The study of asymptotically fine wrinkling in nonlinear elasticity using a boundary layer analysis

doi:10.1016/j.jmps.2013.04.003

Journal of the Mechanics and Physics of Solids

Volume 61, Issue 8, August 2013, Pages 1691-1711

https://doi.org/10.1016/j.jmps.2013.04.003 Get rights and content

Highlights

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We study nonlinear elastic wrinkling using a linear-stability analysis.
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An analytic postbuckling analysis is performed to determine the magnitude of buckle.
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Perform a boundary layer analysis to study the asymptotic behaviour.

Abstract

Fine sinusoidal wrinkling on the surfaces of mechanically compressed objects has been observed in many contexts over many years. In this paper we investigate such wrinkling through the application of a boundary layer analysis to an elastostatic problem in nonlinear elasticity. We determine the onset of buckling using a linear-stability analysis, and the leading-order postbuckling behaviour through consideration of higher order terms of the energy. The object is assumed to (initially) ‘preserve’ its shape, so that these equations reduce to ordinary differential equations. We then apply a boundary-layer analysis to this problem, determining (in the asymptotic limit of large wavenumbers) the leading order behaviours of the eigenmode, the critical parameter, and the magnitude of the buckle. We find that, to leading order, the shape of the buckle and the time of buckling are independent of the local geometry, however the magnitude of the buckle is dependent on the local geometry. Indeed we find that the magnitude of a buckle with wavenumbers $ς γ_{2}^{(⁎)}$ and $ς γ_{3}^{(⁎)}$ (for fixed $γ_{2}^{(⁎)}$ and $γ_{3}^{(⁎)}$ ) has leading asymptotic order $ς^{- 3 / 2} \sqrt{λ_{(2)}}$ , for an increment $λ_{(2)}$ of the critical parameter beyond the critical time of buckling. We provide electronic supplementary material which extends this analysis to that of incompressible elasticity. Finally we confirm the accuracy of our ansatz on a compressed NeoHookean ring.

Introduction

Wrinkling on the surface of nonlinearly elastic materials has been observed in many contexts over many years. The wrinkling is often observed in biological materials, for example on the surfaces of cells, tumours (Dervaux and Amar, 2011), tubular organs (such as airways, Moulton and Goriely, 2011a, Moulton and Goriely, 2011b; the oesophagus, Li et al., 2011b; intestine and blood vessels, MacLaurin et al., 2012) or brain convolutions. It is also observed in nonbiological materials, such as the wrinkling of stretched elastic sheets with low bending modulus (Brau et al., 2011), the formation of sulci in compressed soft materials (Hohlfeld and Mahadevan, 2011) or the wrinkling on the inner surface of a bent rubber block (Destrade et al., 2009).

These wrinkled patterns all occur in elastic materials in response to a gradual loading $λ$ , whether by external forcing or internal growth fields, with the following characteristics. For small loading the unwrinkled initial state remains mechanically stable. If wrinkling (buckling) does occur with wavenumbers $(γ_{2}, γ_{3})$ associated with the wrinkled pattern (to be defined more precisely in the text below), there must be a critical value $λ^{(γ_{2}, γ_{3})}$ of the loading such that the unwrinkled state is unstable to deformations of this type for loadings above $λ^{(γ_{2}, γ_{3})}$ and stable below. This critical value $λ^{(γ_{2}, γ_{3})}$ is found by superimposing an infinitesimally small sinusoidal deformation with these wavenumbers on the unwrinkled state, and searching for the value of $λ$ for which the wrinkled state has the same energy as the unwrinkled state. This only characterises the solution at the critical point. To find the solution for values of $λ$ in the neighbourhood of the critical value, we need to analyse the higher order terms in the asymptotic expansion of the energy. The equations governing the postbuckled state are often difficult to analyse, but progress can be made if we make simplifying assumptions about the nature of the geometry and the solution.

We analyse the equations in the limit that the wavenumbers $(γ_{2}, γ_{3})$ are large (and hence the wavelength is short), in which case the deformation is largely confined to a boundary layer at the surface. This assumption allows us to obtain a simplified expression for the shape of the buckled configuration in terms of geometric and material parameters. We assume that, if $(γ_{2}, γ_{3})$ are the wavenumbers associated with a wrinkled pattern and $λ^{(γ_{2}, γ_{3})}$ is the value of the critical parameter at which the wrinkled pattern first emerges, then $λ^{(γ_{2}, γ_{3})}$ asymptotes to a finite constant as $(γ_{2}, γ_{3}) \to \infty$ . In other words, we assume that the wrinkled patterns ‘cluster together’ as the wavelength shrinks. Furthermore we assume that the wrinkles are increasingly confined to the surface as $(γ_{2}, γ_{3}) \to \infty$ . This phenomenon has been observed in previous studies of buckling in nonlinear elasticity (Biot, 1966, Coman and Bassom, 2007, Dervaux and Amar, 2011, Hohlfeld and Mahadevan, 2011, MacLaurin et al., 2012, Cao and Hutchinson, 2012). It is analogous to the surface buckling of compressed infinite half-spaces: since there is no characteristic lengthscale, all of the wavelengths go unstable simultaneously (Biot, 1966). It has been studied in various contexts using boundary layer techniques (see particularly the papers by Coman and coworkers, Coman, 2010, Coman and Bassom, 2007). A mathematical analysis using the theory of partial differential equations has also been performed by Negron-Marrero and Montes-Pizarro, 2011, Negron-Marrero and Montes-Pizarro, 2012. These authors conjecture a necessary and sufficient condition – the ‘Complementing Condition’ – for the clustering of the wavenumbers in the above manner, and they use this to investigate the nature of the buckling.

This paper has four main parts. In the first part (Section 2), we outline an elastostatic problem and our assumptions on the pre-buckling behaviour of the system. In the second part (3 Stability analysis, 4 Postbuckling analysis), we apply a linear stability analysis to determine when the system starts to buckle, before applying a postbuckling analysis to determine the leading order of the magnitude of the buckle. In the third part (5 Boundary layer analysis of the eigenmode, 6 Boundary layer analysis of postbuckled solution), we perform a boundary layer analysis to elucidate the behaviour of the buckle for vanishingly small wavelengths. We will find that the magnitude of the buckle is of the order of the square root of the increment in the bifurcation parameter multiplied by $ς^{- 3 / 2}$ (where $ς$ is of the order of the wavenumber). In the fourth part (Section 8) we apply the theory to the wrinkling of a compressed NeoHookean ring, and correlate the predictions of our boundary layer analysis with numerical simulations. We also provide extra supplementary materials in which we provide further explanation of these previous sections and extend our analysis to incompressible elasticity.

Section snippets

Elastostatic problem

Our problem is that of a nonlinear elastic body subject to surface forces. We assume that there is no body force, although this paper could easily be extended to such a case. We let the reference configuration of the body be $B_{0}$ , with current configuration $B$ . Let $(X^{1}, X^{2}, X^{3})$ be coordinates for $B_{0}$ and $(x^{1}, x^{2}, x^{3})$ be coordinates for $B$ . These coordinate systems are in $C^{\infty}$ correspondence with $B_{0}$ and $B$ . Throughout this paper, we denote coordinates in the current configuration with lower case letters and

Stability analysis

There are two basic components of our stability analysis. In the first step, we employ a linear stability analysis to determine the value(s) of $λ$ for which the system bifurcates. In the next step, we perform a perturbation expansion about the point of bifurcation to determine the leading order of the magnitude of the buckle and its stability.

We work in the current configuration $B$ . We recall that the base state is the configuration prior to buckling, with displacement $u_{(0)}$ and $λ = λ_{(0)}$ . We insist

Postbuckling analysis

In this section we perform a perturbation expansion at the critical time of buckling of a mode $(γ_{2}, γ_{3})$ . Thus the expansion is at $λ = λ^{(γ_{2}, γ_{3})}$ , and there exists a solution of the form (22) to (14), (19), (20). We perform the expansion over the current (Eulerian) configuration. The coordinates $x$ and basis vectors $g_{a}$ are fixed and we obtain the displacement field by successively rendering each order of $Δ W$ stationary. We are essentially employing the energy criterion of Lagrange: we are assuming that

Boundary layer analysis of the eigenmode

We assume that a boundary layer develops on the face $\partial B^{(⁎)}$ of $B$ given by $x^{1} = x_{c}^{1}$ . What we mean by this is that, if $(γ_{2}, γ_{3}) = ς (γ_{2}^{(⁎)}, γ_{3}^{(⁎)})$ for some fixed ${γ_{2}^{(⁎)}, γ_{3}^{(⁎)}}$ , then as $ς \to \infty$ , $λ^{(γ_{2}, γ_{3})}$ converges to a finite constant. We denote this constant $λ_{(I)}^{(γ_{2}^{(⁎)}, γ_{3}^{(⁎)})}$ . In the sections to follow, we will perform a boundary layer analysis to determine $λ_{(I)}^{(γ_{2}^{(⁎)}, γ_{3}^{(⁎)})}$ , and also the leading order of $u_{(1)}$ and $u_{(2)}$ .

We have already made assumptions about the nature of the elasticity tensor in Section 3.

Boundary layer analysis of postbuckled solution

We determine the leading asymptotic behaviour of ${u_{(2)}^{(α_{1}, α_{2}, α_{3}, α_{4})}}$ as $ς \to \infty$ . These variables are defined in Section 4.1. Once we have determined the asymptotic behaviour of these variables, we determine the asymptotic behaviour of the integrals governing the magnitude and stability of the buckle.

Boundary layer analysis of the integrals governing the magnitude of the buckle

In this section we determine the asymptotic order of the integrals governing the magnitude of the buckle (as defined in Section 4.2). However before doing this, we summarise our asymptotic analysis thus far. We must reappend the superscripts $(α, β)$ to distinguish between the different Fourier modes. To leading order, the kth component of $u_{(1)}^{(α γ_{2}, β γ_{3})}$ (as defined in (22), and where $α, β \in {\pm 1}$ ) is equal to $u_{(1)}^{(α γ_{2}, β γ_{3}), k} = \sum_{j = 1}^{3} C^{(α, β) j} E_{kj}^{(α, β)} \exp (ρ d_{jj} + i ς (γ_{2}^{(⁎)} x^{2} + γ_{3}^{(⁎)} x^{3})) .$ We have neglected terms of

An application: a compressed NeoHookean annulus

We apply the preceding theory to the following two-dimensional problem. We consider an annulus, with internal radius R₀ and external radius R₁ (we scale R₁ to be 1). We work in polar coordinates, with reference coordinates ${R, Θ}$ and current coordinates ${r, θ}$ . The ring is modelled as an incompressible NeoHookean Material, with strain energy density given by $Ψ (F) = \frac{1}{2} Σ F : F,$ where $Σ$ is the shear modulus (which is scaled to be 1). Since the ring is incompressible, we must include an extra term to

Conclusion

In this paper we have investigated surface-wrinkling in elastostatic problems. We determined the onset of buckling using a linear-stability analysis, and the leading-order postbuckling behaviour through consideration of higher order terms of the energy. We then applied a boundary-layer analysis to this problem, determining the leading order behaviours of the eigenmode, the critical parameter, and the magnitude of the buckle. We found that the magnitude of a buckle with wavenumbers $ς γ_{2}^{(⁎)}$ and $ς γ_{3}$

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View full text

The study of asymptotically fine wrinkling in nonlinear elasticity using a boundary layer analysis

Highlights

Abstract

Introduction

Section snippets

Elastostatic problem

Stability analysis

Postbuckling analysis

Boundary layer analysis of the eigenmode

Boundary layer analysis of postbuckled solution

Boundary layer analysis of the integrals governing the magnitude of the buckle

An application: a compressed NeoHookean annulus

Conclusion

Int. J. Solids Struct.

Int. J. Solids Struct.

J. Mech. Phys. Solids

J. Mech. Phys. Solids

Int. J. Solids Struct.

Int. J. Solids Struct.

Journal of Biomechanics

J. Mech. Phys. Solids

J. Mech. Phys. Solids

J. Comput. Phys.

Multiple-length-scale elastic instability mimics parametric resonance of nonlinear oscillators

Nat. Phys.