Elastoplastic torsion of a cylindrical rod for finite deformations

A solution of the problem of the torsion of a cylindrical rod was obtained in /1/ for a general, isotropic, incompressible elastic material. The present paper gives an analytical solution of the elastoplastic torsion problem for finite deformations, written in terms of quadratures of elliptic functions. The non-linear kinematics of elastoplastic deformation is introduced into the defining equations with the help of a multiplicative decomposition of the deformation gradient into elastic and plastic components /2, 3/. The elastic deformation and rate of plastic deformation are related to the state of stress of the body, in accordance with the defining Mooney-Rivlin equations /4/ and the law of flow for finite deformations associated with the Tresca yield condition /5/. A non-linear first-order partial differential equation and the initial data at the elastoplastic boundary are obtained in order to determine the angle of rotation within the plastic zone of the basis formed from the eigenvectors of the stress tensor, relative to the radial direction. The integration of the resulting equation is reduced to determining the general integral of the Ricatti equation with right-hand side determined from the angular velocity of flow of the material within the plastic zone. It is shown that neglecting the finiteness of the deformation leads to too high an estimate of the rigidity of the rod.     

Some problems of simple finite deformation of viscoelastic bodies

Boltzmann's superposition principle is extended to large strains in the case where the principal strain axes do not change with time and are identically directed at all points of the body. The relations obtained are confirmed by experiment. On the basis of these relations the author examines the problem of large strains of a heated viscoelastic cylinder subjected to the action of time-dependent internal and external pressures, and the analogous problem in the presence of a reinforcing cylindrical shell. The solution of the solving nonlinear integral equation of the latter problem is unique and is obtained in the form of a convergent infinite series.Mekhanika Polimerov, Vol. 2, No. 4, pp. 508–518, 1966     

On inhomogeneous deformations in finite thermoelasticity

For thermoelastic materials with thermal conductivity whichdepends only upon the temperature we present an interestinganalogy with the theory of laminated bodies. Using this factwe compute several families of exact solutions.     

On the plane deformation of fibre-reinforced viscoelastic plates

The realization method of elastic solutions is used to solve the problem of bending of a viscoelastic plate reinforced unidirectionally by elastic fibres. Numerical computations are carried out for three kinds of external load. The plane deformation of this plate is discussed.     

On an analogy between plane finite elastic deformations and certain magnetohydrodynamic flows

Summary The governing non-linear coupled partial differential equations for certain steady magnetohydrodynamic flows are shown to be similar in form to those arising for plane strain finite elastic deformations of a neo-Hookean or Mooney material. Based on this analogy new results are obtained for the magnetohydrodynamic system. Firstly a restricted form of Adkins' reciprocal theorem is given and secondly two new classes of exact solutions are obtained. One of these includes as a special case a known class of spiral flows for geometries wherein the magnetic field and fluid velocity are everywhere inclined at a constant angle.

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Zusammenfassung Es wird gezeigt, daß die Bewegungsgleichungen für gewisse magnetohydrodynamische Flüsse ähnliche Form haben wie die Gleichungen, die im Zusammenhang mit endlicher elastischer ebener Deformation von Mooney Material auftreten. Aufgrund dieser Analogie erhält man neue Ergebnisse für das magnetohydrodynamische System. Zunächst wird eine beschränkte Form von Adkins' Reziprozitätssatz gegeben. Dann werden zwei neue Familien von exakten Lösungen abgeleitet. Eine von diesen enthält als Sonderfall eine bekannte Klasse von spiralförmigen Lösungen für Situationen, in denen das Magnetfeld und die Geschwindigkeit isoklin sind.

     

Small finite deformations of an amorphous polymer material

Small finite deformations of a polymer material are considered. It is assumed that the volume of the specimen is conserved during deformation and that the strains are affine; furthermore, the equilibrium distance between network nodes is not equal to zero in the specimen, and the unit cell of the network is cubic. Various types of deformation are considered. The expressions obtained are compared with the results of the theory of high elasticity and the experimental data.Branch Institute of Chemical Physics, AS USSR, Moscow. Translated from Mekhanika Polimerov, Vol. 4, No. 6, pp. 997–1001, November–December, 1968.     

A variational framework for nonlinear viscoelastic models in finite deformation regime

This work presents a general framework for constitutive viscoelastic models in the finite deformation regime. The approach is qualified as variational since the constitutive updates consist of a minimization problem within each load increment. The set of internal variables is strain-based and uses a multiplicative decomposition of strain in elastic and viscous components. Spectral decomposition is explored in order to accommodate, into analytically tractable expressions, a wide set of specific models. Moreover, it is shown that, through appropriate choices of the constitutive potentials, the proposed formulation is able to reproduce results obtained elsewhere in the literature. Finally, numerical examples are included to illustrate the characteristics of the present formulation.     

On the antiplane shear deformation of finite wedges

Antiplane shear deformation of finite wedges is considered under different boundary conditions. First, the assertions and results of a recent paper, namely Chue and Liu [C.H. Chue, W.J. Liu, Comments on “Analysis of an isotropic finite wedge under antiplane deformation”, Int. J. Solids Struct. 41 (2004) 5023–5034] are invalidated. Then, closed form solutions are extracted for the stress distribution in the wedge. These closed forms have the advantages of showing the possible geometric stress singularity as well as the load singularity explicitly, in addition to the continuity or discontinuity as well as the convergence of the results in the entire region. Finally, the stress intensity factors are extracted in the special case of a circular shaft containing an edge crack under different boundary conditions.     

Grapho-analytical method of calculating the deformation of a viscoelastic rod in axial tension under nonstationary temperature conditions

The strains in the material are calculated on the basis of a compound mechanical model consisting of a system of individual elastico-viscous models of the Maxwell type. This model makes it possible to allow for the time dependence of the temperatures and stresses in the cross sections of bodies of simple shape—rod, plate, thin-walled shell—and to calculate the strains under nonstationary temperature conditions from the creep curves for reinforced plastics with allowance for the previous loading history both under a constant load and for an arbitrary loading law.Mekhanika Polimerov, Vol. 2, No. 3, pp. 413–420, 1966     

Stability of a growing viscoelastic rod subjected to ageing

The stability of a compressed growing rod of viscoelastic material that possesses the property of ageing /1/ is investigated. In conformity with the Chatayev definition of the stability of dynamic systems and the Lyapunov method described in /2/, stability conditions are obtained for a rod growing during a finite time interval, and in finite and semi-infinite time intervals. Some results of a numerical analysis of the behaviour of such a rod are presented in /3/.     

Numerical solution of the impact problem for a rigid mass and a viscoelastic rod of finite length

The problem of the propagation and interaction of longitudinal waves in a rod struck by a rigid mass is considered. A numerical computer solution is obtained for the case when the mechanical behavior of the rod material is expressed by a rectangular relaxation spectrum. A numerical method of integral transform inversion is developed and applied to problems of polymer dynamics.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 450–456, May–June, 1971.     

Kinematics of finite elastoplastic deformations

In this paper we review various approaches to the decomposition of total strains into elastic and nonelastic (plastic) components in the multiplicative representation of the deformation gradient tensor. We briefly describe the kinematics of finite deformations and arbitrary plastic flows. We show that products of principal values of distortion tensors for elastic and plastic deformations define principal values of the distortion tensor for total deformations. We describe two groups of methods for decomposing deformations and their rates into elastic and nonelastic components. The methods of the first group additively decompose specially built tensors defined in a common basis (initial, current, or “intermediate”). The second group implies a certain relation connecting tensors that describe elastic and plastic deformations. We adduce an example of constructing constitutive relations for elastoplastic continuums at large deformations from thermodynamic equations.     

Existence theory to the equations in viscoelasticity at finite deformations

In this paper we prove under the assumption of small initial data the global existence of a classical solution to the equations in viscoelasticity, associated with a free damping boundary condition. We also show that if we choose the initial data large enough, blow up will occur in finite time. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

On extended fundamental classes over a finite field

A partition over a finite field is defined which is an extension of the partition defined previously to minimize the number of the fundamental sets required to carry out the additions over the field. Solutions of some trinomial polynomial equations over finite field will be discussed.