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We study the propagation of shear waves in an anisotropic incompressible medium composed of an elastic fiber-reinforced material. We first derive the general determining equations of such motions when the fibers are arbitrarily arranged with respect to the direction of propagation of the waves. Then, we examine the “standard reinforced model” so as to root the nonlinearity of the dynamic problem directly to the presence of the reinforcing fibers, and finally, we consider an asymptotic model in terms of the limit of finite but small amplitude of the waves. Clearly, if the fibers are arranged along the direction of propagation of the waves, the mechanical behavior of the material will be isotropic. When the fibers are tilted an amount of the same order of the amplitude of the wave with respect to this direction, the asymptotic first-order reduced system is characterized by a peculiar mixed second-/third-order nonlinearity. This contrasts to what occurs for larger tilting angles where only quadratic nonlinearities must be considered. In the latter case, as we may expect, in the asymptotic limit, we reduce the system to an inviscid classical Burger’s equation. Our results are fundamental to the study of the dynamics of fiber-reinforced materials and equally when we consider dissipative and/or dispersive effects.

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.

The choice of an appropriate strain energy function W is key to accurate modeling and computational finite element analysis of the mechanical behavior of unfilled non-crystalizing rubberlike materials. Despite the existing variety of models, finding a suitable model that can capture many deformation modes of a rubber specimen with a single set of parameter values and satisfy the a priori mathematical and structural requirements remains a formidable task. Previous work proposed a new generalized neo-Hookean W (I1) function (doi: 10.1016/j.ijnonlinmec.2020.103626), showing a promising fitting capability and enjoying a structural basis. In this work we use two extended forms of that model which include an I2 term adjunct, W(I1,I2), for application to various boundary value problems commonly encountered in rubber mechanics applications. Specifically, two functional forms of the I2 invariant are considered: a linear function and a logarithmic function. The boundary value problems of interest include the in-plane uniaxial, equi-biaxial, and pure shear deformations and simple shear, inflation, and nonhomogeneous deformations such as torsion. By simultaneous fitting of each model to various deformation modes of rubber specimens, it is demonstrated that a single set of model parameter values favorably captures the mechanical response for all the considered deformations of each specimen. It is further shown that the model with a logarithmic I2 function provides better fits than the linear function. Given the functional simplicity of the considered W (I1, I2) models, the low number of model parameters (three in total), the structurally motivated bases of the models, and their capability to capture the mechanical response for various deformations of rubber specimens, the considered models are recommended as a powerful tool for practical applications and analysis of rubber elasticity.

The Ogden model is often considered as a standard model in the literature for application to the deformation of brain tissue. Here, we show that, in some of those applications, the use of the Ogden model leads to the non-convexity of the strain-energy function and mis-prediction of the correct concavity of the experimental stress–stretch curves over a range of the deformation domain. By contrast, we propose a family of models which provides a favourable fit to the considered datasets while remaining free from the highlighted shortcomings of the Ogden model. While, as we discuss, those shortcomings might be due to the artefacts of the testing protocols, the proposed family of models proves impervious to such artefacts. This article is part of the theme issue 'The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity'.

We place the Ogden model of rubber elasticity, published inProceedings of the Royal Society50 years ago, in the wider context of the theory of nonlinear elasticity. We then follow with a short interview of Ray Ogden FRS and introduce the papers collected for this Theme Issue.This article is part of the theme issue ‘The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity’.

We propose an anisotropic and nonlinear generalization of the Kelvin–Voigt viscoelastic model obtained considering the additive splitting of the Cauchy stress tensor in an elastic and a dissipative part. The former one corresponds to a fiber-reinforced hyperelastic material while the dissipative effect is described by the most general linear transverse-isotropic tensorial function of symmetric part of the velocity gradient. In a such a way we characterize the dissipative contribution via three viscoelastic moduli. We then show, by a detailed analysis of the simple shear quasistatic motion and the corresponding creep phenomena, that this motion may be used to determine experimentally the viscoelastic parameters.

In questo capitolo applicheremo le Leggi della Meccanica enunciate nel Capitolo 7 all’analisi del moto di punti materiali, sia liberi che vincolati, sottoposti a diversi tipi di forze attive.

Scopo della Meccanica e quello di collegare il movimento dei corpi con le forze che su questi possono agire. A tal fine e necessario introdurre principi generali che hanno il compito di spiegare le possibili interazioni tra gli schemi introdotti. Questi principi sono Leggi della Meccanica: un insieme di postulati assunti i quali secondo il procedimento ipotetico-deduttivo delle scienze matematiche si ottengono le relazioni causa-effetto cercate.

The propagation of localised (in space-time) waves is analysed in the context of the dynamic theory of incompressible hyperelastic solids subject to body forces corresponding to a dual power-law substrate potential. A broad class of exact solutions is obtained which, under the assumption of slow modulation, incorporates Helmholtz-type solitary waves. The linear stability of these solutions is studied under the assumption that the speed of propagation of the wave is small enough compared to the speed at which transverse waves travel in the linear regime and in the absence of external actions.

Experimental data from creep tests on polycrystalline ice samples highlight not only the non-Newtonian behavior of ice but also suggest a critical dependence of the various rheological parameters upon the applied hydrostatic pressure. We propose a new modeling framework, based on implicit theories of continuum mechanics, that generalizes two well-known constitutive models by taking into account the effect of the pressure in the description of ice in creep. To ascertain the validity of the proposed models, we fit the physical parameters with experimental data for the elongational flow of ice samples. The results show good agreement with the experimental creep curves. In particular, the proposed generalized models reproduce the increase of the creep rate due to the presence of hydrostatic pressure.

A vivid research on brain tissue has proven that in simple shear tests the relation between the shear stress and shear strain is linear for strains in a range which is significative in the physiological and pathological regime. Since Mooney–Rivlin materials satisfy this peculiar property when subjected to simple shear deformations, the celebrated mathematical model introduced first by Mooney, and then developed by Rivlin, has been often used to describe the mechanical behaviour of brain tissue. Recently, it has been shown that a most general strain energy density for incompressible isotropic elastic materials exhibiting a linear relationship between shear stress and shear strain in simple shear deformation consists of the sum of the Mooney–Rivlin model and an arbitrary function of the difference of the first two principal invariants of the deformation tensor. For this reason, a strain energy function of this form is called a generalised Mooney–Rivlin model. In this note we design theoretically a procedure aiming at the determination of the generalised Mooney–Rivlin model which fits best the experimental data.

Traveling waves of permanent form with compact support are possible in several nonlinear partial nonlinear differential equations and this, mainly, along two pathways: A pure nonlinearity stronger than quadratic in the higher order gradient terms describing the mathematical model of the phenomena or a special inhomogeneity in quadratic gradient terms of the model. In the present note we perform a rigorous analysis of the mathematical structure of compactification via a generalization of a classical theorem by Weierstrass. Our mathematical analysis allows to explain in a rigorous and complete way the presence of compact structures in nonlinear partial differential equations 1 + 1 dimensions.

The application of a newly proposed generalised neo-Hookean strain energy function to the inflation of incompressible rubber-like spherical and cylindrical shells is demonstrated in this paper. The pressure ($P$ P ) – inflation ($\lambda $ λ or $v$ v ) relationships are derived and presented for four shells: thin- and thick-walled spherical balloons, and thin- and thick-walled cylindrical tubes. Characteristics of the inflation curves predicted by the model for the four considered shells are analysed and the critical values of the model parameters for exhibiting the limit-point instability are established. The application of the model to extant experimental datasets procured from studies across 19th to 21st century will be demonstrated, showing favourable agreement between the model and the experimental data. The capability of the model to capture the two characteristic instability phenomena in the inflation of rubber-like materials, namely the limit-point and inflation-jump instabilities, will be made evident from both the theoretical analysis and curve-fitting approaches presented in this study. A comparison with the predictions of the Gent model for the considered data is also demonstrated and is shown that our presented model provides improved fits. Given the simplicity of the model, its ability to fit a wide range of experimental data and capture both limit-point and inflation-jump instabilities, we propose the application of our model to the inflation of rubber-like materials.

An important one-dimensional rheological model for the propagation of a linearly polarized shear wave was recently obtained and proposed by Cormack and Hamilton [(2018). J. Acoust. Soc. Am. 143(2), 1035–1048]. We show that it is possible to embed such a result within a wider and complete set of general three-dimensional models derived within the theoretical framework of rigorous continuum mechanics. We show that, following this approach, we are able to derive in a simple and straightforward way the equations that govern the propagation of circularly polarized shear waves. When the phase of such waves is constant, we find the same equation for linearly polarized shear waves already proposed elsewhere. Moreover, we show that, under appropriate asymptotic assumptions, our results are indifferent with respect to the choice of the objective time derivative used in the constitutive class.

Summary In the framework of linear piezoelectric ceramics, we study the deformations of a circular infinite hollow cylinder, subjected to a potential difference between the inner and outer surfaces. When the poling direction is perfectly aligned with the cylinder axis, the solution to this problem is a trivial axisymmetric anti-plane state. However, when the poling direction has an offset angle with respect to the cylinder axis, the corresponding behavior becomes more complex, with the anti-plane state coupled to an in-plane, potentially non axisymmetric, deformation. In this work, we show that this structural coupling, due to the poling direction, can be fruitfully exploited to achieve non-standard actuation and sensing properties, like the possibility to generate a torque under axially symmetric electromechanical controls.

A local universal relation is an equation between the stress components and the position vector components which holds for any material in an assigned constitutive family. Although universal relations may be of great help to modellers in characterizing the material behaviour, they are often ignored. In this paper we briefly discuss the valuable insights that universal relations may offer in solid mechanics and determine novel universal relations associated with shearing motions in nonlinearly elastic and nonlinearly viscoelastic materials of differential type.

In the framework of linear transversely isotropic elasticity, we study the deformation of a circular infinite hollow cylinder, whose inner face is fixed, while its outer surface is subject to a constant axial surface traction. In isotropic linear elasticity, the solution of this problem is just a state of anti-plane axial shear. In the case of a transversely isotropic material with a chosen arrangement of fibers, the corresponding deformation field becomes more complex and the coupling of axial anti-plane shear with an in-plane deformation may occur. Therefore, it is possible to use an axial tension field to generate, for example, an azimuthal shear deformation. This fact suggests to use anisotropy to design some elastic machines which can combine different deformation modes.

We show that porcine brain matter can be modelled accurately as a very soft rubber-like material using the Mooney-Rivlin strain energy function, up to strains as high as 60%. This result followed from simple shear experiments performed on small rectangular fresh samples (2.5 cm(3) and 1.1 cm(3)) at quasi-static strain rates. They revealed a linear shear stress-shear strain relationship (R-2 > 0.97), characteristic of Mooney-Rivlin materials at large strains. We found that porcine brain matter is about 30 times less resistant to shear forces than a silicone gel. We also verified experimentally that brain matter exhibits the positive Poynting effect of non-linear elasticity, and numerically that the stress and strain fields remain mostly homogeneous throughout the thickness of the samples in simple shear. (C) 2015 Elsevier Ltd. All rights reserved. Partial funding by a “New Foundations” award from the Irish Research Council peer-reviewed 2017-03-13