Incremental Equations for Soft Fibrous Materials
The general theory of nonlinear anisotropic elasticity is extended to describe small-amplitude motions and static deformations that can be superimposed on large pre-strains of fibre-reinforced solids. The linearised governing equations of incremental moti
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Introduction
Consider two rectangular solid blocks, one made of silicone, the other made of mammalian skeletal muscle (‘meat’), and subject them to a large shear. The first block deforms smoothly and its surface remains flat; see Figure 1(a). The second block, however, experiences a form of buckling early on, as small-amplitude wrinkles appear on its surface. From visual inspection and intuition, we can come up with an explanation for these strikingly different behaviours. If we were careful in our moulding of the silicone block, we can safely assume that it is isotropic. On the other hand, the piece of meat is clearly anisotropic, as it is ‘reinforced’ with visible aligned fibres. When sheared, these fibres and/or their entanglements resist compression ∗
I am grateful to Sophie Labat (Bordeaux), Jorge Bruno, Artur Gower and Joanne McCarthy (Galway) for their technical assistance in preparing this chapter.
L. Dorfmann, R. W. Ogden (Eds.), Nonlinear Mechanics of Soft Fibrous Materials, CISM International Centre for Mechanical Sciences DOI 10.1007/978-3-7091-1838-2_5 © CISM Udine 2015
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and crumple early, lifting the surrounding tissue into a wavy pattern, with wavefronts at right-angles to the fibres; see Figure 1(b). Hence, although both blocks are soft and easily deformed, it is likely that they behave differently due to the absence or presence of fibres.
(a)
(b)
Figure 1. (a) A block of silicone subject to a large shear by hand; its surface remains flat and smooth. (b) A piece of meat sheared by hand, in a direction approximatively at 45◦ with respect to the fibres; its surface buckles early, with wrinkles forming at right-angle to the fibres. From a mechanics point of view, we may now ask ourselves whether there exists a way of describing and predicting how the two blocks should behave in shear. As seen in the course of this chapter, it turns out that the simplest models of isotropic (for the silicone) and anisotropic (for the meat) nonlinear incompressible elasticity can indeed capture these effects. Of course the analysis itself is not easy and requires a good grasp of theoretical issues, physically-based modelling, and numerical analysis. In order to model the small-amplitude wrinkles, we need to derive the incremental equations of motion of anisotropic non-linear elasticity. This procedure is described in Section 2; simply put, it relies on linearising the equations of motion in the neighbourhood of a static state of equilibrium corresponding to a large homogeneous deformation. These equations can be established in all generality, and in Section 3 we use them to study the propagation of bulk waves in deformed soft solids. Indeed, wave propagation is a straightforward tool for figuring out if a solid is isotropic or not. Consider, for example, the experimental results displayed in Figure 2: they clearly show two privileged directions, along which a mechanical signal travels at different speeds than in other directions. Again, this can be captured by very simple models of nonlinear anisotropic elasti
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