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We investigate how two finite-amplitude, transverse, plane body waves may be superposed to propagate in a deformed hyperelastic incompressible solid. We find that the equations of motion reduce to a well-determined system of partial differential equations, making the motion controllable for all solids. We find that in deformed Mooney-Rivlin materials, they may travel along any direction and be polarised along any transverse direction, an extension of a result by Boulanger and Hayes [Quart. J. Mech. Appl. Math. 45 (1992) 575]. Furthermore, their motion is governed by a linear system of partial differential equations, making the Mooney-Rivlin special in that respect. We select another model to show that for other materials, the equations are nonlinear. We use asymptotic equations to reveal the onset of nonlinearity for the waves, paying particular attention to how close the propagation direction is to the principal axes of pre-deformation.

Soft materials such as rubbers, silicones, gels and biological tissues have a nonlinear response to large deformations, a phenomenon which in principle can be captured by hyperelastic models. The suitability of a candidate hyperelastic strain energy function is then determined by comparing its predicted response to the data gleaned from tests and adjusting the material parameters to get a good fit, an exercise which can be deceptive because of nonlinearity. Here we propose to generalise the approach of Rivlin and Saunders [Phil Trans A 243 (1951) 251-288] who, instead of reporting the data as stress against stretch, manipulated these measures to create the 'Mooney plot', where the Mooney-Rivlin model is expected to produce a linear fit. We show that extending this idea to other models and modes of deformation (tension, shear, torsion, etc.) is advantageous, not only (a) for the fitting procedure, but also to (b) delineate trends in the deformation which are not obvious from the raw data (and may be interpreted in terms of micro-, meso-, and macro-structures) and (c) obtain a bounded condition number \k{appa} over the whole range of deformation; a robustness which is lacking in other plots and spaces.

We consider shear wave propagation in soft viscoelastic solids of rate type. Based on objective stress rates, the constitutive model accounts for finite strain, incompressibility, as well as stress- and strain-rate viscoelasticity. The theory generalises the standard linear solid model to three-dimensional volume-preserving motions of large amplitude in a physically-consistent way. The nonlinear equations governing shear motion take the form of a one-dimensional hyperbolic system with relaxation. For specific objective rates of Cauchy stress (lower- and upper-convected derivatives), we study the propagation of acceleration waves and shock waves. Then we show that both smooth and discontinuous travelling wave solutions can be obtained analytically. We observe that the amplitude and velocity of steady shocks are very sensitive to variations of the stress relaxation time. Furthermore, the existence of steady shocks is conditional. Extension of these results to the case of multiple relaxation mechanisms and of the Jaumann stress rate is attempted. The analysis of simple shearing motions is more involved in these cases.

On one hand, classical Monte Carlo and molecular dynamics (MD) simulations have been very useful in the study of liquids in nanotubes, enabling a wide variety of properties to be calculated in intuitive agreement with experiments. On the other hand, recent studies indicate that the theory of continuum breaks down only at the nanometer level; consequently flows through nanotubes still can be investigated with Navier-Stokes equations if we take suitable boundary conditions into account. The aim of this paper is to study the statics and dynamics of liquids in nanotubes by using methods of non-linear continuum mechanics. We assume that the nanotube is filled with only a liquid phase; by using a second gradient theory the static profile of the liquid density in the tube is analytically obtained and compared with the profile issued from molecular dynamics simulation. Inside the tube there are two domains: a thin layer near the solid wall where the liquid density is non-uniform and a central core where the liquid density is uniform. In the dynamic case a closed form analytic solution seems to be no more possible, but by a scaling argument it is shown that, in the tube, two distinct domains connected at their frontiers still exist. The thin inhomogeneous layer near the solid wall can be interpreted in relation with the Navier length when the liquid slips on the boundary as it is expected by experiments and molecular dynamics calculations., Comment: 27 pages