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The authors present a note on this issue of the journal including fundamental contributions to the field of applied mathematics. They particularly thank Robin Knops for his fundamental contributions on the subject. Also, the editorial board of the journal is thankful to Nigel Scott, for having given them the opportunity to edit this special issue. They further thank Mike Hayes for writing the biographical notes.

Derives families of exact solutions to several boundary-value problem models of thermoelastic materials whose thermal conductivity depends only on the temperature, using an analogy of Wang's theory of laminated elastic bodies. Restricted Fourier law materials; Constitutive partial differential equation relating the heat flux vector to the temperature gradient; Field equations.

Mathematical modelling is a discipline pledged to identify problems, which may arise from virtually any branch of the human knowledge, and formalise them in the language of mathematics by developing suitable methodologies of investigation. To pursue its goals, modelling must build connections with other mathematical sciences and, in particular, with numerics. Three major examples of the efficiency of such combination are industrial mathematics, mathematical biology and biomechanics. At first sight, industrial mathematics is a branch of applied mathematics focusing on problems that come from industry and it aims at determining solutions relevant to manufacturing. Some relevant examples are petroleum engineering, hydrogeology, and the description of sand dynamics in the neighbourhood of railways in desert zones. On the other hand, the adoption of mathematics to formalise problems of biological relevance has attracted scientists working on population dynamics, epidemiology and related fields. Moreover, a strong impact has been given by the combination of modelling with the mechanics of biological tissues, thereby giving rise to biomechanics. Few examples in this respect are the mechanics of cell motion and migration, which relate to kinetic theories, the mechanics of the interactions between cells and the extracellular matrix, the conversion of mechanical signals into chemical stimuli, and "mathematical oncology". Since it is not possible to present a theoretical corpus of all that, the aim of the present special issue is to put together a list of outstanding scientific papers giving clear connections among nonlinear mechanics, industrial mathematics, biomathematics, biomechanics and kinetic theories, in different fields of interest. This special issue of IJNLM is the Festschrift celebrating the 60th birthday of Luigi Preziosi, whose research is a recognised example of how mechanics may be the fuel for interesting applied mathematics. [ABSTRACT FROM AUTHOR]

The aim of this note is to analyze the flow of a mixture of two immiscible fluids whose viscosity depends on pressure, generalizing the classical Buckley--Leverett model. Such extension is mainly motivated by the overwhelming evidence that many fluids have a viscosity that depends on pressure, a typical example being crude oils, where viscosity is extremely sensitive to pressure variations. We model the flow of two immiscible fluids through a channel neglecting nonlinear effects and assuming that the fluids are separated by a smooth interface. The solid matrix is treated as an impervious rigid body and a classical no-slip condition is imposed on its boundary. We end up with a system of equations for the saturation and for the pressure which reduces to the classical Buckley--Leverett equation when both viscosities do not depend on pressure. In doing so, we find that the relative permeabilities depend on the fluid pressure, and when the solid matrix is treated as rigid. Moreover, we study analytically the steady state problem for generic pressure-dependent viscosities and for the general case we provide some numerical simulations. [ABSTRACT FROM AUTHOR]

The theory of Lie remarkable equations, i.e. , differential equations characterized by their Lie point symmetries, is reviewed and applied to ordinary differential equations. In particular, we consider some relevant Lie algebras of vector fields on R k and characterize Lie remarkable equations admitted by the considered Lie algebras. [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]

We study the propagation of two-dimensional finite-amplitude shear waves in a nonlinear pre-strained incompressible solid, and derive several asymptotic amplitude equations in a simple, consistent and rigorous manner. The scalar Zabolotskaya (Z) equation is shown to be the asymptotic limit of the equations of motion for all elastic generalized neo-Hookean solids (with strain energy depending only on the first principal invariant of Cauchy-Green strain). However, we show that the Z equation cannot be a scalar equation for the propagation of two-dimensional shear waves in general elastic materials (with strain energy depending on the first and second principal invariants of strain). Then, we introduce dispersive and dissipative terms to deduce the scalar Kadomtsev- Petviashvili (KP), Zabolotskaya-Khokhlov (ZK) and Khokhlov-Zabolotskaya-Kuznetsov (KZK) equations of incompressible solid mechanics. [ABSTRACT FROM AUTHOR]

Abstract: Planar Kirchhoff elastic rods with non-linear constitutive relations are shown to admit traveling wave solutions with compact support. The existence of planar compact waves is a general property of all non-linearly elastic intrinsically straight rods, while intrinsically curved rods do not exhibit this type of behavior. [Copyright &y& Elsevier]

We investigate the linear stability of a unidirectional flow of a suspension (blood) modeling the latter as a inhomogeneous non-Newtonian fluid in which viscosity depends on both the red blood cell concentration (RBCs) and the shear rate. We consider small vessels like arteries terminal branches, arterioles or venules, where the RBCs do not distribute uniformly on the cross section. The stability analysis is performed applying the classical normal-mode linear analysis. The results obtained indicate that the flow is unconditionally unstable unless the inhomogeneity of the RBCs distribution is neglected. • We investigate the linear stability of a inhomogeneous fluid (blood). • The fluid is modeled as a power-law fluid. • The consistency index depends on the particles' concentration. • Linear stability is studied via Pseudo-spectral method. • We prove that the flow is unconditionally unstable. [ABSTRACT FROM AUTHOR]

We consider three-dimensional nonlinear viscoelastic models that account for both stress relaxation and creep/recovery phenomena. These models are based on different frame indifferent time derivatives: the Oldroyd (or upper-convected) derivative, the Jaumann (or co-rotational) derivative and the Cotter–Rivlin (or lower-convected) derivative. Under a simple tension creep process, these constitutive equations predict the same stress relaxation but lead to different situations. The models based on the Oldroyd and the lower-convected derivative require restrictions on the values of the material parameters as well as on the traction/compression stress. The model based on the Jaumann derivatives does not require any restriction. All the constitutive models examined are used to study the finite amplitude, horizontal oscillatory motion of a mass attached to a rate-type viscoelastic string. In this way we generalize the classical results by Beatty and Zhou (1991). [ABSTRACT FROM AUTHOR]

The continuous softening behavior of the brain tissue, i.e., the softening in the primary loading path with an increase in deformation, is modeled in this work as a state of hyperelasticity up to the onset of failure. That is, the softening behavior is captured via a core hyperelastic model without the addition of damage variables and/or functions. Examples of the application of the model will be provided to extant datasets of uniaxial tension and simple shear deformations, demonstrating the capability of the model to capture the whole-range deformation of the brain tissue specimens, including their softening behavior. Quantitative and qualitative comparisons with other models within the brain biomechanics literature will also be presented, showing the clear advantages of the current approach. The application of the model is then extended to capturing the rate-dependent softening behavior of the tissue by allowing the parameters of the core hyperelastic model to evolve, i.e., vary, with the deformation rate. It is shown that the model captures the rate-dependent and softening behaviors of the specimens favorably and also predicts the behavior at other rates. These results offer a clear set of advantages in favor of the considered modeling approach here for capturing the quasi-static and rate-dependent mechanical properties of the brain tissue, including its softening behavior, over the existing models in the literature, which at best may purport to capture only a reduced set of the foregoing behaviors, and with ill-posed effects., (Copyright © 2024 by ASME.)

A new modelling approach is employed in this work for application to the rate-dependent mechanical behaviour of the brain tissue, as an incompressible isotropic material. Extant datasets encompassing single- and multi-mode compression, tension and simple shear deformation(s) are considered, across a wide range of deformation rates from quasi-static to rates akin to blast loading conditions, in the order of 1000 s -1 . With a simple functional form and a reduced number of parameters, the model is shown to capture the considered rate-dependent behaviours favourably, including in both single- and multi-mode deformation fits, and over all range of deformation rates. The provided modelling results here are obtained from either first fitting the model to the quasi-static data, or/and predicting the behaviour at a different rate than those used for calibrating the model parameters. Given its simplicity, versatility, predictive capability and accuracy, the application of the utilised modelling framework in this work to the rate-dependent mechanical behaviour of the brain tissue is proposed., Competing Interests: Declaration of competing interest The authors have no competing interests to declare., (Copyright © 2024 The Authors. Published by Elsevier Ltd.. All rights reserved.)

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.

We study the propagation of linearly polarized transverse waves in a pre-strained incompressible isotropic elastic solid. Both finite and small-but-finite 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., Competing Interests: Authors declare no competing interests.

We consider a simple mesoscopic model of DNA in which the binding of the RNA polymerase enzyme molecule to the promoter sequence of the DNA is included through a substrate energy term modeling the enzymatic interaction with the DNA strands. We focus on the differential system for solitary waves and derive conditions--in terms of the model parameters--for the occurrence of the parametric resonance phenomenon. We find that what truly matters for parametric resonance is not the ratio between the strength of the stacking and the inter-strand forces but the ratio between the substrate and the inter-strands. On the basis of these results, the standard objection that longitudinal motion is negligible because of the second order seems to fail, suggesting that all the studies involving the longitudinal degree of freedom in DNA should be reconsidered when the interaction of the RNA polymerase with the DNA macromolecule is not neglected.

Collagen fibres play an important role in the mechanical behaviour of many soft tissues. Modelling of such tissues now often incorporates a collagen fibre distribution. However, the availability of accurate structural data has so far lagged behind the progress of anisotropic constitutive modelling. Here, an automated process is developed to identify the orientation of collagen fibres using inexpensive and relatively simple techniques. The method uses established histological techniques and an algorithm implemented in the MATLAB image processing toolbox. It takes an average of 15 s to evaluate one image, compared to several hours if assessed visually. The technique was applied to histological sections of human skin with different Langer line orientations and a definite correlation between the orientation of Langer lines and the preferred orientation of collagen fibres in the dermis (p < 0.001, R(2) = 0.95) was observed. The structural parameters of the Gasser-Ogden-Holzapfel (GOH) model were all successfully evaluated. The mean dispersion factor for the dermis was κ = 0.1404±0.0028. The constitutive parameters μ, k(1) and k(2) were evaluated through physically-based, least squares curve-fitting of experimental test data. The values found for μ, k(1) and k(2) were 0.2014 MPa, 243.6 and 0.1327, respectively. Finally, the above model was implemented in ABAQUS/Standard and a finite element (FE) computation was performed of uniaxial extension tests on human skin. It is expected that the results of this study will assist those wishing to model skin, and that the algorithm described will be of benefit to those who wish to evaluate the collagen dispersion of other soft tissues.

We propose a new elastic constitutive law for arterial tissue in which the limiting polymeric chain extensibility of both collagen and elastin fibres is accounted for. The elastic strain-energy function is separated additively into two parts: an isotropic contribution associated with the matrix (incorporating the elastin fibre network) and an anisotropic one associated with the collagen fibres. Information on the limiting extensibility in each case provides some mesoscopic input into the model. The (logarithm-based) model is compared with the Fung-Demiray exponential model and certain other recently proposed models. Some aspects of the elastic response under extension and inflation of a thin-walled circular cylindrical tube (the artery) are then examined and compared with the corresponding response of a rubber-like tube. We point out that our model, when both isotropic and anisotropic terms are included, can be developed to accommodate changing mechanical properties associated with degradation of the elastin and collagen by considering the material constants that define the limit of chain extensibility to evolve in time.