Uniaxial stretch-release of rubber-plastic bilayers: Strain-dependent transition to stable helices, rolls, saddles, and tubes

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Abstract

Polymeric plastics deform irreversibly (i.e., inelastically) whereas rubbers deform reversibly, i.e., elastically. Thus, uniaxially stretching a rubber-plastic bilayer composite beyond its yield point can create an elastic strain mismatch between the two layers. Upon release, the bilayer may then bend out-of-plane. We quantify the mechanics of such stretch-release-induced shape changes in rectangular specimens of rubber-plastic bilayers. We show a remarkable dependence of the final shape upon the stretch applied prior to release. At small stretch, all bilayers bend into arch or roll shapes with the plastic on the convex face. At large stretch, the bilayers bend into half-tubes with the plastic, now heavily-wrinkled, becoming the concave face. Thus, the sign and direction of the curvature both flip as applied stretch increases. Between these two extremes, saddle shapes appear which have characteristics of both arches as well as half-tubes. Sufficiently narrow samples show different behavior: they transition from arches to helices as stretch increases. All these shapes are mono-stable. We document numerous ways in which the mechanics of rubber-plastic bilayers differs from that of fully-elastic bilayers. Most importantly, yielding of the plastic layer during the shape change strongly affects the mechanics of the elastic–plastic bilayers, and yielding accompanied by plastic wrinkling has an especially large effect. A strain energy model illustrates how the shape change is dictated by the change in the ratio of elastic strain mismatch in the two directions due to the formation of wrinkles at the rubber-plastic interface.

Introduction

When stretched, an elastomeric sheet stores the work done on it as elastic energy and recovers its original shape upon unloading. In contrast, a sheet of yielding material dissipates almost all the work of stretching, and therefore remains at or near its stretched configuration when unloaded. A composite bilayer composed of these two materials, when stretched and unloaded, can bend out-of-plane, sometimes with interfacial buckling or wrinkling. This article is about the various mono-stable shapes with single or dual curvature formed by a uniaxial stretch and release of a rubber-plastic bilayer. The central focus of this article is to elucidate the role of inelastic deformation in the bending process.

Elastic strain mismatch within a material can trigger complex shape deformations. Hence it is a powerful tool for creating intricate shapes and structures. In nature, plants and animal tissues exploit growth-induced heterogeneous elastic strain to create a myriad of static shapes and dynamic mechanisms. Opening of seed pods by elastic incompatibility [1], growing complex three dimensional features in internal organs [2], [3], twisting of plant tendrils [4], curvatures in leaves [5] and others [6], [7], [8] are all examples of shape changes through inhomogeneous strain generation. Many researchers have sought to exploit such shape changes for engineering [9], [10], [11] and biomedical [2], [12], [13] applications. Heterogeneous strain driven fabrication techniques have also found popularity in micro- and nano-scale fabrication [12], [14], [15], [16], where it is difficult to create intricate 3D shapes with conventional methods. Nano tubes and channels and many other three-dimensional shapes can be created through these techniques. Morphology changes driven by controllable strain gradients are also explored widely in the literature [17], [18], [19], [20], [21].

Much previous research on shape changes has created a strain mismatch by applying a uniform stimulus to a heterogeneous structure. A common example of this mechanism is the bimetallic strip sometimes used in household thermostats. A bimetallic strip is made by bonding two metals with unequal thermal expansion coefficient. When heated, the strip bends into an arch, with the metal with higher thermal expansion coefficient material on the outside. Instead of thermal expansion, other stimuli such as differences in solvent swelling of the layers [16], [22], [23], [24], or shape memory of one of the layers [25], can also be used to induce bending. The rest of this paper focuses on such bilayers in which a large difference in mechanical properties between layers induces shape changes.

Some of the recent fundamental research on bilayer mechanics was conducted by bonding together layers while holding one or both layers prestretched [18], [26], [27], [28]. Upon bonding the layers and then releasing, the composite specimen adopted a bent configuration. When the prestretched layer was under equi-biaxial stretch, the composite bilayers bent into bistable cylinders or spherical bowls depending on thickness and width of the specimen as well as the applied prestretch [26]. Unlike a bimetallic strip where the strain mismatch is inherently isotropic, prestretching allows investigations of anisotropic strain states. In [26] it is found that the curvatures of the system bifurcate upon reducing bilayer thickness, and such bifurcation separates two scaling regimes for the energy of the system. In a related work, upon searching for isometries from a reference surface of the bilayer, DeSimone modeled the effects of bonding a uniaxially prestretched rubber layer to an unstretched rubber layer [27]. This is done in [27] using a novel finite dimensional constrained energy minimization problem. Upon unloading, the composite bilayer bent into an mono-stable roll, with the prestretched layer on the inside [27]. In contrast, Chen et al. bonded rubber layers that were both prestretched, but along perpendicular directions [29]. Upon releasing, the composite bilayers bent into saddle shapes or bistable cylinders depending on thickness and width of the specimens [29]. Such bonding with prestretch also allows investigations of how specimen geometry couples with anisotropy in prestretch, e.g. a rectangular strip where the long direction is at an angle with respect to the prestretch direction forms a helical shape [27].

This article focuses on shape changes occurring due to a deformation-induced strain mismatch. Specifically, we examine shape changes of a rubber-plastic bilayer which has been uniaxially stretched, and then released, as illustrated in Fig. 1B. First consider the free-standing layers of the rubber and the plastic, Fig. 1A. Upon stretch-release, the rubber layer recovers its original configuration, whereas the plastic maintains its stretched shape. Now consider a bilayer in which these two layers are bonded to each other, both under stress-free conditions. Upon stretching, an elastic strain mismatch is created. Thus, upon releasing, the bilayer is expected to bend (Fig. 1B).

Real rubbery materials are not perfectly elastic and may undergo modest inelastic deformations. Similarly, real plastics do not maintain their deformed shape perfectly, but instead show modest shape recovery after deformation. Nevertheless, provided that the degree of inelastic deformation of the two layers is unequal, bending is expected. Excellent examples of such bending are shown by Wisinger et al. [30]. Incidentally note that Fig. 1, which is a 2-dimensional illustration, only shows simple arch shapes, whereas Wisinger et al. showed arches as well as helical shapes, depending on the magnitude of elastic strain mismatch. This paper will show an even wider range of complexity including the saddle shapes mentioned in the previous paragraph for fully elastic bilayers. Further, we will show that even the direction and sign of the curvature depend on the applied stretch, a complexity not reflected in Fig. 1.

The deformation-induced strain mismatch of Fig. 1B is a powerful means to realize shape-morphing materials. Changes in shape can be induced by simply stretching the material to the desired extent, instead of stimuli such as light or heat [28], [30]. Irreversible deformation is common amongst a variety of materials including metals, polymers, or even amorphous particulate materials, and hence this approach can be applied to a variety of systems. Indeed, an excellent example is of helically-coiled synthetic muscle fibers which were created by simply stretching an elastic–plastic composite [13].

Despite the potential power of elastic–plastic composites to realize complex shape changes, their mechanics is poorly understood, even for the case of bilayers. At first glance, it is tempting to interpret such bending as being analogous to the shape change of an elastic–elastic bilayer with a strain mismatch. This analogy is illustrated in Fig. 1C: the stretched state of the elastic–plastic bilayer is regarded as equivalent to bonding a stress-free elastic layer to a rubber layer pre-stretched to the same dimensions [30]. As per this analogy, the only role of inelastic deformation is to create a strain mismatch when the sample is stretched; the mechanics after release presumes that both materials behave elastically. Indeed, this “elastic-after-release” is exactly the modeling approach followed by Wisinger et al. [30]. Yet, although the analogy of Fig. 1C is useful, inelastic deformation may have several consequences beyond merely creating the strain mismatch upon stretching.

First, upon releasing, the plastic layer experiences compression, and if the compressive stress exceeds the yield stress of the plastic, it will deform inelastically. Fig. 1B presumes that the plastic remains at its stretched dimensions upon release, but if it yields in compression, the degree of bending will reduce. This is well-recognized in bimetallic strips in which, if the yield stress of one of the layers is exceeded, the curvature becomes much less sensitive to temperature changes [31], [32]. An example will be shown in this paper.

Second, if the elastic layer is relatively soft and thick, the plastic face of the bilayer may buckle in compression and develop wrinkles at sufficiently large strain mismatch [33], [34], [35], [36]. In fact, wrinkles may develop even for a purely elastic bilayer in the geometry of Fig. 1C [37], [38], [39], [40], [41], [42], [43], but in elastic–plastic composites, the wrinkling is coupled with yielding, and wrinkles can form before or after yielding [36]. This paper will show that yielding occurs before wrinkling, and wrinkles persist even when the plastic layer is debonded from the rubber, which indicates the formation of plastic hinges. The effect of wrinkles on shape changes have not been explored in this context before.

Third, in fully-elastic bilayers, bistable shapes are encountered commonly. Bistability can be desirable, e.g. Venus flytrap relies on the snap-through of a double curvature surface [44] or snapping mechanical wires [45] and many others reviewed in [46]. Similarly, a biaxial elastic strain mismatch can induce bistability such that the bilayer can bend in one of two possible directions [26], [27]. However, bistability may not be desirable and considerable research been done on guiding the direction along which curvature develops [25], [47], [48]. This article shows that rubber-plastic bilayers tend to form single stable shapes. Thus, no external control other than the magnitude of applied stretch is required to control the final morphology.

Finally, even during the stretching process itself, inelastic deformation may introduce complexities beyond merely introducing a strain mismatch. Free-standing plastic layers are prone to necking behavior in tension, and free-standing polymeric plastic layers may also show stable neck propagation [49]. Bonding an elastic layer to a plastic reduces the tendency for necking, nevertheless, if the elastic layer is sufficiently thin or soft, an elastic–plastic bilayer may show necking and/or stable neck propagation [50], [51], [52], [53], [54], [55]. Thus, due to non-homogeneous stretching, in-plane strain gradients may develop during the stretching phase, which would then affect subsequent shape changes. An example of this will also be shown in this paper.

In summary, the above arguments suggest that an “elastic-after-release” framework is tenable only if (1) the elastic–plastic bilayer deforms homogeneously during stretching, (2) compressive stress developed in the plastic layer after release is lower than its yield stress and (3) surface instabilities like wrinkles do not appear. The second and third condition suggests that elastic strain mismatch must depend, even qualitatively, on the applied stretch. Upon releasing from a low stretch, the rubber imposes only a modest compressive stress on the plastic layer, and hence the plastic may not yield. However, upon releasing after a large applied stretch, the rubber must impose a correspondingly large stress on the plastic, and yielding and wrinkling become likely. Indeed, this paper confirms a significant dependence of shape changes on the applied stretch.

This article is organized as follows. Section 2 describes the materials and methods, including a list of symbols in Table 1. Section 3.1 describes experimental observations of rubber-plastic bilayers subjected to stretch and release. We show that even the qualitative nature of shape changes in such composites depend on the applied strain. Sections 3.2 Effect of specimen width, 3.3 Effect of rubber thickness show how the sample geometry affects the results, in particular that narrow specimens are susceptible to twisting deformations. Finally, Section 3.4 shows non-homogeneous bending that occurs because the plastic layer is prone to necking. An analytical model is described in Section 4 which shows how modification in elastic strain mismatch due to inelastic deformation and wrinkling during release can lead to flip in sign and direction of the curvature with increasing stretch. Section 5 summarizes the paper.

Section snippets

Experimental

Sheets of natural rubber (McMaster Carr) of undeformed thickness, Hr=250μm, 500 μm and 750 μm were used as the elastomer layer. Sheets of linear low-density polyethylene (LLDPE) of Hp=50μm thickness was used as the plastic. Rubber-plastic bilayers of different thickness ratios were prepared by bonding the LLDPE sheets to the rubber sheets. The bond was made by passing the two layers together through heated mechanical rollers to eliminate air pockets, and then heating to 150°C and holding for 10

Results

The schematics of stretch-release and subsequent morphology change is shown in Fig. 3A. The initial sample dimensions are denoted L, W, and H along the X, Y, and Z directions respectively. When a stretch λx is applied, the length increases but the width, w and thickness, h in the stretched configuration both reduce due to the Poisson effect.

The final morphology after unloading can have two curvatures. The curvature about the Y-axis (i.e., in the XZ plane) is marked as κy, whereas the curvature

Discussion: Strain-dependent arch to half-tube transition

In a bilayer with an elastic strain mismatch, unidirectional bending, saddle formation, wrinkling, and helical twisting all offer competing pathways for energy minimization. In fully-elastic bilayers, uniaxial elastic strain mismatch induces stable unidirectional bending [27]. Meanwhile, a equi-biaxial elastic strain mismatch creates dual curvature into bowl shapes at small mismatch, but bistable unidirectional bending into cylindrical shapes at large mismatch [60]. This paper shows that for

Summary and conclusions

In summary, we have explored shape changes induced by stretching rectangular strips of rubber-plastic bilayer uniaxially, and then releasing them back to a stress-free state. Such shape changes result from deformation-induced strain mismatch: the plastic deforms permanently, whereas the rubber deforms elastically. Accordingly, upon stretching, the stress-free shape of the plastic is longer and narrower than of the rubber, and hence upon releasing, the specimen bends. We found that the bilayers

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This research was supported by the grant NSF-CMMI 1636064, NSF-2036164 and NSF-CMMI-1561789. We are grateful to Dr. Steven Abramowitch for the use of his tensile testing equipment.

L.D. gratefully acknowledges the partial support of the following grants: (i) The Italian MIUR with the “Departments of Excellence” Grant L. 232/2016, (ii) BOHME, FET EU grant No. 86317, (iii) PRIN 2017 20177TTP3S, (iv) FET Proactive (Neurofibres) grant No. 732344. L.D. also acknowledges the participation to the NIH,

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