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Under given prestress conditions, solitary waves in fluid-filled elastic tubes are confined to a rather narrow set of combinations of the background fluid velocity and the wave speed. This set, which we call the domain of existence, is bounded by several curves that represent various physical and mathematical restrictions. Remarkably, these restrictions can be cast as purely algebraic conditions to be imposed upon the governing system of differential equations. Paramount among the physical restrictions are the avoidance of wrinkles and the self-impenetrability of the wave profile. In particular, the existence of a critical wave speed of impending wrinkling, independent of the background fluid velocity, is established rigorously.

The non-homogeneous, anisotropic material properties, and triphasic nature of articular cartilage enables diarthrodial joints to withstand large and complex physiological loading conditions. To develop biomaterials that provide similar functional properties as those found in articular cartilage, it is vital to have knowledge of the strain distributions in cartilage for a large range of loading conditions. Applied stress vs. strain properties of articular cartilage have been measured primarily for static conditions, but the dynamic properties are thought to be more relevant for joint function and cartilage biosynthesis. Furthermore, the dynamic stress-strain properties are expected to vary significantly from those obtained for static, steady-state conditions. Here, we present a method for the determination of axial strain fields throughout the depth of articular cartilage for static loading conditions and dynamic conditions performed at different loading rates. For the conditions tested here, the strain distributions throughout the cartilage depth were more uniform for the dynamic than the static loading conditions, and more uniform for high compared to low strain rates.

Starting from the observation that the growth of a body breaks the time translation symmetry of the body’s dynamics, we determine a scalar field, called internal time, that defines an indicator of the intrinsic time scale of the growth-related body’s structural evolution. By recasting the theory of growth for monophasic media within a variational framework, we obtain the internal time as the solution of a partial differential equation descending from Noether’s Theorem. We do this by considering two approaches, one formulated in terms of internal variables and one adopting the concept of augmented kinematics.

This study focusses on the remodelling of biological tissues in the framework of the theory of material uniformity. A constitutive evolution model is introduced, including fibre recruitment and reorientation, and subjected to the entropy inequality, which enforces the Second Principle of Thermodynamics. The model is applied to a numerical example describing a pressurised fibre-reinforced cylinder, roughly representing an artery, and is able to capture the major characteristics of remodelling in arteries, as reported in the literature.

Here, an asymmetric double-layer membrane has been designed and fabricated by electrospinning as a tool for a potential wound healing application. A hydrophobic layer has been produced by using a polyurethane-polycaprolactone (PU-PCL) copolymer and loaded with the antibacterial ciprofloxacin whereas an ion responsive hydrophilic layer has been produced by using an octyl derivative of gellan gum (GG-C8) and polyvinyl alcohol (PVA) and loaded with the growth factor FGF-2. This study investigated how the properties of this asymmetric membrane loaded with actives, were influenced by the ionotropic crosslinking of the hydrophilic layer. In particular, the treatment in DPBS and the crosslinking in CaCl2 0.1 or 1 M of the hydrophilic layer affected the release profile of the bioactive molecules allowing to modulate both the antimicrobial effect, as assayed by logarithmic reduction of the Staphylococcus aureus viable count, and the chemoattractant properties on NIH 3 T3 cell line, as assayed by scratch test and cell chemoattraction assay. © 2021 Elsevier B.V.

A systematic representation of the fourth-order magnetostriction tensor for all crystal classes is presented, based on the formalism proposed by Walpole ( Proc R Soc Lond Ser A 1984; 391: 149–179). This representation allows the general, unconstrained case, as well as the case in which the magnetostrictive strain is assumed to be isochoric, to be studied. The knowledge of the fourth-order magnetostriction tensor enables the stress-free magnetostrictive strain tensor as well as the scalar strain in a given direction to be calculated.

The aim of this work is to propose a recruitment and damage constitutive model for collagen fibres in soft biological tissues. Similarly to other published models, our model employs probability distribution functions in order to capture the progressive recruitment and damage of fibrils in a collagen fibre. We rigorously investigate the continuum mechanical treatment of recruitment and damage using a multiplicative decomposition of the deformation gradient. Our proposed model stems from the correction, generalisation and extension to damage of the recruitment model proposed by Martufi and Gasser (2011, J. Biomech. , 44, 2544–2550). We demonstrate that the generalised model is equivalent to the recruitment and damage model proposed by Hurschler et al. (1997, J. Biomech. Eng. , 119, 392–399). Finally, we explore the sensitivity of the proposed model to the parameters describing recruitment and damage, implement the model into Finite Elements and show an example of application, which gives good agreement with published experimental data.

Building on the celebrated works by Toupin (1963) on non-linear elastic dielectric materials and by Noll (1967) on the theory of material uniformity, we use the method of linear perturbations to obtain a generalised form of the equations of piezoelectricity in the linear regime, i.e., below the ferroelectric threshold. More specifically, in their most general form, the piezoelectric constitutive equations that we obtain include an anelastic strain and an anelastic polarisation, both arising from the intrinsic inhomogeneity of the body and according to the theory of material uniformity. These are distinct from the irreversible strain and polarisation that occur at values of stress and electric field above the ferroelectric threshold. Some of the known standard forms of the piezoelectricity equations are retrieved as particular cases.

The influence of an electric field on the propagation of axially symmetric solitary waves in a fluid-filled thin-walled tube made of a dielectric elastomer is analysed. The electric field is applied by means of soft flexible electrodes lining the internal and external surfaces of the tube and connected to a power source with a given constant difference of potential. For a specific hyperelastic constitutive law, numerical results are obtained and discussed.

The flux of a certain extensive physical quantity across a surface is often represented by the integral over the surface of the component of a pseudo-vector normal to the surface. A pseudo-vector is in fact a possible representation of a second-order differential form, i.e. a skew-symmetric second-order covariant tensor, which follows the regular transformation laws of tensors. However, because of the skew-symmetry of differential forms, the associated pseudo-vector follows a transformation law that is different from that of proper vectors, and is named after the Italian mathematical physicist Gabrio Piola (1794–1850). In this work, we employ the methods of Differential Geometry and the representation in terms of differential forms to demonstrate how the flux of an extensive quantity transforms from the spatial to the material point of view. After an introduction to the theory of differential forms, their transformation laws, and their role in integration theory, we apply them to the case of first-order transport laws such as Darcy’s law and Ohm’s law.

With a universally accepted abuse of terminology, materials having much larger stiffness for volumetric than for shear deformations are called incompressible. This work proposes two approaches for the evaluation of the correct form of the linear elasticity tensor of so-called incompressible materials, both stemming from non-linear theory. In the approach of strict incompressibility, one imposes the kinematical constraint of isochoric deformation. In the approach of quasi-incompressibility, which is often employed to enforce incompressibility in numerical applications such as the Finite Element Method, one instead assumes a decoupled form of the elastic potential (or strain energy), which is written as the sum of a function of the volumetric deformation only and a function of the distortional deformation only, and then imposes that the bulk modulus be much larger than all other moduli. The conditions which the elasticity tensor has to obey for both strict incompressibility and quasi-incompressibility have been derived, regardless of the material symmetry. The representation of the linear elasticity tensor for the quasi-incompressible case differs from that of the strictly incompressible case by one parameter, which can be conveniently chosen to be the bulk modulus. Some important symmetries have been studied in detail, showing that the linear elasticity tensors for the cases of isotropy, transverse isotropy and orthotropy are characterised by one, three and six independent parameters, respectively, for the case of strict incompressibility, and two, four and seven independent parameters, respectively, for the case of quasi-incompressibility, as opposed to the two, five and nine parameters, respectively, of the general compressible case.

This work is aimed at emphasising the relationship between metric and deformation, under the light of a novel formalism for the Polar Decomposition Theorem. All results are first presented in the classical formalism of Cauchy’s celebrated theorem, and then in the proposed alternative formalism. Although the latter requires a little more work to be established, it allows for directly defining all strain tensors as “covariant”, i.e. with both feet being covectors. Emphasis is also placed on how, in the absence of the metric structure, the available mathematical tools are restricted to the deformation gradient alone. Along with these main results, and in the didactical intention that permeates this work, several hints are given, which could be useful in teaching Continuum Mechanics, e.g. the rigorous definition of the determinant of the deformation gradient in Riemannian manifolds, and a caveat on the definition of the spatial Hencky logarithmic strain. The setting is that of modern Continuum Mechanics, based on the description given by Differential Geometry in terms of differentiable manifolds. However, passing to the simpler case of affine spaces takes almost no effort, paying attention to keeping the distinction between vectors and covectors, and therefore allowing the matrices representing the metric tensors to differ from the unit matrix.

Under suitable conditions, the motion of a fluid in a porous medium can be studied by assuming the validity of Darcy’s law. Since many biological tissues can be thought of as porous media, Darcy’s law is invoked in several biomechanical contexts, like the transport of the chemical species needed for the metabolism of tissue cells. Although Darcy’s law supplies physically sound results in many circumstances, there may be cases in which the dynamic behaviour of a biological fluid deviates from the Darcian one. The scope of this work is to analyse some possible consequences of such deviations, with emphasis on the fluid velocity and pressure, which, in turn, influence the health and correct functioning of the tissue cells. In particular, our study addresses the flow of an interstitial fluid through a fibre-reinforced tissue, in which the fibres are oriented statistically. We take articular cartilage as a representative tissue of this type, and study the deviation from Darcy’s law known as “Forchheimer’s correction”. Moreover, we introduce two models of tissue permeability, which lead to discrepant results when the fluid velocity is described by Darcy’s law. We show, however, that the discrepancies in the description of the flow can be reduced if Forchheimer’s correction is applied.

This work aims at obtaining a covariant representation of the elasticity tensor of a hyperelastic material when the elastic strain energy potential is written employing the volumetric–distortional decomposition of the deformation. This requires the careful definition of some fundamental fourth-order tensors: the identity, the spherical operator, and the deviatoric operator, which appear in the material and spatial expressions of the elasticity tensor. These operators can be defined in the spatial or the material setting and admit pulled-back and pushed-forward forms, respectively. These forms are intimately related to the pulled-back and pushed-forward metric tensors, and are somewhat awkward to derive in Cartesian coordinates, because of the loss of the distinction between a vector space and its dual, and therefore between objects having contravariant and covariant components, which obey to different transformation laws. The relationship between the deformation and the various forms of the identity, spherical, and deviatoric operators can be entirely clarified only within a covariant theory, where the central role played by the spatial and material metric tensors, and their pulled-back and pushed-forward counterparts, which are deformation tensors, can be emphasised.

We represent a biological tissue by a multi-constituent, fiber-reinforced material, in which we consider two phases: fluid, and a fiber-reinforced solid. Among the various processes that may occur in such a system, we study growth, mass transfer, and remodeling. To us, mass transfer is the reciprocal exchange of constituents between the phases, growth is the variation of mass of the system in response to interactions with the surrounding environment, and remodeling is the evolution of its internal structure. We embrace the theory according to which these events, which lead to a structural reorganization of the system and anelastic deformations, require the introduction of balance laws, which complete the physical picture offered by the standard ones. The former are said to be non-standard. Our purposes are to determine the rates of anelastic deformation related to mass transfer and growth, and to study fiber reorientation in the case of a statistical distribution of fibers. In particular, we discuss the use of the non-standard balance laws in modeling transfer of mass, and compare our results with a formulation in which such balance laws are not introduced.

The aim of this work is to propose a model of the chemical stress arising in a polyelectrolyte. To do so, we consider the latter as a continuum and we suppose that the mechanical properties of the polyelectrolyte derive from its underlying microstructure. Calling upon principles of theories of generalized continua, we write the macroscopic constitutive assignments of the model in terms of variables that are characteristic of the network of macromolecules, such as orientation, solid fraction and number density of the macromolecules.

In this work, a porous material is represented as a composite made of an isotropic matrix and spheroidal inclusions with zero stiffness, describing the pores. The shape of the pores ranges from flat to spherical to fibre-like, and their orientation is assumed to obey a given probability distribution function. As an example of straightforward physical interpretation, the isotropic case of randomly oriented voids-pores is studied as a function of the matrix Poisson's ratio and the porosity (void volumetric fraction). The results are in good agreement with published experimental data on porous metals, and with previous isotropic solutions in the literature.

The deformation gradient admits a multiplicative decomposition into a purely volumetric component and a purely distortional component. For a hyperelastic material, based on this decomposition, the elastic strain energy potential, the stress, and the elasticity tensor can be expressed in general as a function of both the volumetric deformation and the distortional deformation. However, the volumetric—distortional decomposition of deformation has often been employed in a fully decoupled form of the elastic strain energy potential, which is expressed as the sum of a term depending solely on the volumetric deformation and a term depending solely on the distortional deformation. This work has three main objectives. First, to derive the elasticity tensor in the general (non-decoupled) case, in its material, spatial, and linear forms; this is achieved by extensive use of fourth-order tensor algebra, and in particular of the properties of the so-called spherical operator, which is largely used, but very seldom given the dignity of being assigned a symbol and a name, in the literature. Second, to show that a fully decoupled potential gives rise to an elasticity tensor which may be inconsistent with its linearized counterpart, as some components of the linear elasticity tensor in general do not have a corresponding term in nonlinear decoupled elasticity tensor. Third, to obtain the conditions under which a purely hydrostatic stress causes a purely volumetric deformation, by means of the developed theory; the results show that this condition is satisfied if and only if the elastic potential is fully decoupled. While the whole approach is completely independent of the material symmetry, the cases of isotropy and transverse isotropy are shown as an example.

Biological tissues can be considered as composite materials comprised of a porous matrix filled with interstitial fluid and reinforced by impermeable collagen fibres. Motivated by studies on fluid flow in articular cartilage, we would like to quantify the undeformed configuration permeability of fibre-reinforced composite materials. If there is a sufficient scale separation between the internal structure of the porous matrix and the arrangement of the fibres, the matrix can be taken as a porous continuum at the fibre scale. In this case, the fibres can be treated as inclusions in a porous continuum, and the overall permeability of the composite can be evaluated using homogenisation procedures. For an isotropic homogeneous matrix, the symmetry of the system is governed by the orientation of the fibres. Here, we propose to retrieve the overall permeability through geometrical considerations and directional averaging methods. The special case of transverse isotropy is discussed in detail, with particular attention to the sub-cases of aligned fibres and fibres lying on a plane.

Biological materials can be regarded as composites with spheroidal and fibre-like inclusions, representing cells and collagen fibres, respectively. The orientation and arrangement of the inclusions in a biological tissue is crucial to the determination of the mechanical properties of the material. Furthermore, the reorientation and rearrangement of the inclusions due to the deformation and external forces is of primary interest when dealing with growth and remodelling. We propose to look at the presence of inclusions as a source of internal hyperstaticity: when the material undergoes deformation, a generic inclusion is drifted by the deformation, but at the same time it “feels” the stress field and tends to carry a portion of stress proportional to its stiffness relative to that of the surrounding matrix. With this assumption, we can extend the classical “drift” evolution law for the unit vector field, in order to take the hyperstaticity into account. This method might be used in the description of remodelling in disordered media, such as biological tissues, and may be extended to investigate the reorientation of preferred directions of micro-structural elements in media described with a continuum approach.

In the literature, the determination of global elastic properties of composites with ellipsoidal inclusions is based on the averaged stress, strain and elastic-energy fields (e.g. Compos. Sci. Technol. 27 (1986) 111). These are related to the local fields of the inclusion, the matrix, and the inclusion-matrix interface. In this study, we propose a method to obtain the global elastic properties of any transversely isotropic composite directly from the elastic properties of the matrix and the inclusions. Thus, it is not necessary to refer to the stress and strain applied globally or generated locally. The inclusions can have any transversely isotropic probability distribution of orientation. The problem is entirely geometrized and is treated in terms of averages of Walpole's (Adv. Appl. Mech. 21 (1981) 169) components of the fourth-order tensors describing the problem. We give a general numerical solution for any transversely isotropic statistical distribution of orientation, and also provide a validation of our method by applying it to some known cases and by retrieving the known exact solutions from the literature.

We present a mathematical model of structural reorganisation in a fibre-reinforced composite material in which the fibres are oriented statistically, i.e. obey a probability distribution of orientation. Such a composite material exemplifies a biological tissue (e.g. articular cartilage or a blood vessel) whose soft matrix is reinforced by collagen fibres. The structural reorganisation of the composite takes place as fibres reorient, in response to mechanical stimuli, in order to optimise the stress distribution in the tissue. Our mathematical model is based on the Principle of Virtual Powers and the study of dissipation. Besides incompressibility, our main hypothesis is that the composite is characterised by a probability density distribution that measures the probability of finding a family of fibres aligned along a given direction at a fixed material point. Under this assumption, we describe the reorientation of fibres as the evolution of the most probable direction along which the fibres are aligned. To test our theory, we compare our simulations of a benchmark problem with selected results taken from the literature.

Soft biological tissues with collagen reinforcement can be represented by a porous matrix saturated by a fluid and reinforced by a network of statistically oriented, impermeable fibres. This paper aims at determining the effect of the fibres on both the elastic properties and the permeability of the system, under large deformations, and represents the unification and generalisation of previous works in which the elasticity was studied for a pure solid, in the absence of the pore fluid, and the permeability was studied in the neighbourhood of the undeformed configuration. Throughout this work, the saturation constraint is assumed to hold, and the solid and fluid phases are assumed to be intrinsically incompressible. These hypotheses imply that the volumetric fraction of one of the two phases (e.g., the solid) is sufficient to determine the distribution of solid and fluid mass at every point of the homogenised porous medium. Overall incompressibility is achieved at compaction, i.e., when pores are closed and all the fluid has escaped. A new form of the elastic strain energy potential is proposed, based on the sum of a given “base” potential and a “correction” potential, function of the volumetric deformation, which serves solely to impose the incompressibility constraint at compaction. The large-strain overall permeability is obtained by employing a pull-back of the structure tensor to the reference configuration. This gives rise to an integral form of the permeability that needs to be calculated at each increment of deformation. The presentation is entirely covariant, so that general curvilinear coordinates can immediately be employed, if needed.

Confined compression is a common experimental technique aimed at gaining information on the properties of biphasic mixtures comprised of a solid saturated by a fluid, a typical example of which are soft hydrated biological tissues. When the material properties (elastic modulus, permeability) are assumed to be homogeneous, the governing equation in the axial displacement reduces to a Fourier equation which can be solved analytically. For the more realistic case of inhomogeneous material properties, the governing equation does not admit, in general, a solution in closed form. In this work, we propose a semi-analytical alternative to Finite Element analysis for the study of the confined compression of linearly elastic biphasic mixtures. The partial differential equation is discretised in the space variable and kept continuous in the time variable, by use of the Finite Difference Method, and the resulting system of ordinary differential equations is solved by means of the Laplace Transform method.

We demonstrate that for the class of anisotropic hyperelastic materials with stiffening behaviour (i.e., the stiffness increases for increasing strain), it is possible to find an approximation by means of the linear superposition of an anisotropic quadratic potential, generated by the true linear elasticity tensor of the target material, and a suitable correction potential that is isotropic and hyperelastic. The proposed method can be implemented into commercially available Finite Element software by use of featured options only. This approach is intended to provide the solution to a stress–strain problem, based entirely on energetic considerations, which ensures the convexity of the potentials, and provides a simple material characterisation procedure.

The anisotropic, non-linear elastic behavior of biological soft tissue is typically accounted for by the hypothesis of hyperelasticity, i.e., the existence of an elastic potential. Fung-type potentials, based on the exponential of a quadratic form in the components of the Green-Lagrange strain, have been widely used in soft tissue modeling, and have inspired potentials in which the exponential was replaced by other monotonically increasing functions. It has been shown that simple fitting of the parameters of a Fung-type potential to experimental stress-strain curves may lead to non-convexity, with undesirable effects on the reliability of the algorithms used in Finite Element simulations. In this paper, we prove that the necessary and sufficient condition for the strict convexity of a Fung-type potential is that the quadratic form in the exponential is positive definite. This result provides a clear physical meaning for the parameters featuring in the quadratic form, and their relationship with the small-strain elastic moduli. This consistency relationship must be respected in order to guarantee that the Fung-type potential correctly reduces to the quadratic potential of classic linear elasticity in the small-strain approximation. Furthermore, we show that, when the conditions of convexity and consistency with the linear theory are respected, Fung-type potentials become a one-parameter family, and we discuss the consequences of this result for when fitting experimental data.