Kink band instability in layered structures

doi:10.1016/j.jmps.2003.09.026

Journal of the Mechanics and Physics of Solids

Volume 52, Issue 5, May 2004, Pages 1071-1091

https://doi.org/10.1016/j.jmps.2003.09.026 Get rights and content

Abstract

A recent two-dimensional prototype model for the initiation of kink banding in compressed layered structures is extended to embrace the two propagation mechanisms of band broadening and band progression. As well as interlayer friction, overburden pressure and layer bending energy, the characteristics of transverse layer compressibility and foundation stiffness are now included. Experiments on constrained layers of paper show good agreement with the predictions of angle of orientation, kink band width and post-kink load–deflection response obtained from the model.

Introduction

Kink banding is a phenomenon seen on a variety of scales across the physical sciences. It should be considered as a potential failure mode for any layered or fibrous material, held together by external pressure or some form of internal “glue”, and subjected to layer-parallel compression. Examples can be found in the deformation of geological strata (Anderson, 1964; Hobbs et al., 1976; Price and Cosgrove, 1990), wood and fibre composites (Kyriakides et al., 1995; Reid and Peng, 1997; Fleck, 1997; Hull and Clyne, 1996; Byskov et al., 2002; Vogler and Kyriakides, 2001), and internally in wire and fibre ropes (Hobbs et al., 2000). There have been many attempts to reproduce kink banding theoretically, from early mechanical models (Rosen, 1965; Argon, 1972), to more sophisticated formulations coming from both continuum mechanics (Budiansky, 1983) and numerical perspectives (Vogler et al., 2001). Notable work on layered structures include the theoretical models of Johnson (1995), and the experiments of Ghosh (1968).

Perhaps because composite materials have had such a high profile, most formulations to date have been aimed at fibrous rather than layered structures. Extra problems are then encountered in the modelling process. First, although two-dimensional models are commonly employed (Budiansky et al., 1998), modelling into the third dimension adds a significant extra component. It necessarily involves a mix of fibres and either voids or matrix material, and usually has to be handled by some kind of smeared approximation. Secondly, failure is likely to be governed by plastic shear in the matrix material (Fleck, 1997), and this is considerably less easy to measure or control than the combination of overburden pressure and friction considered here.

The formulation follows naturally from earlier work (Hunt 2000, Hunt 2001), with a shift of emphasis from initial instability to subsequent propagation. The motivation is found in structural geology, specifically in the formation of kink bands and related chevron folding in compressed sedimentary rocks, as seen for example in the exposed cliff face at Millook Haven in Cornwall, just south of Bude. Insight into such mechanisms can be obtained from laboratory experiments on layers of paper constrained by transversely applied overburden pressure and compressed in one of the layer-parallel directions in a loading device. The inclusion of transverse compressibility has added a significant new component to the formulation; the suggestion is made for instance that release of the initial compression caused by the overburden pressure is instrumental in selecting the orientation of the band across the specimen.

For comparison with experiments, quantitative measures of the coefficient of friction, overburden pressure and compressive load are easy to obtain. More difficult is the extra resistance to band formation coming from the stiffness of the supporting foundation, but this is usefully inferred from the experimental transverse load response. The bending energy incorporated in the layers is again difficult to measure; bands form with near-straight limbs and near-sharp corners, and the curvature of the corner is chosen as a compromise between work done in bending and against overburden pressure. A tailored “corner analysis”, based on minimizing the energy contribution of this compromise, allows the distributed corner energy to be replaced with an equivalent rotational spring. Experimental and theoretical loading curves are then found to agree well over the full loading range.

The paper starts with a description of the loading history of a typical experiment, involving both the initial formation and subsequent propagation of kink bands. This is followed by identification of the significant components of either energy or pseudo-energy (Hunt et al., 2000) that make up the full nonlinear potential function. The experimental response has two distinct phases, instability and propagation, and each is fully reflected in the nonlinear response that follows from the potential energy description. Both kink band rotation and kink band width are included as degrees of freedom, and two different forms of propagation are thus identified, band broadening and band progression. At the point of instability, there is a sudden appearance of a kink band of nonzero width, which broadens under increasing load until formation of a second band, then a third and so on, with progression continuing as new bands form in zig-zag fashion along the length. The body of the paper is devoted to extensions to the mechanical model, and it closes with comparison between the resulting predictions and a few simple experiments.

Section snippets

A typical experiment

The following set of experiments was conducted in the Department of Civil and Environmental Engineering at Imperial College London. Layers of A4 size paper $(210 mm ×297 mm)$ or smaller were held together transversely under a rigid screw device, and then compressed in one of the layer-parallel directions by a second loading system which applied end-shortening at constant rate over a portion of the available layers, as shown in Fig. 1. A photograph of the loading detail and resulting deformation

Model characteristics

The formulation used here is developed from earlier models of Hunt 2000, Hunt 2001, with the important added ingredient of transverse compressibility. A kink band is assumed to be made from straight limbs and sharp corners, as shown in Fig. 4, with the band width b allowed to vary and each layer being of thickness t. Bending energy in each layer is concentrated into two rotational elastic springs of stiffness c at each end of the inclined portion, and the layers are offset relative to one

Total potential energy function

The elements that make up the analytical tool of the total potential energy function are next introduced in turn. A similar set of energy contributions also appear in a recent two-layer model for the related problem of parallel folding (Budd et al., 2003). The difference between the formulations lies primarily in the assumption that all layers involved in kink banding behave identically; in parallel folding they necessarily deform to different curvatures. The governing parameters in the energy

Comparison with experiments

A set of four experimental results from the rig of Fig. 1 is given in Table 1. Here Experiment 1 is that of Fig. 2, with the response shown in Fig. 3. Paper of grade $80 g / m^{2}$ was used in each case. Direct comparison with the model requires independent estimates of k,E,q,μ and k_f, obtained in the following manner.

Concluding remarks

The comparisons between the experiments and the theory demonstrate that good agreement can be achieved from this simple mechanical model, provided certain important characteristics are included, viz:

(1)
Bending energy: As with models of the continuous strut on elastic foundation (Hunt et al., 1989; Budd et al., 2003), a balance between bending energy and other energy contributions (quasi-friction and/or foundation energy terms) sets a length scale to the buckle pattern (here band width b). Bending

Acknowledgements

The authors would like to thank Ron Millward of the Structures Laboratory in the Department of Civil and Environmental Engineering at Imperial College London for constructing and helping to develop the testing rig. This work has been supported by the UK Engineering and Physical Sciences Research Council (EPSRC) through Grant GR/R37173.

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Kink band instability in layered structures

Abstract

Introduction

Section snippets

A typical experiment

Model characteristics

Total potential energy function

Comparison with experiments

Concluding remarks

Acknowledgements

Treatise Mater. Sci. Technol.

Comput. Struct.

J. Mech. Phys. Solids

Int. J. Solids Struct.

Adv. Appl. Mech.

Tectonophysics

Tectonophysics

Int. J. Solids Struct.

Int. J. Impact Eng.

Int. J. Solids Struct.

Int. J. Solids Struct.