Asymptotic analysis of a junction of hyperelastic rods

Abstract

In this article we obtain a 1-dimensional asymptotic model for a junction of thin hyperelastic rods as the thickness goes to zero. We show, under appropriate hypotheses on the loads, that the deformations that minimize the total energy weakly converge in a Sobolev space towards the minimum of a $1 D$ -dimensional energy for elastic strings by using techniques from Γ-convergence.

Keywords

Junctions thin structures hyperelasticity nonlinear elasticity thin rods asymptotic analysis

1. Introduction

The asymptotic modeling for thin structures (plates, shells, beams, rods etc) by Γ-convergence from $3 D$ -nonlinear elasticity equations has been of special interest for the mathematical community during the last four decades (see, for instance [1,2,5,7,8,14,15,25–27,31]) motivated by applications in engineering. Complex thin structures which are obtained by a junction of much simpler thin structures are of special relevance in this study. Examples of such thin multistructures are bridges (where, for instance, cables are connected to the board of the bridge) and $T -$ or L-shaped junctions of rods.

There exists an extensive literature on dimension reduction problems related to multi-structures in the context of nonlinear hyperelasticity for which we refer to [3,12,13,20–24,30] starting from the pioneering works of Le Dret [23,24]. For literature on multi-structures in contexts other than nonlinear hyperelasticity, we refer to Gaudiello et al. [16–20], Slastikov et al. [29] and, more recently, [6] for a number of interesting problems involving ferroelectricity, electromagnetism, diffusion, etc.

In this article, starting from the $3 D$ -model of a T-junction of two orthogonal nonlinearly hyperelastic thin rods joined to each other (see Fig. 1), we obtain a $1 D$ -model. For obtaining the limit model we shall closely follow the arguments used in Le Dret and Raoult [25] which are classical and consist of rescaling the problem and obtaining the variational limit of rescaled energies, in the framework of Γ-convergence. An important part of the analysis of thin multi-structures consists in establishing the junction condition and for this we shall employ ideas from Gaudiello et al. [16,17].

The paper is organized as follows. We start Section 2 by providing the basic background about the multidomain in the context of 3D elasticity. We then do a rescaling of the problem and fix the appropiate Sobolev spaces for the study of the rescaled problem. The main result (Theorem 2) of the chapter is stated in Section 2.4. We begin Section 3 by proving some crucial results, Lemma 3 and Propositions 4, 5 and then prove Theorem 2 with their help. Finally, in Section 4, we end by computing the 1-d stored energy when the T-shaped structure is made of a Saint Venant–Kirchhoff material. (Proposition 6).

Fig. 1.

The set $Ω_{ε}$ .

2. Preliminaries

2.1. Notation and definitions

Throughout this paper, we denote by $N, R$ , $R^{+}$ and $R^{m \times n}$ the sets of natural, real, nonnegative real numbers and the space of real $m \times n$ matrices endowed with the usual Euclidean norm $‖ F ‖ = \sqrt{tr F^{T} F}$ respectively. For all $z_{i} \in R^{3}$ , $i = 1, 2, 3$ , we note $(z_{1} | z_{2} | z_{3})$ the matrix whose i-th column is $z_{i}$ . In the sequel, $x = (x_{1}, x_{2}, x_{3})$ denotes a generic point in $R^{3}$ .

For all $\overline{F} = (z_{1} | z_{2}) \in R^{3 \times 2}$ and $z_{3} \in R^{3}$ , we also note $(\overline{F} | z_{3})$ the matrix whose first two columns are $z_{1}$ and $z_{2}$ and whose third column is $z_{3}$ . Analogously, for all $\tilde{F} = (z_{2} | z_{3}) \in R^{3 \times 2}$ and $z_{1} \in R^{3}$ , we also note $(z_{1} | \tilde{F})$ the matrix whose first column is $z_{1}$ and the last two columns are $z_{2}$ and $z_{3}$ .

We assume that $W : R^{3 \times 3} ⟶ R$ is a continuous function that satisfies the following growth and coercivity hypotheses: $\begin{array}{l} (2.1) & \exists C > 0, \exists p \in (1, + \infty), \forall F \in R^{3 \times 3}, | W (F) | ⩽ C (1 + ‖ F ‖^{p}), \\ (2.2) & \exists α > 0, \exists β ⩾ 0, \forall F \in R^{3 \times 3}, W (F) ⩾ α ‖ F ‖^{p} - β . \end{array}$

Given such $W : R^{3 \times 3} \to R$ , we introduce two functions $W_{a}, W_{b} : R^{3} ⟶ R$ $\begin{matrix} (2.3) & W_{a} (z_{3}) = inf_{\overline{F} \in R^{3 \times 2}} W ((\overline{F} | z_{3})), W_{b} (z_{1}) = inf_{\tilde{F} \in R^{3 \times 2}} W ((z_{1} | \tilde{F})) . \end{matrix}$ Due to the coercivity assumption (2.2), it is clear that these functions are well defined. Besides, since W is continuous, the infimum for both are attained. Let us briefly state a few properties of $W_{a}$ and $W_{b}$ . The continuity of $W_{a}$ and $W_{b}$ on $R^{3}$ is a consequence of (2.2) (see e.g. [1,25]) and these functions satisfy the growth and coecivity estimates $\begin{array}{l} (2.4) & \exists C^{'} > 0, \forall z \in R^{3} | W_{a} (z) | ⩽ C^{'} (1 + ‖ z ‖^{p}), | W_{b} (z) | ⩽ C^{'} (1 + ‖ z ‖^{p}), \\ (2.5) & W_{a} (z) ⩾ α ‖ z ‖^{p} - β, W_{b} (z) ⩾ α ‖ z ‖^{p} - β . \end{array}$

2.2. Setting up of the three-dimensional problem

For all $ε > 0$ , we introduce the thin multidomain $Ω_{ε} : = Ω_{ε}^{a} \cup Ω_{ε}^{b}$ (see Fig. 1), where $\begin{matrix} Ω_{ε}^{a} = ((- \frac{ε}{2}, \frac{ε}{2}) \times (- \frac{ε}{2}, \frac{ε}{2})) \times [0, 1), Ω_{ε}^{b} = (- \frac{1}{2}, \frac{1}{2}) \times ((- \frac{ε}{2}, \frac{ε}{2}) \times (- ε, 0)) . \end{matrix}$ The multidomain $Ω_{ε}$ models a nonlinearly hyperelastic body consisting of two joined orthogonal rods $Ω_{ε}^{a}$ and $Ω_{ε}^{b}$ with small thickness ε which are joined along the surface $ε {(- \frac{1}{2}, \frac{1}{2})}^{2} \times {0}$ .

Let $\begin{array}{l} Σ_{ε}^{a} = {\pm \frac{ε}{2}} \times (- \frac{ε}{2}, \frac{ε}{2}) \times (0, 1) \cup (- \frac{ε}{2}, \frac{ε}{2}) \times {\pm \frac{ε}{2}} \times (0, 1), \\ Σ_{ε}^{b} = (- \frac{1}{2}, \frac{1}{2}) \times {\pm \frac{ε}{2}} \times (- ε, 0) \cup (- \frac{1}{2}, \frac{1}{2}) \times (- \frac{ε}{2}, \frac{ε}{2}) \times {- ε}, \\ T_{ε}^{a} = {(- \frac{ε}{2}, \frac{ε}{2})}^{2} \times {1}, S_{ε}^{b} = {\pm \frac{1}{2}} \times ((- \frac{ε}{2}, \frac{ε}{2}) \times (- ε, 0)), \end{array}$ where $T_{ε}^{a}$ is the top of the vertical rod, $S_{ε}^{b}$ corresponds to the ends of the horizontal rod; $Σ_{ε}^{a}$ corresponds to the sides of the vertical rod and $Σ_{ε}^{b}$ corresponds to the sides of the horizontal rod except the top portion $((- \frac{1}{2}, \frac{1}{2}) \times (- \frac{ε}{2}, \frac{ε}{2}) ∖ {(- \frac{ε}{2}, \frac{ε}{2})}^{2}) \times {0}$ (we exclude this solely in order to lighten the notations and the proofs).

We consider $Ω_{ε}$ to be the reference configuration of a three-dimensional body made of a nonlinearly hyperelastic homogeneous material and whose stored energy function is denoted by W. We suppose that the structure is solely submitted to the action of dead loading on $Σ_{ε}^{a} \cup Σ_{ε}^{b}$ of traction densities $g^{ε}$ of small order (see (2.11) and (2.16) for the precise assumptions on $g^{ε}$ ) whereas the multidomain does not deform on $T_{ε}^{a} \cup S_{ε}^{b}$ .

To obtain the equilibrium position one has to consider the energy minimization problem $\begin{matrix} (2.6) & inf_{ψ \in Φ_{ε}} I_{ε} (ϕ), \end{matrix}$ for the total energy $I_{ε}$ $\begin{matrix} (2.7) & I_{ε} (ϕ) = \int_{Ω_{ε}} W (D ϕ (x)) d x - \int_{Σ_{ε}^{a} \cup Σ_{ε}^{b}} g^{ε} (x) \cdot ϕ (x) d σ, \end{matrix}$ over the set of admissible deformations $\begin{matrix} (2.8) & Φ_{ε} = {ϕ \in W^{1, p} (Ω_{ε}; R^{3}); ϕ (x) = x on T_{ε}^{a} and on S_{ε}^{b}} . \end{matrix}$ In the above, $d σ$ is the surface element on $Σ_{ε}^{a} \cup Σ_{ε}^{b}$ . Under assumptions (2.1)–(2.2), the energy functional $I_{ε}$ is coercive on $Φ_{ε}$ . Further, as is well known, it will also be sequentially weakly lower semi-continuous on $W^{1, p} (Ω_{ε}; R^{3})$ if W is quasiconvex (see Dacorogna [9]). However, we shall not assume the polyconvexity or quasiconvexity of W since we do not want to rule out important classes of elastic materials such as the Saint-Venant–Kirchhoff which are neither polyconvex nor quasiconvex (see [28]). The convergence results will apply to approximate minimizing sequences as in [25].

2.3. The rescaled problem

As is usual, the problem (2.6) is reformulated on a reference domain $Ω^{a} \cup Ω^{b}$ independent of ε where $\begin{matrix} (2.9) & Ω^{a} = (- \frac{1}{2}, \frac{1}{2}) \times (- \frac{1}{2}, \frac{1}{2}) \times (0, 1), Ω^{b} = (- \frac{1}{2}, \frac{1}{2}) \times (- \frac{1}{2}, \frac{1}{2}) \times (- 1, 0) \end{matrix}$ and we denote by $\begin{array}{l} Σ^{a} = {\pm \frac{1}{2}} \times (- \frac{1}{2}, \frac{1}{2}) \times (0, 1) \cup (- \frac{1}{2}, \frac{1}{2}) \times {\pm \frac{1}{2}} \times (0, 1), \\ Σ^{b} = (- \frac{1}{2}, \frac{1}{2}) \times {\pm \frac{1}{2}} \times (- 1, 0) \cup {(- \frac{1}{2}, \frac{1}{2})}^{2} \times {- 1}, \\ T^{a} = {(- \frac{1}{2}, \frac{1}{2})}^{2} \times {1}, S^{b} = {\pm \frac{1}{2}} \times (- \frac{1}{2}, \frac{1}{2}) \times (- 1, 0) . \end{array}$ The scaling $r_{ε} : (y_{1}, y_{2}, x_{3}) \in Ω^{a} \mapsto (ε y_{1}, ε y_{2}, x_{3}) \in Int Ω_{ε}^{a}$ maps $Ω^{a}, Σ^{a}$ and $T^{a}$ , respectively, to $Ω_{ε}^{a}, Σ_{ε}^{a}$ and $T_{ε}^{a}$ and the scaling $s_{ε} : (x_{1}, y_{2}, y_{3}) \in Ω^{b} \mapsto (x_{1}, ε y_{2}, ε y_{3}) \in Ω_{ε}^{b}$ maps $Ω^{b}, Σ^{b}$ , and $S^{b}$ , respectively, to $Ω_{ε}^{b}, Σ_{ε}^{b}$ and $S_{ε}^{b}$ .

For every $ϕ \in W^{1, p} (Ω_{ε}; R^{3})$ , we define $\begin{matrix} (2.10) & \{\begin{matrix} ψ^{a} (ε) (y_{1}, y_{2}, x_{3}) : = ϕ (r_{ε} (y_{1}, y_{2}, x_{3})) & (y_{1}, y_{2}, x_{3}) \in Ω^{a}, \\ ψ^{b} (ε) (x_{1}, y_{2}, y_{3}) : = ϕ (s_{ε} (x_{1}, y_{2}, y_{3})) & (x_{1}, y_{2}, y_{3}) \in Ω^{b}, \end{matrix} \end{matrix}$ and for any surface density $g^{ε}$ on $Σ_{ε}^{a} \cup Σ_{ε}^{b}$ , we define a rescaled surface density $g (ε)$ defined componentwise on $Σ^{a}$ and $Σ^{b}$ through $\begin{array}{l} (2.11) & \{\begin{matrix} g^{a} (ε) (y_{1}, y_{2}, x_{3}) = ε^{- 1} g^{ε} (ε y_{1}, ε y_{2}, x_{3}), for (y_{1}, y_{2}, x_{3}) \in Σ^{a}, \\ g^{b} (ε) (x_{1}, y_{2}, y_{3}) = ε^{- 1} g^{ε} (x_{1}, ε y_{2}, ε y_{3}), for (x_{1}, y_{2}, y_{3}) \in Σ^{b} . \end{matrix} \end{array}$ Define the set $\begin{array}{l} Ψ (ε) = & {ψ = (ψ^{a}, ψ^{b}) \in W^{1, p} (Ω^{a}; R^{3}) \times W^{1, p} (Ω^{b}; R^{3}) : ψ^{a} = r_{ε} on T^{a}, ψ^{b} = s_{ε} on S^{b}, \\ (2.12) & ψ^{a} (y_{1}, y_{2}, 0) = ψ^{b} (ε y_{1}, y_{2}, 0) in {(- \frac{1}{2}, \frac{1}{2})}^{2}} . \end{array}$ We observe that, if $ϕ \in Φ_{ε}$ then $(ψ^{a} (ε), ψ^{b} (ε))$ given by (2.10) belongs to $Ψ (ε)$ and this defines a bijection between $Φ_{ε}$ and $Ψ_{ε}$ . Also observe that, through a change of variables, we have $\begin{array}{l} I_{ε} (ϕ) = & ε^{2} (\int_{Ω^{a}} W (\frac{1}{ε} \frac{\partial ψ^{a} (ε)}{\partial y_{1}} | \frac{1}{ε} \frac{\partial ψ^{a} (ε)}{\partial y_{2}} | \frac{\partial ψ^{a} (ε)}{\partial x_{3}}) d x \\ + \int_{Ω^{b}} W (\frac{\partial ψ^{b} (ε)}{\partial x_{1}} | \frac{1}{ε} \frac{\partial ψ^{b} (ε)}{\partial y_{2}} | \frac{1}{ε} \frac{\partial ψ^{b} (ε)}{\partial y_{3}}) d x \\ (2.13) & - \int_{Σ^{a}} g^{a} (ε) ψ^{a} (ε) d ζ - \int_{Σ^{b}} g^{b} (ε) ψ^{b} (ε) d ζ) . \end{array}$ In view of the above calculation, we define the rescaled functional $\begin{array}{l} I (ε) (ψ) : = & \int_{Ω^{a}} W (\frac{1}{ε} \frac{\partial ψ^{a}}{\partial x_{1}} | \frac{1}{ε} \frac{\partial ψ^{a}}{\partial x_{2}} | \frac{\partial ψ^{a}}{\partial x_{3}}) d x + \int_{Ω^{b}} W (\frac{\partial ψ^{b}}{\partial x_{1}} | \frac{1}{ε} \frac{\partial ψ^{b}}{\partial x_{2}} | \frac{1}{ε} \frac{\partial ψ^{b}}{\partial x_{3}}) d x \\ (2.14) & - \int_{Σ^{a}} g^{a} (ε) ψ^{a} d ζ - \int_{Σ^{b}} g^{b} (ε) ψ^{b} d ζ . \end{array}$ By these considerations, we are able to establish a correspondence between the minimization problem (2.6) and the following minimization problem $\begin{matrix} (2.15) & inf_{ψ \in Ψ (ε)} I (ε) (ψ) . \end{matrix}$ The goal of this paper is to study the asymptotic behaviour, as $ε \to 0$ , of problem (2.15), under the following assumptions $\begin{matrix} (2.16) & g^{a} (ε) ⇀ g^{a} weakly in L^{q} (Σ^{a}; R^{3}), g^{b} (ε) ⇀ g^{b} weakly in L^{q} (Σ^{b}; R^{3}) \end{matrix}$ where q is the dual exponent to p given by $q = p / (p - 1)$ . As in [25], we can rewrite the problem (2.15) in terms of displacements on the rescaled domain. For this end, we define appropiate spaces. Let, $W_{a}^{1, p} (Ω^{a}; R^{3}) = {u^{a} \in W^{1, p} (Ω^{a}; R^{3}) : u^{a} = 0 on T^{a}}$ and $W_{b}^{1, p} (Ω^{b}; R^{3}) = {u^{b} \in W^{1, p} (Ω^{b}; R^{3}) : u^{b} = 0 on S^{b}}$ and we define $\begin{matrix} (2.17) & Z^{p} = W_{a}^{1, p} (Ω^{a}; R^{3}) \times W_{b}^{1, p} (Ω^{b}; R^{3}) . \end{matrix}$ There is a natural bijection between the deformations in $Ψ (ε)$ and the displacements in $\begin{matrix} (2.18) & V (ε) = {v = (v^{a}, v^{b}) \in Z^{p} : v^{a} (x_{1}, x_{2}, 0) = v^{b} (ε x_{1}, x_{2}, 0) in {(- \frac{1}{2}, \frac{1}{2})}^{2}}, \end{matrix}$ given by $\begin{matrix} (2.19) & v^{a} = ψ^{a} - r_{ε} on Ω^{a}, v^{b} = ψ^{b} - s_{ε} on Ω^{b} . \end{matrix}$ Therefore, in terms of displacements, the rescaled energy (2.14) may be written as: $\begin{array}{l} J (ε) (v^{a}, v^{b}) = & \int_{Ω^{a}} W ((e_{1} + \frac{1}{ε} \frac{\partial v^{a}}{\partial x_{1}} | e_{2} + \frac{1}{ε} \frac{\partial v^{a}}{\partial x_{2}} | e_{3} + \frac{\partial v^{a}}{\partial x_{3}})) d x \\ + \int_{Ω^{b}} W ((e_{1} + \frac{\partial v^{b}}{d x_{1}} | e_{2} + \frac{1}{ε} \frac{\partial v^{b}}{\partial x_{2}} | e_{3} + \frac{1}{ε} \frac{\partial v^{b}}{\partial x_{3}})) d x \\ (2.20) & - \int_{Σ^{a}} g^{a} (ε) \cdot ((0, 0, x_{3}) + v^{a}) d ζ - \int_{Σ^{b}} g^{b} (ε) \cdot ((x_{1}, 0, 0) + v^{b}) d ζ \end{array}$ for $(v^{a}, v^{b}) \in V (ε)$ . Thus the minimization problem (2.15) is equivalent to the following: $\begin{matrix} (2.21) & inf_{v = (v^{a}, v^{b}) \in V (ε)} J (ε) (v) . \end{matrix}$

2.4. The main result

Let us consider the extension $\tilde{J} (ε)$ of the energies $J (ε)$ , defined on $v \in L^{p} (Ω^{a}; R^{3}) \times L^{p} (Ω^{b}; R^{3})$ by $\begin{matrix} (2.22) & \tilde{J} (ε) (v) = \{\begin{matrix} J (ε) (v) & if v \in V (ε), \\ + \infty & otherwise . \end{matrix} \end{matrix}$ The asymptotic behaviour of the energies (2.15) or, equivalently, (2.21) will be obtained through the $Γ -$ limit of the sequence $\tilde{J} (ε)$ for the strong topology on $L^{p} (Ω^{a}; R^{3}) \times L^{p} (Ω^{b}; R^{3})$ when $ε \to 0$ . We refer to Dal Maso [10] and Braides [4] for the definition and main properties of Γ-convergence.

We introduce the following functional space: $\begin{array}{l} V_{J} = {(v^{a}, v^{b}) \in Z^{p} : v^{a} is independent of (x_{1}, x_{2}), v^{b} is independent of (x_{2}, x_{3}), v^{a} (0) = v^{b} (0)} \end{array}$ $V_{J}$ is called the space of displacements on the T-shaped structure, the subscript J refers to the junction condition $v^{a} (0) = v^{b} (0)$ . Note that this condition is meaningful in the trace sense. Indeed, if we take $W_{a}^{1, p} ((0, 1); R^{3}) = {\overline{w} \in W^{1, p} ((0, 1); R^{3}) : \overline{w} (1) = 0}$ and $W_{b}^{1, p} ((- \frac{1}{2}, \frac{1}{2}); R^{3}) = W_{0}^{1, p} ((- \frac{1}{2}, \frac{1}{2}); R^{3})$ , then $V_{J}$ is canonically isomorphic to $\begin{matrix} (2.23) & {\overline{V}}_{J} = {({\overline{v}}^{a}, {\overline{v}}^{b}) \in W_{a}^{1, p} ((0, 1); R^{3}) \times W_{b}^{1, p} ((- \frac{1}{2}, \frac{1}{2}); R^{3}) : {\overline{v}}^{a} (0) = {\overline{v}}^{b} (0)}, \end{matrix}$ The functions of a single variable $({\overline{v}}^{a}, {\overline{v}}^{b}) \in {\overline{V}}_{J}$ are continuous and we denote by $\overline{v} = ({\overline{v}}^{a}, {\overline{v}}^{b})$ the element of ${\overline{V}}_{J}$ that is associated with $v = (v^{a}, v^{b}) \in V_{J}$ through this isomorphism. We also need the following density result for which we refer to Proposition 4.1 in Carbone et al. [6].

Proposition 1.
Let ${\overline{V}}_{J}$ and $V_{reg}$ be defined in ( 2.23 ) and ( 2.24 ) respectively. Then $V_{reg}$ is dense in ${\overline{V}}_{J}$ where $\begin{array}{l} V_{reg} = & {(v^{a}, v^{b}) \in C^{1} ([0, 1]; R^{3}) \times C^{1} ([- \frac{1}{2}, \frac{1}{2}]; R^{3}) : \\ (2.24) & v^{a} (1) = 0, v^{b} (\pm \frac{1}{2}) = 0, v^{a} (0) = v^{b} (0)} . \end{array}$

Let $W_{a}$ and $W_{b}$ be as defined in (2.3) and let $W_{a}^{* }$ and $W_{b}^{ }$ be their respective convex envelopes on $R^{3}$ . Consider the functional $\begin{array}{l} J (0) (v^{a}, v^{b}) = & \int_{0}^{1} W_{a}^{ } (e_{3} + \frac{d {\overline{v}}^{a}}{d x_{3}}) d x_{3} + \int_{- 1 / 2}^{1 / 2} W_{b}^{ *} (e_{1} + \frac{d {\overline{v}}^{b}}{d x_{1}}) d x_{1} \\ (2.25) & - \int_{0}^{1} {\overline{g}}^{a} \cdot ((0, 0, x_{3}) + {\overline{v}}^{a}) d x_{3} - \int_{- 1 / 2}^{1 / 2} {\overline{g}}^{b} \cdot ((x_{1}, 0, 0) + {\overline{v}}^{b}) d x_{1} \end{array}$ for $(v^{a}, v^{b}) \in V_{J}$ , where ${\overline{g}}^{a}, {\overline{g}}^{b}$ are defined below: $\begin{array}{l} (2.26) & {\overline{g}}^{a} (x_{3}) : = \sum_{i = 1}^{2} \int_{- 1 / 2}^{1 / 2} g^{a} (y_{1}, {(- 1)}^{i} \frac{1}{2}, x_{3}) d y_{1} + \sum_{i = 1}^{2} \int_{- 1 / 2}^{1 / 2} g^{a} ({(- 1)}^{i} \frac{1}{2}, y_{2}, x_{3}) d y_{2} \\ (2.27) & {\overline{g}}^{b} (x_{1}) : = \sum_{i = 1}^{2} \int_{- 1}^{0} g^{b} (x_{1}, {(- 1)}^{i} \frac{1}{2}, y_{3}) d y_{3} + \int_{- 1 / 2}^{1 / 2} g^{b} (x_{1}, y_{2}, - 1) d y_{2} \end{array}$ We then extend $J (0)$ to $L^{p} (Ω^{a}; R^{3}) \times L^{p} (Ω^{b}; R^{3})$ as follows $\begin{matrix} (2.28) & \tilde{J} (0) (v) = \{\begin{matrix} J (0) (v), & if v \in V_{J}, \\ + \infty, & otherwise . \end{matrix} \end{matrix}$

The following theorem is our main result.
Theorem 2.
Assume ( 2.16 ), and that there exist $C > 0$ , $α > 0$ , $β ⩾ 0$ , and some $p > 1$ such that the stored energy function $W : R^{3 \times 3} \to R$ of the hyperelastic material satisfies the growth and coercivity conditions ( 2.1 )–( 2.5 ). Then, the sequence of energies $\tilde{J} (ε)$ given in ( 2.22 ) and ( 2.20 ) Γ-converges, as $ε \to 0$ , to $\tilde{J} (0)$ for the strong topology of $L^{p} (Ω^{a}; R^{3}) \times L^{p} (Ω^{b}; R^{3})$ . □

The main consequence of the theorem, as is well known in the theory of Γ-convergence, is that the Γ-convergence and the equicoercivity of the functionals $\tilde{J} (ε)$ guarantee that a sequence of minimizers (in the case that $\tilde{J} (ε)$ admits a minimizer) or even any sequence of approximate minimizers (or the so-called, diagonalizing minimizing sequence) converges (up to a subsequence) to a minimum of $\tilde{J} (0)$ which does exist. The reader may refer to Section 4 of Le Dret and Raoult [26] for more details about such arguments.
3. Proof of the main theorem

For establishing the limit model using classical arguments from Γ-convergence we will closely follow the arguments laid out in [25, Section 3]. As is to be expected, while dealing with multi-structures, obtaining the junction conditions is another aspect which needs to be dealt with and for this we follow Gaudiello and Hadiji [17]. For clarity, we decompose the proof of Theorem 2 into a series of various comparatively simple results. The assumptions on W are as in the theorem for the following lemma.

Lemma 3.
Consider any sequence $u (ε) = (u^{a} (ε), u^{b} (ε)) \in L^{p} (Ω^{a}; R^{3}) \times L^{p} (Ω^{b}; R^{3})$ for which $\tilde{J} (ε) (u (ε)) ⩽ C < + \infty$ where C does not depend on ε. Then $u (ε)$ is uniformly bounded in $Z^{p}$ and any limit point for the weak topology of $Z^{p}$ belongs to $V_{J}$ .
Proof.
Let $u (ε) = (u^{a} (ε), u^{b} (ε)) \in L^{p} (Ω^{a}; R^{3}) \times L^{p} (Ω^{b}; R^{3})$ be such that $\tilde{J} (ε) (u (ε)) ⩽ C < + \infty$ for all $ε > 0$ . Then, the definition (2.22) implies that $u (ε) = (u^{a} (ε), u^{b} (ε)) \in V (ε)$ for all $ε > 0$ . Let us call $ψ^{a} (ε) = u^{a} (ε) + r_{ε}$ and $ψ^{b} (ε) = u^{b} (ε) + s_{ε}$ , the deformations that are associated with the displacements $u^{a} (ε)$ and $u^{b} (ε)$ , respectively, where $r_{ε}$ and $s_{ε}$ are as defined in Section 2.3. The boundedness of the sequence $\tilde{J} (ε) (u (ε))$ , the weak convergence hypothesis (2.16) which gives bounds in $L^{q}$ , together with the growth and coercivity estimates (2.1)–(2.2) of the function W, imply that $\begin{array}{l} α \int_{Ω^{a}} {| (\frac{1}{ε} \partial_{1} ψ^{a} (ε) | \frac{1}{ε} \partial_{2} ψ^{a} (ε) | \partial_{3} ψ^{a} (ε)) |}^{p} d x - β \\ ⩽ \int_{Ω^{a}} W ((\frac{1}{ε} \partial_{1} ψ^{a} (ε) | \frac{1}{ε} \partial_{2} ψ^{a} (ε) | \partial_{3} ψ^{a} (ε))) d x \\ (3.1) & ⩽ C^{″} (1 + {‖ ψ^{a} (ε) ‖}_{W^{1, p} (Ω^{a}; R^{3})}), \\ α \int_{Ω^{b}} {| (\partial_{1} ψ^{b} (ε) | \frac{1}{ε} \partial_{2} ψ^{b} (ε) | \frac{1}{ε} \partial_{3} ψ^{b} (ε)) |}^{p} d x - β \\ ⩽ \int_{Ω^{b}} W ((\partial_{1} ψ^{b} (ε) | \frac{1}{ε} \partial_{2} ψ^{b} (ε) | \frac{1}{ε} \partial_{3} ψ^{b} (ε))) d x \\ (3.2) & ⩽ C^{″} (1 + {‖ ψ^{b} (ε) ‖}_{W^{1, p} (Ω^{b}; R^{3})}), \end{array}$ where $C^{″}$ does not depend on ε. It is clear that for $ε > 0$ such that $ε ⩽ 1$ , we have $\begin{array}{l} ‖ (ε^{- 1} z_{1} | ε^{- 1} z_{2} | z_{3}) ‖ ⩾ ‖ (z_{1} | z_{2} | z_{3}) ‖, ‖ (z_{1} | ε^{- 1} z_{2} | ε^{- 1} z_{3}) ‖ ⩾ ‖ (z_{1} | z_{2} | z_{3}) ‖ . \end{array}$ Therefore (3.1) and (3.2) imply that $\begin{array}{l} (3.3) & α {‖ D ψ^{a} (ε) ‖}_{L^{p} (Ω^{a}; R^{3})}^{p} ⩽ C^{″} (1 + {‖ ψ^{a} (ε) ‖}_{W^{1, p} (Ω^{a}; R^{3})}), \\ (3.4) & α {‖ D ψ^{b} (ε) ‖}_{L^{p} (Ω^{b}; R^{3})}^{p} ⩽ C^{″} (1 + {‖ ψ^{b} (ε) ‖}_{W^{1, p} (Ω^{b}; R^{3})}), \end{array}$ which, together with the clamped conditions $\begin{matrix} (3.5) & ψ^{a} (ε) = r_{ε} on T^{a} and ψ^{b} (ε) = s_{ε} on S^{b} \end{matrix}$ yield the desired uniform bound for $ψ^{a} (ε)$ and $ψ^{b} (ε)$ in $W^{1, p} (Ω^{a}; R^{3})$ and $W^{1, p} (Ω^{b}; R^{3})$ , respectively, by Poincaré inequality.

Then, going back to (3.1), (3.2) and using a priori bounds on $ψ^{a} (ε)$ and $ψ^{b} (ε)$ , we obtain $\begin{array}{l} (3.6) & {‖ \partial_{1} ψ^{a} (ε) ‖}_{L^{p} (Ω^{a}; R^{3})} ⩽ C^{‴} ε, {‖ \partial_{2} ψ^{a} (ε) ‖}_{L^{p} (Ω^{a}; R^{3})} ⩽ C^{‴} ε and, \\ (3.7) & {‖ \partial_{2} ψ^{b} (ε) ‖}_{L^{p} (Ω^{b}; R^{3})} ⩽ C^{‴} ε, {‖ \partial_{3} ψ^{b} (ε) ‖}_{L^{p} (Ω^{b}; R^{3})} ⩽ C^{‴} ε . \end{array}$ Therefore, $\begin{array}{l} (3.8) & \begin{aligned} \partial_{1} ψ^{a} (ε) \to 0 strongly in L^{p} (Ω^{a}; R^{3}), \\ \partial_{2} ψ^{a} (ε) \to 0 strongly in L^{p} (Ω^{a}; R^{3}) and, \\ \partial_{2} ψ^{b} (ε) \to 0 strongly in L^{p} (Ω^{b}; R^{3}), \\ \partial_{3} ψ^{b} (ε) \to 0 strongly in L^{p} (Ω^{b}; R^{3}) . \end{aligned} \end{array}$ If we denote by $ψ^{a}$ and $ψ^{b}$ , the limit points of the sequences $ψ^{a} (ε)$ and $ψ^{b} (ε)$ , respectively, in the weak topology of the corresponding spaces $W^{1, p} (Ω^{a}; R^{3})$ and $W^{1, p} (Ω^{b}; R^{3})$ , then it follows at once, from (3.8), that $\begin{matrix} (3.9) & \partial_{1} ψ^{a} = 0, \partial_{2} ψ^{a} = 0, \partial_{2} ψ^{b} = 0, \partial_{3} ψ^{b} = 0 . \end{matrix}$ Thus, $ψ^{a}$ is independent of $(x_{1}, x_{2})$ and $ψ^{b}$ is independent of $(x_{2}, x_{3})$ . From the by boundary conditions (3.5) for $ψ = (ψ^{a}, ψ^{b})$ and the weak convergences of the traces in $L^{p} (T^{a})$ and $L^{p} (S^{b})$ it follows that $ψ^{a} (x) = (0, 0, x_{3})$ on $T^{a}$ and $ψ^{b} (x) = (x_{1}, 0, 0)$ on $S^{b}$ .

It remains to prove the junction condition $\begin{matrix} (3.10) & ψ^{a} (0) = ψ^{b} (0) . \end{matrix}$

Depending on the model or the problem being considered, the junction condition which appears in the limit can be quite different and at the same time the techniques needed to establish it also could vary. In general, to achieve this requires the handling of the traces around the junctions in the 3-d models in a suitable way. For the current analysis, we refer to the approach used in [16] and [17]. Remember that we have $ψ^{a} (ε) (x_{1}, x_{2}, 0) = ψ^{b} (ε) (ε x_{1}, x_{2}, 0)$ in ${(- \frac{1}{2}, \frac{1}{2})}^{2}$ . To begin with, the sequences $ψ^{a} (ε)$ and $ψ^{b} (ε)$ are bounded in $W^{1, p} (Ω^{a})$ and $W^{1, p} (Ω^{b})$ respectively and, as a consequence of the compactness of the trace map at $x_{3} = 0$ from $W^{1, p} (Ω^{a})$ or $W^{1, p} (Ω^{b})$ into $L^{1} ({(- \frac{1}{2}, \frac{1}{2})}^{2})$ for any $p > 1$ , we have that the sequence of traces $ψ^{a} (ε) (\cdot, 0)$ and $ψ^{b} (ε) (\cdot, 0)$ converge strongly in $L^{1} ({(- \frac{1}{2}, \frac{1}{2})}^{2})$ . The $L^{1}$ convergence of $ψ^{a} (ε) (\cdot, 0)$ is enough to pass to the limit in the relation $ψ^{a} (ε) (x_{1}, x_{2}, 0) = ψ^{b} (ε) (ε x_{1}, x_{2}, 0)$ in ${(- \frac{1}{2}, \frac{1}{2})}^{2}$ whereas the $L^{1}$ convergence of $ψ^{b} (ε) (\cdot, 0)$ is not strong enough due to the concentration of the first argument near $x_{1} = 0$ . We require a stronger form of convergence, for example, the strong convergence of the traces $ψ^{b} (ε) (\cdot, 0)$ in $C ({(- \frac{1}{2}, \frac{1}{2})}^{2})$ . This cannot be guaranteed, but in view of the last estimate in (3.7), it turns out that it is sufficient to find at least a height, $x_{3}^{} \in [- 1, 0]$ , at which the traces $ψ^{b} (ε) (\cdot, x_{3}^{})$ are compact in $C ({(- \frac{1}{2}, \frac{1}{2})}^{2})$ . Then using this we can pass to the limit in $\begin{matrix} (3.11) & \int_{{(- \frac{1}{2}, \frac{1}{2})}^{2}} ψ^{a} (ε) (x_{1}, x_{2}, 0) d x_{1} d x_{2} = \int_{{(- \frac{1}{2}, \frac{1}{2})}^{2}} ψ^{b} (ε) (ε x_{1}, x_{2}, 0) d x_{1} d x_{2} . \end{matrix}$ Passing to the limit on the left-hand side we obtain $\begin{matrix} (3.12) & lim_{ε \to 0} \int_{{(- \frac{1}{2}, \frac{1}{2})}^{2}} ψ^{a} (ε) (x_{1}, x_{2}, 0) d x_{1} d x_{2} = ψ^{a} (0), \end{matrix}$ since, we have observed, following (3.9), that $ψ^{a}$ is constant with respect to $x_{1}$ and $x_{2}$ .

Then, to calculate the limit on the right hand side of (3.11) we write $\begin{array}{l} \int_{{(- \frac{1}{2}, \frac{1}{2})}^{2}} ψ^{b} (ε) (ε x_{1}, x_{2}, 0) d x_{1} d x_{2} \\ = \int_{{(- \frac{1}{2}, \frac{1}{2})}^{2}} (ψ^{b} (ε) (ε x_{1}, x_{2}, 0) - ψ^{b} (ε) (ε x_{1}, x_{2}, x_{3}^{})) d x_{1} d x_{2} \\ (3.13) & + \int_{{(- \frac{1}{2}, \frac{1}{2})}^{2}} ψ^{b} (ε) (ε x_{1}, x_{2}, x_{3}^{}) d x_{1} d x_{2} . \end{array}$ Then, it can be shown that the first term on the right hand side in the above goes to 0 as $ε \to 0$ using in a crucial way the last estimate in (3.7). Whereas, the second integral converges to $ψ^{b} (0)$ since we have the strong convergence of $ψ^{b} (ε) (ε x_{1}, x_{2}, x_{3}^{})$ to $ψ^{b} (0, x_{2}, x_{3}^{})$ in $C ({(- \frac{1}{2}, \frac{1}{2})}^{2})$ and $ψ^{b} (0, x_{2}, x_{3}^{})$ is constant in $x_{2}$ and $x_{3}$ . At last, we obtain $\begin{matrix} (3.14) & lim_{ε \to 0} \int_{{(- \frac{1}{2}, \frac{1}{2})}^{2}} ψ^{b} (ε) (ε x_{1}, x_{2}, 0) d x_{1} d x_{2} = ψ^{b} (0) . \end{matrix}$ Thus, from (3.12) and (3.14) the desired conclusion (3.10) follows. For more details we refer to the arguments in [16] or [17].

Finally, if $u^{a}$ and $u^{b}$ denote the corresponding limits of the sequences $u^{a} (ε)$ and $u^{b} (ε)$ respectively, since $u^{a} (x) = ψ^{a} (x) - (0, 0, x_{3})$ , $u^{b} (x) = ψ^{b} (x) - (x_{1}, 0, 0)$ , then $u^{a}$ is independent of $(x_{1}, x_{2})$ , $u^{b}$ is independent of $(x_{2}, x_{3})$ , $u^{a} (x) = 0$ on $T^{a}$ , $u^{b} (x) = 0$ on $S^{b}$ and $u^{a} (0) = u^{b} (0)$ follows directly from (3.10). Therefore, $(u^{a}, u^{b}) \in V_{J}$ . □

We now prove the so-called Γ-liminf inequality. Proposition 4.
Let* (2.16) hold. Then, for any sequence $(v^{a} (ε), v^{b} (ε)) \in L^{p} (Ω^{a}; R^{3}) \times L^{p} (Ω^{b}; R^{3})$ , such that $(v^{a} (ε), v^{b} (ε)) \to (v^{a}, v^{b}) : = v$ strongly in $L^{p} (Ω^{a}; R^{3}) \times L^{p} (Ω^{b}; R^{3})$ we have that $\begin{matrix} (3.15) & \underset{ε \to 0}{lim inf} \tilde{J} (ε) ((v^{a} (ε), v^{b} (ε))) ⩾ \tilde{J} (0) (v) \end{matrix}$
Proof.
If $lim {inf}_{ε \to 0} \tilde{J} (ε) ((v^{a} (ε), v^{b} (ε))) = + \infty$ there is nothing to prove. Without loss of generality we assume that $\tilde{J} (ε) ((v^{a} (ε), v^{b} (ε)))$ is bounded from above. We have seen, in Lemma 3, that this implies that the sequence $(v^{a} (ε), v^{b} (ε)) \in V (ε)$ , is bounded in $Z^{p}$ and any weak $W^{1, p}$ limit, which necessarily coincides with $(v^{a}, v^{b})$ , belongs to $V_{J}$ . To obtain the inequality (3.15) we use the fact that $\tilde{J} (ε) ((v^{a} (ε), v^{b} (ε)))$ can be written as in (2.20) and observe the following facts. The weak convergence of $v^{a} (ε)$ and $v^{b} (ε)$ in $W^{1, p}$ implies the strong convergence in $L^{p}$ of their traces which, by (2.16), implies that, $\begin{array}{l} \int_{Σ^{a}} g^{a} (ε) \cdot ((0, 0, x_{3}) + v^{a} (ε)) d ζ + \int_{Σ^{b}} g^{b} (ε) \cdot ((x_{1}, 0, 0) + v^{b} (ε)) d ζ \\ \overset{ε \to 0}{⟶} \int_{Σ^{a}} g^{a} \cdot ((0, 0, x_{3}) + v^{a}) d ζ + \int_{Σ^{b}} g^{b} \cdot ((x_{1}, 0, 0) + v^{b}) d ζ \\ (3.16) & = \int_{0}^{1} {\overline{g}}^{a} \cdot ((0, 0, x_{3}) + {\overline{v}}^{a}) d x_{3} - \int_{- 1 / 2}^{1 / 2} {\overline{g}}^{b} \cdot ((x_{1}, 0, 0) + {\overline{v}}^{b}) d x_{1} . \end{array}$ For the elastic energy, we have that (with $ψ^{a} (ε) : = v^{a} (ε) + r_{ε}$ and $ψ^{b} (ε) : = v^{b} (ε) + s_{ε}$ , as usual) $\begin{array}{l} \int_{Ω^{a}} W ((\frac{1}{ε} \partial_{1} ψ^{a} (ε) | \frac{1}{ε} \partial_{2} ψ^{a} (ε) | \partial_{3} ψ^{a} (ε))) d x \\ ⩾ \int_{Ω^{a}} W_{a} ((\partial_{3} ψ^{a} (ε))) d x \\ (3.17) & ⩾ \int_{Ω^{a}} W_{a}^{* } ((\partial_{3} ψ^{a} (ε))) d x : = G^{a} (ψ^{a} (ε)) \\ \int_{Ω^{b}} W ((\partial_{1} ψ^{b} (ε) | \frac{1}{ε} \partial_{2} ψ^{b} (ε) | \frac{1}{ε} \partial_{3} ψ^{b} (ε))) d x \\ ⩾ \int_{Ω^{b}} W_{b} ((\partial_{1} ψ^{b} (ε))) d x \\ (3.18) & ⩾ \int_{Ω^{b}} W_{b}^{ } ((\partial_{1} ψ^{b} (ε))) d x : = G^{b} (ψ^{b} (ε)) . \end{array}$ The integral functionals $G^{a}$ and $G^{b}$ , defined on $W^{1, p} (Ω^{a}; R^{3})$ and $W^{1, p} (Ω^{b}; R^{3})$ respectively, are weakly lower semicontinuous and we have the weak convergences $ψ^{a} (ε) ⇀ ψ^{a} = v^{a} + r_{0}$ in $W^{1, p} (Ω^{a}; R^{3})$ and $ψ^{b} (ε) ⇀ ψ^{b} = v^{b} + s_{0}$ in $W^{1, p} (Ω^{b}; R^{3})$ . Therefore, $\begin{array}{l} \underset{ε \to 0}{lim inf} \int_{Ω^{a}} W ((\frac{1}{ε} \partial_{1} ψ^{a} (ε) | \frac{1}{ε} \partial_{2} ψ^{a} (ε) | \partial_{3} ψ^{a} (ε))) d x \\ (3.19) & ⩾ G^{a} (ψ^{a}) = \int_{0}^{1} W_{a}^{ } (e_{3} + \partial_{3} {\overline{v}}^{a}) d x_{3} \\ \underset{ε \to 0}{lim inf} \int_{Ω^{b}} W ((\partial_{1} ψ^{b} (ε) | \frac{1}{ε} \partial_{2} ψ^{b} (ε) | \frac{1}{ε} \partial_{3} ψ^{b} (ε))) d x \\ (3.20) & ⩾ G^{b} (ψ^{b}) = \int_{- 1 / 2}^{1 / 2} W_{b}^{ } (e_{1} + \partial_{1} {\overline{v}}^{b}) d x_{1} . \end{array}$ So, while observing that $\tilde{J} (ε) ((v^{a} (ε), v^{b} (ε)))$ can be decomposed into three parts, that the lim inf of a sum is greater than the sum of the lim inf and using the observed facts (3.16), (3.19) and (3.20), the desired conclusion (3.15) follows. □
Proposition 5.
Given $({\overline{v}}^{a}, {\overline{v}}^{b})$ in $V_{r e g}$ , $w_{1}, w_{2}$ in $C^{1} ([0, 1]; R^{3})$ vanishing at 1 and $ζ_{2}, ζ_{3}$ in $C^{1} ([- \frac{1}{2}, \frac{1}{2}]; R^{3})$ vanishing at $- 1 / 2$ and $1 / 2$ , there exists $(v^{a} (ε), v^{b} (ε)) \in L^{p} (Ω^{a}; R^{3}) \times L^{p} (Ω^{b}; R^{3})$ which converges strongly to $({\overline{v}}^{a}, {\overline{v}}^{b})$ and also such that $\begin{array}{l} lim_{ε \to 0} \tilde{J} (ε) ((v^{a} (ε), v^{b} (ε))) \\ = \int_{0}^{1} W ((e_{1} + w_{1} (x_{3}) | e_{2} + w_{2} (x_{3}) | e_{3} + \frac{d \overline{v^{a}}}{d z} (x_{3}))) d x_{3} \\ + \int_{- 1 / 2}^{1 / 2} W ((e_{1} + \frac{d \overline{v^{b}}}{d z} (x_{1}) | e_{2} + ζ_{2} (x_{1}) | e_{3} + ζ_{3} (x_{1}))) d x_{1} \\ (3.21) & - \int_{0}^{1} {\overline{g}}^{a} \cdot ((0, 0, x_{3}) + {\overline{v}}^{a}) d x_{3} - \int_{- 1 / 2}^{1 / 2} {\overline{g}}^{b} \cdot ((x_{1}, 0, 0) + {\overline{v}}^{b}) d x_{1} . \end{array}$
Proof.
Given $({\overline{v}}^{a}, {\overline{v}}^{b}) \in V_{r e g}$ , $w_{1}, w_{2} \in C^{1} ([0, 1]; R^{3})$ which vanish at 1 and $ζ_{2}, ζ_{3} \in C^{1} ([- \frac{1}{2}, \frac{1}{2}]; R^{3})$ vanishing at $- 1 / 2$ and $1 / 2$ , we define the displacements: $\begin{array}{l} (3.22) & v^{a} (ε) (x) = \{\begin{matrix} {\overline{v}}^{a} (x_{3}) + ε x_{2} w_{2} (x_{3}) + ε x_{1} w_{1} (x_{3}), & if x \in {(- \frac{1}{2}, \frac{1}{2})}^{2} \times (ε, 1), \\ \frac{x_{3}}{ε} ({\overline{v}}^{a} (ε) + ε x_{2} w_{2} (ε) + ε x_{1} w_{1} (ε)) \\ + \frac{ε - x_{3}}{ε} [{\overline{v}}^{b} (ε x_{1}) + ε x_{2} ζ_{2} (ε x_{1}) + ε x_{3} ζ_{3} (ε x_{1})] & if x \in {(- \frac{1}{2}, \frac{1}{2})}^{2} \times [0, ε] \end{matrix} \\ (3.23) & v^{b} (ε) (x) = {\overline{v}}^{b} (x_{1}) + ε x_{2} ζ_{2} (x_{1}) + ε x_{3} ζ_{3} (x_{1}) if x \in {(- \frac{1}{2}, \frac{1}{2})}^{2} \times (- 1, 0) . \end{array}$ whose choice is inspired by those made in Gaudiello et al. [16,17]. The construction of $v (ε) = (v^{a} (ε), v^{b} (ε))$ and the boundary conditions on $(w_{2}, ζ_{2}), (w_{1}, ζ_{3})$ , allow to conclude that $v (ε) \in V (ε)$ .

The strong convergence of $(v^{a} (ε), v^{b} (ε))$ , in $L^{p} (Ω^{a}; R^{3}) \times L^{p} (Ω^{b}; R^{3})$ , to $({\overline{v}}^{a}, {\overline{v}}^{b})$ is obtained without much difficulty. In fact, it is easy to check the almost everywhere convergence and the boundedness in $L^{p}$ for which the regularity (up to the boundary) of $w_{1}$ , $w_{2}$ , $ζ_{2}$ and $ζ_{3}$ is useful.

We now proceed to show that (3.21) holds. For this we begin by calculating the expressions for the partial derivatives of $(v^{a} (ε) and v^{b} (ε))$ and obtain, $\begin{array}{l} \frac{\partial v^{a} (ε)}{\partial x_{1}} (x) = \{\begin{matrix} ε w_{1} (x_{3}), & if x \in {(- \frac{1}{2}, \frac{1}{2})}^{2} \times (ε, 1), \\ x_{3} w_{1} (ε) + (ε - x_{3}) [\frac{d {\overline{v}}^{b}}{d z} + ε x_{2} \frac{d ζ_{2}}{d z} + ε x_{3} \frac{d ζ_{3}}{d z}] (ε x_{1}) & if x \in {(- \frac{1}{2}, \frac{1}{2})}^{2} \times [0, ε], \end{matrix} \\ \frac{\partial v^{a} (ε)}{\partial x_{2}} (x) = \{\begin{matrix} ε w_{2} (x_{3}), & if x \in {(- \frac{1}{2}, \frac{1}{2})}^{2} \times (ε, 1), \\ x_{3} w_{2} (ε) + (ε - x_{3}) ζ_{2} (ε x_{1}) & if x \in {(- \frac{1}{2}, \frac{1}{2})}^{2} \times [0, ε], \end{matrix} \\ \frac{\partial v^{a} (ε)}{\partial x_{3}} (x) = \{\begin{matrix} \frac{d {\overline{v}}^{a}}{d z} (x_{3}) + ε x_{2} \frac{d w_{2}}{d z} (x_{3}) + ε x_{1} \frac{d w_{1}}{d z} (x_{3}) & if x \in {(- \frac{1}{2}, \frac{1}{2})}^{2} \times (ε, 1), \\ \frac{1}{ε} ({\overline{v}}^{a} (ε) + ε x_{2} w_{2} (ε) + ε x_{1} w_{1} (ε)) \\ - \frac{1}{ε} ({\overline{v}}^{b} (ε x_{1}) + ε x_{2} ζ_{2} (ε x_{1}) + ε x_{3} ζ_{3} (ε x_{1})) \\ + (ε - x_{3}) ζ_{3} (ε x_{1}) & if x \in {(- \frac{1}{2}, \frac{1}{2})}^{2} \times [0, ε], \end{matrix} \end{array}$ and also $\begin{array}{l} \frac{\partial v^{b} (ε)}{\partial x_{1}} (x) = \frac{d {\overline{v}}^{b}}{d z} (x_{1}) + ε x_{2} \frac{d ζ_{2}}{d z} (x_{1}) + ε x_{3} \frac{d ζ_{3}}{d z} (x_{1}), \\ \frac{\partial v^{b} (ε)}{\partial x_{2}} (x) = ε ζ_{2} (x_{1}), \\ \frac{\partial v^{b} (ε)}{\partial x_{3}} (x) = ε ζ_{3} (x_{1}) . \end{array}$ The calculated expressions for the derivatives of $v^{b} (ε)$ allows us to see that the sequence $v^{b} (ε)$ converges strongly to ${\overline{v}}^{b}$ in $W^{1, p} (Ω^{b}; R^{3})$ . In the upper part, $Ω_{a}$ , using the assumed regularity of ${\overline{v}}^{b}$ , $w_{1}$ , $w_{2}$ , $ζ_{2}$ and $ζ_{3}$ we can see that $\frac{\partial v^{a} (ε)}{\partial x_{1}} (x)$ and $\frac{\partial v^{a} (ε)}{\partial x_{2}} (x)$ are of the order ε in $L^{\infty}$ . To deal with $\frac{\partial v^{a} (ε)}{\partial x_{3}} (x)$ , we use the condition ${\overline{v}}^{a} (0) = {\overline{v}}^{b} (0)$ , the regularity of ${\overline{v}}^{a}$ , ${\overline{v}}^{b}$ and the mean value theorem to obtain the estimate $\begin{array}{l} | ε^{- 1} {\overline{v}}^{a} (ε) - ε^{- 1} {\overline{v}}^{b} (ε x_{1}) | & = | ε^{- 1} ({\overline{v}}^{a} (ε) - {\overline{v}}^{a} (0)) - ε^{- 1} ({\overline{v}}^{b} (ε x_{1}) - {\overline{v}}^{b} (0)) | \\ (3.24) & ⩽ sup_{z \in [0, ε]} \frac{d {\overline{v}}^{a}}{d z} (z) + sup_{y_{1} \in [- 1 / 2, 1 / 2]} \frac{d {\overline{v}}^{b}}{d z} (ε y_{1}) . \end{array}$ From this it can be concluded that $\frac{\partial v^{a} (ε)}{\partial x_{3}} (x)$ converges strongly in $L^{p}$ to $\frac{d {\overline{v}}^{a}}{d z} (x_{3})$ although the convergence is not in $L^{\infty}$ . So, putting together all this information we conclude that the sequence $(v^{a} (ε), v^{b} (ε))$ converges strongly to $({\overline{v}}^{a}, {\overline{v}}^{b})$ in $W^{1, p} (Ω^{a}; R^{3}) \times W^{1, p} (Ω^{b}; R^{3})$ . From this we have the convergence (3.16) of the boundary terms like in the earlier proposition.

The convergence obtained for $(v^{a} (ε), v^{b} (ε))$ in $W^{1, p} (Ω^{a}; R^{3}) \times W^{1, p} (Ω^{b}; R^{3})$ is not enough for the volume terms. To this end, we note that the sequence $(e_{1} + \frac{1}{ε} \frac{\partial v^{a} (ε)}{\partial x_{1}} | e_{2} + \frac{1}{ε} \frac{\partial v^{a} (ε)}{\partial x_{2}} | e_{3} + \frac{\partial v^{a} (ε)}{\partial x_{3}})$ is bounded in $L^{\infty} (Ω^{a}; R^{3 \times 3})$ and we have the strong convergence in $L^{p}$ $\begin{array}{l} (3.25) & (e_{1} + \frac{1}{ε} \frac{\partial v^{a} (ε)}{\partial x_{1}} | e_{2} + \frac{1}{ε} \frac{\partial v^{a} (ε)}{\partial x_{2}} | e_{3} + \frac{\partial v^{a} (ε)}{\partial x_{3}}) & \overset{ε \to 0}{⟶} (e_{1} + w_{1} | e_{2} + w_{2} | e_{3} + \frac{d {\overline{v}}^{a}}{d z} (x_{3})) . \end{array}$ From this it follows, using the growth and continuity assumptions on W and by the dominated convergence theorem, that $\begin{array}{l} \int_{Ω^{a}} W ((e_{1} + \frac{1}{ε} \frac{\partial v^{a} (ε)}{\partial x_{1}} | e_{2} + \frac{1}{ε} \frac{\partial v^{a} (ε)}{\partial x_{2}} | e_{3} + \frac{\partial v^{a} (ε)}{\partial x_{3}})) d x \\ \to \int_{0}^{1} W ((e_{1} + w_{1} (x_{3}) | e_{2} + w_{2} (x_{3}) | e_{3} + \frac{d \overline{v^{a}}}{d z} (x_{3}))) d x_{3} . \end{array}$ Similarly, $(e_{1} + \frac{\partial v^{b} (ε)}{d x_{1}} | e_{2} + \frac{1}{ε} \frac{\partial v^{b} (ε)}{\partial x_{2}} | e_{3} + \frac{1}{ε} \frac{\partial v^{b} (ε)}{\partial x_{3}})$ is bounded in $L^{\infty} (Ω^{a}; R^{3 \times 3})$ and we have the strong convergence in $L^{p}$ $\begin{array}{l} (3.26) & (e_{1} + \frac{\partial v^{b} (ε)}{d x_{1}} | e_{2} + \frac{1}{ε} \frac{\partial v^{b} (ε)}{\partial x_{2}} | e_{3} + \frac{1}{ε} \frac{\partial v^{b} (ε)}{\partial x_{3}}) & \overset{ε \to 0}{⟶} (e_{1} + \frac{d {\overline{v}}^{b}}{d z} (x_{1}) | e_{2} + ζ_{2} | e_{3} + ζ_{3}) . \end{array}$ So, similarly, on the lower part, we obtain $\begin{array}{l} \int_{Ω^{b}} W ((e_{1} + \frac{\partial v^{b} (ε)}{d x_{1}} | e_{2} + \frac{1}{ε} \frac{\partial v^{b} (ε)}{\partial x_{2}} | e_{3} + \frac{1}{ε} \frac{\partial v^{b} (ε)}{\partial x_{3}})) d x \\ \to \int_{- 1 / 2}^{1 / 2} W ((e_{1} + \frac{d \overline{v^{b}}}{d z} (x_{1}) | e_{2} + ζ_{2} (x_{1}) | e_{3} + ζ_{3} (x_{1}))) d x_{1} . \end{array}$ The conclusion of the proposition follows from these observations. □
The proof of Theorem 2.
It is well known that in a separable metric space, the sequence $\tilde{J} (ε)$ always has a Γ-convergent subsequence (see [4]). We shall now show that any Γ-limit coincides with $\tilde{J} (0)$ defined in Section 2.4. It follows by the Urysohn property of Γ-convergence [4, Proposition 1.44] that the entire sequence Γ-converges to $\tilde{J} (0)$ proving the theorem. For the sake of convenience we will split the proof into several steps.

Step 1: To begin with, we consider a subsequence of $\tilde{J} (ε)$ which Γ-converges and for convenience, index it by ε. Also let $\overline{J}$ be the Γ-limit of this subsequence which is then characterized by the following equalities (see Braides [4]): $\begin{array}{l} \overline{J} (v) & = inf {\underset{ε \to 0}{lim inf} \tilde{J} (ε) (v (ε)) : v (ε) \to v \in L^{p} (Ω^{a}; R^{3}) \times L^{p} (Ω^{b}; R^{3})} \\ = inf {\underset{ε \to 0}{lim sup} \tilde{J} (ε) (v (ε)) : v (ε) \to v \in L^{p} (Ω^{a}; R^{3}) \times L^{p} (Ω^{b}; R^{3})} . \end{array}$ Indeed, the first expression on the right hand side is called the Γ-lower limit (Γ-lim inf for short) and the second expression on the right hand side is called the Γ-upper limit (Γ-lim sup for short) and the equality between the two corresponds to the existence of the Γ-limit which has been denoted by $\overline{J}$ .

Step 2: From the Γ-lim inf representation of $\overline{J}$ , it follows, using Proposition 4, by taking the infimum over all sequences $v (ε)$ that converge to v in $L^{p} (Ω^{a}; R^{3}) \times L^{p} (Ω^{b}; R^{3})$ , that $\begin{array}{l} (3.27) & \overline{J} (v) ⩾ \tilde{J} (0) (v) for all v \in L^{p} (Ω^{a}; R^{3}) \times L^{p} (Ω^{b}; R^{3}) . \end{array}$ Step 3: Now, it remains to prove the reverse inequality $\begin{array}{l} (3.28) & \overline{J} (v) ⩽ \tilde{J} (0) (v) for all v \in L^{p} (Ω^{a}; R^{3}) \times L^{p} (Ω^{b}; R^{3}) . \end{array}$ Clearly, it is enough to prove this for $v = (v^{a}, v^{b}) \in V_{J}$ because for $v = (v^{a}, v^{b}) \notin V_{J}$ we have $\tilde{J} (0) (v) = + \infty$ and there is nothing to prove.

Step 4: But at first, we shall prove (3.28) for $({\overline{v}}^{a}, {\overline{v}}^{b})$ in $V_{r e g}$ . For any fixed, $w_{1}, w_{2}$ in $C^{1} ([0, 1]; R^{3})$ vanishing at 1 and $ζ_{2}, ζ_{3}$ in $C^{1} ([- \frac{1}{2}, \frac{1}{2}]; R^{3})$ vanishing at $- 1 / 2$ and $1 / 2$ , we use Proposition 5 to obtain a sequence $(v^{a} (ε), v^{b} (ε)) \in L^{p} (Ω^{a}; R^{3}) \times L^{p} (Ω^{b}; R^{3})$ converging strongly to $({\overline{v}}^{a}, {\overline{v}}^{b})$ such that $\begin{array}{l} lim_{ε \to 0} \tilde{J} (ε) ((v^{a} (ε), v^{b} (ε))) = & \int_{0}^{1} W ((e_{1} + w_{1} (x_{3}) | e_{2} + w_{2} (x_{3}) | e_{3} + \frac{d \overline{v^{a}}}{d z} (x_{3}))) d x_{3} \\ + \int_{- 1 / 2}^{1 / 2} W ((e_{1} + \frac{d \overline{v^{b}}}{d z} (x_{1}) | e_{2} + ζ_{2} (x_{1}) | e_{3} + ζ_{3} (x_{1}))) d x_{1} \\ - \int_{0}^{1} {\overline{g}}^{a} \cdot ((0, 0, x_{3}) + {\overline{v}}^{a}) d x_{3} - \int_{- 1 / 2}^{1 / 2} {\overline{g}}^{b} \cdot ((x_{1}, 0, 0) + {\overline{v}}^{b}) d x_{1} . \end{array}$ Therefore, using the above characterization of the Γ-lim sup (exploiting the sequence $(v^{a} (ε), v^{b} (ε))$ ), it follows that $\begin{array}{l} \overline{J} ({\overline{v}}^{a}, {\overline{v}}^{b}) ⩽ & (\int_{0}^{1} W ((e_{1} + w_{1} | e_{2} + w_{2} | e_{3} + \frac{d \overline{v^{a}}}{d z})) d x_{3} \\ + \int_{- 1 / 2}^{1 / 2} W ((e_{1} + \frac{d \overline{v^{b}}}{d z} | e_{2} + ζ_{2} | e_{3} + ζ_{3})) d x_{1} - \int_{0}^{1} {\overline{g}}^{a} \cdot ((0, 0, x_{3}) + {\overline{v}}^{a}) d x_{3} \\ (3.29) & - \int_{- 1 / 2}^{1 / 2} {\overline{g}}^{b} \cdot ((x_{1}, 0, 0) + {\overline{v}}^{b}) d x_{1}) \end{array}$ for all $w_{1}, w_{2}$ in $C^{1} ([0, 1]; R^{3})$ vanishing at 1 and $ζ_{2}, ζ_{3}$ in $C^{1} ([- \frac{1}{2}, \frac{1}{2}]; R^{3})$ which vanish at $- 1 / 2$ and $1 / 2$ .

Step 5: Now, if $v = (v^{a}, v^{b}) \in V_{J}$ , we recall that, by Proposition 1, we can find a sequence $v_{n} = (v_{n}^{a}, v_{n}^{b})$ in $V_{r e g}$ which converges strongly to $v = (v^{a}, v^{b}) \in V_{J}$ and moreover this convergent sequence can be dominated. The left hand side in (3.29) is lower semicontinuous (since it is a Γ-limit) whereas the right hand side is continuous for this convergence. Therefore, we also have (3.29) for every $v = (v^{a}, v^{b}) \in V_{J}$ and for any fixed, $w_{1}, w_{2}$ in $C^{1} ([0, 1]; R^{3})$ which vanish at 1 and $ζ_{2}, ζ_{3}$ in $C^{1} ([- \frac{1}{2}, \frac{1}{2}]; R^{3})$ which vanish at $- 1 / 2$ and $1 / 2$ .

Step 6: Now, given $v = (v^{a}, v^{b}) \in V_{J}$ , we use the measurable selection lemma cf. [11] to obtain measurable functions $w_{1}^{}, w_{2}^{}, ζ_{2}^{}$ and $ζ_{3}^{}$ such that $\begin{array}{l} (3.30) & \{\begin{matrix} W_{a} (e_{3} + \frac{d \overline{v^{a}}}{d z} (x_{3})) = W ((e_{1} + w_{1}^{} (x_{3}) | e_{2} + w_{2}^{} (x_{3}) | e_{3} + \frac{d \overline{v^{a}}}{d z} (x_{3}))) \\ for almost all x \in (0, 1), \\ W_{b} (e_{1} + \frac{d \overline{v^{b}}}{d z} (x_{1})) = W ((e_{1} + \frac{d \overline{v^{b}}}{d z} (x_{1}) | e_{2} + ζ_{2}^{} (x_{1}) | e_{3} + ζ_{3}^{} (x_{1}))) \\ for almost all x \in (- \frac{1}{2}, \frac{1}{2}) . \end{matrix} \end{array}$ Since, it can be seen that $W_{a} (e_{3} + \frac{d \overline{v^{a}}}{d z} (x_{3}))$ and $W_{b} (e_{1} + \frac{d \overline{v^{b}}}{d z} (x_{1}))$ are in $L^{p}$ and, given the coercivity condition on W, we can deduce that $(w_{1}^{}, w_{2}^{}, ζ_{2}^{}, ζ_{3}^{}) \in L^{p} {((0, 1); R^{3})}^{2} \times L^{p} {((- \frac{1}{2}, \frac{1}{2}); R^{3})}^{2}$ .

Step 7: Let $(w_{1}^{}, w_{2}^{}, ζ_{2}^{}, ζ_{3}^{}) \in L^{p} {((0, 1); R^{3})}^{2} \times L^{p} {((- \frac{1}{2}, \frac{1}{2}); R^{3})}^{2}$ as in the previous step. We can find sequences $w_{1}^{(n)}, w_{2}^{(n)}$ in $C^{1} ([0, 1]; R^{3})$ vanishing at 1 and $ζ_{2}^{(n)}, ζ_{3}^{(n)}$ in $C^{1} ([- \frac{1}{2}, \frac{1}{2}]; R^{3})$ vanishing at $- 1 / 2$ and $1 / 2$ and which converge strongly, in a dominated manner, in $L^{p}$ strongly to $w_{1}^{}, w_{2}^{}$ and $ζ_{2}^{}, ζ_{3}^{}$ respectively. This comes from the fact that $W_{a}^{1, p} ((0, 1); R^{3})$ and $W_{b}^{1, p} ((- \frac{1}{2}, \frac{1}{2}); R^{3})$ are dense in $L^{p} ((0, 1); R^{3})$ and $L^{p} ((- \frac{1}{2}, \frac{1}{2}); R^{3})$ , respectively and that $W_{a}^{1, p}$ functions with a vanishing trace can be approximated by $C^{1}$ functions with an appropriate vanishing condition. Therefore, using the convergence of the sequence $w_{1}^{(n)}, w_{2}^{(n)}$ and the growth conditions on W, it can be concluded that $\begin{array}{l} \overline{J} (v^{a}, v^{b}) ⩽ & \int_{0}^{1} W ((e_{1} + w_{1}^{} | e_{2} + w_{2}^{} | e_{3} + \frac{d \overline{v^{a}}}{d z})) d x_{3} \\ + \int_{- 1 / 2}^{1 / 2} W ((e_{1} + \frac{d \overline{v^{b}}}{d z} | e_{2} + ζ_{2}^{} | e_{3} + ζ_{3}^{})) d x_{1} - \int_{0}^{1} {\overline{g}}^{a} \cdot ((0, 0, x_{3}) + {\overline{v}}^{a}) d x_{3} \\ - \int_{- 1 / 2}^{1 / 2} {\overline{g}}^{b} \cdot ((x_{1}, 0, 0) + {\overline{v}}^{b}) d x_{1} \\ = & (\int_{0}^{1} W_{a} (e_{3} + \frac{d {\overline{v}}^{a}}{d x_{3}}) d x_{3} + \int_{- 1 / 2}^{1 / 2} W_{b} (e_{1} + \frac{d {\overline{v}}^{b}}{d x_{1}}) d x_{1} \\ - \int_{0}^{1} {\overline{g}}^{a} \cdot ((0, 0, x_{3}) + {\overline{v}}^{a}) d x_{3} - \int_{- 1 / 2}^{1 / 2} {\overline{g}}^{b} \cdot ((x_{1}, 0, 0) + {\overline{v}}^{b}) d x_{1}) \end{array}$ Step 8: Now, let $\begin{matrix} G (v^{a}, v^{b}) = \{\begin{matrix} (\int_{0}^{1} W_{a} (e_{3} + \frac{d {\overline{v}}^{a}}{d x_{3}}) d x_{3} + \int_{- 1 / 2}^{1 / 2} W_{b} (e_{1} + \frac{d {\overline{v}}^{b}}{d x_{1}}) d x_{1} \\ - \int_{0}^{1} {\overline{g}}^{a} \cdot ((0, 0, x_{3}) + {\overline{v}}^{a}) d x_{3} \\ - \int_{- 1 / 2}^{1 / 2} {\overline{g}}^{b} \cdot ((x_{1}, 0, 0) + {\overline{v}}^{b}) d x_{1}) & if (v^{a}, v^{b}) \in V_{J} \\ + \infty & otherwise . \end{matrix} \end{matrix}$ Observe that from the previous step we have $\begin{matrix} \overline{J} ({\overline{v}}^{a}, {\overline{v}}^{b}) ⩽ G (v^{a}, v^{b}) for all G (v^{a}, v^{b}) for all (v^{a}, v^{b}) in L^{p} (Ω^{a}; R^{3}) \times L^{p} (Ω^{b}; R^{3}) . \end{matrix}$ By, the properties of the Γ-limit, $\overline{J}$ is lower semi-continuous on $L^{p} (Ω^{a}; R^{3}) \times L^{p} (Ω^{b}; R^{3})$ whereas it is classical that the lower semicontinuous envelope of G on the same is $\begin{matrix} \tilde{J} (0) (v^{a}, v^{b}) = \{\begin{matrix} (\int_{0}^{1} W_{a}^{ } (e_{3} + \frac{d {\overline{v}}^{a}}{d x_{3}}) d x_{3} + \int_{- 1 / 2}^{1 / 2} W_{b}^{ *} (e_{1} + \frac{d {\overline{v}}^{b}}{d x_{1}}) d x_{1} \\ - \int_{0}^{1} {\overline{g}}^{a} \cdot ((0, 0, x_{3}) + {\overline{v}}^{a}) d x_{3} \\ - \int_{- 1 / 2}^{1 / 2} {\overline{g}}^{b} \cdot ((x_{1}, 0, 0) + {\overline{v}}^{b}) d x_{1}) & if (v^{a}, v^{b}) \in V_{J} \\ + \infty & otherwise . \end{matrix} \end{matrix}$ Thus, we have proved (3.28). □

4. An example

We conclude the article by examining the case of the Saint Venant–Kirchhoff material in detail similarly as in [25, Section 5]. Recall that the Saint Venant–Kirchhoff stored energy function is given by $\begin{matrix} W (F) = \frac{μ}{4} tr {(F^{T} F - I)}^{2} + \frac{λ}{8} {(tr (F^{T} F - I))}^{2}, \forall F \in R^{3 \times 3}, \end{matrix}$ where μ and λ are the Lamé moduli, which we assume to be such that $μ > 0$ and $λ ⩾ 0$ .

This W has the following properties required of hyperelastic three-dimensional bodies which are:

Objectivity or material frame-indifference principle: $\begin{matrix} (4.1) & \forall F \in R^{3 \times 3}, \forall R \in S O (3), W (R F) = W (F) . \end{matrix}$

Natural state: $W (I) = min W = 0$ .

We did not consider these properties during the process of obtaining the variational limit of the three dimensional model since it does not affect the convergence analysis. However, these properties have a consequence on the stored energy of the one dimensional model. In fact, it has been shown in Acerbi et al. [1] that when these hold for W then

W_{a} (z)

and

W_{b} (z)

will depend only on

| z |

and moreover,

W_{a}^{* *}

and

W_{b}^{* *}

necessarily vanish on the unit ball in

R^{3}

. We now obtain the explicit expressions of

W_{a}^{* *}

and

W_{b}^{* *}

Proposition 6.

For the Saint Venant–Kirchhoff stored energy function, we have $\begin{matrix} (4.2) & W_{b} (z) = W_{a} (z) = \frac{μ (3 λ + 2 μ)}{8 (λ + μ)} {(| z |^{2} - 1)}^{2} + \frac{1}{8 (λ + μ)} {({[λ (| z |^{2} - 1) - 2 (λ + μ)]}_{+})}^{2}, \end{matrix}$ where the effective energies are given by $\begin{array}{l} (4.3) & W_{a}^{* *} (z) = W_{b}^{* *} (z) = \frac{E}{8} {({[| z |^{2} - 1]}_{+})}^{2} + \frac{E}{4 (1 + ν) (1 - 2 ν)} {({[ν | z |^{2} - (1 + ν)]}_{+})}^{2} \end{array}$ where $| z |$ is the norm of $z \in R^{3}$ and $E = \frac{μ (3 λ + 2 μ)}{λ + μ}$ is the Young modulus and $ν = \frac{λ}{2 (μ + λ)}$ is the Poisson’s ratio.

Proof.

Obviously, it suffices to compute $W_{a}$ and $W_{a}^{* *}$ . For calculating $W_{a}$ , let us first express $W (F)$ in terms of the column vectors of F: $\begin{array}{l} W (F) = & \frac{μ}{4} (\sum_{i, j = 1}^{3} {(z_{i} \cdot z_{j} - δ_{i j})}^{2}) + \frac{λ}{8} {(\sum_{i = 1}^{3} (| z_{i} |^{2} - 1))}^{2} \\ = & \frac{μ}{4} (\sum_{α, β = 1}^{2} {(z_{α} \cdot z_{β} - δ_{α β})}^{2}) + \frac{λ}{8} {(\sum_{α = 1}^{2} (| z_{α} |^{2} - 1))}^{2} + \frac{μ}{2} (\sum_{α = 1}^{2} {(z_{α} \cdot z_{3})}^{2}) \\ (4.4) & + \frac{(2 μ + λ)}{8} {(| z_{3} |^{2} - 1)}^{2} + \frac{λ}{4} (| z_{3} |^{2} - 1) (\sum_{α = 1}^{2} | z_{α} |^{2} - 1) . \end{array}$ By an inspection of (4.4), it is clear that in order to minimize $W ((\overline{F} | z_{3}))$ with respect to $\overline{F} = (z_{1} | z_{2})$ , we need to choose ${z_{1}, z_{2}, z_{3}}$ as a orthogonal set. If we now set $t = | z_{1} |$ and $s = | z_{2} |$ , then we are left with minimizing the function $\begin{array}{l} f (s, t) = & \frac{2 μ + λ}{8} [{(s^{2} - 1)}^{2} + {(t^{2} - 1)}^{2}] + \frac{λ}{4} [(s^{2} - 1) + (t^{2} - 1)] (| z_{3} |^{2} - 1) \\ (4.5) & + \frac{λ}{4} (s^{2} - 1) (t^{2} - 1) \end{array}$ over the set ${(s, t) \in R^{2} : s ⩾ 0 \land t ⩾ 0}$ with $| z_{3} |$ as a parameter. Letting $x = s^{2} - 1$ , $y = t^{2} - 1$ and $h = (| z_{3} |^{2} - 1)$ , we need to minimize the function $\begin{matrix} (4.6) & g (x, y) = \frac{λ + 2 μ}{8} (x^{2} + y^{2}) + \frac{λ}{4} (x + y) h + \frac{λ}{4} x y \end{matrix}$ over the set $Λ = {(x, y) \in R^{2} : x ⩾ - 1 \land y ⩾ - 1}$ . By writting g as $\begin{matrix} \frac{λ}{8} {(x + y)}^{2} + \frac{μ}{4} (x^{2} + y^{2}) + \frac{λ}{4} (x + y) h, \end{matrix}$ it is clear that g is a strictly convex coercive function and this has to be minimized over the closed convex set Λ. Therefore, a unique minimizer exists. In addition, we can observe that since the function is symmetric in x and y, the unique minimizer must be of the form $(x, x)$ . So, the problem reduces to calculating the minimum of $\begin{matrix} (4.7) & w (x) = g (x, x) = \frac{λ + 2 μ}{4} x^{2} + \frac{λ}{2} x h + \frac{λ}{4} x^{2} = \frac{λ + μ}{2} x^{2} + \frac{λ}{2} x h \end{matrix}$ for $x ⩾ - 1$ . The critical point of w is $\begin{matrix} (4.8) & x_{0} = - \frac{λ}{2 (λ + μ)} h . \end{matrix}$ We now need to analyze two cases. When $x_{0} ⩾ - 1$ , which happens if and only if $h ⩽ \frac{2 (λ + μ)}{λ}$ , the minimum of w is attained at the critical point $x_{0}$ . In the remaining case, $h > \frac{2 (λ + μ)}{λ}$ , it can be shown that w is monotone increasing on $[- 1, \infty)$ and so it’s minimimum is attained at $- 1$ . Therefore, we calculate the minumum value of g as $\begin{matrix} (4.9) & min g = \{\begin{matrix} g (x_{0}, x_{0}) = - \frac{λ^{2}}{8 (λ + μ)} h^{2} & if h ⩽ \frac{2 (λ + μ)}{λ}, \\ g (- 1, - 1) = \frac{λ + μ}{2} - \frac{λ}{2} h & if h > \frac{2 (λ + μ)}{λ} . \end{matrix} \end{matrix}$ Then, the value of $W_{a} (z_{3})$ as stated in (4.2) is obtained by adding the term $\frac{(2 μ + λ)}{8} h^{2}$ to $min g$ in (4.9) which only depends on $z_{3}$ and there after performing a simple calculation.

We now compute the convex envelope $W_{a}^{* *}$ of $W_{a}$ . We observe that $W_{a}$ is non-negative and takes its minimum value at all $z \in R^{3}$ with norm 1 (which can be seen by evaluating $W_{a}$ explicitly). Moreover, it is convex for $| z | ⩾ 1$ . So, the convex envelope is given by the expression (4.3) after rewriting $E, ν$ in terms of $λ, μ$ . □

Footnotes

Acknowledgements

The author thanks Professors A. Gaudiello, R. Mahadevan, R. Prakash and I. Velčić for some fruitful discussions and comments. This work was supported by Comisión Nacional de Investigación Científica y Tecnológica de Chile (CONICYT) Doctorado Nacional 2015 Grant 21150001. The author also would like to thank the referees for their careful reading of the manuscript and their valuable suggestions which helped improve the paper greatly.

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