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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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.     

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.     

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.     

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.     

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.     

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.     

Non-singular terms for multiple cracks in anisotropic elastic solids

The non-singular and bounded terms for the stresses near the crack tip are investigated. This paper deals with the multiple crack problem for an infinite plate. The original problem is decomposed into three elementary subproblems: (1) the problem for remote uniform stress filed without cracks; (2) the single crack problem with traction applied along the crack face and (3) the problem for the influence of the other cracks. Several examples for collinear and non-collinear cracks are discussed and the results are shown.     

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.     

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.     

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.     

Plane-strain crack problems in microstructured solids governed by dipolar gradient elasticity

The present study aims at determining the elastic stress and displacement fields around the tips of a finite-length crack in a microstructured solid under remotely applied plane-strain loading (mode I and II cases). The material microstructure is modeled through the Toupin-Mindlin generalized continuum theory of dipolar gradient elasticity. According to this theory, the strain-energy density assumes the form of a positive-definite function of the strain tensor (as in classical elasticity) and the gradient of the strain tensor (additional term). A simple but yet rigorous version of the theory is employed here by considering an isotropic linear expression of the elastic strain-energy density that involves only three material constants (the two Lamé constants and the so-called gradient coefficient). First, a near-tip asymptotic solution is obtained by the Knein-Williams technique. Then, we attack the complete boundary value problem in an effort to obtain a full-field solution. Hypersingular integral equations with a cubic singularity are formulated with the aid of the Fourier transform. These equations are solved by analytical considerations on Hadamard finite-part integrals and a numerical treatment. The results show significant departure from the predictions of standard fracture mechanics. In view of these results, it seems that the classical theory of elasticity is inadequate to analyze crack problems in microstructured materials. Indeed, the present results indicate that the stress distribution ahead of the crack tip exhibits a local maximum that is bounded. Therefore, this maximum value may serve as a measure of the critical stress level at which further advancement of the crack may occur. Also, in the vicinity of the crack tip, the crack-face displacement closes more smoothly as compared to the standard result and the strain field is bounded. Finally, the J-integral (energy release rate) in gradient elasticity was evaluated. A decrease of its value is noticed in comparison with the classical theory. This shows that the gradient theory predicts a strengthening effect since a reduction of crack driving force takes place as the material microstructure becomes more pronounced.     

Minimum principles for incompressible viscoelastic solids

Summary Four minimum principles are established, governing the quasi-static deformations of linear, isotropic, incompressible viscoelastic solids.

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Sommario Si stabiliscono quattro principi di minimo che governano le deformazioni quasi statiche di solidi viscoelastici, lineari, isotropici e incompressibili.

     

The slowness surfaces of incompressible and nearly incompressible elastic materials

The slowness surface of a compressible elastic material has three sheets whilst that of an incompressible elastic material has only two sheets. The explanation for this qualitative difference is found to be that as the material approaches an incompressible limit the inmost sheet becomes a small sphere collapsing to the origin whilst the other two sheets tend to the two sheets of the limiting incompressible solid. The theory of nearly incompressible materials is developed here because of its important applications to rubberlike solids. Some results on the wave polarisations and on the convexity of the slowness surfaces are also given.     

Propagation of acceleration waves in incompressible saturated porous solids

Within the framework of the incompressible porous media model, the propagation properties of acceleration waves in liquid-filled porous solids is discussed. The incompressibility of the two constituents in the model forces the amplitudes of the longitudinal waves in the skeleton and in the liquid to satisfy a certain relation. The two propagation speeds are presented by examination for the existence of acceleration waves and only longitudinal and transverse waves are realizable in the incompressible two-phase porous materials.