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Antonio Signorini's contribution to the constitutive theory of non-linear elasticity is reconstructed and analyzed. Some uninformed opinions suggesting he had a minor role, lacking of significant results, are discussed and refuted. It is shown that Signorini should be rightly credited for being among the first scholars aware of the central problem of non-linear elasticity: the determination of the general form of the elastic potential. [ABSTRACT FROM AUTHOR]

We analyze small amplitude shearing motions superimposed onto a harmonic extension of a string made of an isotropic rate-type viscoelastic material. In particular, we consider also the case in which the harmonic extension is perturbed by adding stochastic noise. We study the onset of resonance as a function of the characteristic parameters both in the absence and in the presence of noise. In the first case, we use the renormalization group (RG) method, while in the second case, we make use of a " deterministic" approach that implies replacing the noise by high-frequency excitations. The main conclusion of our investigation is that the substitution of a classical elastic rope with a viscoelastic stress–relaxing string allows us to reproduce the parametric resonance phenomena observed by Melde. The presence of noise can change the well-known 2:1 resonance. In particular, the 2:1 resonance is reduced proportionally to the square of the amplitude ratio, frequency of the noise, and to the ratio between the characteristic viscous stress and the elastic stress. [ABSTRACT FROM AUTHOR]

Due to the formal resemblance of some models for the Cauchy stress tensor of elastic solids and viscous fluids, some classes of exact solutions for the equations governing the flows in Navier–Stokes fluids have been generalized to linear and nonlinear elastodynamics. In this paper, we study the conditions under which two special classes of generalized Beltrami flows, the vortices in lattice form and Kelvin's cat's eye solutions, are solutions of the equations governing the motions in a linearly elastic transversely isotropic solid. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We propose paradigmatic examples to show how material damage phenomena can be efficiently described as a solid-solid phase transition. Starting from the pioneering work of J.L. Ericksen (J. Elast. 5(3):191–201, 1975) and the extensions of R.L. Fosdick and other authors to three-dimensional non linear elasticity, we describe the insurgence of damage as a hard → soft transition between two material states (damage and undamaged) characterized by two different energy wells. We consider the two separate constitutive assumptions of a simple Neo-Hookean type damageable material and a more complex microstructure inspired damageable Gent type material with variable limit threshold of the first invariant. In both cases we study two different deformation shear classes, one homogeneous and the other one inhomogeneous and obtain fully analytic description of the system damage response under cyclic loading. The considered constitutive assumptions and deformation classes are aimed at attaining fully analytic descriptions. On the other hand, we remark that the proposed, Griffith type, variational approach of damage, based on two different energy density functions for the damaged and undamaged material phases, and a resulting non (rank-one) convex energy, can be extended to systems with more complex energy functions, possibly with a larger number of wells representing an increasing degree of damage. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We place the Ogden model of rubber elasticity, published in Proceedings of the Royal Society 50 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'. [ABSTRACT FROM AUTHOR]

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'. [ABSTRACT FROM AUTHOR]

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 (Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci. 243:251–288, 1951) 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 κ over the whole range of deformation, a robustness which is lacking in other plots and spaces. [ABSTRACT FROM AUTHOR]

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 ) – inflation (λ or 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study the oscillations of a spherical shell of a rate-type viscoelastic solid subject to a pressure difference between the inner and the outer surface. The stable equilibrium configurations, in the class of spherically symmetric deformations, correspond to the minima of the elastic energy function. Numerical simulations indicate that the way in which the equilibrium state is reached, strongly depends on the material parameters. [ABSTRACT FROM AUTHOR]

Third- and fourth-order weakly nonlinear theories of elasticity are widely used by applied mathematicians, physicists and acousticians. Although their introduction is traced back to a paper by Signorini dated 1930, some aspects related to the rigorous mathematical derivation of the strain energy for quasi-incompressible isotropic elastic solids are not satisfactorily clear. In this paper we address these aspects and use our results to discuss the incompressible limit of some fourth-order nonlinear constitutive parameters. [ABSTRACT FROM AUTHOR]

Structure evolution during deformation of isotropic auxetic foams is investigated by simple compression experiments. It is shown that the main feature observed in the experimental data can be accurately described by an isotropic hyperelastic model with an Ogden-type strain energy function. The model can be easily implemented in a finite element code and, hence, can be used to simulate foams undergoing complex structural deformations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider nonclassical symmetries of partial differential equations (PDEs) in 1+1 dimensions. Given a nth‐order ordinary differential equation G=0 in the unknown w=w(z) we are able to find the most general scalar PDE of a given order which can be reduced via a nonclassical symmetry to G=0. [ABSTRACT FROM AUTHOR]

We provide some spatial estimates for the nonlinear partial differential equation governing anti-plane motions in a nonlinear viscoelastic theory of Kelvin–Voigt type when the viscosity is a function of the strain rate. The spatial estimates we prove are an alternative of Phragmen–Lindelöf type. These estimates are possible when a precise balance between the elastic and viscoelastic nonlinearities holds. [ABSTRACT FROM AUTHOR]

Given a local one parameter Lie group of transformations G, we determine the most general scalar partial differential equation in (1 + 1)‐independent variables of a given order admitting G as nonclassical symmetry in the Bluman and Cole sense. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study elastic shear waves of small but finite amplitude, composed of an anti-plane shear motion and a general in-plane motion. We use a multiple scales expansion to derive an asymptotic system of coupled nonlinear equations describing their propagation in all isotropic incompressible nonlinear elastic solids, generalizing the scalar Zabolotskaya equation of compressible nonlinear elasticity. We show that for a general isotropic incompressible solid, the coupling between anti-plane and in-planemotions cannot be undone and thus conclude that linear polarization is impossible for general nonlinear twodimensional shear waves. We then use the equations to study the evolution of a nonlinear Gaussian beam in a soft solid: we show that a pure (linearly polarized) shear beam source generates only odd harmonics, but that introducing a slight in-plane noise in the source signal leads to a second harmonic, of the same magnitude as the fifth harmonic, a phenomenon recently observed experimentally. Finally, we present examples of some special shear motions with linear polarization. [ABSTRACT FROM AUTHOR]

We study the propagation of linearly polarized transverse waves in a pre-strained incompressible isotropic elastic solid. Both finite and small-butfinite amplitude waves are examined. Irrespective of the magnitude of the wave amplitude, these waves may propagate only if the (unit) normal to the plane spanned by the directions of propagation and polarization is a principal direction of the left Cauchy-Green deformation tensor associated with the pre-strained state. A rigorous asymptotic analysis of the equations governing the propagation of waves of small but finite amplitude reveals that the time scale over which the nonlinear effects become significant depends on the direction along which the wave travels. Moreover, we design theoretically an experimental procedure to determine the Landau constants of the fourth-order weakly nonlinear theory of elasticity. [ABSTRACT FROM AUTHOR]

We consider the superimposition of two shear waves on a pseudo-plane motion of the first kind with elliptical streamlines. If the shear waves satisfy some special assumptions it is possible to establish a recurrence relation among the Rivlin-Ericksen tensors associated with the flow at hand. This remarkable kinematical result allows to determine new exact solutions for a large class of materials and to generalize some well known solutions modelling special flows (such as the celebrated Berker’s solution for a Navier-Stokes fluid in an orthogonal rheometer). [ABSTRACT FROM AUTHOR]

A nonlinear elastodynamic system is investigated which is descriptive of transverse wave propagation in an isotropic, incompressible, hyperelastic material subject to body forces associated with a nonlinear substrate potential. Notably, by introducing an appropriate ansatz, both cnoidal and gausson-type exact solutions are isolated. [ABSTRACT FROM AUTHOR]

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A great variety of models can describe the nonlinear response of rubber to uniaxial tension. Yet an indepth understanding of the successive stages of large extension is still lacking. We show that the response can be broken down in three steps, which we delineate by relying on a simple formatting of the data, the so-called Mooney plot transform. First, the small-to-moderate regime, where the polymeric chains unfold easily and the Mooney plot is almost linear. Second, the strain-hardening regime, where blobs of bundled chains unfold to stiffen the response in correspondence to the 'upturn' of the Mooney plot. Third, the limiting-chain regime, with a sharp stiffening occurring as the chains extend towards their limit. We provide strain-energy functions with terms accounting for each stage that (i) give an accurate local and then global fitting of the data; (ii) are consistent with weak nonlinear elasticity theory and (iii) can be interpreted in the framework of statistical mechanics.We apply our method to Treloar's classical experimental data and also to some more recent data. Our method not only provides models that describe the experimental data with a very low quantitative relative error, but also shows that the theory of nonlinear elasticity is much more robust that seemed at first sight. [ABSTRACT FROM AUTHOR]

Small- and large-amplitude oscillatory shear tests are widely used by experimentalists to measure, respectively, linear and nonlinear properties of viscoelastic materials. These tests are based on the quasi-static approximation according to which the strain varies sinusoidally with time after a number of loading cycles. Despite the extensive use of the quasi-static approximation in solid mechanics, few attempts have been made to justify rigorously such an approximation. The validity of the quasi-static approximation is studied here in the framework of the Mooney-Rivlin Kelvin-Voigt viscoelastic model by solving the equations of motion analytically. For a general nonlinear model, the quasi-static approximation is instead derived by means of a perturbation analysis. [ABSTRACT FROM AUTHOR]

In mean-field theory, the non-local state of fluid molecules can be taken into account using a statistical method. The molecular model combined with a density expansion in Taylor series of the fourth order yields an internal energy value relevant to the fourth-gradient model, and the equation of isothermal motions takes then density's spatial derivatives into account for waves travelling in both liquid and vapour phases. At equilibrium, the equation of the density profile across interfaces is more precise than the Cahn and Hilliard equation, and near the fluid's critical point, the density profile verifies an Extended Fisher-Kolmogorov equation, allowing kinks, which converges towards the Cahn-Hillard equation when approaching the critical point. Nonetheless, we also get pulse waves oscillating and generating critical opalescence. [ABSTRACT FROM AUTHOR]