The Mullins effect in the wrinkling behavior of highly stretched thin films
Introduction
The need for finite-deformation nonlinear elasticity in the accurate modeling of wrinkling phenomena in highly axially stretched thin elastomer sheets was recently demonstrated, cf. Healey et al. (2013), Li and Healey (2016), and Sipos and Fehér (2016). The 2D membrane model is that of geometrically exact nonlinear elasticity, while the fine thickness of such sheets manifests itself in an extremely small bending stiffness. This is in contrast to the well-known Flöppl-von Kármán (FvK) model (von Kármán, 1910), employing linear infinitesimal elasticity in the membrane part that also incorporates a nonlinear term in the gradient of the out-of-plane displacement. The FvK model has a long and successful track record in the prediction of buckling and initial post-buckling behavior of classical plates and shells.
Of course the wrinkling of a flat sheet is also a buckling phenomenon. Nonetheless, as shown in Healey et al. (2013) and Li and Healey (2016), the FvK model leads to qualitative errors in the prediction of wrinkles for that particular class of problems. A bifurcation analysis in the macroscopic (applied) strain indicates that, for an apparently semi-bounded interval of aspect-ratio values, the FvK theory predicts a super-critical pitch-fork bifurcation indicating the initiation of wrinkles. In particular, it fails to predict the disappearance of wrinkles as the macroscopic strain is increased beyond initiation. In contrast, the geometrically exact models of Healey et al. (2013), and Li and Healey (2016) predict an isola-center bifurcation, indicating that for a bounded interval of aspect ratios only, stable wrinkles appear and then disappear as the macroscopic strain is increased. To summarize, the FvK model not only predicts the wrong phenomena (wrinkles do not disappear), but it also predicts the initiation of wrinkles for a vast range of aspect ratios not exhibiting wrinkling.
In recent experimental work (Fehér, Sipos, 2014, Sipos, Fehér, 2016), the appearance and subsequent disappearance of wrinkles for a bounded range of aspect ratios was unambiguously verified for the problem addressed in Healey et al. (2013) and Li and Healey (2016). This underscores the importance of a geometrically exact mathematical model. However, the experiments reported in Sipos and Fehér (2016) reveal inelastic behavior corresponding to a permanent change in length and shape of sheets upon first loading with little change upon subsequent loadings. We also point out that the recent work (Zhu et al., 2018) also calls for an understanding of the role of inelasticity in wrinkling behavior.
Here we report on further experimental results for this problem, carried out on finely thin, rectangular polyurethane sheets. The specimens were subjected to cyclical “hard” loading and unloading. To fix terminology, the first macroscopic stretch of the virgin sheet is called the first loading. The first unloading and all subsequent loadings and unloadings are referred to collectively as cyclic loading. Among other observations, our work here is motivated by the following striking experimental observation: For certain aspect ratios for which no wrinkling occurred upon the first loading, wrinkles then appeared during unloading and again during all subsequent cyclic loading. (See Fig. 1. and the attached video in the “supplementary material”). Moreover, residual strain and significant stress softening was observed after the first unloading, while all subsequent load cycles exhibited elastic behavior, as pointed out earlier in Sipos and Fehér (2016). Wrinkling aside, the latter is consistent with the so-called Mullins effect in elastomers, e.g., Dorfmann and Ogden (2004) and Ogden and Roxburgh (1999).
We can give a heuristic explanation of the observed phenomenon, based upon our earlier results for purely elastic models. A typical stability boundary of the type obtained in Li and Healey (2016) is a simple closed curve in the plane of aspect ratio, denoted vs. macroscopic strain, denoted . Here L and W are the length and width, respectively, of the rectangular sheet, while ΔL is the change in length under externally imposed, longitudinal stretching. In Fig. 2 we depict such a computed curve for the specimens studied in this work, employing a finite-elasticity model from Li and Healey (2016). The intersection of a vertical line, corresponding to a fixed aspect ratio, with the closed curve gives the locations where stable wrinkles first emerge and then disappear as ε is steadily increased. In particular, no wrinkling occurs for aspect ratios below or above the closed curve. With that in mind, suppose that a given sheet has aspect ratio μ1 that is just below but sufficiently close to the stability boundary. Now assume that the first loading produces a permanent change of length such that the aspect ratio now increases to μ2 as shown in Fig. 2. Then wrinkling occurs upon unloading.
Of course, the qualitative explanation above is a gross oversimplification. The purely elastic model neither captures the developed orthotropy observed in Sipos and Fehér (2016), nor does it account for the change in the stability boundary due to damage. In particular, the entire specimen goes slack at a small positive value of the macroscopic strain upon unloading, corresponding to the residual strain in the specimen. Moreover, it was observed that the unloaded specimens were wrinkled at the residual strain level. This is illustrated in the top right panel of Fig. 1; the residual strain is about . Our goal here is to present a simple pseudo-elastic model, accounting for the longitudinal stress softening and residual strain observed in the experiments, that accurately captures the correct wrinkling behavior just described. In particular for specific aspect ratios, the model correctly predicts the scenario of no wrinkling during first loading with wrinkles developing upon unloading and during all subsequent cyclic loading.
The outline of the paper is as follows. In Section 2 we present our model incorporating a single state variable to capture the Mullins effect as in Dorfmann and Ogden (2004) for the 2D Mooney-Rivlin elasticity model with small bending employed in Li and Healey (2016). The dominantly uni-axial Mullins effect is accounted for solely in the membrane part of the model, while in the presence of small elastic bending. In Section 3 we present the pertinent experimental data, some of which are used to first “tune” the mathematical model in Section 4. We then compare the experimental results for the appearance and disappearance of wrinkles for a large set of aspect ratios with our numerical predictions, demonstrating the accuracy of the mathematical model. We present some concluding remarks in Section 5.
Section snippets
Model development
We let denote a rectangular domain of length of L and width of W, which we choose as a stress-free reference configuration (Fig. 3). The displacement field is denoted by For convenience, the in-plane displacement field is denoted by i.e. . We henceforth employ the notation . The sheet is subjected to hard-loading: Displacements are prescribed on opposite ends via for while the
Experimental results
Based upon elastic model predictions predictions, we expect either two critical values of the macroscopic strain, at which wrinkles appear and disappear, or no critical values – depending upon aspect ratio (Li and Healey, 2016). In the first case, the critical values are denoted by 0 < ε1 < ε2, respectively. Two series of experiments were carried out on h = 32µm thick polyurethane sheets:
- 1.
Traditional displacement controlled pull-tests to obtain a stress-strain diagram for the material.
- 2.
A series
Model vs. experiments
As mentioned earlier, our problem is essentially uniaxial, and values of the material parameters α, β, c1 and c2 can be inferred from our uniaxial-test data. In light of incompressibility, this is facilitated by first introducing where is the principal stretch assumed to be aligned with the x1 axis, and then setting With these simplifications, the engineering stress (force per unit reference area) in the x direction, denoted by T0 is found from (5) to be
Conclusion
A finite-deformation pseudo-elastic model accounting for the Mullins effect is introduced to qualitatively and quantitatively explain experimental data measured on highly stretched, thin elastomer sheets. Recognizing the anisotropic nature of damage, a simple model, characterized by a single state variable, is tuned to capture the measured residual strain and stress softening behavior, as well as the measured re-emergent wrinkling behavior observed upon unloading. Our motivating experimental
Author contributions
T.J.H. initiated the research, E.F. carried out the experiments, T.J.H. and A.Á.S. developed the model. E.F., T.J.H. and A.Á.S. wrote the paper.
Acknowledgments
The work of A.A.S was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. The work of T.J.H. was supported in part by the National Science Foundation through grant DMS-1613753. We thank Károly P. Juhász and Ottó Sebestyén for their valuable help in the experimental work. The Zwick Z150 material testing machine was provided by the TÁMOP 4.2.1/B-09/1/KMR-2010-0002 grant.
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