Acceleration wave propagation in hyperelastic rods of variable cross-section

It is shown that when an acceleration wave propagates in hyperelastic rod with slowly varying cross-section, the transport equation for the wave intensity is a generalized Riccati equation. The three coefficients in the equation all depend on the material properties, but only the coefficient of the quadratic term is independent of the effect of area change. Three theorems are proved, based on the use of comparison equations, which establish that in general the acceleration wave intensity will become infinite (escape) after the wave has propagated only a finite distance along the rod. The existence of thresholds for the initial intensity are also established in certain cases, with their most notable property being that as the intial intensity decreases towards the threshold, so the distance the wave propagates to escape increases without bound.     

Exact travelling-wave solutions of an integrable equation arising in hyperelastic rods

In this paper, we study an integrable nonlinear evolution equation which arises in the context of nonlinear dispersive waves in hyperelastic rods. To consider bounded travelling-wave solutions, we conduct a phase plane analysis. A new feature is that there is a vertical singular line in the phase plane. By considering equilibrium points and the relative position of the singular line, we find that there are in total three types of phase planes. The trajectories which represent bounded travelling-wave solutions are studied one by one. In total, we find there are 12 types of bounded travelling waves, both supersonic and subsonic. While in literature solutions for only two types of travelling waves are known, here we provide explicit solution expressions for all 12 types of travelling waves. Also, it is noted for the first time that peakons can have applications in a real physical problem.     

Variational theory for spatial rods

The simplest theory of spatial rods is presented in a variational setting and certain necessary conditions for minimizers of the potential energy are derived. These include the Weierstrass and Legendre inequalities, which require that the vector describing curvature and twist belong to a domain of convexity of the strain energy function.     

A second order constitutive theory for hyperelastic materials

The second order constitutive equation for a hyperelastic material with arbitrary symmetry is derived. In developing a second order theory, it is necessary to be discriminating in the choice of measures of deformation. Here the derivation is done in terms of the Biot strain, which has a direct physical interpretation in that its eigenvalues are the principal extensions of the deformation. The constitutive equation is specialized for the cases of isotropy and transverse isotropy. The isotropic equation derived here is compared with equations obtained by other authors in terms of the displacement gradient and the Green strain.     

Remarks on the theory of rods

A rod theory, defined as a curve at every point of which is attached a rotation vector, is shown to be a special constrained case of a rod theory in which two deformable directors are attached to each point of a curve.

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Zusammenfassung Eine Theorie von Stäben, die als Kurve bezeichnet wird, an jedem Punkt deren ein Rotationsvektor angebracht ist, wird als besonderer Zwangsfall einer Theorie von Stäben dargestellt, in der zwei verformbare Direktoren an jeden Punkt einer Kurve angebracht werden.

     

DYNAMICAL FORMATION OF CAVITY FOR COMPOSED THERMAL HYPERELASTIC SPHERES IN NONUNIFORM TEMPERATURE FIELDS

Dynamical formation and growth of cavity in a sphere composed of two incompressible thermal-hyperelastic Gent-Thomas materials were discussed under the case of a non-uniform temperature field and the surface dead loading. The mathematical model was first presented based on the dynamical theory of finite deformations. An exact differential relation between the void radius and surface load was obtained by using the variable transformation method. By numerical computation, critical loads and cavitation growth curves were obtained for different temperatures. The influence of the temperature and material parameters of the composed sphere on the void formation and growth was considered and compared with those for static analysis. The results show that the cavity occurs suddenly with a finite radius and its evolvement with time displays a non-linear periodic vibration and that the critical load decreases with the increase of temperature and also the dynamical critical load is lower than the static critical load under the same conditions.     

A direct theory of affine rods

A direct theory of affine rods is developed from first principles. To concentrate on the central aspects of the model, we use an axiomatic format and tools from Lie group theory. To facilitate comparisons with other theories, we propose an identification procedure to derive the constitutive relations of the affine rod from those of a rod modeled as a three-dimensional body.     

On the theory of porous elastic rods

We consider the direct approach to the theory of rods, in which the thin body is modelled as a deformable curve with a triad of rigidly rotating orthonormal vectors attached to every material point. In this context, we employ the theory of elastic materials with voids to describe the mechanical behavior of porous rods. First, we derive the dynamical nonlinear field equations of the model. Then, in the framework of linear theory, we prove the uniqueness of the solution to the associated boundary-initial-value problem. We identify the relevant field quantities from the theory of directed curves by comparison with the three-dimensional equations of straight porous rods. Finally, for orthotropic and homogeneous rods, we determine the constitutive coefficients in terms of the three-dimensional elasticity constants by solving several problems in the two different approaches.     

A linear theory of straight elastic rods

This paper is based on the work of Green & Laws who have given a general thermodynamical theory of rods which is valid for any material. Here, starting with the general non-linear theory of elastic rods, we derive a linear theory allowing for thermal effects. The resulting free energy as a quadratic function of kinematic variables is restricted by certain symmetry conditions. The basic equations then separate into four groups, two for flexure, one for torsion and one for extension of the rod with temperature effects occurring only in the latter group. Wave propagation along an infinite rod is considered. There are two wave speeds for each type of flexure, two for torsion and three for isothermal extension and all wave speeds depend on the wave length.     

An exact solution in a nonlinear theory of rods

A three dimensional nonlinear equilibrium theory of elastic rods, applicable to large displacements and small strains, and accounting for extensibility and shear deformation is developed. Integrals of the governing equations are determined for the case of specified end force and moment. A class of solutions is obtained for an initially straight, untwisted rod and compared to the classical solution. The effects of extensibility and shear deformation are discussed.     

On universal deformations in analysis of Signorini’s nonlinear theory of hyperelastic media

It is shown that uniform compression/tension and simple shear as universal deformations are quite useful in studying Signorini’s nonlinear theory of hyperelastic materials. They make it possible to formulate restrictions for the elastic constants of the theory and to explain the Poynting effect __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 12, pp. 54–60, December 2007.     

Neutral nano-inhomogeneities in hyperelastic materials with a hyperelastic interface model

We examine the existence of neutral nano-inhomogeneities in a hyperelastic inhomogeneity-matrix system subjected to finite plane deformations when uniform (in-plane) external loading is imposed on the matrix. We incorporate nanoscale interface effects by representing the material interface as a separate hyperelastic membrane, perfectly bonded to the surrounding bulk material. We show that for any type of hyperelastic bulk material and practically any type of hyperelastic membrane representing the interface, neutral nano-inhomogeneities do exist but are necessarily circular in shape. We show further that the radius of the circular neutral nano-inhomogeneity is determined by the (uniform) external loading (which must be hydrostatic) and the respective strain energy density functions associated with the hyperelastic bulk and interface materials.