is the upper-left \(2\times 2\) submatrix of \(\mathbb{E}(\cdot )\). By arguing as in the proof of Theorem 2.7 of [22], if
then the radial function defined by \(w(r):=\int _{0}^{r}\eta _{*}(t)\,dt\) belongs to \(H^{2}(B)\) and \(\nabla w\) minimizes
over \(H^{1}(B)\), hence \(w\) minimizes
among all \(v\) in \(H^{2}(B)\), where \(D^{2}\) denotes the Hessian in the \(x\), \(y\) variables. Therefore for every \(0 < \varepsilon <1\) and for every \(\psi \in H^{2}_{0}(B\setminus B_{\varepsilon })\)
that is, \(w\) solves the biharmonic equation \(8\Delta ^{2} w=-\Delta \phi \) in \(B\setminus B_{\varepsilon }\) and since \(\Delta \phi \) is not identically zero then \(\Delta w\) is not identically zero as well. This implies, by Young inequality,
Notice that the inequality is strict, since the Young inequality \(2(w_{yy}-w_{xx})^{2}\le 4w_{xx}^{2}+4w_{yy}^{2}\) holds with equality if and only if \(w_{xx}=-w_{yy}\), and we have just checked that \(\Delta w\) does not vanish identically on \(B\). In particular, \(2(w_{xx}-w_{yy})^{2}< 4w_{xx}^{2}+4w_{yy}^{2}\) on a set of positive measure in \(B\). From Lemma 5.1 and from (5.6) we infer
as claimed. In particular, Corollary 2.8 applies to this example.
References
Agostiniani, V., Dal Maso, G., De Simone, A.: Linear elasticity obtained from finite elasticity by Gamma-convergence under weak coerciveness conditions. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 29, 715–735 (2012)
Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63, 337–403 (1976/77)
Beatty, M.F.: Some static and dynamic implications of the general theory of elastic stability. Arch. Ration. Mech. Anal. 19, 167–188 (1965)
Charrier, P., Dacorogna, B., Hanouzet, B., Laborde, P.: An existence theorem for slightly compressible material in nonlinear elasticity. SIAM J. Math. Anal. 19, 70–86 (1988)
Chillingworth, D., Marsden, J., Wan, Y.: Symmetry and bifurcation in three-dimensional elasticity. I. Arch. Ration. Mech. Anal. 80, 295–331 (1982)
Chillingworth, D., Marsden, J., Wan, Y.: Symmetry and bifurcation in three-dimensional elasticity. II. Arch. Ration. Mech. Anal. 83, 363–395 (1983)
Ciarlet, P.G.: Mathematical Elasticity, Volume I: Three-Dimensional Elasticity. Elsevier, Amsterdam (1988)
Ciarlet, P.G., Gratie, L., Mardare, C.: Intrinsic methods in elasticity: a mathematical survey. Discrete Contin. Dyn. Syst. 23, 133–164 (2009)
Dal Maso, G., Negri, M., Percivale, D.: Linearized elasticity as \(\Gamma \)-limit of finite elasticity. Set-Valued Anal. 10(2–3), 165–183 (2002)
Fosdick, R.L.: Elastic stability and the zero moment condition. J. Elast. 1, 19–28 (1971)
Fosdick, R.L., MacSithigh, G.P.: Minimization in incompressible nonlinear elasticity theory. J. Elast. 16, 267–301 (1986)
Friesecke, G., James, R.D., Müller, S.: A theorem on geometric rigidity and the derivation of non linear plate theory from three dimensional elasticity. Commun. Pure Appl. Math. 55, 1461–1506 (2002)
Friesecke, G., James, R.D., Müller, S.: A hierarchy of plate models from nonlinear elasticity by Gamma-convergence. Arch. Ration. Mech. Anal. 180, 183–236 (2006)
Holzapfel, G.A.: Nonlinear Solid Mechanics: A Continuum Approach for Engineering. Wiley, Chichester (2000)
Jesenko, M., Schmidt, B.: Geometric linearization of theories for incompressible elastic materials and applications. Math. Models Methods Appl. Sci. 31(4), 829–860 (2021)
Lanza De Cristoforis, M., Valent, T.: On Neumann’s problem for a quasilinear differential system of the finite elastostatics type. Local theorems of existence and uniqueness. Rend. Semin. Mat. Univ. Padova 68, 183–206 (1983)
Le Dret, H.: Structure of the set of equilibrated loads in nonlinear elasticity and applications to existence and nonexistence. J. Elast. 17, 123–141 (1987)
Maddalena, F., Percivale, D., Tomarelli, F.: A new variational approach to linearization of traction problems in elasticity. J. Optim. Theory Appl. 182, 383–403 (2019)
Maddalena, F., Percivale, D., Tomarelli, F.: The gap in pure traction problems between linear elasticity and variational limit of finite elasticity. Arch. Ration. Mech. Anal. 234, 1091–1120 (2019)
Mainini, E., Percivale, D.: Variational linearization of pure traction problems in incompressible elasticity. Z. Angew. Math. Phys. 71, 146 (2020)
Mainini, E., Percivale, D.: Sharp conditions for the linearization of finite elasticity. Calc. Var. Partial Differ. Equ. 60, 164 (2021)
Mainini, E., Percivale, D.: Linearization of elasticity models for incompressible materials. Z. Angew. Math. Phys. 73, 132 (2022)
Maor, C., Mora, M.G.: Reference configurations vs. optimal rotations: a derivation of linear elasticity from finite elasticity for all traction forces. J. Nonlinear Sci. 31, 62 (2021)
Nitsche, J.A.: On Korn second inequality. RAIRO. Anal. Numér. 15, 237–248 (1981)
Schmidt, B.: Linear Gamma-limits of multiwell energies in nonlinear elasticity theory. Contin. Mech. Thermodyn. 20(6), 375–396 (2008)
Stoppelli, F.: Un teorema di esistenza ed unicità relativo alle equazioni dell’elastostatica isoterma per deformazioni finite. Ric. Mat. 3, 247–267 (1954)
Stoppelli, F.: Sulla sviluppabilità in serie di potenze di un parametro delle soluzioni delle equazioni dell’Elastostatica isoterma. Ric. Mat. 4, 58–73 (1955)
Truesdell, C., Noll, W.: The Non-linear Field Theories of Mechanics, 3rd edn. Springer, Berlin (2004)
Valent, T.: Boundary Value Problems of Finite Elasticity. Springer, New York (1988)
Yeoh, O.H.: Characterization of elastic properties of carbon-black-filled rubber vulcanizates. Rubber Chem. Technol. 63(5), 792–805 (1990)
Yeoh, O.H.: Some forms of the strain energy function for rubber. Rubber Chem. Technol. 66(5), 754–771 (1993)
Acknowledgements
The authors wish to warmly thank professor Giuseppe Puglisi for his kind suggestions.
The authors acknowledge support from the MIUR-PRIN project No 2017TEXA3H. The authors are members of the GNAMPA group of the Istituto Nazionale di Alta Matematica (INdAM).
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Mainini, E., Ognibene, R. & Percivale, D. Asymptotic Behavior of Constrained Local Minimizers in Finite Elasticity. J Elast 152, 1–27 (2022). https://doi.org/10.1007/s10659-022-09946-9
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DOI: https://doi.org/10.1007/s10659-022-09946-9
Keywords
- Calculus of variations
- Linear elasticity
- Finite elasticity
- Traction problem
- Gamma-convergence
- Equilibrium equations
- Constrained minimizers