Wrinkling of a compressed hyperelastic half-space with localized surface imperfections

Highlights

• Compression induced wrinkling of a hyperelastic half-space with surface imperfections is studied using asymptotic methods.

• It is found that surface imperfection always increases the critical stretch whether it is a trench or ridge.

• It is shown that the increase in the critical stretch is proportional to the square of the maximum gradient of surface profile.

Abstract

We consider a variant of the classical Biot problem concerning the wrinkling of a compressed hyperelastic half-space. The traction-free surface is no longer flat but has a localized ridge or trench that is invariant in the $x_1$-direction along which the wrinkling pattern is assumed to be periodic. With the $x_2$-axis aligned with the depth direction, the localized imperfection is assumed to be slowly varying and localized in the $x_3$-direction, and an asymptotic analysis is conducted to assess the effect of the imperfection on the critical stretch for wrinkling. The imperfection introduces a length scale so that the critical stretch is now weakly dependent on the wave number. It is shown that the imperfection increases the critical stretch (and hence reduces the critical strain) whether the imperfection is a ridge or trench, and the amount of increase is proportional to the square of the maximum gradient of the surface profile.

Introduction

Biot [1] was the first to examine the problem of possible surface wrinkling of a compressed hyperelastic half-space. The problem was further studied by Nowinski [2], Usmani and Beatty [3], Reddy [4], [5], Dowaikh and Ogden [6], Fu and Mielke [7], Destrade and Scott [8], Murphy and Destrade [9], and Chen et al. [10]. Some of these studies are in the context of surface waves in a pre-stressed hyperelastic half-space. For the case of a neo-Hookean half-space in a state of plane strain, the critical stretch was found to be 0.544. Since the wrinkling modes are non-dispersive due to lack of a natural length scale in the problem, a small-amplitude monochromatic mode will induce all higher harmonics at second order through nonlinear interactions. Based on this fact Ogden and Fu [11] attempted to determine the post-buckling solution through Fourier expansion and concluded that a convergent post-buckling solution (and hence a solution with enough regularity) cannot exist. Fu [12] then considered the deformation of a corrugated half-space and concluded that any post-buckling solution should probably contain static shocks. The corrugated half-space problem was also investigated by Cao and Hutchinson [13] with focus on interactions of a finite number of modes.

By bending a rectangular rubber block, Gent and Cho [14] demonstrated that creases, instead of periodic wrinkles, form on the compressed inner surface when the local stretch reaches 0.65, much earlier than what Biot predicted for periodic wrinkles. Subsequently, Gent and Cho’s observation has been confirmed by numerical [15], [16], experimental [17], [18], and analytical studies [19], [20], [21].

Although Biot’s buckling mode does not seem realizable in practice, it has nonetheless provided a major reference point in stability and bifurcation analysis of a variety of soft materials and structures. For instance, it often appears as the large wave number limit in a bifurcation analysis [22], and is closely associated with the complementing condition for the existence of a unique solution for boundary value problems in nonlinear elasticity [23]. Two variants of the Biot problem have received a lot of attention in recent years. The first is concerned with the buckling of a compressed half-space with material properties varying with depth [24], [25], [26], [27], [28], [29]. The second variant is concerned with the buckling of a coated half-space (or a film/substrate bilayer). For the latter there now exists a huge body of literature, driven by a variety of applications. We refer to the review articles by Yang et al. [30], Li et al. [31], Wang and Zhao [32], and Dimmock et al. [33] for a comprehensive list of the literature and discussion of applications from different perspectives.

In this paper, we propose and study another variant of the Biot problem by taking into account a geometrical imperfection on the free surface. The imperfection takes the form of a localized ridge or trench that varies slowly in the direction perpendicular to the direction of periodic wrinkling; see Fig. 1. Our aim is to assess how such an imperfection affects the critical stretch for periodic wrinkling. The other extreme variant of the Biot problem is concerned with the case when the imperfection is fast-varying such that it is wedge-like. The latter problem has recently been studied by Lestringant et al. [34].

The current problem has a counterpart in the dynamical setting of topography-guided surface waves; see Samuel et al. [35] and Fu et al. [36]. As the half-space is compressed, the surface wave speed will change with respect to the stretch. When the stretch is such that the surface wave speed vanishes, the surface wave mode becomes the wrinkling mode that is studied in the current paper.

The rest of this paper is divided into six sections as follows. After problem formulation in Section 2, the next three sections are concerned with the asymptotic solutions at leading-, second-, and third-orders. The leading-order problem recovers Biot’s classical problem, the second-order problem is mainly concerned with the determination of the anti-plane displacement component, and it is at the third order that we derive an eigenvalue problem that determines the leading-order correction to the critical stretch due to surface imperfections. The eigenvalue problem is solved numerically and asymptotically in Section 6, and a summary and further discussions are presented in the concluding section.