Relaxation for an optimal design problem in BD(Ω)

1. Introduction

In optimal design one aims to find an optimal shape which minimizes a cost functional. The optimal shape is a subset $E$ of a bounded, open set $\Omega \subset \mathbb {R}^N$ which is described by its characteristic function $\chi : \Omega \to \{0,1\}$, $E = \{\chi = 1\}$, and, in the linear elasticity framework, the cost functional is usually a quadratic energy, so we are lead to the problem

(1.1)\begin{equation} \min_{(\chi,u)} \int_{\Omega}\chi(x) W_1({\mathcal{E}} u(x))+ (1 - \chi(x))W_0({\mathcal{E}} u(x))\,{\rm d}x, \end{equation}

where $W_0$ and $W_1$ are two elastic densities, with $W_0 \geq W_1$, and ${\mathcal {E}} u$ denotes the symmetrized gradient of the displacement $u$. We refer to the seminal papers [Reference Allaire and Lods1, Reference Kohn and Strang35–Reference Kohn and Strang37, Reference Murat and Tartar43], among a wide literature (see, for instance, the recent contributions [Reference Babadjian, Iurlano and Rindler7, Reference Babadjian, Iurlano and Rindler8]).

However, as soon as plasticity comes into play, the observed stress–strain relation is no longer linear and, due to the linear growth of the stored elastic energy and to the lack of reflexivity of the space $L^1$, a suitable functional space is necessary to account for fields $u$ whose strains are measures. The space of special fields with bounded deformation, $BD(\Omega )$, was first proposed in [Reference Kohn and Strang36, Reference Kohn and Strang37, Reference Suquet47–Reference Suquet50] and starting from these pioneering papers a vast literature developed.

Indeed, already in the case where $\chi \equiv \chi _\Omega$, the search for equilibria in the context of perfect plasticity leads naturally to the study of lower semicontinuity properties, and eventually relaxation, for energies of the type

(1.2)\begin{equation} \int_{\Omega}f({\mathcal{E}} u(x))\,{\rm d}x \end{equation}

where $f$ is the volume energy density. As mentioned above, $u$ belongs to the space $BD(\Omega )$ of functions of bounded deformation composed of integrable vector-valued functions for which all components $E_{ij}$, $i,j = 1,\ldots,N$, of the deformation tensor $Eu := {(Du + Du^T)}/{2}$ are bounded Radon measures and ${\mathcal {E}} u$ stands for the absolutely continuous part, with respect to the Lebesgue measure, of the symmetrized distributional derivative $Eu$.

Lower semicontinuity for (1.2) was established in [Reference Bellettini, Coscia and Dal Maso16] under convexity assumptions on $f$ and in [Reference Ebobisse27] for symmetric quasiconvex integrands, under linear growth conditions and for $u \in LD(\Omega )$, the subspace of $BD(\Omega )$ comprised of functions for which the singular part $E^su$ of the measure $Eu$ vanishes. For a symmetric quasiconvex density $f$ with an explicit dependence on the position in the body and satisfying superlinear growth assumptions, lower semicontinuity properties were established in [Reference Ebobisse28] for $u \in SBD(\Omega )$.

In the case where the energy density takes the form $\|{\mathcal {E}} u\|^2$ or $\|{\mathcal {E}}^D u\|^2 + (\text {div}\, u)^2$ (where $A^D$ stands for the deviator of the $N \times N$ matrix $A$ given by $A^D := A - ({1}/{N})\text {tr} (A)I$), and the total energy also includes a surface term, a first relaxation result was proved in [Reference Braides, Defranceschi and Vitali18]. We also refer to [Reference Kosiba and Rindler39] for the relaxation in the case where there is no surface energy and to [Reference Jesenko and Schmidt33, Reference Matias, Morandotti and Santos40, Reference Mora42] for related models concerning evolutions and homogenization, among a wider list of contributions.

For general energy densities $f$, Barroso, Fonseca and Toader [Reference Barroso, Fonseca and Toader12] studied the relaxation of (1.2) for $u \in SBD(\Omega )$ under linear growth conditions but placing no convexity assumptions on $f$. They showed that the relaxed functional admits an integral representation where a surface energy term arises naturally. The global method for relaxation due to Bouchitté, Fonseca and Mascarenhas [Reference Bouchitté, Fonseca and Mascarenhas17] was used to characterize the density of this term, whereas the identification of the relaxed bulk energy term relied on the blow-up method [Reference Fonseca and Müller31] together with a Poincaré-type inequality.

Ebobisse and Toader [Reference Ebobisse and Toader29] obtained an integral representation result for general local functionals defined in $SBD(\Omega )$ which are lower semicontinuous with respect to the $L^1$ topology and satisfy linear growth and coercivity conditions. The functionals under consideration are restrictions of Radon measures and are assumed to be invariant with respect to rigid motions. Their work was extended to the space $SBD^p(\Omega )$, $p > 1$, which arises in connection with the study of fracture and damage models, by Conti, Focardi and Iurlano [Reference Conti, Focardi and Iurlano24] in the two-dimensional setting. A crucial and novel ingredient of their proof is the construction of a $W^{1,p}$ approximation of an $SBD^p$ function $u$ using finite-elements on a countable mesh which is chosen according to $u$ (recall that $SBD^p$ denotes the space of fields with bounded deformation such that the symmetrized gradient is the sum of an $L^p$ field and a measure supported on a set of finite $\mathcal {H}^{N-1}$ measure).

The analysis of an integral representation for a variational functional satisfying lower semicontinuity, linear growth conditions and the usual measure theoretical properties, was extended to the full space $BD(\Omega )$ by Caroccia, Focardi and Van Goethem [Reference Caroccia, Focardi and Van Goethem23]. In this work, the invariance of the studied functional with respect to rigid motions, required in [Reference Ebobisse and Toader29], is replaced by a weaker condition stating continuity with respect to infinitesimal rigid motions. Their result relies, as in papers mentioned above, on the global method for relaxation, as well as on the characterization of the Cantor part of the measure $Eu$, due to De Philippis and Rindler [Reference De Philippis and Rindler25], which extends to the $BD$ case the result of Alberti's rank-one theorem in $BV$.

In the study of the minimization problem (1.1) one usually prescribes the volume fraction of the optimal shape, leading to a constraint of the form $\displaystyle \frac {1}{{\mathcal {L}}^N(\Omega )}\int _\Omega \chi (x)\,{\rm d}x= \theta,\;\theta \in (0,1)$. It is sometimes convenient to replace this constraint by inserting, instead, a Lagrange multiplier in the modelling functional which, in the optimal design context, becomes

(1.3)\begin{equation} F(\chi,u;\Omega) : =\displaystyle \int_{\Omega}\chi(x) W_1({\mathcal{E}} u(x))+ (1 - \chi(x))W_0({\mathcal{E}} u(x)) + \int_\Omega k \chi(x)\,{\rm d}x. \end{equation}

Despite the fact that we have compactness for $u$ in $BD(\Omega )$ for functionals of the form (1.3), it is well known that the problem of minimizing (1.3) with respect to $(\chi,u)$, adding suitable forces and/or boundary conditions, is ill-posed, in the sense that minimizing sequences $\chi _n \in L^\infty (\Omega ;\{0,1\})$ tend to highly oscillate and develop microstructure, so that in the limit we may no longer obtain a characteristic function. To avoid this phenomenon, as in [Reference Ambrosio and Buttazzo2, Reference Kohn and Lin34], we add a perimeter penalization along the interface between the two zones $\{\chi = 0\}$ and $\{\chi = 1\}$ (see [Reference Carita and Zappale21] for the analogous analysis performed in $BV$, and [Reference Barroso and Zappale14, Reference Barroso and Zappale15, Reference Carita and Zappale20] for the Sobolev settings, also in the presence of a gap in the growth exponents).

Thus, with an abuse of notation (i.e. denoting $W_1+ k$, in (1.3), still by $W_1$), our aim in this paper is to study the energy functional given by

(1.4)\begin{equation} F(\chi,u;\Omega) : =\displaystyle \int_{\Omega}\chi(x) W_1({\mathcal{E}} u(x))+ (1 - \chi(x))W_0({\mathcal{E}} u(x))\,{\rm d}x + |D\chi|(\Omega), \end{equation}

where $u \in BD(\Omega )$, $\chi \in BV(\Omega ;\{0,1\})$ and the densities $W_i$, $i = 0,1$, are continuous functions satisfying the following linear growth conditions from above and below:

(1.5)\begin{equation} \exists \, \alpha, \beta > 0 \text{ such that }\alpha |\xi| \leq W_i(\xi) \leq \beta (1 + |\xi|),\quad \forall \xi \in {\mathbb{R}}^{N \times N}_s. \end{equation}

We point out that no convexity assumptions are placed on $W_i$, $i = 0,1$.

To simplify the notation, in the sequel, we let $f: \{0,1\} \times \mathbb {R}^{N \times N}_s \to [0,+\infty )$ be defined as

(1.6)\begin{equation} f\left( q,\xi\right):=q W_1(\xi)+ (1-q)W_0(\xi), \end{equation}

and for a fixed $q \in \{0,1\}$, we recall that the recession function of $f$, in its second argument, is given by

(1.7)\begin{equation} f^{\infty}(q,\xi) := \limsup_{t\rightarrow+\infty}\frac{f(q,t\xi)}{t}. \end{equation}

Since we place no convexity assumptions on $W_i$, we consider the relaxed localized functionals arising from the energy (1.4), defined, for an open subset $A \subset \Omega$, by

(1.8)\begin{align} \mathcal{F}\left(\chi,u;A\right)& :=\inf\left\{ \liminf_{n\rightarrow +\infty} F(\chi_n,u_n;A): u_{n} \in W^{1,1}(A;\mathbb{R}^{N}),\chi_{n} \in BV(A;\{0,1\}),\right.\nonumber\\ & \quad \vphantom{\liminf_{n\rightarrow +\infty}}\left.u_{n}\to u\text{ in }L^{1}(A;\mathbb{R}^{N}),\chi_{n}\to\chi\text{ in }L^1(A;\{0,1\})\vphantom{\liminf_{n\rightarrow +\infty}}\right\}, \end{align}

and

(1.9)\begin{align} \mathcal{F}_{LD}\left(\chi,u;A\right)& :=\inf\left\{\liminf_{n\rightarrow +\infty} F(\chi_n,u_n;A): u_{n} \in LD(A), \chi_{n} \in BV(A;\{0,1\}),\right.\nonumber\\ & \quad \vphantom{\liminf_{n\rightarrow +\infty}}\left.u_{n}\to u\text{ in }L^{1}(A;\mathbb{R}^{N}),\chi_{n}\to\chi\text{ in }L^1(A;\{0,1\})\vphantom{\liminf_{n\rightarrow +\infty}}\right\}, \end{align}

where $LD(\Omega ):=\{u \in BD(\Omega ): E^s u=0\}$.

Due to the expression of (1.4), and to the fact that $\chi _n \mathop {\rightharpoonup }\limits ^{\ast }\chi$ in $BV$ if and only if $\{\chi _n\}$ is uniformly bounded in $BV$ and $\chi _n \to \chi$ in $L^1$, it is equivalent to take $\chi _n \mathop {\rightharpoonup }\limits ^{\ast }\chi$ in $BV$ or $\chi _n \to \chi$ in $L^1$ in the definitions of the functionals (1.8) and (1.9), obtaining for each of them the same infimum regardless of the considered convergence.

As a simple consequence of the density of smooth functions in $LD(\Omega )$ we show in remark 3.5 that, under the above growth conditions on $W_0, W_1$,

\[ \mathcal{F}\left(\chi,u;A\right) = \mathcal{F}_{LD}\left(\chi,u;A\right),\;\text{for every} \; \chi \in BV(A;\left\{0,1\right\}), u \in BD(\Omega), A \in {\mathcal{O}}(\Omega). \]

We prove in proposition 3.8 that $\mathcal {F}\left (\chi,u;\cdot \right )$ is the restriction to the open subsets of $\Omega$ of a Radon measure, the main result of our paper concerns the characterization of this measure.

Theorem 1.1 Let $f:\{0,1\} \times \mathbb {R}^{N \times N}_s\to [0, + \infty )$ be a continuous function as in (1.6), where $W_0$ and $W_1$ satisfy (1.5), and consider $F:BV(\Omega ;\{0,1\})\times BD(\Omega )\times \mathcal {O}(\Omega )$ defined in (1.4). Then

(1.10)\begin{align} \mathcal{F}\left(\chi,u;A\right)& =\int_A SQf(\chi(x),{\mathcal{E}} u(x))\,{\rm d}x\notag\\ & \quad +\int_{A \cap (J_\chi \cup J_u)} g(x,\chi^+(x),\chi^-(x),u^+(x),u^-(x),\nu(x))\,{\rm d}{\mathcal{H}}^{N-1}(x)\nonumber\\ & \quad + \int_A(SQf)^{\infty}(\chi(x), \frac{{\rm d} E^c u}{{\rm d} |E^c u|}(x))\,{\rm d}|E^c u|(x), \end{align}

where $SQf$ is the symmetric quasiconvex envelope of $f$ and $(SQf)^{\infty }$ is its recession function $($cf. § 2.3 and (1.7), respectively$)$. The relaxed surface energy density is given by

\[ g(x_0,a,b,c,d,\nu) := \limsup_{\varepsilon \to 0^+}\frac{m(\chi_{a,b,\nu}({\cdot}{-} x_0),u_{c,d,\nu}({\cdot}{-} x_0);Q_\nu(x_0,{\varepsilon}))} {{\varepsilon}^{N-1}} \]

where $Q_{\nu }(x_0,{\varepsilon })$ stands for an open cube with centre $x_0$, sidelength ${\varepsilon }$ and two of its faces parallel to the unit vector $\nu$,

\begin{align*} & m(\chi,u;V) := \inf\left\{\mathcal{F} (\theta,v;V) :\theta \in BV(\Omega;\{0,1\}), v \in BD(\Omega),\right.\\ & \left.\theta = \chi \text{ on } \partial V, v = u \text{ on } \partial V\right\}, \end{align*}

for any $V$ open subset of $\Omega$ with Lipschitz boundary, and, for $(a,b,c,d,\nu ) \in \{0,1\} \times \{0,1\} \times \mathbb {R}^{N} \times \mathbb {R}^{N} \times S^{N-1},$ the functions $\chi _{a,b,\nu }$ and $u_{c,d,\nu }$ are defined as

\[ \chi_{a,b,\nu}(y) := \begin{cases} a, & {\rm if\ }\; y \cdot \nu > 0\\ b, & {\rm if\ }\; y \cdot \nu < 0 \end{cases}\quad{\rm and}\quad u_{c,d,\nu}(y) := \begin{cases} c, & {\rm if\ }\; y \cdot \nu > 0\\ d, & {\rm if\ }\; y \cdot \nu < 0. \end{cases} \]

For the notation regarding the jump sets $J_\chi$, $J_u$ and the corresponding vectors $\chi ^+(x)$, $\chi ^-(x)$, $\nu _\chi (x)$, $u^+(x)$, $u^-(x)$ and $\nu _u(x)$ we refer to § 2.1, 2.2 and 4.3.

The above expression for the relaxed surface energy density arises as an application of the global method for relaxation [Reference Bouchitté, Fonseca and Mascarenhas17]. However, as we will see in § 4.3, in the case where $f$ satisfies the additional hypothesis (3.9), this density can be described more explicitly, leading to an integral representation for (1.8), in the $BD$ setting, entirely similar to the one in $BV$, obtained in [Reference Carita and Zappale21], when $W_0$ and $W_1$ depend on the whole gradient $\nabla u$. Indeed, under this assumption, we show that

\[ g(x_0,a,b,c,d,\nu) = K(a,b,c,d,\nu) \]

where

(1.11)\begin{align} & K(a,b,c,d,\nu)\notag\\ & \quad :=\inf\left\{\int_{Q_{\nu}}(SQf)^{\infty}(\chi(x),{\mathcal{E}} u(x))\,{\rm d}x+|D\chi|(Q_{\nu}):\left(\chi,u\right) \in\mathcal{A}(a,b,c,d,\nu)\right\}, \end{align}

and, for $(a,b,c,d,\nu ) \in \{0,1\} \times \{0,1\} \times \mathbb {R}^{N} \times \mathbb {R}^{N} \times S^{N-1},$ the set of admissible functions is

\begin{align*} \mathcal{A}(a,b,c,d,\nu)& :=\left\{ \left(\chi,u\right)\in BV_{\rm loc}\left(S_{\nu};\{0,1\}\right) \times W^{1,1}_{\rm loc}\left(S_{\nu};\mathbb{R}^{N}\right):\right.\nonumber\\ & (\chi(y),u(y)) = (a,c) \text{ if } y\cdot\nu=\frac{1}{2},\ (\chi(y),u(y)) = (b,d) \text{ if } y\cdot\nu={-}\frac{1}{2},\nonumber\\ & \left.(\chi, u)\text{ are 1-periodic in the directions of }\nu_{1},\dots,\nu_{N-1}\right\}, \end{align*}

$\left \{\nu _{1},\nu _{2},\dots,\nu _{N-1},\nu \right \}$ is an orthonormal basis of $\mathbb {R}^{N}$ and $S_\nu$ is the strip given by

\[ S_\nu = \left\{x \in {\mathbb{R}}^N : |x \cdot \nu| < \frac{1}{2}\right\}. \]

As an application of the result of Caroccia, Focardi and Van Goethem, obtained in the abstract variational functional setting in [Reference Caroccia, Focardi and Van Goethem23], the authors proved an integral representation for the relaxed functional, defined in $BD(\Omega ) \times \mathcal {O}(\Omega )$,

\[ \mathcal{F}_{0}(u; A) := \inf \left\{\liminf_{n \to + \infty}F_0(u_n;A) : u_n \in W^{1, 1}(A; {\mathbb{R}}^N), u_n \to u \; {\rm in} \; L^1(A;{\mathbb{R}}^N) \right\}, \]

where

\[ F_0(u;A) :=\begin{cases} \int_A f_{0} (x,u(x),{\mathcal{E}} u(x))\,{\rm d}x, & {\rm if}\ u \in W^{1,1}(\Omega;{\mathbb{R}}^N)\\ +\infty, & {\rm otherwise} \end{cases} \]

and the density $f_0$ satisfies linear growth conditions from above and below

\[ \frac{1}{C}|A| \leq f_0(x,u,A) \leq C\left(1 + |A|\right),\ \forall (x,u,A) \in \Omega \times {\mathbb{R}}^N \times {\mathbb{R}}^{N \times N}_s, \]

as well as a continuity condition with respect to $(x,u)$. This generalizes to the full space $BD(\Omega )$, and to the case of densities $f_0$ depending explicitly on $(x,u)$, the results obtained in [Reference Barroso, Fonseca and Toader12]. We will make use of their work in § 4.2 to prove both lower and upper bounds for the density of the Cantor part of the measure $\mathcal {F}(\chi,u;\cdot )$, by means of an argument based on Chacon's biting lemma which allows us to fix $\chi$ at an appropriately chosen point $x_0$, as in [Reference Matias, Morandotti and Zappale41].

The contents of this paper are organized as follows. In § 2 we fix our notation and provide some results pertaining to $BV$ and $BD$ functions and notions of quasiconvexity which will be used in the sequel. Section 3 contains some auxiliary results which are needed to prove our main theorem. In particular, in proposition 3.8 we show that $\mathcal {F}(\chi,u;\cdot )$ is the restriction to the open subsets of $\Omega$ of a Radon measure $\mu$. Section 4 is dedicated to the proof of our main theorem, which characterizes this measure. In each of § 4.1, 4.2 and 4.3 we prove lower and upper bounds of the densities of $\mu$ with respect to the bulk and Cantor parts of $Eu$, as well as with respect to a surface measure which is concentrated on the union of the jump sets of $\chi$ and $u$.

The fact that our functionals have an explicit dependence on the $\chi$ field prevented us from applying existing results (such as [Reference Arroyo-Rabasa, De Philippis and Rindler5, Reference Breit, Diening and Gmeineder19]) directly and required us to obtain direct proofs.