Incompressible Nonlinearly Elastic Solids Research Articles

Necking of thin-walled cylinders via bifurcation of incompressible nonlinear elastic solids.

Necking localization under quasi-static uniaxial tension is experimentally observed in ductile thin-walled cylindrical tubes, made of soft polypropylene. Necking nucleates at multiple locations along the tube and spreads throughout, involving the occurrence of higher-order modes, evidencing trefoil and fourth-foiled (but rarely even fifth-foiled) shaped cross-sections. No evidence of such a complicated necking occurrence and growth was found in other ductile materials for thin-walled cylinders under quasi-static loading. With the aim of modelling this phenomenon, as well as all other possible bifurcations, a two-dimensional formulation is introduced, in which only the mean surface of the tube is considered, paralleling the celebrated Flügge 's treatment of axially-compressed cylindrical shells. This treatment is extended to include tension and a broad class of nonlinear-hyperelastic constitutive law for the material, which is also assumed to be incompressible. The theoretical framework leads to a number of new results, not only for tensile axial force (where necking is modelled and, as a particular case, the classic Considère formula is shown to represent the limit of very thin tubes), but also for compressive force, providing closed-form formulae for wrinkling (showing that a direct application of the Flügge equation can be incorrect) and for Euler buckling. It is shown that the J2-deformation theory of plasticity (the simplest constitutive assumption to mimic through nonlinear elasticity the plastic branch of a material) captures multiple necking and occurrence of higher-order modes, so that experiments are explained. The presented results are important for several applications, ranging from aerospace and automotive engineering to the vascular mechanobiology, where a thin-walled tube (for instance an artery, or a catheter, or a stent) may become unstable not only in compression, but also in tension.

Generalization of the Zabolotskaya equation to all incompressible isotropic elastic solids.

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-plane motions cannot be undone and thus conclude that linear polarization is impossible for general nonlinear two-dimensional 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.

Simple shearing of rubberlike materials filled with parallel fibers at large strain

The purpose of this paper is to investigate the mechanical behavior of fiber-reinforced incompressible nonlinearly elastic solids under large simple shear deformations. Two different rubberlike materials, with distinct properties of adhesion, were reinforced by a single family of parallel fibers of nylon. Fibers of nylon 6 monofilament fishing line with diameters of 0.25, 0.45, and 0.80 mm were used. Fiber-reinforced specimens were tested under monotonic load at constant temperature, and values of amount of shear were obtained by the digital image correlation method. A phenomenological constitutive model is proposed to predict the mechanical behavior of the transversely isotropic materials. The proposed model takes into account the shear and stretch in the fiber, and fiber-matrix iterations. These iterations are related to the quality of the fiber-matrix bonding and fibers pullout. The obtained results can be useful in understanding the mechanical behavior of fiber-reinforced rubberlike solids and fibrous soft tissues.

Distribution Regularities of Shear Deformations in Incompressible Nonlinear Elastic Solids

The process of shock disturbances in incompressible nonlinear elastic medium is considered. For the case of one-dimensional deformation the two types of plane shock waves are shown, which carry shift deformations in the medium. The properties of these waves are described, their propagation velocities are calculated. The solution of the boundary problem of the interaction of two plane waves with different shear polarization is presented to demonstrate ways of using the obtained results.

Non-principal Rayleigh waves in deformed transversely isotropic incompressible non-linearly elastic solids

Non-principal Rayleigh waves in deformed transversely isotropic incompressible non-linearly elastic solids

Inhomogeneous shear of orthotropic incompressible non-linearly elastic solids: Singular solutions and biomechanical interpretation

The paper presents a detailed study of rectilinear shear deformation in the framework of orthotropic nonlinear elasticity, under Dirichlet and mixed boundary conditions. Here the shear takes place in a slab made of a soft matrix reinforced with two families of extensible fibers, along the bisectrix of the angle between the two privileged directions aligned with the fibers. An analytic approach coupled to a careful computational treatment reveal that if the two families of parallel fibers are mechanically equivalent, then only smooth solutions are possible, whereas if the mechanical differences between the two families of fibers are pronounced, then strain singularities may develop. For the standard reinforcing orthotropic model, it is possible to determine the precise conditions for the existence of singular solutions. For an orthotropic constitutive law used for artery modeling, the existence of singular solutions can have repercussions in biomechanical applications.

The rectilinear shear of fiber-reinforced incompressible non-linearly elastic solids

In this paper we study the problem of rectilinear shear of a slab of transversely isotropic incompressible non-linearly elastic material. In particular, the material under consideration is a base neo-Hookean model augmented with a function that accounts for the existence of a unidirectional reinforcement. The slab is of infinite length in two dimensions and finite thickness in the other one and is clamped to two rigid plates. Closed form analytic solutions are found for this problem. It is shown that, depending on the reinforcement strength and the fiber orientation in the undeformed configuration, weak solutions, i.e. solutions for which the smoothness required by the differential equations is relaxed, are to be expected. These solutions give rise to fiber kinking. It is shown that: (i) both sides of the kink involve fiber contraction; (ii) a suitable intermediate deformation between the two conjoined kink deformation states is non-elliptic.

Mechanical response of neo-Hookean fiber reinforced incompressible nonlinearly elastic solids

The mechanical behavior of an incompressible neo-Hookean material, directionally reinforced by neo-Hookean fibers, is examined under homogeneous deformations. A composite model for this transversely isotropic material is developed based on a multiplicative decomposition of the deformation gradient which considers interaction between the fiber and the matrix. The so-called standard reinforcing model exhibits non-monotonic behavior in compression. The present composites-based approach leads to a modification of the standard reinforcing model in which monotonic behavior in compression is observed. This stems from the micromechanical basis of the model in which the fiber is treated as a neo-Hookean material. The conditions for loss of monotonicity and positivity in the stress-shear behavior in off-axis simple 2D shear are also obtained.

Remarks on cavity formation in fiber-reinforced incompressible non-linearly elastic solids

In this paper we focus on cavity formation in fiber-reinforced incompressible non-linearly elastic solids. In particular, the material under consideration is a base neo-Hookean augmented by a function that accounts for the existence of a unidirectional reinforcing. This function characterizes the anisotropy of the material and is referred to as reinforcing model. Previous works has dealt with the analysis of a specific reinforcing model, the so called standard reinforcing model, that is a quadratic function that depends only on the fiber reinforcement stretch. Here, two different reinforcing models are examined: a power law that depends only on the fiber stretch and a quadratic function that depends simultaneously on the fiber stretch and fiber shearing. Closed form analytic solutions are found for the classical problem of cavity formation in a sphere under uniform radial tensile dead-load with the fiber in the radial direction. It is shown that for some of the new reinforcing models under study the cavitation instabilities obtained in previous works for the standard reinforcing model are not possible.

A new constitutive theory for fiber-reinforced incompressible nonlinearly elastic solids

We consider an incompressible nonlinearly elastic material in which a matrix is reinforced by strong fibers, for example fibers of nylon or carbon aligned in one family of curves in a rubber matrix. Rather than adopting the constraint of fiber inextensibility as has been previously assumed in the literature, here we develop a theory of fiber-reinforced materials based on the less restrictive idea of limiting fiber extensibility. The motivation for such an approach is provided by recent research on limiting chain extensibility models for rubber. Thus the basic idea of the present paper is simple: we adapt the limiting chain extensibility concept to limiting fiber extensibility so that the usual inextensibility constraint traditionally used is replaced by a unilateral constraint. We use a strain-energy density composed with two terms, the first being associated with the isotropic matrix or base material and the second reflecting the transversely isotropic character of the material due to the uniaxial reinforcement introduced by the fibers. We consider a base neo-Hookean model plus a special term that takes into account the limiting extensibility in the fiber direction. Thus our model introduces an additional parameter, namely that associated with limiting extensibility in the fiber direction, over previously investigated models. The aim of this paper is to investigate the mathematical and mechanical feasibility of this new model and to examine the role played by the extensibility parameter. We examine the response of the proposed models in some basic homogeneous deformations and compare this response to those of standard models for fiber reinforced rubber materials. The role of the strain-stiffening of the fibers in the new models is examined. The enhanced stability of the new models is then illustrated by investigation of cavitation instabilities. One of the motivations for the work is to apply the model to the biomechanics of soft tissues and the potential merits of the proposed models for this purpose are briefly discussed.

On tensile instabilities and ellipticity loss in fiber-reinforced incompressible non-linearly elastic solids

Loss of ellipticity and associated failure in fiber-reinforced non-linearly elastic solids is examined for uniaxial plane deformations. We consider separately fiber reinforcement that either endows the material with additional stiffness only in the fiber direction or introduces additional stiffness under shear deformations. In the first case it is shown that loss of ellipticity under tensile loading in the fiber direction corresponds to a turning point of the nominal stress and requires concavity of the Cauchy stress–stretch curve. For the second example loss of ellipticity occurs after the nominal stress maximum and prior to a turning point of the Cauchy stress.

Mechanical response of fiber-reinforced incompressible non-linearly elastic solids

The mechanical response of some fiber-reinforced incompressible non-linearly elastic solids is examined under homogeneous deformation. In particular, the materials under consideration are neo-Hookean models augmented with a function that accounts for the existence of a unidirectional reinforcement. This function endows the material with its anisotropic character and is referred to as a reinforcing model. The nature of the anisotropy considered has a particular influence on the shear response of the material, in contrast to previous analyses in which the reinforcing model was taken to depend only on the stretch in the fiber direction.

Remarks on the behavior of simple directionally reinforced incompressible nonlinearly elastic solids

The effect of directional reinforcing in generating qualitative changes in the mechanical response of a base neo-Hookean material is examined in the context of homogenous deformation. Single axis reinforcing giving transverse isotropy is the major focus, in which case a standard reinforcing model is characterized by a single constitutive reinforcing parameter. Various qualitative changes in the mechanical response ensue as the reinforcing parameter increases from the zero-value associated with neo-Hookean response. These include (in order): the existence of a limiting contractive stretch for transverse-axis tensile load; loss of monotonicity in off-axis simple shear; loss of monotonicity in on-axis compression; loss of positivity in the stress-shear product in off-axis simple shear; and loss of monotonicity for plane strain in on-axis compression. The qualitative changes in the simple shear response are associated with stretch relaxation in the reinforcing direction due to finite rotation.

Cavity Formation at the Center of a Composite Incompressible Nonlinearly Elastic Sphere

In this paper, the effect of material inhomogeneity on void formation and growth in incompressible nonlinearly elastic solids is examined. A bifurcation problem is considered for a solid composite sphere composed of two arbitrary homogeneous isotropic incompressible elastic materials perfectly bonded across a spherical interface. Under a uniform radial tensile dead load, a branch of radially symmetric configurations involving a traction-free internal cavity centered at the origin bifurcates from the undeformed configuration. In contrast to the situation for a homogeneous neo-Hookean sphere, bifurcation here may occur either locally to the right or to the left. In the latter case, the cavity has finite radius on first appearance. This discontinuous change in stable equilibrium configurations is reminiscent of the snap-through buckling phenomenon observed in certain structural mechanics problems. Explicit conditions determining the type of bifurcation are established for the general composite sphere. An analysis of the stress distribution is carried out and the effect of cavitation at the center on possible interface debonding is explored for the special case when the constituent materials are both neo-Hookean. It is shown that, in a quasi-static loading process, cavitation has the effect of preventing debonding at the interface.

Void nucleation in tensile dead-loading of a composite incompressible nonlinearly elastic sphere

In this paper, the effect of material inhomogeneity on void formation and growth in incompressible nonlinearly elastic solids is examined. A bifurcation problem is considered for a solid composite sphere composed of two neo-Hookean materials perfectly bonded across a spherical interface. Under a uniform radial tensile dead-load, a branch of radially symmetric configurations involving a traction-free internal cavity bifurcates from the underformed configuration. Such a configuration is the only stable solution for sufficiently large loads. In contrast to the situation for a homogeneous neo-Hookean sphere, bifurcation here may occur either locally to the right orto the left. In the latter case, the cavity has finite radius on first appearance. This discontinuous change in stable equilibrium configurations is reminiscent of the snap-through buckling phenomenon observed in certain structural mechanics problems. Since this paper was written, the authors have carried out further analysis of the class of problems of concern here [11]. In particular the stress distribution in the composite neo-Hookean sphere has been described in [11].

Penalty function formulations for incompressible nonlinear elastostatics

The analysis of incompressible nonlinear elastic solids is considered by a penalty function approach. In particular the necessary compatibility conditions for developing penalty forms for isotropic nonlinear elasticity are addressed, and the commonly used form of strain energy functional leading to a penalty formulation of the incompressibility constraint extended. The Mooney-Rivlin model is used to show how previous developments can lead to physically meaningless situations. For a certain class of incompressible materials, which includes the important Mooney-Rivlin model, a new and particularly simple formulation is proposed.

Numerical modelling of nonlinear behaviour of soft biological materials

This paper deals with the formulation of rigorous numerical methods for the stress analysis, at finite strains, of soft biological materials that can be characterized as incompressible nonlinear elastic solids.

Plane elastic standing waves of finite amplitude

A plane standing wave solution is obtained for homogeneous, isotropic, incompressible nonlinearly elastic solids. The motion describes the nonlinear interaction of two oppositely propagating, circularly polarized waves. It is used to obtain exact steady state solutions for nonlinear vibrations of a plate and for reflection and transmission of finite amplitude, circularly polarized waves at a plane interface.

Non-linear deformations of a mixture of two isotropic elastic solids

The consequences of the entropy production inequality are investigated for a mixture of two incompressible isotropic non-linear elastic solids. Some isothermal elastostatic problems involving large deformations of such a mixture are shown to be maintainable by surface tractions alone and are solved for a general form of theHelmholtz free energy function.

On controllable deformations in a mixture of two non‐linear elastic solids

Recently there has been much interest in theories describing the behaviour of a mixture of two or more continua, which are allowed to diffuse through each other and which interact mechanically and thermally. In the present note we consider such a mixture consisting of two isotropic incompressible non-linear elastic solids; physically this may describe a composite of two rubber-like materials. Such a mixture possesses internal friction between the constituents when deformed (called diffusive resistance or diffusive force), which acts as a damping mechanism. This damping mechanism has a static part and a dynamic part, and we here show that the exact static solutions which exist for an isotropic incompressible non-linear elastic solid are such that the damping mechanism is expressible as a gradient, and hence are also controllable for such a mixture. We also discuss the corresponding dynamic solutions. From a practical viewpoint, some experiments carried out on vulcanized synthetic rubber-like materials may involve a mixture rather than a pure solid.