Shell design from planar pre-stressed structures

Abstract

The interplay between mechanics and geometry is used to construct a theoretical framework able to describe the class of three-dimensional objects that can be fabricated from suitable planar designs by using relaxation of pre-strains/stress in ultra-thin films. Small deformations and large rotations are used here to model the elastic relaxation into various three-dimensional shapes. Over the kinematics associated with the designed mid-surface, a small perturbation of Love–Kirchhoff type is considered in order to deduce the design plate-to-shell equations for orthotropic materials with an important pre-stress/strain heterogeneity. The resulting equations for the efforts average and efforts moments provide the supplementary equations to compute the in-plane pre-strain/stress. In particular, for materials with a weak material transversal heterogeneity the moments equations involve only the thickness, the curvature tensor, and the pre-strain/stress moments. Special attention is devoted to materials that can be obtained by layer-by-layer crystal growth (molecular beam epitaxy), which posses an in-plane isotropic pre-strain. We have found that a rectangular plate could relax both into a cylindrical surface or on a part of a sphere in which case it should have a small diameter with respect to the sphere radius. In both cases, the theoretical estimates have been compared with the experimental realizations and finite-element numerical computations and we found very good agreement among all of them. In addition, for the cone geometry we found that the design is not possible from an isotropic pre-stress with an in-plane homogeneity. However, the 3D finite-element computation of the relaxed surface with a (necessary) non-isotropic pre-stress obtained from the theoretical estimates matches remarkably well the designed conical surface.

1. Introduction

Among the advanced methods for semiconductor crystal growth, molecular beam epitaxy (MBE) provides very fine tuning of the thickness and/or material composition, but in order to obtain almost defect-free materials, the lattice mismatch between the planar template and the grown crystal should be as small as possible. Even when this mismatch is small, the pre-strain induced by the lattice mismatch can manifest through the creation of defects and/or bending of the layered structure, a common situation when the respective thicknesses of the layers are of the same order of magnitude. From a practical point of view, the bending process may be beneficial at very low thickness as one can use it to design various three-dimensional (3D) objects starting from planar pre-strained templates. Initiated by the group of Prinz [1, 2] the control of the bending process was essentially restricted to ribbons, curls, and/or similar constant curvature objects [3 –5], mainly one-dimensional (1D).

As semiconductors are anisotropic elastic/fragile materials, the study of the equilibrium shape for a bilayer plate with X₃ -dependent pre-strain/stress can be formulated as a classical linear elasticity problem as long as one is interested in small perturbations theory (small displacement, small strain, and small rotations) of a given but fixed reference configuration. However, this classical approach has at least two main drawbacks: (i) it cannot recover experimental results illustrated for instance in [5] for the relaxed configurations that do not fit the small perturbation theory, which is a situation frequently encountered in applications; and (ii) it cannot answer the important practical question which is the two-dimensional (2D) shape that relaxes toward a given 3D given surface/object? While the former drawback should be a benchmark for various approximate and/or exact nonlinear models for elastic plates/shells, the latter question fits the framework of topological optimization within the framework of shell theory.

Recent work in [6, 7] on various approximations of the 3D elasticity with incompatible pre-strain/stress provided a hierarchy of nonlinear elastic models that fulfill the requirement (i) but which cannot solve the design problem stated in (ii). From a different perspective, the pioneering work in [8 –10] built a theory of incompatible surface growth that includes geometric nonlinearities and finite stretch, both core ingredients for a realistic description of the relaxed shapes.

There are several approaches in deducing (2D) plate models from (3D) elasticity equations. While some of them are based on ad hoc hypotheses (see Ciarlet [11] for a review), others use a mathematical formalism. The latter are based on Gamma convergence (see, for instance, [12 –15]) or use an asymptotic expansions on the 3D weak formulation (as in [16 –18]) or on the 3D differential formulation. This last method was used to derive various consistent finite-strain plate models for different magnitudes of the in-plane and normal surface loads (as in [19 –21]) or without any assumption on magnitudes of surface loads (see [22, 23]). Another way of deducing the asymptotic model, used by Steigmann and his coworkers [24 –30], is based on the expansion of the potential energy for conservative boundary-value problems. In all of these approaches there are two main assumptions that prevent the use of these models for pre-stressed structures: (i) the a priori scaling of the in-plane and normal surface loads and (ii) the regularity assumptions on stresses supposed to be smooth with respect to the thickness variable. Indeed, the deformation of the structure is due to the pre-stress only with vanishing surface loads, while for technological reasons the pre-stress is discontinuous with respect to the thickness.

The aim of this article is to construct a theoretical framework able to enlarge the class of 3D objects that can be fabricated from suitable planar design by using relaxation of pre-strains/stress. More precisely, given a shape of a shell’s mid-surface (described as a 3D image of a plate mid-surface) we are looking for the shell’s thickness and for the pre-strain/stress such that the plate will relax (with small deformations) into the given shell as the effect of pre-strain/stress exclusively. As we focus on brittle–elastic materials (as semiconductors) the small deformations assumption is here a technological restriction and not a mathematical simplification. However, to describe complex 3D relaxed shapes, large rotations are mandatory.

This article is organized as follows. Section 2 introduces the kinematics of the plate-to-shell deformation. In particular, over the kinematics associated with the (exact) target mid-surface, we consider a small perturbation of Love–Kirchhoff type, which ensures an approximative designed kinematics. In Section 3, we present the plate-to-shell equations for orthotropic materials with a general dependence of the elastic coefficients on the thickness variable and with an important pre-stress/strain heterogeneity. As for the materials we discuss in this article the material properties are almost homogeneous, we also discuss the case of a weak material transversal heterogeneity, situation in which the system of equation is considerably simpler. In Section 4, we apply the proposed theoretical framework to the small-diameter spherical surfaces, the (arbitrary) cylindrical surfaces and the conical surfaces, all of them in the case of isotropic elastic materials with isotropic in-plane pre-stress. We compare the experimental realizations of the first of these cases both with the theoretical estimates and with 3D finite-element numerical computations and find a good agreement between all of them.

2. Kinematics of the plate-to-shell deformation

In this section, we introduce the kinematics of the plate-to-shell map, further discussed in detail in this article. The starting point is the designed mid-surface of the shell that will be given through a purely geometric transformation. We point out that not only is the surface designed but also its associated transformation. The main restriction is that the corresponding (membrane) deformation should be small in a certain sense, precisely defined in the following. The kinematics of the plate deformation involves the classical Love–Kirchhoff assumption, that is, the normal to the plate mid-plane remains normal to the designed mid-surface, but in a finite deformation context (thus, including large rotations). In addition, the transversal deformation is affine with respect to the plate thickness. Over the kinematics associated with the exact design that reproduces the target mid-surface, we also consider a small perturbation of Love–Kirchhoff type in order to compensate for the small (membrane) deformation of the proposed geometric transformation. As a consequence, the mid-surface of this overall kinematics will be close to the designed mid-surface, and for this reason we called it the approximative designed kinematics. Thus, our goal is to provide conditions that ensure that an “approximative designed configuration” could be reached from a pre-strained plate.

2.1. Designed shell’s mid-surface

Let $ω_{0} \subset ℝ^{3}$ be the designed surface, given by the given geometric transformation $x_{0} : Ω_{0} \to ℝ^{3}$ (i.e., $ω_{0} = x_{0} (Ω_{0})$ , see Figure 1), where $Ω_{0} \subset ℝ^{2}$ is the mid-surface in the Lagrangian description. We denote by $\nabla_{2}$ the gradient with respect to $\bar{X} \in Ω_{0}$ and by $F_{0} = \nabla_{2} x_{0}$ the gradient deformation tensor acting, in each point $\bar{X} = (X_{1}, X_{2}) \in Ω_{0}$ , from $ℝ^{2}$ in the tangent plane $T (x_{0} (\bar{X}))$ , denoted by $T = T (\bar{X})$ , of the designed surface $ω_{0}$ .

Figure 1.

Schematic representation of the geometric deformation of a planar mid-surface.

In what follows $δ ≪ 1$ will be a small parameter. The main assumption of the article is that the associated deformation of the geometric transformation is small, that is, there exists a tensor $ε_{0} = ε_{0} (\bar{X})$ , called small design strain (or small membrane strain) such that

| ε_{0} | = O (δ), C_{0} (\bar{X}) = : F_{0}^{T} (\bar{X}) F_{0} (\bar{X}) = I_{2} + 2 ε_{0} (\bar{X}) + O (δ^{2}), for all \bar{X} \in Ω_{0},

(2.1)

where we have denoted by $I_{2} = c_{1} \otimes c_{1} + c_{2} \otimes c_{2}$ the identity tensor on $ℝ^{2}$ and by ${c_{1}, c_{2}, c_{3}}$ the Cartesian basis in the Lagrange description.

Let $e_{3} = e_{3} (\bar{X})$ be the unit normal to the Eulerian surface $ω_{0}$ in $x_{0} (\bar{X}) \in ω_{0}$ . If $K = K (\bar{X})$ denotes the curvature tensor acting from $T (\bar{X})$ into itself and $K_{0}$ is the Lagrangian curvature tensor acting from $ℝ^{2}$ into itself, we have

\nabla_{2} e_{3} = K F_{0}, K_{0} = F_{0}^{T} K F_{0} .

(2.2)

2.2. Large rotations and small deformations of the designed shell

Let $S = {X = (\bar{X}, X_{3}); \bar{X} \in Ω, X_{3} \in (- H (\bar{X}) ∕ 2, H (\bar{X}) ∕ 2)}$ be the plate with mid-surface $Ω \subset ℝ^{2}$ and thickness $H = H (\bar{X})$ (see Figure 2) in the Lagrangian configuration. We shall assume in what follows that the thickness is small with respect to the characteristic length in the Lagrangian description, further denoted $L_{c}$ , and to the characteristic curvature radius in Eulerian configuration, that is,

\frac{H}{L_{c}} = O (δ), | \nabla_{2} H | = O (δ), H | K | = O (δ) .

(2.3)

We shall also assume that

H | \nabla_{2} F_{0} | = O (δ), H | \nabla_{2} ε_{0} | = O (δ^{2}) .

(2.4)

Figure 2.

Schematic representation of the deformation of a plate.

Restricting attention to orthotropic materials, which are a subclass of materials exhibiting reflection symmetry with respect to the mid-surface considered by Steigman [26], we consider the expansion of the 3D transformation with respect to the thickness as

x_{D} (\bar{X}, X_{3}) = x_{0} (\bar{X}) + X_{3} (1 + ϵ_{3}) e_{3} (\bar{X}) + \frac{1}{2} X_{3}^{2} \frac{β}{H} e_{3} (\bar{X}),

where $ϵ_{3}, β : Ω \to ℝ$ are two scalar functions, to be found later, which express the transverse deformation $ϵ_{3} + X_{3} \frac{β}{H}$ . In this case, normals to the undeformed mid-surface move to normals of the deformed mid-surface and we assume further that

| ϵ_{3} | = O (δ), H | \nabla_{2} ϵ_{3} | = O (δ^{2}), β = O (δ), H | \nabla_{2} β | = O (δ^{2}) .

(2.5)

2.3. Small perturbations of the designed shell

To define a small perturbation of the previous kinematics we consider the following.

An in-plane displacement $u : Ω \to ℝ^{2}$ such that $Ω_{0}$ is an in-plane perturbation of $Ω$ that is small and has small gradients, that is, $\bar{X} + u (\bar{X}) \in Ω_{0}$ , for all $\bar{X} \in Ω$ , and

| u | = O (H), | \nabla_{2} u | = O (δ), H | \nabla_{2} (\nabla_{2} u) | = O (δ^{2}) .

(2.6)

An out-of-plane displacement $w : Ω \to ℝ^{2}$ that is small and has small gradients, that is,

| w | = O (H), | \nabla_{2} w | = O (δ), H | \nabla_{2} (\nabla_{2} w) | = O (δ^{2}) .

(2.7)

The plate perturbation, defined as

x_{LK} (\bar{X}, X_{3}) = \bar{X} + u (\bar{X}) - X_{3} \nabla_{2} w (\bar{X}) + (X_{3} + w (\bar{X})) c_{3},

(2.8)

involves a kinematics of Love–Kirchhoff type. However, we have to point out that the out-of-plane displacement w does not have the same scaling as in the Love–Kirchhoff theory. This was necessary because the mid-surface of the shell has to be close to the designed surface $ω_{0}$ (see the following proposition) and the curvature associated with w has to be negligible with respect to the curvature of the designed surface.

The plate-to-shell kinematics considered in this article superpose the exact design kinematics $x_{D}$ with the small perturbation $x_{LK}$ as

x (\bar{X}, X_{3}) = x_{D} (x_{LK} (\bar{X}, X_{3})), for all \bar{X} \in Ω, X_{3} \in (- \frac{H (\bar{X})}{2}, \frac{H (\bar{X})}{2}) .

(2.9)

In conclusion, the designed surface $x_{0}$ , the in-plane displacement $u$ , the out-of-plane displacement w, and the transversal strains $ϵ_{3}$ and $β$ will completely define the kinematics of the plate-to-shell deformation. The Eulerian configuration of the shell will be denoted by $s = x (S)$ .

The following result estimates the distance between the mid-surface of the shell $x (Ω, 0)$ and the designed mid-surface $x_{0} (Ω)$ and shows that with this specific kinematics the deformed shell will be close to the desired (designed) shell.

Proposition 2.1. If $x_{0}, u$ , and w are smooth and the scaling assumptions (2.3)–(2.7) hold then we have the following estimation for all $p \in [1, + \infty]$

| | x (\cdot, 0) - x_{0} | |_{p} \leq | | F_{0} | |_{\infty} | | u | |_{p} + | | w | |_{p} + c O (δ^{2}) = c O (δ),

(2.10)

| | \nabla_{2} x (\cdot, 0) - F_{0} | |_{p} \leq | | F_{0} | |_{\infty} | | \nabla_{2} u | |_{p} + | | \nabla_{2} w | |_{p} + | | K F_{0} | |_{\infty} ∥ w | |_{p} + c O (δ^{2}) = c O (δ),

(2.11)

where $c = L_{c} | Ω |^{1 ∕ p}$ and $| | \cdot | |_{p}$ is the norm in $L^{p} (Ω)$ .

Proof. As $x (\bar{X}, 0) = x_{0} (\bar{X} + u (\bar{X})) + [(1 + ϵ_{3}) w + \frac{β}{2 H} w^{2}] e_{3}$ we use the scaling assumptions to obtain $x (\bar{X}, 0) - x_{0} (\bar{X}) = F_{0} (\bar{X}) u + w e_{3} + L_{c} O (δ^{2})$ so that (2.10) results from the Schwartz inequality in the $L^{p} (Ω)$ -norm. Bearing in mind that $\nabla_{2} x (\cdot, 0) = F_{0} (I_{2} + \nabla_{2} u) + e_{3} \otimes \nabla_{2} w + w \nabla_{2} e_{3} + O (δ^{2})$ , we can use (2.2) and (2.3)–(2.7) to obtain (2.11).

2.4. First-order deformations

The deformation gradient tensor $F$ , associated with the kinematics (2.9), can be computed from $F (\bar{X}, X_{3}) = \nabla x (\bar{X}, X_{3}) = \nabla x_{K} (\tilde{X}, X_{3} + w) \nabla x_{LK} (\bar{X}, X_{3})$ , where $\tilde{X} = \bar{X} + u (\bar{X}) - X_{3} \nabla_{2} w (\bar{X})$ and

\nabla x_{K} = F_{0} + e_{3} \otimes (X_{3} \nabla_{2} ϵ_{3} + \frac{1}{2} X_{3}^{2} \nabla_{2} \frac{β}{H}) + ((1 + ϵ_{3}) X_{3} + \frac{β}{2 H} X_{3}^{2}) K F_{0} + (1 + ϵ_{3} + \frac{β}{H} X_{3}) e_{3} \otimes c_{3},

\nabla x_{LK} = I_{2} + \nabla_{2} u - X_{3} \nabla_{2} \nabla_{2} w + c_{3} \otimes \nabla_{2} w - \nabla_{2} w \otimes c_{3} + c_{3} \otimes c_{3} .

Having in mind the asymptotic assumptions (2.3)–(2.5), the deformation gradient tensor $F$ can be written at different orders of $δ$ as

F = F^{0} + O (δ), F = F^{0} + F^{1} + O (δ^{2}),

(2.12)

where $F^{0} = F_{0} + e_{3} \otimes c_{3}$ and

F^{1} = (X_{3} + w) K F_{0} + F_{0} \nabla_{2} u + (ϵ_{3} + \frac{β}{H} (X_{3} + w)) e_{3} \otimes c_{3} + e_{3} \otimes \nabla_{2} w - (F_{0} \nabla_{2} w) \otimes c_{3},

Finally, using assumption (2.1) we obtain the (finite) strain tensor $E = \frac{1}{2} (F^{T} F - I_{3})$ at order $δ$ as

E = E_{2} + (ϵ_{3} + \frac{β}{H} (X_{3} + w)) c_{3} \otimes c_{3} + O (δ^{2}), E_{2} = ε_{0} + ε (u) + (X_{3} + w) K_{0},

(2.13)

where $E_{2}$ is the (finite) in-plane strain tensor and $ε (u) = \frac{1}{2} (\nabla_{2}^{T} u + \nabla_{2} u)$ is the small strain operator.

3. Plate-to-shell equations for design

In this section, we introduce the plate-to-shell equations for orthotropic materials. The principal unknowns of the problem are the small perturbation $(u, w)$ , introduced in (2.8) and the pre-stress/strain which will generate the deformation of the plate to a configuration close to the designed mid-surface. For the applications we have in mind, the pre-stress/strain has an important thickness heterogeneity, but the material properties are almost homogeneous. That is why, even if we start with a rather general dependence of the elastic coefficients on the thickness variable X₃, we will focus mainly on a weak transversal heterogeneity. In this case, among the six equations for the design problem, three involve the small perturbation $(u, w)$ , which has to be computed from the membrane deformation and the pre-stress resultant. The three others involve the relation between the three components of the pre-stress moments and the membrane curvature.

3.1. Constitutive assumptions

We consider here a pre-stressed hyper-elastic material. For small deformations $| E | = O (δ)$ and the second Piola–Kirchhoff stress tensor, further denoted $S$ , is given by

S = C E + S^{*} + Σ O (δ^{2}),

(3.1)

where $C$ is the fourth-order tensor of elastic coefficients, $S^{*} = S^{*} (\bar{X}, X_{3})$ is the pre-stress with $| S^{*} | = Σ O (δ)$ and $Σ$ the characteristic stress.

In what follows, we consider only orthotropic materials with the elastic coefficients $C_{ij} = C_{ij} (\bar{X}, X_{3})$ , that is,

C A = C_{2} A_{2} + A_{33} C_{3} + (C_{3} : A_{2} + C_{33} A_{33}) c_{3} \otimes c_{3} + C_{44} A_{23} {(c_{2} \otimes c_{3})}_{S} + C_{55} A_{13} {(c_{1} \otimes c_{3})}_{S},

where $C_{3} = C_{13} c_{1} \otimes c_{1} + C_{23} c_{2} \otimes c_{2}, and C_{2} A_{2} = (C_{11} A_{11} + C_{12} A_{22}) c_{1} \otimes c_{1} + (C_{12} A_{11} + C_{22} A_{22}) c_{2} \otimes c_{2} + C_{66} A_{12} {(c_{1} \otimes c_{2})}_{S}$ . If, in addition, the material is isotropic, then we obtain the Saint-Venant–Kirchhoff law, that is, (Voigt notation),

C_{2} A = λtrace (A) I_{2} + 2 μ A, C_{3} = λ I_{2}, C_{33} = λ + 2 μ, C_{44} = C_{55} = μ,

(3.2)

where $λ = λ (\bar{X}, X_{3})$ and $μ = μ (\bar{X}, X_{3})$ are the Lamé elastic moduli.

Concerning the pre-stress $S^{*}$ , we assume that the tangential pre-stresses acting on the surfaces parallel to the mid-surface vanish, that is,

S^{*} = S_{2}^{*} + S_{33}^{*} c_{3} \otimes c_{3} + Σ O (δ^{2}),

(3.3)

where $S_{2}^{*}$ is the in-plane pre-stress, and $S_{33}^{*}$ is the transversal pre-stress. Two cases will be considered in the following: “plane pre-strain” and “plane pre-stress.” In the first case, we consider a small plane deformation $ε_{2}^{*}$ with $| ε_{2}^{*} | = O (δ)$ , which induce $S_{2}^{*} = C_{2} ε_{2}^{*}$ and $S_{33}^{*} = C_{3} : ε_{2}^{*}$ . In the second case $S_{33}^{*} = 0$ .

From (2.13) and the assumption (3.3) the constitutive equation (3.1) reads

S = S_{2} + S_{33} c_{3} \otimes c_{3} + Σ O (δ^{2}),

(3.4)

where the in-plane stress $S_{2}$ and the normal stress $S_{33}$ are given by

S_{2} = C_{2} E_{2} + (ϵ_{3} + \frac{β}{H} (X_{3} + w)) C_{3} + S_{2}^{*}, S_{33} = C_{3} : E_{2} + (ϵ_{3} + \frac{β}{H} (X_{3} + w)) C_{33} + S_{33}^{*} .

(3.5)

Let $Π = F S$ be the first Piola–Kirchhoff stress tensor. Then using (2.12) we obtain

Π = Π_{2} + S_{33} e_{3} \otimes c_{3} + Σ O (δ^{2}), Π_{2} = F_{0} S_{2} .

(3.6)

3.2. Boundary conditions

In order to formulate the boundary conditions let us denote by $∂Ω$ the boundary of the mid-surface and by $\bar{ν} \in ℝ^{2}$ its exterior unit normal. The Lagrangian lateral surface of the shell is denoted by $Γ_{lat} = {(\bar{X}, X_{3}); \bar{X} \in ∂Ω, X_{3} \in (- H (\bar{X}) ∕ 2, H (\bar{X}) ∕ 2)}$ and its normal is $(\bar{ν}, 0)$ . The upper and lower boundaries of the shell will be denoted by $Γ_{+} = {(\bar{X}, X_{3}); \bar{X} \in Ω, X_{3} = H (\bar{X}) ∕ 2}$ and by $Γ_{-} = {(\bar{X}, X_{3}); \bar{X} \in Ω, X_{3} = - H (\bar{X}) ∕ 2}$ , respectively.

It follows that the stress-free boundary condition on the lateral boundary can be written as $Π (X) (\bar{ν}, 0) = 0 for all \bar{X} \in ∂Ω$ , and from (3.6) we obtain

F_{0} S_{2} \bar{ν} = 0 + Σ O (δ^{2}), for all \bar{X} \in ∂Ω .

(3.7)

On the upper and lower boundaries $Γ_{+}$ and $Γ_{-}$ we consider traction-free boundary conditions, that is, $Π c_{3} = 0, for X_{3} = \pm H (\bar{X}) ∕ 2$ . We will replace these boundary equations by $\bar{Π} c_{3} = 0, \hat{Π} c_{3} = 0$ , where we have denoted by $\bar{A}, \hat{A}, \overset{⌣}{A}$ the thickness average and first- and second-order momentum of $A$ , that is,

\bar{A} = \frac{1}{H} \int_{- H ∕ 2}^{H ∕ 2} A (X_{3}) d X_{3}, \hat{A} = \frac{1}{H^{2}} \int_{- H ∕ 2}^{H ∕ 2} X_{3} A (X_{3}) d X_{3}, \overset{⌣}{A} = \frac{1}{H^{3}} \int_{- H ∕ 2}^{H ∕ 2} X_{3}^{2} A (X_{3}) d X_{3} .

From the normal components ${\bar{S}}_{33} = 0$ and $Ŝ_{33} = 0$ we can compute the transversal deformation functions $α = ϵ_{3} + w \frac{β}{H}$ and $β$ from the following linear system

\begin{matrix} {\begin{matrix} α {\bar{C}}_{33} + β Ĉ_{33} = & - {\bar{C}}_{3} : (ε_{0} + ε (u) + w K_{0}) - H {\hat{C}}_{3} : K_{0} - {\bar{S}}_{33}^{*} \\ α Ĉ_{33} + β {\overset{⌣}{C}}_{33} = & - {\hat{C}}_{3} : (ε_{0} + ε (u) + w K_{0}) - H {\overset{⌣}{C}}_{3} : K_{0} - Ŝ_{33}^{*} . \end{matrix} \end{matrix}

(3.8)

The solution of the above linear system will be denoted by $α = α (u, w)$ , $β = β (u, w)$ , where $α, β$ are functions defined on $Ω$ , with an affine dependence on $u$ and w.

3.3. Plate-to-shell equations

We consider here the plate-to-shell equations for design associated with the kinematics introduced in Section 2. These equations, deduced in the Appendix using a classical asymptotic expansion method, involve the in-plane stress resultants and couples. If $N = N (u, w) = H \bar{S_{2}}$ is the in-plane stress resultant,

N (u, w) = H {\bar{C}}_{2} (ε (u) + ε_{0} + w K_{0}) + α (u, w) H {\bar{C}}_{3} + β (u, w) H {\hat{C}}_{3} + H^{2} {\hat{C}}_{2} K_{0} + H \bar{S_{2}^{*}},

then, from the tangential components (in the deformed configuration) of the equilibrium equations (see the Appendix), we obtain

di v_{2} (N (u, w)) = 0, in Ω, (N (u, w)) \bar{ν} = 0, on ∂Ω,

(3.9)

while the normal component (in the deformed configuration) of the equilibrium becomes

N (u, w) : K_{0} = 0, in Ω .

(3.10)

If we introduce the in-plane stress couples $M = M (u, w) = H^{2} \hat{S_{2}}$ , given by

M (u, w) = H^{2} {\hat{C}}_{2} (ε (u) + ε_{0} + w K_{0}) + α (u, w) H^{2} {\hat{C}}_{3} + β (u, w) H^{2} {\overset{⌣}{C}}_{3} + H^{3} {\overset{⌣}{C}}_{2} K_{0} + H^{2} \hat{S_{2}^{*}},

then, the tangential components (in the deformed configuration) of the equilibrium equations read

di v_{2} (M (u, w)) = 0, in Ω, (M (u, w)) \bar{ν} = 0, on ∂Ω,

(3.11)

while the normal one becomes

M (u, w) : K_{0} = 0, in Ω .

(3.12)

To summarize, we have derived six linear equations (three for the resultants (3.9)–(3.10) and three for the couples (3.11)–(3.12)) for the three small-perturbation unknown functions $u$ and w, three components of pre-stress resultant $H \bar{S_{2}^{*}}$ and three components of the pre-stress couple $H^{2} \hat{S_{2}^{*}}$ .

3.4. Weak transversal heterogeneity

We assume further that the material has a weak transversal heterogeneity, that is,

{\hat{C}}_{2} = Σ O (δ), {\overset{⌣}{C}}_{2} = \frac{1}{12} {\bar{C}}_{2} + Σ O (δ),

(3.13)

{\hat{C}}_{3} = Σ O (δ), {\overset{⌣}{C}}_{3} = \frac{1}{12} {\bar{C}}_{3} + Σ O (δ) Ĉ_{33} = Σ O (δ), {\overset{⌣}{C}}_{33} = \frac{1}{12} {\bar{C}}_{33} + Σ O (δ) .

In this case the system (3.8) has a simpler form and we obtain

α (u, w) = - \frac{1}{{\bar{C}}_{33}} ({\bar{C}}_{3} : (ε_{0} + ε (u) + w K_{0}) + {\bar{S}}_{33}^{*}), β = - \frac{1}{{\bar{C}}_{33}} (H {\bar{C}}_{3} : K_{0} + 12 Ŝ_{33}^{*})

N^{*} = H \bar{S_{2}^{*}} - \frac{H {\bar{S}}_{33}^{*}}{{\bar{C}}_{33}} {\bar{C}}_{3}

denotes the pre-stress resultants then, the in-plane stress resultant is $N (u, w) = H M_{2} (ε (u) + ε_{0} + w K_{0}) + N^{*}$ , where we use $M_{2}$ for the in-plane elastic fourth-order tensor

M_{2} A = {\bar{C}}_{2} A - \frac{{\bar{C}}_{3} : A}{{\bar{C}}_{33}} {\bar{C}}_{3} .

The average shell equations (3.9)–(3.10) become a linear partial differential equation for $u$ and w

di v_{2} (H M_{2} (ε (u) + ε_{0} + w K_{0}) + N^{*}) = 0, in Ω,

(3.14)

(H M_{2} (ε (u) + ε_{0} + w K_{0}) + N^{*}) ν = 0, on ∂Ω,

(3.15)

while the normal component (in the deformed configuration) gives

(H M_{2} (ε (u) + ε_{0} + w K_{0}) + N^{*}) : K_{0} = 0, in Ω .

(3.16)

Let $M^{*}$ denotes the pre-stress couple

M^{*} = H^{2} \hat{S_{2}^{*}} - \frac{H^{2} Ŝ_{33}^{*}}{{\bar{C}}_{33}} {\bar{C}}_{3} .

Then the stress couple $M = \frac{H^{3}}{12} M_{2} K_{0} + M^{*}$ does not depend on the small perturbation $u, w$ and the moments shell equations (3.11)–(3.12) read

di v_{2} (\frac{H^{3}}{12} M_{2} K_{0} + M^{*}) = 0, in Ω, (\frac{H^{3}}{12} M_{2} K_{0} + M^{*}) \bar{ν} = 0, on ∂Ω

(3.17)

(\frac{H^{3}}{12} M_{2} K_{0} + M^{*}) : K_{0} = 0, in Ω .

(3.18)

We end this section with several remarks.

We note that the small perturbation $u, w$ can be obtained by solving the linear system (3.14)–(3.16) for any $N^{*}$ . Meanwhile, necessary conditions for the existence of a solution $u, w$ of the system (3.14)–(3.16) impose some restrictions on $ε_{0}, K_{0}$ , and $N^{*}$ . In addition, sufficient conditions for the existence of a solution $u, w$ could also involve $K_{0}$ and $N^{*}$ . For instance, let us remark that if $w \equiv 0$ and $H (M_{2} (ε (u) + ε_{0})) + N^{*} = 0$ in $Ω$ , then (3.14)–(3.16) are satisfied. This means that

ε (u) = - ε_{0} - \frac{1}{H} M_{2}^{- 1} N^{*}

(3.19)

and the compatibility conditions to be satisfied for the right-hand side of (3.19) are sufficient conditions for the existence of a solution $u, w = 0$ of the system (3.14)–(3.16).

We also note that w can be computed explicitly from (3.16) and we have only two (degenerate) equations for $u$ , that is,

di v_{2} (H M_{2}^{K_{0}} (ε (u) + ε_{0}) + N_{K_{0}}^{*}) = 0, in Ω, (M_{2}^{K_{0}} (ε (u) + ε_{0}) + N_{K_{0}}^{*}) ν = 0, on ∂Ω,

(3.20)

where we have denoted by $N_{K_{0}}^{*}$ the reduced pre-stress and by $M_{2}^{K_{0}}$ the elastic operator

M_{2}^{K_{0}} A = M_{2} (A - \frac{M_{2} A : K_{0}}{M_{2} K_{0} : K_{0}} K_{0}), N_{K_{0}}^{*} = N^{*} - \frac{M_{2} N^{*} : K_{0}}{M_{2} K_{0} : K_{0}} M_{2} K_{0} .

By solving (3.17)–(3.18) we obtain the three components of the unknown pre-stress couple $M^{*}$ , able to deform the plate into a shell close to the designed one. Let us also remark that the equations and boundary conditions (3.17)–(3.18) are satisfied if

M = \frac{H^{3}}{12} M_{2} K_{0} + M^{*} = 0, in Ω,

(3.21)

which means that $K_{0}$ and $M_{2}^{- 1} M^{*}$ are co-axial tensors. The above equation is only a sufficient condition for (3.17)–(3.18) but not a necessary one. Indeed, we will see in the next section for cylindrical surfaces that (3.21) is not satisfied but (3.17)–(3.18) hold.

For isotropic materials the condition of weak transversal heterogeneity (3.13) becomes

\hat{λ} = Σ O (δ), \hat{μ} = Σ O (δ), \overset{⌣}{λ} = \frac{1}{12} \bar{λ} + Σ O (δ), \overset{⌣}{μ} = \frac{1}{12} \bar{μ} + Σ O (δ),

(3.22)

the elastic in-plane stress tensor $M_{2}$ is

M_{2} A = \frac{2 \bar{μ} \bar{λ}}{\bar{λ} + 2 \bar{μ}} trace (A) I_{2} + 2 \bar{μ} A

(3.23)

and we can see that (3.14)–(3.15) and (3.17)–(3.18) are membrane-type equations.

4. Applications to some classical designed surfaces

In this section, we apply the plate-to-shell equations for design, introduced in the previous section, to three classical surfaces: a sphere, a cylinder, and a cone. In this section, we consider only isotropic materials with a weak heterogeneity, that is, (3.2) and (3.22) hold.

Special attention will be paid for the plane pre-stress configuration induced by a small pre-strain $ϵ^{*} I_{2} + ϵ_{3}^{*} c_{3} \otimes c_{3}$ , which is isotropic in the in-plane directions. In this case $ϵ_{3}^{*}$ could be computed from $S_{33}^{*} = 0$ and the in-plane pre-stress $S_{2}^{*}$ is given by

S_{2}^{*} = σ^{*} I_{2}, where σ^{*} = ϵ^{*} \frac{2 μ (3 λ + 2 μ)}{λ + 2 μ},

(4.1)

and the pre-stress resultant and pre-stress couple are $N^{*} = {\bar{σ}}^{*} H I_{2}, M^{*} = {\hat{σ}}^{*} H^{2} I_{2}$ .

A particular case of a plane pre-stress configuration induced by a small isotropic pre-strain pre-strained elastic layer is the following two layers system. One layer of thickness h₁ is grown under stress-free conditions on an elastic layer of thickness h₂, that is,

\begin{matrix} h_{1} + h_{2} = H, ϵ^{*} (\bar{X}, X_{3}) = {\begin{matrix} m, & for \frac{H}{2} - h_{1} < X_{3} < \frac{H}{2}, \\ 0, & for \frac{H}{2} < X_{3} < - \frac{H}{2} + h_{2} . \end{matrix} \end{matrix}

(4.2)

4.1. Small parts of spherical surfaces

Let $(r, θ, φ)$ be the spherical coordinates in the Eulerian description and denote by $e_{r} = e_{r} (θ, φ), e_{θ} = e_{θ} (θ, φ), e_{φ} = e_{φ} (φ)$ the local physical basis in the Eulerian description. For the Lagrangian description we use the polar coordinates $R, Φ$ with the local physical basis denoted by $e_{R} = e_{R} (Φ), e_{Φ} = e_{Φ} (Φ)$ . If $R_{*}$ is the radius of the Eulerian sphere and we put

x_{0} (\bar{X}) = R_{*} e_{r} (θ (R), φ (Φ)), with r = r (R), φ = φ (Φ),

(4.3)

then after some algebra we have

F_{0} = R_{*} \frac{∂θ}{∂R} e_{θ} \otimes e_{R} + \frac{R_{*} sin (θ)}{R} \frac{∂φ}{∂Φ} e_{φ} \otimes e_{Φ} .

The unit normal of the Eulerian surface is $e_{3} (R, Φ) = e_{r} (θ (R), φ (Φ))$ and the curvature tensor is

K = - \frac{1}{R_{*}} (e_{θ} \otimes e_{θ} + e_{φ} \otimes e_{φ}) .

Let us show that if $Ω_{0}$ has small enough diameter, then the radial projection $R_{*} tan (θ) = R$ and $φ = Φ$ (see Figure 3) gives a geometric deformation $x_{0}$ that satisfies (2.1). If $Ω_{0} = B_{R_{0}} (0)$ for R₀, then we have

F_{0} = {cos}^{2} (θ) e_{θ} \otimes e_{R} + cos (θ) e_{φ} \otimes e_{Φ} .

Figure 3.

Illustration of the radial projection $R_{*} tan (θ) = R, φ = Φ$ of the map (4.3).

We can now compute $C_{0}$ and $K_{0}$ to find that

C_{0} (\bar{X}) = I_{2} - \frac{R^{2}}{R^{2} + R_{*}^{2}} [e_{Φ} \otimes e_{Φ} + \frac{2 R_{*}^{2} + R^{2}}{R^{2} + R_{*}^{2}} e_{R} \otimes e_{R}], H K_{0} = - \frac{H}{R_{*}} C_{0} .

If $δ = \frac{H}{R_{*}}$ , then for $\frac{R_{0}}{R_{*}} = O (δ)$ (i.e., $R_{0} = O (H)$ ) we obtain (2.1) with $ε_{0} = 0$ , while for $\frac{R_{0}^{2}}{R_{*}^{2}} = O (δ)$ we obtain (2.1) with

ε_{0} (R) = - \frac{R^{2}}{R_{*}^{2}} [e_{Φ} \otimes e_{Φ} + 2 e_{R} \otimes e_{R}], H K_{0} = - \frac{H}{R_{*}} I_{2} for R < R_{0} = \sqrt{H R_{*}} .

(4.4)

We note that the small perturbation $u, w$ can be found by solving the linear system (3.14)–(3.16), but from (3.18) we obtain

0 = M_{RR} + M_{ΦΦ} = - 2 \frac{H^{3}}{12 R_{*}} \frac{2 \bar{μ} (3 \bar{λ} + 2 \bar{μ})}{\bar{λ} + 2 \bar{μ}} + M_{RR}^{*} + M_{ΦΦ}^{*},

(4.5)

and if we suppose that $M_{RΦ}^{*} = 0$ and $M_{RR}^{*}, M_{ΦΦ}^{*}$ depend only on R then the equilibrium equation (3.17) reads $\frac{d}{dR} (M_{RR}) + 2 \frac{M_{RR}}{R} = 0$ and we obtain

R^{2} M_{RR} = - \frac{H^{3} R^{2}}{12 R_{*}} \frac{2 \bar{μ} (3 \bar{λ} + 2 \bar{μ})}{\bar{λ} + 2 \bar{μ}} + R^{2} M_{RR}^{*} = const .,

(4.6)

while the boundary conditions read

M_{RR} ν_{R} = 0, M_{ΦΦ} ν_{Φ} = 0, on ∂Ω .

(4.7)

Finally, from (4.5)–(4.7) we found the pre-stress couples $M_{RR}^{*}, M_{ΦΦ}^{*}$ , needed for designing a small part of a spherical surface.

For isotropic pre-stress/strain given by (4.1) equations (4.5)–(4.7) reads

\frac{H}{12 R_{*}} \frac{2 \bar{μ} (3 \bar{λ} + 2 \bar{μ})}{\bar{λ} + 2 \bar{μ}} = {\hat{σ}}^{*},

(4.8)

and the equilibrium equations for the moments are satisfied for all the geometries $Ω \subset Ω_{0}$ .

In conclusion, we found that: a spherical configuration of radius $R_{*}$ could be designed from a material with weak transversal heterogeneity and an in-plane homogeneity if the isotropic pre-stress $σ^{*}$ (or pre-strain $ϵ^{*}$ ) satisfies (4.8) and the in-plane surface $Ω$ has a diameter of order of $2 \sqrt{H R_{*}}$ .

In the particular case when an isotropically pre-strained elastic layer of thickness h₁ is grown under stress-free conditions on an elastic layer of thickness h₂, that is, $ϵ^{*}$ is given by (4.2), and if additionally the contrast between the elasticities of the two layers is small, from (4.8) we obtain

\frac{1}{R_{*}} = \frac{6 mξ}{h_{2} {(1 + ξ)}^{3}},

(4.9)

where we introduced $ξ = h_{1} ∕ h_{2}$ . The result in (4.9) can also be obtained [31, 32] from classical Love–Kirchhoff theory of plates in plane-stress. This is expected since for a small diameter domain the linearized small-perturbation theory applies.

A refinement of this result may be obtained from (4.8) using $(λ_{i}, μ_{i})$ for the elastic moduli in the layer of thickness $h_{i}$ ( $i = 1, 2$ ). Then we obtain

\frac{1}{R_{*}} = \frac{6 mξ (ξ + \frac{λ_{2} + 2 μ_{2}}{λ_{1} + 2 μ_{1}})}{h_{2} {(1 + ξ)}^{2} (ξ + \frac{3 λ_{2} + 2 μ_{2}}{3 λ_{1} + 2 μ_{1}}) (ξ + \frac{μ_{2}}{μ_{1}})},

(4.10)

which, obviously, reduces to (4.9) when $λ_{2} = λ_{1}$ and $μ_{2} = μ_{1}$ .

The comparison between the theoretical, numerical, and experimental results is illustrated in Figure 4. Let us describe first the experimental results. We have grown a bi-layer semiconductor material that consist of a $h_{1} = 200$ nm thick ${In}_{0.88} {Ga}_{0.12} P$ (lattice parameter 5.8185 ) layer on a $h_{2} = 200$ nm thick $InP$ layer (lattice parameter $5.8687$ ). Both materials are cubic and the misfit strain is isotropic in the growth plane with $m = 0.0085$ . A $10 μ m \times 10 μ$ m domain was then released from the substrate by the under-etching of a 500 nm sacrificial layer grown prior to the bi-layer. The photo of the relaxed square is given in Figure 4(top right). We see that the relaxed shell has a shape close to a spherical cap, but its radius is difficult to measure.

Figure 4.

Right, top: Experimental evidence of a $a = 10 μ$ m length square plate with an isotropic in-plane pre-stress that relaxes into a shape close to a spherical cap (reprinted with permission from IOP [32]). Right, bottom: 3D finite-element computations of the relaxation of the same finite elastic square plate with the same pre-stress. Left: The designed sphere computed with formula (4.9) versus two 3D finite-element computations of the relaxation of an elastic square plates with the same pre-stress: the first corresponds to the experimental length a and the second, which is smaller, to the theoretical estimated length $a_{*} = 2 R_{0} ∕ \sqrt{2}$ given by formula (4.4).

Figure 5.

Illustration of the cylindrical projection $θ (\bar{X}) = \frac{X_{1}}{R_{*}}, z (\bar{X}) = X_{2}$ of map (4.11).

If we suppose that the elastic coefficients are homogeneous, then the radius $R_{*}$ of the designed sphere can obtained from (4.9) independent of the elastic coefficients. If we use the real elastic coefficients, then the factor between the right-hand side in (4.10) and (4.9) is 1.034, hence the difference between the curvature radius $R_{*}$ obtained with two formula is negligible. The designed mid-surface computed from (4.10) or (4.9) is plotted in Figure 4(left). The theoretical curvature radius $R_{*}$ is, thus, estimated to be around $31 μ$ m, while the diameter $2 R_{0}$ of the pre-strained plate is estimated from (4.4) to be of order of $2 \sqrt{H R_{*}} \approx 7 μ$ m.

The 3D finite-element numerical computations of the relaxation of the same finite elastic square ( $a = 10 μ$ m) plate with a pre-strain given by (4.2) is shown in Figure 4(bottom right and left). We remark that there are some (small) differences between the computed mid-surface and the designed surface in the corners of the square. We deduce that the computed relaxed shell has not exactly a spherical shape. In the same figure we have plotted the finite-element computations for a smaller square $a_{*} \times a_{*}$ such that the plate is included in theoretical disk of radius $R_{0} = \sqrt{R_{*} H}$ , that is, $a_{*} = 2 R_{0} ∕ \sqrt{2} \approx 4.94 μ$ m. We remark that the computed relaxed mid-surface matches very well with the designed sphere.

4.2. Cylindrical surfaces

We use $(r, θ, z)$ for the cylindrical coordinates in the Eulerian description and $e_{r} = e_{r} (θ), e_{θ} = e_{θ} (θ), e_{z}$ for the local physical basis in the Eulerian description. If $R_{*}$ is the radius of the Eulerian cylinder and we define

x_{0} (\bar{X}) = R_{*} e_{r} (θ (\bar{X})) + z (\bar{X}) e_{z},

(4.11)

then, after some algebra, we obtain

F_{0} = R_{*} [\frac{∂θ}{\partial X_{1}} e_{θ} \otimes c_{1} + \frac{∂θ}{\partial X_{2}} e_{θ} \otimes c_{2}] + \frac{∂z}{\partial X_{1}} e_{z} \otimes c_{1} + \frac{∂z}{\partial X_{2}} e_{z} \otimes c_{2} .

Obviously, the unit normal of the Eulerian surface is $e_{3} (\bar{X}) = e_{r} (θ (\bar{X}))$ and the curvature tensor is

K = - \frac{1}{R_{*}} e_{θ} \otimes e_{θ} .

Let $Ω \subset Ω_{0} = (0, L) \times ℝ$ be an arbitrary domain, with $L < 2 π R_{*}$ and consider the cylindrical projection defined by $θ (\bar{X}) = X_{1} ∕ R_{*}$ and $z (\bar{X}) = X_{2}$ . Then, $F_{0} = e_{θ} \otimes c_{1} + e_{z} \otimes c_{2}$ so that we obtain

H K_{0} = - \frac{H}{R_{*}} c_{1} \otimes c_{1}, C_{0} (\bar{X}) = : F_{0}^{T} (\bar{X}) F_{0} (\bar{X}) = I_{2},

(4.12)

and (2.1) follows with $ε_{0} = 0$ .

Once again, the small perturbation $u, w$ can be found by solving the linear system (3.14)–(3.16) and from (3.18) we obtain

0 = M_{11} = - \frac{H^{3}}{12 R_{*}} \frac{4 \bar{μ} (\bar{λ} + \bar{μ})}{\bar{λ} + 2 \bar{μ}} + M_{11}^{*} .

(4.13)

If we assume further that $M_{12}^{*} = 0$ and that, for symmetry reasons, the mechanical setting are invariant with respect to X₂, then the equilibrium equation (3.17) reads $M_{22} ν_{2} = 0$ . This means that

0 = M_{22} = - \frac{H^{3}}{12 R_{*}} \frac{2 \bar{μ} \bar{λ}}{\bar{λ} + 2 \bar{μ}} + M_{22}^{*} or ν_{2} = 0 .

(4.14)

Finally, from (4.13) and (4.14) we found the pre-stress couples $M_{11}^{*}, M_{22}^{*}$ , needed for designing a cylindrical surface.

For isotropic pre-stress/strain given by (4.1) the first alternative in (4.14) cannot be true and (4.13) reads

\frac{H}{12 R_{*}} \frac{4 \bar{μ} (\bar{λ} + \bar{μ})}{\bar{λ} + 2 \bar{μ}} = {\hat{σ}}^{*} and ν_{2} = 0 .

(4.15)

In conclusion, a cylindrical configuration of radius $R_{*}$ can be designed from a material with weak transversal heterogeneity and an in-plane homogeneity if the isotropic pre-stress $σ^{*}$ (or pre-strain $ϵ^{*}$ ) satisfies (4.15) and the boundary of $Ω$ is parallel to the cylinder’s axis.

Let us analyze here the case of a bi-layer material that is isotropically pre-strained/stressed in the upper part (i.e., (4.2) holds). If it has equal elastic coefficients, from (4.15) we find the cylinder curvature

\frac{1}{R_{*}} = (1 + ν) \frac{6 mξ}{h_{2} {(1 + ξ)}^{3}},

(4.16)

where, as previously, $ξ = h_{1} ∕ h_{2}$ is the aspect ratio of the bi-layer. Once again, a refinement of (4.16) that accounts for the week transversal homogeneity can be obtained from (4.15) and gives

\frac{1}{R_{*}} = \frac{3 mξ (ξ + \frac{λ_{2} + 2 μ_{2}}{λ_{1} + 2 μ_{1}})}{h_{2} {(1 + ξ)}^{2} (ξ + \frac{μ_{2}}{μ_{1}}) (ξ \frac{λ_{1} + μ_{1}}{3 λ_{1} + 2 μ_{1}} + \frac{λ_{2} + μ_{2}}{3 λ_{1} + 2 μ_{1}})},

(4.17)

which again reduces to (4.16) when $λ_{2} = λ_{1}$ and $μ_{2} = μ_{1}$ .

As in the previous subsection, we present here a comparison between the theoretical, numerical, and experimental results. Figure 6(right) shows a scanning electron microscopy (SEM) image of an experimental realization of a cylindrical shape starting from a pre-strained bi-layered material. It concerns a 50 $μ$ m long cylinder obtained by relaxation (using the under-etching of a sacrificial layer) of a bi-layer semiconductor material obtained by MBE growth of a $h_{1} = 57$ nm thick ${In}_{0.88} {Ga}_{0.12} P$ layer on a $h_{2} = 168$ nm thick layer of $InP$ (with $ν = ν_{InP} = 0.36$ ). In the two-layer material, the ${In}_{0.88} {Ga}_{0.12} P$ layer is grown isotropically pre-strained in the horizontal plane at $m = 0.0085$ and released by under-etching. The pattern of holes in the bi-layer material plays no particular role in the problem but was designed in the experimental setting in order to reduce the under-etching time. The (approximate) SEM measure of the cylinder diameter, is estimated at $2 R_{\exp} \approx 35.8 μ$ m.

Figure 6.

Right: Experimental evidence of a large surface with vertically non-homogeneous but isotropic in-plane pre-strain whose relaxed shape is a cylinder and the experimental value of the diameter. Left: The designed cylinder computed with formula (4.16) versus 2D finite-element computations of the relaxation of an elastic plate with a pre-strain/stress. Zoom on the upper and bottom part.

The diameter of the designed cylinder, predicted by the simplified formula (4.16), is $2 R_{*} \approx 34.29 μ$ m and the value of the correction factor between the right-hand side of (4.17) and that of (4.16) is 1.02, which is not significant with respect to the experimental error. We remark that there is very good agreement between the theoretical predictions and the experimental measures.

The 2D finite-element numerical computations of the relaxation of the plate with a pre-strain given by (4.2) versus the designed cylinder, predicted by the simplified formula (4.16) is shown in Figure 6(left). The error between the numerical diameter $2 R_{num} \approx 34.18 μ$ m and the designed one $2 R_{*} \approx 34.29 μ$ m is very small with respect to the discretization numerical error.

4.3. Conical surfaces

Let us consider the spherical coordinates $(r, θ, φ)$ for the Eulerian description and we denote by $e_{r} = e_{r} (θ, φ), e_{θ} = e_{θ} (θ, φ), e_{φ} = e_{φ} (φ)$ the local physical basis in the Eulerian description. For the Lagrangian description we use the polar coordinates $R, Φ$ with the local physical basis denoted by $e_{R} = e_{R} (Φ), e_{Φ} = e_{Φ} (Φ)$ . If $θ_{0}$ denotes the angle of the Eulerian cone we put (see Figure 7)

x_{0} (R, Φ) = r (R) e_{r} (θ_{0}, φ (Φ)),

(4.18)

and a straightforward computation gives

F_{0} = \frac{dr}{dR} e_{r} \otimes e_{R} + \frac{r sin (θ_{0})}{R} \frac{dφ}{dΦ} e_{φ} \otimes e_{Φ} .

Figure 7.

Illustration of the map (4.18).

The unit normal of the Eulerian surface is $e_{3} (Φ) = e_{θ} (θ_{0}, φ (Φ))$ and the curvature tensors are

K = - \frac{1}{r sin (θ_{0})} e_{φ} \otimes e_{φ}, K_{0} = - \frac{r sin (θ_{0})}{R^{2}} {(\frac{dφ}{dΦ})}^{2} e_{Φ} \otimes e_{Φ} .

Let $Ω \subset Ω_{0} = ℝ^{2} \ B_{R_{0}} (0)$ be an arbitrary domain and let us consider the conical map $r (R) = R$ and $φ (Φ) = Φ ∕ sin (θ_{0})$ , then $F_{0} = e_{r} \otimes e_{R} + e_{φ} \otimes e_{Φ}$ and we obtain

H K_{0} = - \frac{H}{R sin (θ_{0})} e_{Φ} \otimes e_{Φ}, C_{0} = F_{0}^{T} F_{0} = I_{2} .

(4.19)

Then (2.1) follows with $ε_{0} = 0$ , while the assumption (2.3) reads

\frac{H}{R sin (θ_{0})} = O (δ), in Ω_{0} .

(4.20)

As previously, the small perturbation $u, w$ can be obtained by solving the linear system (3.14)–(3.16) and from (3.18) we obtain

M_{ΦΦ} = - \frac{H^{3}}{12 R sin (θ_{0})} \frac{4 \bar{μ} (\bar{λ} + \bar{μ})}{\bar{λ} + 2 \bar{μ}} + M_{ΦΦ}^{*} = 0, in Ω .

(4.21)

For materials with a circular symmetry (i.e., $H = H (R), λ = λ (R, X_{3}), μ = μ (R, X_{3}), M^{*} = M^{*} (R)$ ) if we assume that $M_{RΦ}^{*} = 0$ , then the equilibrium equation (3.17) reads $\frac{d}{dR} (M_{RR}) + \frac{M_{RR}}{R} = 0$ and we obtain

R M_{RR} = - \frac{H^{3}}{12 sin (θ_{0})} \frac{2 \bar{μ} \bar{λ}}{\bar{λ} + 2 \bar{μ}} + R M_{RR}^{*} = const . and M_{RR} ν_{R} = 0 .

(4.22)

Finally, from (4.21) and (4.22) we obtain the pre-stress couples $M_{RR}^{*}, M_{ΦΦ}^{*}$ needed for the design of a conical surface.

For isotropic pre-stress/strain given by (4.1) the necessary conditions (4.21) and (4.22) are

{\hat{σ}}^{*} = \frac{H}{12 R sin (θ_{0})} \frac{4 \bar{μ} (\bar{λ} + \bar{μ})}{\bar{λ} + 2 \bar{μ}}, H^{3} \bar{μ} = const . and ν_{R} = 0 .

(4.23)

Finally, we conclude that a conical configuration of angle $θ_{0}$ can be designed from a material with weak transversal heterogeneity and an isotropic pre-stress $σ^{*}$ (or pre-strain $ϵ^{*}$ ) if the elastic coefficients’ average and the thickness satisfy (4.23) and the boundary of $Ω$ is a radial line. Moreover, a conical relaxed surface cannot be designed from an in-plane homogeneous plate with a constant thickness and with a homogeneous pre-strain/stress.

Since, for technological reasons, an experimental realization of a cone relaxation is not yet possible, we restrict ourselves to a comparison between theoretical and numerical results. Let us describe first the mechanical settings. The plate is a pre-stressed bi-layered homogeneous material of constant thickness $H = 2 h_{1} = 2 h_{2} = 400$ nm. The pre-stress is given by

\begin{matrix} S^{*} = {\begin{matrix} {\tilde{S}}_{RR}^{*} (R) e_{R} \otimes e_{R} + {\tilde{S}}_{ΦΦ}^{*} (R) e_{Φ} \otimes e_{Φ}, for & 0 < X_{3} < \frac{H}{2} \\ 0, for & - \frac{H}{2} < X_{3} < 0, \end{matrix} \end{matrix}

(4.24)

where ${\tilde{S}}_{RR}^{*}$ and ${\tilde{S}}_{ΦΦ}^{*}$ where are given from (4.21) and (4.22) such that $M_{RR} = M_{ΦΦ} = 0$ , that is,

{\tilde{S}}_{RR}^{*} (R) = \frac{4 H}{3 R sin (θ_{0})} \frac{2 \bar{μ} \bar{λ}}{\bar{λ} + 2 \bar{μ}}, {\tilde{S}}_{ΦΦ}^{*} (R) = \frac{8 H}{3 R sin (θ_{0})} \frac{4 \bar{μ} (\bar{λ} + \bar{μ})}{\bar{λ} + 2 \bar{μ}} .

The Lagrangian plate is a the angular sector ${R \in (R_{1}, R_{2}), Φ \in (0, Φ_{\max})}$ with $R_{1} = 12 μ$ m and $R_{1} = 16 μ$ m such that (4.20) holds. To have a reasonable computational time on a desktop computer (i.e., less than 250,000 degrees of freedom), we have chosen $Φ_{\max} = sin (θ_{0}) π ∕ 4$ , hence the expected relaxed configuration will cover only $1 ∕ 8$ of the designed cone. In Figure 8, we have plotted the Lagrangian plate with normalized $S_{22}^{*}$ in the color scale. We note that the relaxed configuration, computed with a 3D finite-element code, matches very well with the designed cone.

Figure 8.

The designed cone of angle $θ_{0} = π ∕ 6$ versus 3D finite-element computations of the relaxation of an elastic plate with a non-homogeneous and non-isotropic pre-stress given by (4.24). In color scale, the distribution of the normalized pre-stress $S_{22}^{*}$ in the non-deformed configuration.

5. Conclusions

We have investigated the interplay between mechanics and geometry in the case of a pre-strained/stressed ultra-thin film. Our aim was to construct a theoretical framework able to enlarge the class of 3D objects that can be fabricated from suitable planar design by using relaxation of pre-strains/stress. Toward the applications in the field of light-engineering (photonics), we have focused on semiconductor materials only, which are brittle–elastic materials. For this practical reason the mechanical relaxation problem has to deal with the small deformations only. This is actually a technological restriction and not a mathematical simplification of the problem. As is the case of buckling, large rotations are mandatory in order to describe the complex 3D relaxed shapes of various planar geometries. The design problem considered in this article can be formulated as follows: given a shape of a shell’s mid-surface (described as a 3D image of a plate mid-surface) find some sufficient conditions on the shell’s thickness, on the pre-strain/stress and on the shape of the plate mid-surface such that the pre-strained/stressed plate could relax with small deformations into the given shell without any other loading effort.

As the above design problem is very complex we will split it into two simpler problems. The first, called the geometric design problem, consists of finding a small strain transformation of planar surface into a 3D surface. Significant effort was done (see, for instance, [33 –36]) on isometric transformations (no strain deformation), which represent a special class of small-strain transformations (2.1) considered here. We do not give here any general mathematical treatment of the problem, but we take it like a geometrical input. Moreover, we provide three classical geometries examples, the small-diameter spherical surfaces, the (arbitrary) cylindrical surface, and the conical surfaces, but as expected, many others could also be considered. While the last ones are isometric transformation, the first one is not.

The second design problem, called the mechanical design problem, is related to the relaxation of a plate with a non-homogeneous (in thickness) pre-strain/stress. The deformation of the mid-surface is supposed to be known (see the first geometric problem) and characterized through the small deformation tensor and the curvature tensor. Over this geometric design problem (which reproduces exactly the target mid-surface) we consider a small perturbation of Love–Kirchhoff type in order to compensate for the small (membrane) deformation of the proposed geometric transformation. As a consequence, the mid-surface is close to the designed geometric mid-surface, and we deal with an approximative design problem. The resulting six equations for efforts’ average and moments for three unknowns (two in-plane and one out-of-plane displacements) will provide three supplementary equations to compute the in-plane pre-strain/stress moments.

A special attention was devoted to orthotropic materials with weakly transversal heterogeneity and with an isotropic pre-strain. From a practical perspective, these materials can be obtained by modern methods for crystal growth (MBE) where the pre-strain/pre-stress can be controlled by the composition and the heterogeneity is purely vertical as a consequence of the layer-by-layer growth mode. In this case, the system of equations is simplified and the efforts–moments system involves only the pre-strain moments and the thickness. In the special case of isotropic materials the three examples considered here (sphere, cylinder, and cone) can be easily handled to obtain the formula of the pre-strain moments as a function of the curvature radius and the thickness.

For a given isotropic pre-stress/strain, we have found that a rectangular plate could relax both into a cylindrical surface (if the boundary is parallel to cylinder axis) or on a part of a sphere in which case it should have a small diameter with respect to the sphere radius. In both cases, the theoretical estimates have been compared with the experimental realizations and finite-element numerical computations and we found very good agreement between all of them. For the cone geometry, we found that the design is not possible from an isotropic pre-stress with an in-plane homogeneity. For a non-isotropic pre-stress, given from the theoretical expressions, the 3D finite-element computation of the relaxed surface matches very well with the designed conical surface.

Footnotes

Appendix: Deduction of the plate-to-shell equations

In this appendix, we use a classical asymptotic expansion method to deduce the plate-to-shell equations for design (3.9)–(3.12) associated with the kinematics introduced in Section 2.

The equilibrium equation in the absence of the body forces reads

(A.1)

\sum_{J = 1}^{2} \frac{\partial}{\partial X_{J}} Π c_{J} + \frac{\partial}{\partial X_{3}} Π c_{3} = 0 .

Acknowledgements

We are grateful to Ph. Regreny, P. Cremillieu, and J.-L. Leclercq for the fabrication of pre-strained thin multi-layered films and associated clean-room technology at the NANO-Lyon fabrication facility at the Institute of Nanotechnologies in Lyon.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was partially supported the French Research Agency (Grant Number ANR-17-CE24-0027) and the Romanian National Authority for Scientific Research and Innovation, CNCS-UEFISCDI (Project Number PN-III-P2-2.1-PED-2016-0436), within PNCDI III.

ORCID iD

Ioan R. IONESCU

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