Swelling-induced wrinkling in layered gel beams

Abstract

Gels are widely employed in smart mechanical devices and biomedical applications. Swelling-induced bending actuation can be obtained by means of a simple bilayer gel beam. We show that this system can also exhibit wrinkling patterns of potential interest for structural morphing and sensing. We study swelling-induced wrinkling at the extrados of a bilayer gel beam with the softer layer on top. The bent configuration at finite strain is recovered first and, starting from it, a linear perturbation analysis is performed. We delimit the zone corresponding to wrinkling modes in a parameter plane encompassing a mechanical stiffness ratio and a geometric top layer to total height ratio. Interestingly, we observe that surface instability precedes and envelopes wrinkling modes of finite wavelength. Finally, we discuss the effect of changes in stiffness and of the Flory–Huggins parameters χ on the size of the wrinkling domain.

1. Introduction

Gels, as an instance of soft matter, are increasingly studied and employed in the design of actuating and sensing devices (see [1,2], and references therein). Actuation is a configurational and functional change of a structure produced by an external stimulus. There is great choice both in the type of stimuli to employ, i.e. thermal, electrical, chemical [3], mechanical, and in the arrangements of material and geometric properties, a.k.a. geometric composites, which can be used to attain a desired configuration [4]. In sensing configurational changes are exploited to provide quantitative information on the state of a system or of its environment. The ability of gels to elastically undergo large deformations and bifurcations widens the spectrum of attainable configurations and effects [5,6]. This marks a paradigm reversal with respect to traditional structural design in which both large displacements and instabilities are kept at a specified safety distance.

In this work, we consider an archetypal actuator, i.e. a bilayer beam [7], and, motivated by findings made in the Literature [8–11], we investigate the conditions for the occurrence of wrinkling on the extrados of a bent beam embedded into a solvent bath. Potential applications include design of channels for micro-fluidics applications [12] or sensing of the systems status [13–15] such as changes in the chemical potential of the solvent.

There is already a number of works on the wrinkling bifurcations of beams bent in plane strain and made of incompressible materials [16–18] and we have drawn from them the techniques used here. The reconstruction of the bent configuration rests on the seminal work by Rivlin [19]; then a linear perturbation analysis is performed upon it and the search for roots of the ensuing determinantal equation is made more robust by means of the compound matrix method [20,21]. The extension of the investigations to gels has required some additional assumptions, such as the position that each layer of the beam is homogeneously swollen of the amount it would swell if it were not tied to the other layer. An assessment of these restrictions can be found in [22]. Another element differentiating the current work with respect to the previous ones is the choice not to discuss the dependence of the bifurcation condition on the slenderness ratio of the beam. Based on remarks already made in [22] and on the results of [17,18], the number of wrinkles grows linearly with the length of a beam of constant height while the critical wavelength remains approximately constant. The attention is therefore here focused on the dependence of the number of wrinkles on the ratio between the stiffness moduli of the layers and on the ratio between the heights of the layers. When the gel layers have the same affinity with the solvent, the first of the two ratios determines the mismatch between the characteristic free swelling stretches of the layers on the basis of the bending of the beam. The second ratio determines the amount of bending. The last difference concerns the main result. Wrinkling is here observed at the extrados of the beam and not at the intrados. Moreover, while in previous studies wrinkles of finite wavelength occurred before the condition for surface instability was met [23,24], here they occur afterwards.

This result is novel in the present context but, in hindsight, it is not so unexpected. It has recently been observed in horizontal gel layers growing on top of a rigid substrate [25]. The result in [25] is neat in that an asymptotic relationship is established between the growth parameter and the critical wavenumber. Also it is shown that the introduction of surface tension in the model regularizes the wavenumber which thus no longer attains an infinite value.

An important aspect that is left for future works is the possible formation of folds or creases. Recent literature suggests that in some instances creases may form ahead of wrinkles [26,27], immediately after [28] or sometime later,¹ if ever, depending on the thickness of the layers and stiffness mismatch [29], on the amount of pre-stretch [30] and on the chemical potential of the solvent [31]. Also, the effect of imperfection on the wrinkling modes has been left for future investigations, being especially important when considering applications [28].

The paper is structured in the following way. In §§2 and 3, the stress–diffusion model of gels and the geometry, assumptions and formulation of the problem are shortly summed up [22,32,33], respectively. The recovery of the bent configuration of the swollen bilayer is concisely reported in §4 [22]. All prerequisites having been introduced, the incremental elastic problem is tackled in §5, and the results are presented and discussed in §6.

2. Background: stress–diffusion continuum model

Swelling-induced deformation processes can be fully described within a stress–diffusion continuum model based on the balance equations for forces and solvent, on the thermodynamics inequalities restricting the class of admissible constitutive prescriptions, and on the choice of a free energy density which accounts for both the elastic and mixing contributions. Following [32], we identify the gel body at the dry state with its reference configuration ${\mathcal{B}}_{\mathrm{d}}$, a region of the three-dimensional Euclidean space $\mathcal{E}$ with boundary $\mathrm{\partial }{\mathcal{B}}_{\mathrm{d}}$ of outward unit normal m. We aim to analyse swelling-driven deformation processes induced by boundary tractions t and/or by the immersion of the dry body in a bath of chemical potential μ_ext. In this case, the balance equations of forces and solvent at steady state are

$$\mathrm{div}\hspace{0.17em}\mathbf{\text{S}}=0\text{and}\mathrm{div}\hspace{0.17em}\mathbf{\text{h}}=0\text{on}\hspace{0.17em}{\mathcal{B}}_{\mathrm{d}}$$ 2.1

and

$$\mathbf{\text{S}}\hspace{0.17em}\mathbf{\text{m}}=\mathbf{\text{t}}\text{and}\mu ={\mu }_{\mathrm{ext}}\text{on}\hspace{0.17em}\mathrm{\partial }{\mathcal{B}}_{\mathrm{d}},$$ 2.2

with S, h and μ as the reference stress ([S]=J m⁻³), the reference solvent flux ([h]=mol m⁻² s⁻¹) and the chemical potential of the solvent within the body ([μ]=J mol⁻¹), respectively. State variables of the problem are the displacement field u from the reference configuration ${\mathcal{B}}_{\mathrm{d}}$ ([u]=m) and the solvent concentration c_d per unit reference volume ([c_d]=mol m⁻³); choosing the free energy of the gel in the Flory–Rehner form [34,35], the constitutive equations for the stress S, the chemical potential μ and the solvent flux h come from standard thermodynamical requirements and have the form (see [32,36] for details)

$$\begin{array}{r}\mathbf{\text{S}}={\mathbf{\text{S}}}_{\mathrm{d}}({\mathbf{\text{F}}}_{\mathrm{d}})-p{\mathbf{\text{F}}}_{\mathrm{d}}^{\star }=G{\mathbf{\text{F}}}_{\mathrm{d}}-p{\mathbf{\text{F}}}_{\mathrm{d}}^{\star },\end{array}$$ 2.3

$$\begin{array}{r}\mu =\mu (c_{\mathrm{d}})+p\mathrm{\Omega }=\frac{\mathcal{R}T}{\mathrm{\Omega }}{h}^{\prime }(c_{\mathrm{d}})+p\mathrm{\Omega }\end{array}$$ 2.4

$$\begin{array}{r}\mathrm{and}\mathbf{\text{h}}=\widehat{\mathbf{\text{h}}}({\mathbf{\text{F}}}_{\mathrm{d}},p,c_{\mathrm{d}})=-\frac{D{c}_{\mathrm{d}}}{RT}{\mathbf{\text{C}}}_{\mathrm{d}}^{-1}\mathrm{\nabla }(\mu (c_{\mathrm{d}})+p\mathrm{\Omega }),{\mathbf{\text{C}}}_{\mathrm{d}}={\mathbf{\text{F}}}_{\mathrm{d}}^{\mathrm{T}}{\mathbf{\text{F}}}_{\mathrm{d}},\end{array}$$ 2.5

where

$$\begin{array}{rl}{\mathbf{\text{F}}}_{\mathrm{d}}& =\mathbf{\text{I}}+\mathrm{\nabla }{\mathbf{\text{u}}}_{\mathrm{d}},J_{\mathrm{d}}=\text{det}\hspace{0.17em}{\mathbf{\text{F}}}_{\mathrm{d}},{\mathbf{\text{F}}}_{\mathrm{d}}^{\star }=J_{\mathrm{d}}\hspace{0.17em}{\mathbf{\text{F}}}_{\mathrm{d}}^{-\mathrm{T}}\\ \mathrm{and}h(c_{\mathrm{d}})& =\mathrm{\Omega }{c}_{\mathrm{d}}\hspace{0.17em}\text{log}\frac{\mathrm{\Omega }{c}_{\mathrm{d}}}{1+\mathrm{\Omega }{c}_{\mathrm{d}}}+\chi (T)\frac{\mathrm{\Omega }{c}_{\mathrm{d}}}{1+\mathrm{\Omega }{c}_{\mathrm{d}}},\end{array}\}$$ 2.6

with $\log$ indicating the natural logarithm, in equation (2.6) and in the following. The pressure p represents the reaction to the volumetric constraint, which maintains the volume change due to the displacement equal to the one due to solvent absorption or release:

$$J_{\mathrm{d}}=\text{det}\hspace{0.17em}{\mathbf{\text{F}}}_{\mathrm{d}}={\hat{J}}_{\mathrm{d}}(c_{\mathrm{d}})=1+\mathrm{\Omega }{c}_{\mathrm{d}},$$ 2.7

being Ω ([Ω]=m³ mol⁻¹) the solvent molar volume. Moreover, G ([G]=J m⁻³) is the shear modulus, T ([T]=K) the temperature, $\mathcal{R}$ ($[\mathcal{R}]={\mathrm{J}\hspace{0.17em}\mathrm{K}}^{-1}\hspace{0.17em}{\mathrm{mol}}^{-1}$) the universal gas constant, D ([D]=m²s⁻¹) the diffusivity and χ the dimensionless measure of the dis-affinity between gel and solvent, possibly depending on the temperature.

The problem (2.1)–(2.7) admits a distinguished solution, in the form of a free–swelling solution: the polymer is assumed to be immersed in a solvent bath of chemical potential μ_ext under load-free and constraint-free conditions and it can freely swell or shrink. In this case, mechanical and chemical balance laws prescribe S_d=0 and μ=μ_ext, that is, the swollen and stress-free state ${\mathcal{B}}_{\mathrm{o}}$, attained from ${\mathcal{B}}_{\mathrm{d}}$ is completely defined by the value μ_ext of the bath’s chemical potential. The difference between μ(X) and μ_ext determines the driving force of the swelling (shrinking) process and μ_ext is the control parameter of the deformative process. For homogeneous materials [32]

$${\mathbf{\text{F}}}_{\mathrm{d}}=J_{\mathrm{o}}^{1/3}\mathbf{\text{I}}\text{and}\frac{G\mathrm{\Omega }}{J_{\mathrm{o}}^{1/3}}+\mathcal{R}T(\log \left(\frac{J_{\mathrm{o}}-1}{J_{\mathrm{o}}}\right)+\frac{1}{J_{\mathrm{o}}}+\frac{\chi (T)}{J_{\mathrm{o}}^2})={\mu }_{\mathrm{ext}}.$$ 2.8

Assuming that the temperature T and the external chemical potential μ_ext are fixed, the swelling ratio J^1/3_o depends on the shear modulus G: to a lower shear modulus G, there corresponds a higher swelling stretch ratio J^1/3_o at equilibrium. Let us note that the equilibrium values J_o can be approximated by estimating the leading-order term in the Maclaurin asymptotic expansion in 1/J_o of equation (2.8) in which μ_ext has been set equal to zero, yielding

$$J_{\mathrm{o}}={\left(\frac{\mathcal{R}T}{\mathrm{\Omega }}\frac{1/2-\chi }{G}\right)}^{3/5}.$$ 2.9

The stress–diffusion theory briefly reviewed above is based on a strong coupling between the physics founded on the diffusion equations and the physics based on the elasticity equations. In general, neglecting this coupling brings an incorrect evaluation of the swelling-induced deformative process. However, in a few cases, when thin structures are involved, decoupling the problem gives an approximate solution which is in fairly good agreement with the outcomes of the complete stress–diffusion problem, as previously shown in [9,22].

3. Bilayer gel beams: an uncoupled approach

The focus of our investigations are thin-layered gel beams, with beam-like homogeneous components; the reference dry configuration ${\mathcal{B}}_{\mathrm{d}}$ of a beam segment is shown in figure 1a. By fixing an orthonormal basis {e₁,e₂,e₃} of $T\mathcal{E}$, we assume that e₁ spans the longitudinal axis of the beam, whereas (e₂,e₃) identify the principal axes of the beam’s cross section, and denote as l_d, w_d and h_d the length, depth and thickness of the beam, respectively. The top and bottom layers have a thickness h_t=βh_d and h_b=(1−β)h_d, respectively; moreover, we denote as α=G_t/G_b the ratio between the shear moduli of the two layers. Both α and β range from 0 to 1.

Figure 1.

Reference and current configuration of the dry bilayer beam. (Online version in colour.)

When the bilayer beam is immersed in a solvent bath with chemical potential μ_ext, each of the two layers would swell uniformly but differently from each other in the solvent bath if they were not tied to each other. Actually, the beam resolves the frustration due to the swelling mismatch and the kinematic constraint by assuming a bent configuration which can be determined solving the complete stress–diffusion problem.

Here, following the uncoupled approach proposed in [9] based on the theory of finite elasticity with growth, we assume that the visible deformation F_d can be split into two components F_o and F_e such that F_d=F_eF_o. We assume that the deformation F_o is the free swelling-induced deformation that would take place in each layer if it were free from the rest of the beam, and that has, in each part, the representation form (2.8)₁ F_ol=(J_ol)^1/3I, with subscript l denoting a dummy layer placeholder l=t (top) or l=b (bottom layer), the determinant J_ol being determined by means of equation (2.8)₂ in which the corresponding shear modulus G=G_l has to be taken into account.² Owing to F_ol, the integrity of the body may not be preserved; the elastic deformation F_e takes place to accommodate the elastic distortion F_o and realizes the final visible deformation F_d. We assume that $det{\mathbf{\text{F}}}_{\mathrm{e}}=1$, in such a way that $det{\mathbf{\text{F}}}_{\mathrm{d}}=J_{\mathrm{ol}}$ and the change in volume in the total deformation process is only due to solvent absorption ($det{\mathbf{\text{F}}}_{\mathrm{d}}>1$) or release ($det{\mathbf{\text{F}}}_{\mathrm{d}}<1$). While F_d is generally continuous across the composite beam’s thickness, both F_o and F_e are not, and there is no placement whose gradient corresponds to them. This means that, for the bilayer beam, ${\mathcal{B}}_{\mathrm{o}}$ is an ideal configuration which can never be realized, corresponding to the two layers both swollen and stress-free. A depiction of the multiplicative decomposition F_d=F_eF_o for the bilayer beam and of the corresponding configurations ${\mathcal{B}}_{\mathrm{d}}$, ${\mathcal{B}}_{\mathrm{o}}$ and $\mathcal{B}$ is given in figure 2.

Figure 2.

A sketch of the multiplicative decomposition of the deformation gradient F_d, which shows the reference configuration (dry configuration), the ideal stress-free configuration, which would be attained by the two layers if they werenot glued one to another (ideal swollen configuration), and the current bent configuration (deformed configuration). (Online version in colour.)

As is standard in finite elasticity with growth, we assume that the stress is only related to the elastic component ${\mathbf{\text{F}}}_{\mathrm{e}}={\mathbf{\text{F}}}_{\mathrm{d}}\hspace{0.17em}{\mathbf{\text{F}}}_{\mathrm{o}}^{-1}$, which determines an elastic energetic expenditure φ(F_d) which can be represented, assuming a neo–Hookean material law, as

$$\phi ({\mathbf{\text{F}}}_{\mathrm{d}})=J_{\mathrm{o}}{\phi }_{\mathrm{o}}({\mathbf{\text{F}}}_{\mathrm{e}})\mathrm{and}{\phi }_{\mathrm{o}}({\mathbf{\text{F}}}_{\mathrm{e}})=\frac{1}{2}G({\mathbf{\text{F}}}_{\mathrm{e}}\cdot {\mathbf{\text{F}}}_{\mathrm{e}}-3),$$ 3.1

in terms of the elastic energy density φ_o per unit volume of the freely swollen configuration ${\mathcal{B}}_{\mathrm{o}}$. Deriving φ_o with respect to F_e, we obtain S_o(F_e)=GF_e, which maps vectors from ${\mathcal{B}}_{\mathrm{o}}$ to $\mathcal{B}$. The constitutively determined part S_d(F_d) of S in equation (2.3) is obtained through the relation ${\mathbf{\text{S}}}_{\mathrm{d}}({\mathbf{\text{F}}}_{\mathrm{d}})=J_{\mathrm{o}}\hspace{0.17em}G{\mathbf{\text{F}}}_{\mathrm{e}}{\mathbf{\text{F}}}_{\mathrm{o}}^{-T}$, and the Cauchy stress T is obtained from S through $\mathbf{\text{T}}=J_{\mathrm{d}}^{-1}\mathbf{\text{S}}{\mathbf{\text{F}}}_{\mathrm{d}}^{\mathrm{T}}$ (see e.g. [24]). Remembering that $J_{\mathrm{d}}=det{\mathbf{\text{F}}}_{\mathrm{d}}=J_{\mathrm{o}}$, that F_o is diagonal and using equation (2.3), we obtain the expressions of S and T:

$$\mathbf{\text{S}}=J_{\mathrm{o}}G{\mathbf{\text{F}}}_{\mathrm{e}}{\mathbf{\text{F}}}_{\mathrm{o}}^{-\mathrm{T}}-J_{\mathrm{o}}p\hspace{0.17em}{\mathbf{\text{F}}}_{\mathrm{d}}^{-\mathrm{T}}=J_{\mathrm{o}}^{1/3}G\hspace{0.17em}{\mathbf{\text{F}}}_{\mathrm{d}}-J_{\mathrm{o}}p{\mathbf{\text{F}}}_{\mathrm{d}}^{-\mathrm{T}}$$ 3.2

and

$$\mathbf{\text{T}}=\frac{1}{J_{\mathrm{o}}}\mathbf{\text{S}}{\mathbf{\text{F}}}_{\mathrm{d}}^{\mathrm{T}}=J_{\mathrm{o}}^{-2/3}G\hspace{0.17em}{\mathbf{\text{B}}}_{\mathrm{d}}-p\hspace{0.17em}\mathbf{\text{I}},{\mathbf{\text{B}}}_{\mathrm{d}}={\mathbf{\text{F}}}_{\mathrm{d}}{\mathbf{\text{F}}}_{\mathrm{d}}^{\mathrm{T}}.$$ 3.3

The Cauchy stress T_l in each layer is written as

$${\mathbf{\text{T}}}_{\mathrm{l}}=J_{\mathrm{ol}}^{-2/3}{G}^{\mathrm{l}}\hspace{0.17em}{\mathbf{\text{B}}}_{\mathrm{d}}-p^{\mathrm{l}}\mathbf{\text{I}},$$ 3.4

with p_l the reaction due to the elastic incompressibility constraint which holds separately in each layer, and l=t,b. In equation (3.4) above and in the following, summation is not assumed over repeated layer indices l unless explicitly specified.

To explicitly compute a finite bending solution starting from well-known results based on the plane strain approach [19], we only consider the in-plane differential swelling of the two layers of the beam, neglecting any issues related to the kinematical compatibility of the flexure plane. We start adapting the notation through the introduction of the restrictions ${\widehat{\mathbf{\text{F}}}}_{\mathrm{d}}$, ${\widehat{\mathbf{\text{F}}}}_{\mathrm{o}}$, ${\widehat{\mathbf{\text{F}}}}_{\mathrm{e}}$ and $\widehat{\mathbf{\text{T}}}$ to the flexure plane of unit normal e₂ of the corresponding fields F_d, F_o, F_e and T, being, for example, $\widehat{\mathbf{\text{A}}}=\mathbf{\text{P}}\mathbf{\text{A}}\mathbf{\text{P}}$ with P=I−e₂⊗e₂. Given that $det\widehat{\mathbf{\text{A}}}=0$ for all $\widehat{\mathbf{\text{A}}}$, we define the determinant of the in-plane 2×2 tensor as $\underset{2}{det}\widehat{\mathbf{\text{A}}}=(\widehat{\mathbf{\text{A}}}{\mathbf{\text{e}}}_3)\times (\widehat{\mathbf{\text{A}}}{\mathbf{\text{e}}}_1)$.

We assume that

$${\mathbf{\text{F}}}_{\mathrm{e}}={\widehat{\mathbf{\text{F}}}}_{\mathrm{e}}+{\mathbf{\text{e}}}_2\otimes {\mathbf{\text{e}}}_2\text{and}{\text{det}}_2\hspace{0.17em}{\widehat{\mathbf{\text{F}}}}_{\mathrm{e}}=1,$$ 3.5

that is, the elastic incompressibility constraint holds in the flexure plane. In addition, equation (3.5) implies that the kinematic incompatibility of the intermediate configuration ${\mathcal{B}}_{\mathrm{o}}$ is only addressed in the flexure plane and not out of plane. Assumption (3.5) is a necessary approximation in order to be able to resort to Rivlin’s 1949 closed-form solution [19], which applies to plane strain bending. An assessment of this approximation can be found in [22].

Finally, we write

$${\mathbf{\text{F}}}_{\mathrm{o}}=J_{\mathrm{o}}^{1/3}\mathbf{\text{I}}=j_{\mathrm{o}}^{1/2}\mathbf{\text{P}}+j_{\mathrm{o}}^{1/2}{\mathbf{\text{e}}}_2\otimes {\mathbf{\text{e}}}_2,j_{\mathrm{o}}^{3/2}=J_{\mathrm{o}},$$ 3.6

in such a way that the change of area in the flexure plane for each layer of the beam, due to swelling, be exactly J_o. With this, under plane strain hypothesis, the Cauchy stress is written as

$${\mathbf{\text{T}}}^{\mathrm{l}}=\frac{1}{j_{\mathrm{ol}}}{G}_{\mathrm{l}}\hspace{0.17em}{\mathbf{\text{B}}}_{\mathrm{d}}-p_{\mathrm{l}}\mathbf{\text{I}},l=\mathrm{t},\mathrm{b}.$$ 3.7

In the end, we also note that, being $det{\mathbf{\text{F}}}_{\mathrm{d}}=J_{\mathrm{ol}}$ for each layer, $j_{\mathrm{ol}}=\underset{2}{det}{\widehat{\mathbf{\text{F}}}}_{\mathrm{d}}=j$, measuring the visible change of area in the flexure plane. Hence, within the two layers, j takes different values j=j_ot and j=j_ob in the top and bottom layer, respectively.

4. Swelling-induced bending

Once the bilayer gel beam is immersed in the solvent bath, it eventually reaches a steady-state configuration. The steady deformed configuration $\mathcal{B}$ of the beam corresponds to an arc of angle amplitude $2\overline{\theta }$, and is depicted in figure 1b: each line x₃=const. in the reference configuration is assumed to map into a circular arc, while lines x₁=const. map into radial segments [22]. We denote with r^l_e and $r_i^l$ the curvature radii of the extrados and intrados of each layer (l=t,b), respectively, and assume that

$$j_{\mathrm{ob}}{l}_{\mathrm{d}}(1-\beta )h_{\mathrm{d}}=\overline{\theta }({(r_{\mathrm{e}}^{\mathrm{b}})}^2-{(r_i^{\mathrm{b}})}^2)\mathrm{and}{j}_{\mathrm{ot}}{l}_{\mathrm{d}}\beta h_{\mathrm{d}}=\overline{\theta }({(r_{\mathrm{e}}^{\mathrm{t}})}^2-{(r_i^{\mathrm{t}})}^2),$$ 4.1

stating that the area is only changed by the layer-wise isotropic growth. Introducing the polar basis {e_r,e_θ}, with ${\mathbf{\text{e}}}_r={\mathbf{\text{e}}}_1\sin \theta +{\mathbf{\text{e}}}_3\cos \theta$ and ${\mathbf{\text{e}}}_{\theta }={\mathbf{\text{e}}}_1\cos \theta -{\mathbf{\text{e}}}_3\sin \theta$, and the corresponding polar coordinates (r,θ), the deformation gradient ${\widehat{\mathbf{\text{F}}}}_{\mathrm{d}}$ and the left Cauchy–Green tensor ${\widehat{\mathbf{\text{B}}}}_{\mathrm{d}}$ read

$${\widehat{\mathbf{\text{F}}}}_{\mathrm{d}}={\lambda }_r{\mathbf{\text{e}}}_r\otimes {\mathbf{\text{e}}}_3+{\lambda }_{\theta }{\mathbf{\text{e}}}_{\theta }\otimes {\mathbf{\text{e}}}_1\hspace{0.17em}\text{and}{\widehat{\mathbf{\text{B}}}}_{\mathrm{d}}={\lambda }_r^2{\mathbf{\text{e}}}_{\mathrm{r}}\otimes {\mathbf{\text{e}}}_{\mathrm{r}}+{\lambda }_{\theta }^2{\mathbf{\text{e}}}_{\theta }\otimes {\mathbf{\text{e}}}_{\theta },$$ 4.2

where the radial stretch λ_r and the hoop stretch λ_θ can be integrated from (4.1), yielding

$${\lambda }_r(r)=j_{\mathrm{ol}}\frac{l_{\mathrm{d}}}{2\overline{\theta }r}l=\mathrm{t},\mathrm{b}\text{and}{\lambda }_{\theta }(r)=\frac{2\overline{\theta }r}{l_{\mathrm{d}}}.$$ 4.3

As evident from (3.7) and (4.2), shear stress components are zero everywhere and the Cauchy stress tensor $\widehat{\mathbf{\text{T}}}$ is diagonal. The pressure term p^l in (3.7) can be integrated from the equilibrium equations in the bulk $\mathrm{div}\hspace{0.17em}{\widehat{\mathbf{\text{T}}}}_{\mathrm{l}}=0$ and determined uniquely in each layer but for an integration constant A_l,

$$p_{\mathrm{l}}=\frac{G^{\mathrm{l}}}{2j_{\mathrm{ol}}}({\lambda }_r^2-{\lambda }_{\theta }^2)+A_{\mathrm{l}}.$$ 4.4

The problem has then seven unknowns, i.e. $r_i^{\mathrm{b}}$, r^b_e, $r_i^{\mathrm{t}}$, r^t_e, A_b, A_t and $\overline{\theta },$ which can be determined using the boundary conditions on the radial stress components at the external boundaries $r=r_i^{\mathrm{b}}$ and r=r^t_e:

$$T_r^{\mathrm{t}}(r_{\mathrm{e}}^{\mathrm{t}})=0\text{and}{T}_r^{\mathrm{b}}(r_i^{\mathrm{b}})=0,$$ 4.5

the radial stress and deformation map continuity at the interface

$$T_r^t(r_i^{\mathrm{t}})=T_r^{\mathrm{b}}(r_{\mathrm{e}}^{\mathrm{b}})\mathrm{and}{r}_i^{\mathrm{t}}=r_{\mathrm{e}}^{\mathrm{b}},$$ 4.6

and the condition of zero resultant torque M (per unit beam depth) at the end bases $\theta =\pm \overline{\theta }$

$$M={\int }_{r_i^{\mathrm{t}}}^{r_{\mathrm{e}}^{\mathrm{t}}}r{T}_{\theta }^{\mathrm{t}}(r)\hspace{0.17em}\mathrm{d}r+{\int }_{r_i^{\mathrm{b}}}^{r_{\mathrm{e}}^{\mathrm{b}}}r{T}_{\theta }^{\mathrm{b}}(r)\hspace{0.17em}\mathrm{d}r=0,$$ 4.7

together with the two volumetric constraints in (4.1). The condition of zero resultant axial force per unit beam depth,

$$N={\int }_{r_i^{\mathrm{b}}}^{r_{\mathrm{e}}^{\mathrm{b}}}{T}_{\theta }^{\mathrm{b}}(r)\hspace{0.17em}\mathrm{d}r+{\int }_{r_i^{\mathrm{t}}}^{r_{\mathrm{e}}^{\mathrm{t}}}{T}_{\theta }^{\mathrm{t}}(r)\hspace{0.17em}\mathrm{d}r=0,$$ 4.8

is always trivially satisfied.

Integration constants A_b and A_t are determined from (4.5), volumetric constraints (4.1) are used to express $r_i^{\mathrm{b}}$ in terms of $\overline{\theta }$, r^b_e and r^t_e in terms of $\overline{\theta }$, $r_i^{\mathrm{t}}$, while condition (4.6)₂ reduces the unknowns to $\overline{\theta }$ and $r_i^{\mathrm{t}}$ only. As shown in [22], by introducing the auxiliary non-dimensional variables

$$\overline{\vartheta }=4\frac{h_{\mathrm{d}}}{l_{\mathrm{d}}}\overline{\theta }\text{and}t=\frac{j_{\mathrm{ot}}{h}_{\mathrm{d}}{l}_{\mathrm{d}}}{{(r_i^{\mathrm{t}})}^2\overline{\theta }},$$ 4.9

it is possible to further reduce the problem to just one nonlinear algebraic equation in the unknown t,

$$\begin{array}{rl}& t(\frac{(1-\beta ){\mathrm{\Gamma }}^4}{1-(1-\beta ){\mathrm{\Gamma }}^{2}t}+\frac{\alpha \beta }{1+\beta t})=\frac{2(1-\beta +\alpha \beta )}{4\alpha \beta +4(1-\beta )+t(\alpha {\beta }^2-{(1-\beta )}^2{\mathrm{\Gamma }}^2)}\\ & \cdot (\alpha (1+\log (1+\beta t)-\frac{1}{1+\beta t})-{\mathrm{\Gamma }}^2(1+\log (1-(1-\beta ){\mathrm{\Gamma }}^{2}t)-\frac{1}{1-(1-\beta ){\mathrm{\Gamma }}^{2}t})),\end{array}$$ 4.10

with $\mathrm{\Gamma }=\sqrt{j_{\mathrm{ob}}/j_{\mathrm{ot}}}$ ranging from 0 to 1. It is evident that t only depends on the elastic moduli ratio α and the thickness ratio β, and on the ratio Γ² between the swelling-induced areal expansions j_ob and j_ot, which in turn depend on G_t and G_b, respectively, through equations (2.8)₂ and (3.6). Hence, the actual number of governing parameters is 3, say β, G_b and G_t. Once t has been computed, the rescaled bending rotation $\overline{\vartheta }$ can be computed in terms of α, β, and Γ from a recast form of (4.6)₁ which reads

$${\overline{\vartheta }}^2=\frac{t^2}{1-\beta +\alpha \beta }(\frac{(1-\beta ){\mathrm{\Gamma }}^4}{1-(1-\beta ){\mathrm{\Gamma }}^{2}t}+\frac{\alpha \beta }{1+\beta t}).$$ 4.11

Then the actual bending rotation $\overline{\theta }$ and the actual position $r_i^{\mathrm{t}}$ of the interface are computed from (4.9). Let us note that, from equations (4.9) and (4.1), it is easy to verify that: $r_i^{\mathrm{t}}=r_{\mathrm{e}}^{\mathrm{b}}$, r^t_e, and $r_i^{\mathrm{b}}$ scale with h_d and do not depend on l_d; $\overline{\theta }$ scales linearly with l_d/h_d and is the only quantity depending on the slenderness ratio. For this reason the critical condition for wrinkling will not be discussed w.r.t. the slenderness ratio l_d/h_d but, based on the above results, with respect to parameters α and β, given that, at least for gels, Γ² is tied to α.

To appreciate the compression at the extrados, we evaluate the elastic component of the hoop stretch in r=r^t_e:

$${\lambda }_{\mathrm{e}\hspace{0.17em}\theta }(r=r_{\mathrm{e}}^{\mathrm{t}})=\frac{{\lambda }_{\theta }(r=r_{\mathrm{e}}^{\mathrm{t}})}{\sqrt{j_{\mathrm{ot}}}}=\sqrt{\frac{\overline{\vartheta }}{t}(1+\beta t)}.$$ 4.12

Since $\overline{\vartheta }$ and t depend only on α, β and Γ, so does the expression in equation (4.12). The elastic hoop stretch at the extrados is computed for G_b=45 kPa and the contour plotted for different values of α and β in figure 3. We observe that when α and β are both very small, contraction ensues at the extrados, eventually leading to wrinkling. Also we note that, for a given value of β, the elastic hoop stretch is monotonous w.r.t. to α.

Figure 3.

Contour lines of the elastic component of the hoop stretch ${\lambda }_{\theta }/\sqrt{j_{\mathrm{ot}}}$ computed at the extrados r=r^t_e for G_b=45 kPa and for different values of α in the ordinate and β in the abscissa.

5. The incremental elastic problem

Here, swelling growth is completely driven by the chemical potential μ_ext of the solvent bath, and it is not influenced by the stress state of the system. We imagine to keep μ_ext=0 and investigate the bending determined by the ratios α and β of the stiffness moduli and of the heights of the two layers. Therefore, at this level of description α and β can be seen as the control parameters that change the geometry of the body. The bent solution obtained in §4 represents a mechanical equilibrium whose stability we aim to investigate by considering a wider class of deformations. We follow the general procedure to test the stability of the solution, based on a linear perturbation analysis (see diagram in figure 4) in which the incremental deformations are assumed to be infinitesimal [18,24,37]. In the following, unless explicitly required, the layer identifier l=t,b is dropped for brevity; likewise, we simplify our notation by avoiding the ‘hat’ to specify restricted fields.

Figure 4.

Schematic of the linear perturbation analysis. (Online version in colour.)

(a). Linearized constitutive relationship

We introduce the increment Σ of the first Piola–Kirchhoff stress tensor taking the current configuration $\mathcal{B}$ as reference. It can be proved that (see e.g. [38], eq. 4.1)

$$\mathbf{\Sigma }=j^{-1}\dot{\mathbf{\text{S}}}{\mathbf{\text{F}}}^{\mathrm{T}},$$ 5.1

where the dot indicates an incremental quantity and j is the layer-wise constant change of area, equal to j_ob and j_ot in the bottom and top layers, respectively. Using equation (3.7) and the relation S=jTF^−T, the constitutive relationship for S reads

$$\mathbf{\text{S}}=G\hspace{0.17em}\mathbf{\text{F}}-jp{\mathbf{\text{F}}}^{-\mathrm{T}}.$$ 5.2

Let us denote by $\mathbf{\text{L}}=\mathrm{grad}\hspace{0.17em}\dot{\mathbf{\text{u}}}$ the gradient, computed in the bent configuration $\mathcal{B}$, of the incremental displacement $\dot{\mathbf{\text{u}}}$ from $\mathcal{B}$. Knowing that j is layer-wise constant and using the following identities:

$$\dot{\mathbf{\text{F}}}=\mathbf{\text{L}}\mathbf{\text{F}},{\dot{\mathbf{\text{F}}}}^{-1}=-{\mathbf{\text{F}}}^{-1}\dot{\mathbf{\text{F}}}{\mathbf{\text{F}}}^{-1},$$ 5.3

the increment of S is obtained by derivation of equation (5.2), yielding

$$\dot{\mathbf{\text{S}}}=-j\dot{p}{\mathbf{\text{F}}}^{-\mathrm{T}}+jp{\mathbf{\text{F}}}^{-\mathrm{T}}{\dot{\mathbf{\text{F}}}}^{\mathrm{T}}{\mathbf{\text{F}}}^{-\mathrm{T}}+G\dot{\mathbf{\text{F}}},$$ 5.4

whence

$$\mathbf{\Sigma }=-\dot{p}\mathbf{\text{1}}+p{\mathbf{\text{L}}}^{\mathrm{T}}+j^{-1}G\mathbf{\text{L}}\mathbf{\text{B}}.$$ 5.5

To compute the components of Σ, we remind that the principal directions of both T in (3.7) and B_d in (4.2) coincide with e_r and e_θ. Thus we obtain

$$\begin{array}{rl}{\mathrm{\Sigma }}_{rr}& =-\dot{p}+(p+j^{-1}G{\lambda }_r^2)L_{rr},{\mathrm{\Sigma }}_{\theta \theta }=-\dot{p}+(p+j^{-1}G{\lambda }_{\theta }^2)L_{\theta \theta }\\ \mathrm{and}{\mathrm{\Sigma }}_{r\theta }& =p{L}_{\theta r}+j^{-1}G{\lambda }_{\theta }^2{L}_{r\theta },{\mathrm{\Sigma }}_{\theta r}=p{L}_{r\theta }+j^{-1}G{\lambda }_r^2{L}_{\theta r}.\end{array}\}$$ 5.6

With the aim of comparing (5.6) with expressions available in the literature, we recall that

$$T_r=\frac{1}{j}G{\lambda }_r^2-p\mathrm{and}{T}_{\theta }=\frac{1}{j}G{\lambda }_{\theta }^2-p,$$ 5.7

and we pose

$$\begin{array}{r}{G}_{\ast }=\frac{1}{j}G\frac{{\lambda }_{\theta }^2+{\lambda }_r^2}{2},\end{array}$$ 5.8

$$\begin{array}{r}\overline{T}=\frac{T_{\theta }+T_r}{2}=G_{\ast }-p\end{array}$$ 5.9

$$\begin{array}{r}\mathrm{and}\overline{\mathrm{\Gamma }}=\frac{T_{\theta }-T_r}{2}=\frac{1}{j}G\frac{{\lambda }_{\theta }^2-{\lambda }_r^2}{2}.\end{array}$$ 5.10

Then (5.6) can be rewritten in the form

$$\begin{array}{r}{\mathrm{\Sigma }}_{rr}=-\dot{p}+(2G_{\ast }-\overline{T}-\overline{\mathrm{\Gamma }})L_{rr},\end{array}$$ 5.11

$$\begin{array}{r}{\mathrm{\Sigma }}_{\theta \theta }=-\dot{p}+(2G_{\ast }-\overline{T}+\overline{\mathrm{\Gamma }})L_{\theta \theta },\end{array}$$ 5.12

$$\begin{array}{r}{\mathrm{\Sigma }}_{r\theta }=(G_{\ast }-\overline{T})L_{\theta r}+(G_{\ast }+\overline{\mathrm{\Gamma }})L_{r\theta }\end{array}$$ 5.13

$$\begin{array}{r}\mathrm{and}{\mathrm{\Sigma }}_{\theta r}=(G_{\ast }-\overline{T})L_{r\theta }+(G_{\ast }-\overline{\mathrm{\Gamma }})L_{\theta r}.\end{array}$$ 5.14

Expressions in equations (5.11)–(5.14) are consistent with those appearing in [24, eq. (6.177), 38].

(b). Incremental boundary value problem

We consider the current configuration $\mathcal{B}$ as reference and distinguish in it two subdomains ${\mathcal{B}}^{\mathrm{b}}$ and ${\mathcal{B}}^{\mathrm{t}}$, occupied by the bottom and top layers, respectively, and each having the shape of an annular segment. They share a common interface of equation $r=r_{\mathrm{e}}^{\mathrm{b}}=r_i^{\mathrm{t}}$. Let

$$\dot{\mathbf{\text{u}}}(r,\theta )=\dot{u}(r,\theta ){\mathbf{\text{e}}}_r+\dot{v}(r,\theta ){\mathbf{\text{e}}}_{\theta }$$ 5.15

be the incremental displacement in polar coordinates. The increment $\dot{N}$ of the axial force N and $\dot{M}$ of the bending moment M on the short edges $\theta =\pm \overline{\theta }$ are given by

$$\dot{N}={\int }_{r_{\mathrm{ib}}}^{r_{\mathrm{et}}}{\mathrm{\Sigma }}_{\theta \theta }(r,\pm \overline{\theta })\hspace{0.17em}\mathrm{d}r$$ 5.16

and

$$\dot{M}={\int }_{r_{\mathrm{ib}}}^{r_{\mathrm{et}}}{T}_{\theta }(r)\dot{u}(r,\pm \overline{\theta })\hspace{0.17em}\mathrm{d}r+{\int }_{r_{\mathrm{ib}}}^{r_{\mathrm{et}}}{\mathrm{\Sigma }}_{\theta \theta }(r,\pm \overline{\theta })r\hspace{0.17em}\mathrm{d}r.$$ 5.17

With the above positions, we can now state the complete set of equilibrium equations and boundary conditions governing the incremental boundary value problem in $\mathcal{B}$ as follows:

• — equilibrium equation:

  $$\mathrm{div}\hspace{0.17em}\mathbf{\Sigma }=0\hspace{0.17em}\text{in }{\mathring{\mathcal{B}}}^{\mathrm{b}}\bigcup {\mathring{\mathcal{B}}}^{\mathrm{t}}\text{with}\hspace{0.17em}\mathbf{\Sigma }=-\dot{p}\mathbf{1}+p{\mathbf{\text{L}}}^{\mathrm{T}}+j^{-1}G\mathbf{\text{L}}\mathbf{\text{B}};$$ 5.18
• — boundary condition on the short edges $\theta =\pm \overline{\theta }$:

  $$\dot{N}=0\text{and}\dot{M}=0;$$ 5.19
• — boundary conditions on the edges of constant radius:

  $$\begin{array}{r}{\mathrm{\Sigma }}_{rr}(r_i^{\mathrm{b}})=0{\mathrm{\Sigma }}_{\theta r}(r_i^{\mathrm{b}})=0,\end{array}$$ 5.20

  $$\begin{array}{r}{\mathrm{\Sigma }}_{rr}(r_{\mathrm{e}}^{\mathrm{t}})=0{\mathrm{\Sigma }}_{\theta r}(r_{\mathrm{e}}^{\mathrm{t}})=0,\end{array}$$ 5.21

  $$\begin{array}{r}{\mathrm{\Sigma }}_{rr}(r_{\mathrm{e}}^{\mathrm{b}})={\mathrm{\Sigma }}_{rr}(r_i^{\mathrm{t}}){\mathrm{\Sigma }}_{\theta r}(r_{\mathrm{e}}^{\mathrm{b}})={\mathrm{\Sigma }}_{\theta r}(r_i^{\mathrm{t}})\end{array}$$ 5.22

  $$\begin{array}{r}\mathrm{and}\dot{u}(r_{\mathrm{e}}^{\mathrm{b}})=\dot{u}(r_i^{\mathrm{t}})\dot{v}(r_{\mathrm{e}}^{\mathrm{b}})=\dot{v}(r_i^{\mathrm{t}}).\end{array}$$ 5.23

We seek non-trivial solutions of the incremental boundary value problem (5.18)–(5.23).

(c). Wrinkling Ansatz

Let us introduce positions

$$n=m\frac{\pi }{\overline{\theta }},a=2\frac{\overline{\theta }}{l_{\mathrm{d}}}=\frac{\overline{\vartheta }}{2h_{\mathrm{d}}}\mathrm{and}s=ar={\lambda }_{\theta }(r),$$ 5.24

with $m\in \mathbb{N}$ being the finite number of wavelengths, or wrinkles, along any curve of constant radius, say the extrados.

The following choice of the bifurcation Ansatz ([see, e.g. 38]),

$$\dot{u}(r,\theta )=\frac{1}{a}\cdot f(ar)\sin n\theta ,\dot{v}(r,\theta )=\frac{1}{a}\cdot g(ar)\cos n\theta \mathrm{and}\dot{p}(r,\theta )=G\cdot k(ar)\sin n\theta ,$$ 5.25

ensures that

$$\dot{u}(r,\pm \overline{\theta })={\mathrm{\Sigma }}_{\theta \theta }(r,\pm \overline{\theta })=0,$$ 5.26

thus implying from (5.16) to (5.17) that

$$\dot{N}=\dot{M}=0.$$ 5.27

Such an Ansatz does not imply that the shear stress Σ_rθ is zero everywhere along the faces $\theta =\pm \overline{\theta }$. Using global equilibrium considerations, it is, however, possible to infer that the resultant incremental shear force

$$\dot{T}={\int }_{r_{\mathrm{ib}}}^{r_{\mathrm{et}}}{\mathrm{\Sigma }}_{r\theta }(r,\pm \overline{\theta })\hspace{0.17em}\mathrm{d}r$$ 5.28

is zero on each face $\theta =\pm \overline{\theta }$.

(d). Governing ordinary differential equation and boundary conditions

It is possible to reduce the three unknown functions f,g,k appearing in the Ansatz (5.25) to just one. First by exploiting the incompressibility condition tr L=0, it is possible to express g in terms of f

$$g=\frac{f+ar{f}^{\prime }}{n},$$ 5.29

where the prime denotes derivation w.r.t. the argument s=ar=λ_θ(r). It is then possible to combine the two scalar equilibrium equations div Σ=0 in $\mathcal{B}$ to express k(s) in terms of f(s) (see e.g. [18], eqs. 41–47 for details) and to obtain a governing ODE having only f as the dependent variable.

$$\begin{array}{rl}& -(\hspace{0.17em}{j}^2{s}^4)f^{⁗}(s)-(2j^2{s}^3)f^{‴}(s)+s^2(j^2(n^2+3)+n^2{s}^4)f^{″}(s)\\ & -s(j^2(n^2+3)-3n^2{s}^4)f^{\prime }(s)+(1-n^2)(3j^2+n^2{s}^4)f(s)=0.\end{array}$$ 5.30

The above equation is obtained by substituting into div Σ=0 the incremental constitutive law (5.5), the Ansatz (5.25) and the expressions (4.3)–(4.4) for λ_θ, λ_r and p that characterize the bent configuration $\mathcal{B}$. Interestingly, the above equation depends only on two parameters, i.e. j=j_ol with l=t,b and n, and it is a linear ODE, hence superposition holds.

To prescribe the boundary conditions, the values of displacements $\dot{u}$, $\dot{v}$ and of stress components Σ_rr, Σ_θr need to be computed first. We have

$$\dot{u}(s,\theta )=\frac{\sin (n\theta )}{a}\check{u}(s)\dot{v}(s,\theta )=\frac{\cos (n\theta )}{an}\check{v}(s)$$ 5.31

and

$${\mathrm{\Sigma }}_{rr}(s,\theta )=\frac{\sin (n\theta )}{2n^2{s}^3}{\check{\mathrm{\Sigma }}}_{rr}(s){\mathrm{\Sigma }}_{\theta r}(s,\theta )=\frac{\cos (n\theta )}{2n{s}^3}{\check{\mathrm{\Sigma }}}_{\theta r}(s),$$ 5.32

with

$$\begin{array}{r}\check{u}(s)=f(s),\check{v}(s)=f(s)+s{f}^{\prime }(s),\end{array}$$ 5.33

$$\begin{array}{r}{\check{\mathrm{\Sigma }}}_{rr}(s)=-2Gj{s}^3{f}^{‴}(s)-4Gj{s}^2{f}^{″}(s)+f^{\prime }(s)(2A{n}^2{s}^3+\frac{G{n}^2{s}^5}{j}+Gj(3n^2+2)s)+2Gj(n^2-1)f(s),\end{array}$$ 5.34

$$\begin{array}{r}{\check{\mathrm{\Sigma }}}_{\theta r}(s)=2Gj{s}^2{f}^{″}(s)(-2A{s}^3+\frac{G{s}^5}{j}+3Gjs)f^{\prime }(s)+\frac{1}{j}(n^2-1)(2Aj{s}^2+G(j^2-s^4))f(s).\end{array}$$ 5.35

In addition to j=j_ol with l=t,b and n, boundary conditions depend also on layer-wise constant parameters A=A_l and G=G_l.

We are now in a position to write the set of boundary conditions which accompany the ODE (5.30) governing the incremental elastic problem in each layer:

$$\begin{array}{r}{\check{\mathrm{\Sigma }}}_{rr}(s_i^{\mathrm{b}})=0{\check{\mathrm{\Sigma }}}_{\theta r}(s_i^{\mathrm{b}})=0,\end{array}$$ 5.36

$$\begin{array}{r}{\check{\mathrm{\Sigma }}}_{rr}(s_{\mathrm{e}}^{\mathrm{t}})=0{\check{\mathrm{\Sigma }}}_{\theta r}(s_{\mathrm{e}}^{\mathrm{t}})=0,\end{array}$$ 5.37

$$\begin{array}{r}{\check{\mathrm{\Sigma }}}_{rr}(s_{\mathrm{e}}^{\mathrm{b}})={\check{\mathrm{\Sigma }}}_{rr}(s_i^{\mathrm{t}}){\check{\mathrm{\Sigma }}}_{\theta r}(s_{\mathrm{e}}^{\mathrm{b}})={\check{\mathrm{\Sigma }}}_{\theta r}(s_i^{\mathrm{t}})\end{array}$$ 5.38

$$\begin{array}{r}\mathrm{and}\check{u}(s_{\mathrm{e}}^{\mathrm{b}})=\check{u}(s_i^{\mathrm{t}})\check{v}(s_{\mathrm{e}}^{\mathrm{b}})=\check{v}(s_i^{\mathrm{t}}),\end{array}$$ 5.39

with $s_i^{\mathrm{b}}=a{r}_i^{\mathrm{b}}$ and so forth. The solution to equation (5.30) in each layer is written as the superposition of two solutions

$$f^{\mathrm{t}}(s)=A_1{f}_0^{\mathrm{t}}(s)+A_2{f}_1^{\mathrm{t}}(s)\text{and}{f}^{\mathrm{b}}(s)=A_3{f}_0^{\mathrm{b}}(s)+A_4{f}_1^{\mathrm{b}}(s).$$ 5.40

For the top layer, $f_0^{\mathrm{t}}(s)$ is the solution to the problem that in s=s^t_e has boundary conditions $f_0^{\mathrm{t}}=1$, ${(f_0^{\mathrm{t}})}^{\prime }=0$ and ${(f_0^{\mathrm{t}})}^{″}$, ${(f_0^{\mathrm{t}})}^{‴},$ chosen such that ${\check{\mathrm{\Sigma }}}_{rr}(s_{\mathrm{e}}^{\mathrm{t}})=0$ and ${\check{\mathrm{\Sigma }}}_{\theta r}(s_{\mathrm{e}}^{\mathrm{t}})=0$ are automatically satisfied. On the other hand, $f_1^{\mathrm{t}}(s)$ is the solution to the problem that in s=s^t_e has boundary conditions $f_1^{\mathrm{t}}=0$, ${(f_1^{\mathrm{t}})}^{\prime }=1$ and ${(f_1^{\mathrm{t}})}^{″}$, ${(f_1^{\mathrm{t}})}^{‴}$, chosen such that ${\check{\mathrm{\Sigma }}}_{rr}(s_{\mathrm{e}}^{\mathrm{t}})={\check{\mathrm{\Sigma }}}_{\theta r}(s_{\mathrm{e}}^{\mathrm{t}})=0$. Analogous definitions hold for $f_0^{\mathrm{b}}(s)$ and $f_1^{\mathrm{b}}(s)$ with s^t_e replaced by $s_i^{\mathrm{b}}$.

Numerical integration allows the computation of $f_0^{\mathrm{t}}$, $f_1^{\mathrm{t}}$ and their derivatives up to the third order in $s=s_i^{\mathrm{t}}$, and of $f_0^{\mathrm{b}}$, $f_1^{\mathrm{b}}$ and their derivatives in s=s^b_e. From (5.40) the values of f^t, f^b and their derivatives in $s=s_i^{\mathrm{t}}$ and s=s^b_e are hence known but for four undetermined constants A_i, i=1…4 which are collected in a vector A.

The expressions (5.40) are substituted into boundary conditions (5.38)–(5.39) and the coefficients of constants A_i are collected in a 4×4 coefficient matrix M leading to the algebraic system MA=0 which has non-trivial solution when

$$det\mathbf{\text{M}}=0.$$ 5.41

The existence of adjacent equilibrium configurations characterized by wrinkles is thus determined when condition (5.41) is satisfied. However, from a numerical point of view this method is prone to numerical error and spurious oscillations. For this reason the search for roots of $det\hspace{0.17em}\mathbf{\text{M}}$ has been done here through the implementation of the compound matrix method (see e.g [21], sect. 3).

(e). The surface instability condition

Surface instability [23] is a bifurcation condition that is simultaneously satisfied by wrinkled modes of arbitrary wavelength. The modes are of Rayleigh wave type, i.e. sinusoidal on the surface and vanishing exponentially with depth (see e.g. [39]). The smaller the wavelength, the thinner is the zone close to the surface affected by the bifurcated mode. For this reason it is called surface instability. By the same token, though derived for an elastic half-space, it is also applied to bent, curved domains [16–18], because vanishing wavelengths are always negligible w.r.t to any finite curvature radius.

The result can also be applied to the present case in which distortions are present as long as stretches λ and tensors F_d, B_d are replaced by their elastic counterparts λ_e, F_e, ${\mathbf{\text{B}}}_{\mathrm{e}}={\mathbf{\text{F}}}_{\mathrm{e}}^{\mathrm{T}}{\mathbf{\text{F}}}_{\mathrm{e}}$ stripped of the distortion/swelling contributions. In fact, replacing B^l_e=1/j_ol B_d into (3.7) yields a neo-Hookean constitutive equation

$${\mathbf{\text{T}}}_{\mathrm{l}}=G_{\mathrm{l}}\hspace{0.17em}{\mathbf{\text{B}}}_{\mathrm{e}}-p_{\mathrm{l}}\mathbf{\text{I}},l=\mathrm{t},\mathrm{b}.$$ 5.42

We remind that no summation is carried out over repeated layer index l.

The condition of surface instability is given by the real root of equation

$${\zeta }^3+2{\zeta }^2-2=0\text{with}\hspace{0.17em}\zeta =\frac{1-{\lambda }_{e\theta }^4(r_{\mathrm{e}}^{\mathrm{t}})}{1+{\lambda }_{e\theta }^4(r_{\mathrm{e}}^{\mathrm{t}})}.$$ 5.43

When the elastic component of the extrados hoop stretch λ_eθ in equation (4.12) attains the value 0.544, the surface instability condition is met.

6. Results and discussion

We start by considering the following set of parameters [40]: G_b=45 kPa, h_d=2 mm, l_d=2 cm, χ=0.2, μ_ext=0 J mol⁻¹, T=293 K, Ω=6.023×10⁻⁵ m³ mol⁻¹. Then, for any given β, we seek α such that ${\lambda }_{\mathrm{e}\theta }(r=r_{\mathrm{e}}^{\mathrm{t}})$ in (4.12) equals 0.544. The line of points in the parameter space satisfying the surface instability condition is displayed in figure 5. Then, wrinkling of finite wavelength is considered and a value of m is fixed. For all values of β the critical value of α for which condition (5.41) holds is determined. Curves for m equal to 30, 60, 90, 120, 150, 180 are also shown in figure 5. The results show that larger the m, the closer the corresponding curve is to the surface instability condition. Also the surface instability condition occurs for larger values of α and, according to figure 3, for larger values of ${\lambda }_{\mathrm{e}\theta }(r_{\mathrm{e}}^{\mathrm{t}})$ as well. In short, surface instability corresponds to a less severe condition and occurs before wrinkling with finite wavelength.

Figure 5.

Surface instability condition (surf. inst.) and wrinkling condition for a finite number of wrinkles with m equal to 30, 60, 90, 120, 150, 180 in the parameter plane having β in the abscissa and α in the ordinate. Other parameters: G_b=45 kPa, h_d=2 mm, l_d=2 cm, χ=0.2.

We conjecture that the departure of curves of finite wavelength from the surface instability envelope occurs when the length in the thickness direction affected by the bifurcated mode, presumably of the order of the wrinkle wavelength as in the case of Rayleigh waves, is comparable to the swollen thickness of the top layer $\sqrt{j_{\mathrm{ot}}}\beta h_{\mathrm{d}}$. In fact, as can be seen in figure 5, the smaller the wavelength, i.e. bigger the m, the smaller is the value of β for which the finite wavelength instability curve detaches from the surface instability condition.

The result in figure 5 is interesting and novel in the context of linear bifurcation analysis of beams bent at finite strains in that, to the best of the authors’ knowledge, previous studies (see e.g. [18,41]) did not deal with the presence of swelling or distortions, and wrinkling took place at the intrados. Moreover, wrinkles with finite wavelength occurred ahead of surface instability. However, in a broader context the results in figure 5 are consistent with studies such as [25] dealing with bifurcations of soft compressed layers resting on flat rigid substrates. Finally, there are a number of works (see [42]) on the interplay between growth and bifurcations in which the role of geometric (i.e. growth) and mechanical (i.e. stiffness) factors are nicely elucidated. However, the type of problems and results addressed appear different from the ones presented here.

With a view to applying these results to structural morphing or sensing, there are a number of issues left to clarify.

First, the values of α required for this instability to take place are very small. In fact, as can be inferred from equation (2.9), in order to achieve a in free swelling linear expansion ratio ratio of 2:1, i.e. a value of Γ equal to 0.5, the value of α required is of the order of Γ⁵, that is 1/32≃0.031.

To address this issue, we have explored the possibility to design bilayers with different values of parameter χ, namely χ_b and χ_t in the bottom and top layer, respectively. The smaller the χ, the more the gel swells. Hence we have selected values of χ_t smaller than χ_b.

The effect is apparent in figure 6 where Γ is plotted against α for different values of χ_b, χ_t and G_b. We start with G_b=45 kPa (solid lines) and with equal values of χ in the two layers. We see that reducing the value of χ_b=χ_t from 0.4 to 0.2 reduces Γ, i.e. increases the j_ot/j_ob ratio, but the effect is not pronounced. A much greater reduction of Γ is achieved when cases χ_b=0.4, χ_t=0.1 and χ_b=0.45, χ_t=0.05 are considered. Then, the case in which G_b is raised ninefold to 405 kPa is taken into account (dashed lines). The results are confirmed but the values of Γ are larger, i.e. the j_ot/j_ob ratio is smaller, when the bottom layer is stiffer.

Figure 6.

Γ versus α curves for different values of parameters G_b, χ_b and χ_t.

For the same values of χ_b, χ_t and G_b used in figure 6, the surface instability condition is plotted in figure 7. The effect of the reduction in Γ achieved in figure 6 is apparent. The values of α for which the surface instability is attained are considerably increased. Again results are confirmed for the stiffer bottom layer, i.e. for G_b=405 kPa (dashed lines), but the critical values of α are lower.

Figure 7.

Surface instability condition in the β versus α parameter space for different values of parameters G_b, χ_b and χ_t.

We also looked at how the finite wavelength stability curves changed for different values of χ_b, χ_t and G_b. The case χ_b=0.1, χ_t=0.4 and G_b=45 kPa is displayed in figure 8. The findings of figure 5 are qualitatively confirmed.

Figure 8.

Surface instability condition (surf. inst.) and wrinkling condition for a finite number of wrinkles with m equal to 30, 60, 90, 120, 150, 180 in the parameter plane having β in the abscissa and α in the ordinate. Other parameters: G_b=45 kPa, χ_b=0.4, χ_t=0.1, h_d=2 mm, l_d=2 cm.

7. Conclusion

We have analysed swelling-induced wrinkling in a bilayer gel beam as an example of a prototypical actuator to be used in structural sensing and morphing. The beam is immersed in a solvent bath. The polymeric network of each layer is modelled as a neo-Hookean material having distinct stiffness. First, the bent configuration has been characterized and then a linear stability analysis has been performed to look for wrinkles appearing at the extrados of the beam.

Results have been discussed w.r.t two parameters ranging between 0 and 1 : a material ratio α between the stiffness of the two layers, the softer one being on top, and a geometric ratio β between the dry thickness of the top layer and the dry thickness of the whole beam.

Wrinkling at the extrados has been confirmed and the zone in the space of parameters α, β in which it occurs has been delimited. The size of such a zone is larger when the gel layers are softer and when the Flory–Huggins interaction parameter χ is smaller in the top layer. Moreover, surface instability of vanishing wavelength is seen to precede wrinkles of finite wavelength.

Results appear to be novel both w.r.t. the literature on bifurcations of bent beams at finite strains [18,41] and w.r.t. problems studying the interplay of instability and growth [42]. Results are instead consistent with previous findings concerning bifurcations of soft compressed layers resting on flat rigid substrates [25].

Prospective continuation of the present study may involve the assessment of the sensitivity of configuration changes to variations in the external chemical potential and in the temperature. The latter, in turn, affects the Flory–Huggins parameter χ, i.e. the affinity between solvent and polymer.

However, possible limitations to applications of the results come from crease formation [27] and imperfection sensitivity [28] of wrinkles. An assessment of these aspects is also postponed to a future work.

Data accessibility

This article has no additional data.

Authors' contributions

Both authors contributed to all aspects of this work.

Competing interests

No competing interests.

Funding

E.P. wishes to acknowledge the support of the Italian National Group of Mathematical Physics (GNFM–INdAM).