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.     

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.     

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.     

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.     

Instabilities and loss of ellipticity in fiber-reinforced compressible non-linearly elastic solids under plane deformation

In a recent paper we examined the loss of ellipticity and its interpretation in terms of fiber kinking and other instability phenomena in respect of a fiber-reinforced incompressible elastic material. Here we provide a corresponding analysis for fiber-reinforced compressible elastic materials. The analysis concerns a material model which consists of an isotropic base material augmented by a reinforcement dependent on the fiber direction. The assessment of loss of ellipticity can be cast in terms of the eigenvalues of the acoustic tensors associated with the isotropic and anisotropic parts of the strain-energy function. For the anisotropic part, two different reinforcing models are examined and it is shown that, depending on the choice of model and whether the fiber is under compression or extension, loss of ellipticity may be associated with, in particular, a weak surface of discontinuity normal to or parallel to the deformed fiber direction or at an intermediate angle. Under compression the associated failure interpretations include fiber kinking and fiber splitting, while under extension fiber de-bonding and matrix failure are included.     

Shock waves in incompressible elastic solids

Summary The thermodynamic theory of shock waves in incompressible elastic solids is reviewed, and the Hugoniot relation and the propagation condition for the shock speed are derived. Expanding the equations, for weak shock waves, in powers of the shock strength some well-known results of gasdynamics are generalized to the dynamics of shock waves in incompressible elastic media.

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Zusammenfassung Die thermodynamische Theorie der Stoßwellen in inkompressiblen elastischen Körpern wird zusammenfassend dargestellt, die Hugoniot-Relation und die Ausbreitungsbedingung für die Stoßgeschwindigkeit werden abgeleitet. Durch Reihenentwicklung nach Potenzen der Stoßstärke werden für schwache Stoßwellen einige bekannte Ergebnisse der Gasdynamik für die Dynamik der Stoßwellen in inkompressiblen elastischen Medien verallgemeinert.

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With 2 figures     

Finite extension and torsion of fiber-reinforced non-linearly elastic circular cylinders

In the context of the theory of non-linear elasticity for rubber-like materials, the problem of finite extension and torsion of a circular bar or tube has been widely investigated. More recently, this problem has attracted considerable attention in studies on the biomechanics of soft tissues and has been applied, for example, to examine the mechanical behavior of passive papillary muscles of the heart. A recent study in non-linear elasticity was concerned specifically with the effects of strain-stiffening on the response of solid circular cylinders in the combined deformation of torsion superimposed on axial extension. The cylinders are composed of incompressible isotropic non-linearly elastic materials that undergo severe strain-stiffening in the stress–stretch response. For two specific material models that reflect limiting chain extensibility at the molecular level, it was shown that, in the absence of an additional axial force, a transition value γ=γ_t of the axial stretch exists such that for γ<γ_t, the stretched cylinder tends to elongate on twisting whereas for γ>γ_t, the stretched cylinder tends to shorten on twisting. These results are in sharp contrast with those for classical models for rubber such as the Mooney–Rivlin (and neo-Hookean) models that predict that the stretched circular cylinder always tends to further elongate on twisting. Here we investigate similar issues for fiber-reinforced transversely isotropic circular cylinders. We consider a class of incompressible anisotropic materials with strain-energy densities that are of logarithmic form in the anisotropic invariant. These models reflect limited fiber extensibility and in the biomechanics context model the stretch induced strain-stiffening of collagen fibers on loading. They have been shown to model the mechanical behavior of fiber-reinforced rubber and many fibrous soft biological tissues. The consideration of anisotropy leads to a more elaborate mechanical response than was found for isotropic strain-stiffening materials. The results obtained here have important implications for extension–torsion tests for fiber-reinforced materials, for example in the development of accurate extension–torsion test protocols for determination of material properties of soft tissues.     

Fully non-linear wave models in fiber-reinforced anisotropic incompressible hyperelastic solids

Many composite materials, including biological tissues, are modeled as non-linear elastic materials reinforced with elastic fibers. In the current paper, the full set of dynamic equations for finite deformations of incompressible hyperelastic solids containing a single fiber family are considered. Finite-amplitude wave propagation ansätze compatible with the incompressibility condition are employed for a generic fiber family orientation. Corresponding non-linear and linear wave equations are derived. It is shown that for a certain class of constitutive relations, the fiber contribution vanishes when the displacement is independent of the fiber direction.Point symmetries of the derived wave models are classified with respect to the material parameters and the angle between the fibers and the wave propagation direction. For planar shear waves in materials with a strong fiber contribution, a special wave propagation direction is found for which the non-linear wave equations admit an additional symmetry group. Examples of exact time-dependent solutions are provided in several physical situations, including the evolution of pre-strained configurations and traveling waves.     

Plane-strain buckling of cracks in incompressible elastic solids

The buckling of a crack in an incompressible elastic solid subjected to a crack-parallel compression is studied by using a small-deformation-superposed-on-large-deformation analysis. It is found that for a general incompressible material there exists at least one and at most a finite number of buckling loads. For a Mooney material, a unique buckling load corresponding to a crack-parallel stretch ratio of 0.544 is found to exist.Supported by U.S. Army Research Office-Durham under Grant DAAG-29-76-G-0272.     

Two-dimensional stress wave analysis in incompressible elastic solids

Two-dimensional stress waves in a general incompressible elastic solid are investigated. First, basic equations for simple waves and shock waves are presented for a general strain energy function. Then the characteristic wave speeds and the associated characteristic vectors are deduced. It is shown that there usually exist two simple waves and two shock waves. Finally, two examples are given for the case of plane strain deformation and antiplane strain deformation, respectively. It is proved that, in the case of plane strain deformation the oblique reflection problem of a plane shock is not solvable in general.     

Compressibility effects for nearly incompressible elastic solids

By using the theory of small deformations superposed on large compressibility effects for nearly incompressible materials can be considered. A procedure suggested by Truesdell is outlined and used for the problems of straightening, stretching and shearing of an annular wedge and the telescopic shear of a cylindrical tube.     

Dynamic universal solutions for fiber-reinforced incompressible isotropic elastic materials

We consider the effect of a fiber-reinforcement on the dynamic universality of the following families of motions: bending and shearing of a rectangular block; straightening and shearing of a sector of a circular tube; inflation, eversion, extension, bending and shearing of a sector of a circular tube; inflation, extension, bending and azimuthal shearing of a sector of a circular tube.

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Résumé Nous avons considéré l'effet d'un renforcement par fibre sur l'universalité dynamique des familles de mouvement suivant: courbage et cisaillement d'un bloc rectangulaire; redressement et cisaillement d'un secteur de tube circulaire; gonflement, retournement, allongement, courbage et cisaillement d'un secteur de tube circulaire; gonflement, allongement, courbage et cisaillement azimuthal d'un secteur de tube circulaire.

     

Controllable states of stress for incompressible elastic solids

It is shown that the equilibrium states of Cauchy stress which can exist, in the absence of body force, in every incompressible, homogeneous, isotropic, elastic solid whose deviatoric stress range allows them, must have uniform deviatoric stress invariants. There is at least one such non-uniform stress state. The related problem for incompressible non-Newtonian fluids is also discussed.     

Effects of pre-stress on crack-tip fields in elastic,incompressible solids

A closed-form asymptotic solution is provided for velocity fields and the nominal stress rates near the tip of a stationary crack in a homogeneously pre-stressed configuration of a nonlinear elastic, incompressible material. In particular, a biaxial pre-stress is assumed with stress axes parallel and orthogonal to the crack faces. Two boundary conditions are considered on the crack faces, namely a constant pressure or a constant dead loading, both preserving an homogeneous ground state. Starting from this configuration, small superimposed Mode I or Mode II deformations are solved, in the framework of Biot's incremental theory of elasticity. In this way a definition of an incremental stress intensity factor is introduced, slightly different for pressure or dead loading conditions on crack faces. Specific examples are finally developed for various hyperelastic materials, including the J₂-deformation theory of plasticity. The presence of pre-stress is shown to strongly influence the angular variation of the asymptotic crack-tip fields, even if the nominal stress rate displays a square root singularity as in the infinitesimal theory. Relationships between the solution with shear band formation at the crack tip and instability of the crack surfaces are given in evidence.     

The dynamic response of an incompressible non-linearly elastic membrane tube subjected to a dynamic extension

The dynamic response of an isotropic hyperelastic membrane tube, subjected to a dynamic extension at its one end, is studied. In the first part of the paper, an asymptotic expansion technique is used to derive a non-linear membrane theory for finite axially symmetric dynamic deformations of incompressible non-linearly elastic circular cylindrical tubes by starting from the three-dimensional elasticity theory. The equations governing dynamic axially symmetric deformations of the membrane tube are obtained for an arbitrary form of the strain-energy function. In the second part of the paper, finite amplitude wave propagation in an incompressible hyperelastic membrane tube is considered when one end is fixed and the other is subjected to a suddenly applied dynamic extension. A Godunov-type finite volume method is used to solve numerically the corresponding problem. Numerical results are given for the Mooney-Rivlin incompressible material. The question how the present numerical results are related to those obtained in the literature is discussed.     

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.