Some quasi-static problems for a viscoelastic half-space

Some problems for a viscoelastic half-space are solved in the case of noncommutative operators. A solution of the equilibrium equation analogous to the Boussinesq-Papkovich solution is constructed. The problem of a normal pressure acting on the boundary of a viscoelastic half-space is solved. Two forms of this solution are obtained and both are used in the following problems, the problem of a concentrated load moving over the boundary of a half-space and the problem of a circular rigid stamp. The case of periodic motion of a periodic load is investigated with reference to the example of motion in a circle. At constant Poisson's ratio the solution of the problem of a stamp can be used for determining the creep or relaxation function.Mekhanika Polimerov, Vol. 2, No. 3, pp. 392–402, 1966Presented 12 November 1965 at the Riga Conference on Polymer Mechanics.     

Some optimization problems for elastic bodies

Two optimization problems for finite deformations of an elastic body are formulated. In the first problem, the motion that gives minimum value (maximum value) to a certain functional is chosen from a single family of controllable motions. The necessary conditions for optimality are stated for one specific problem of this type. In the second problem, the minimizing (maximizing) motion is chosen from the class of all possible motions connecting two configurations of the body. The necessary conditions for the optimality of singular solutions are obtained for one specific example of this type.This work was supported by the Pokrajinska Zajednica za Naucni Rad, Novi Sad, Yugoslavia.     

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.     

Some problems concerning centrally symmetric convex bodies

The conjecture that among convex bodies Q in Rⁿ, with a center of symmetry at the origin, for which , the value of is a maximum when Q is the layer between two hyperplanes, is proved for n=2 and n=3. Various approaches to the problem are discussed as well as related unsolved problems. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 45, pp. 75–82, 1974.     

Some key problems on finite semigroups

The paper starts from a historical perspective of the theory of finite semigroups, namely of its most developed part, the theory of pseudovarieties and its connections with the theories of finite automata and rational languages. It goes on to review some of the most important developments and to present some of the key problems that have guided the theory for the past 35 years. Several open problems that may in turn play a significant role in years to come are also discussed. May 31, 2001     

On the deformation of an extended viscoelastic rod subject to finite deformations

The deformation of a polymer material subjected to tensile strain is considered with due allowance for physically linear creep at finite deformations. The deformation process is only stable in time up to a certain value of the load; under heavier loads the deformation is unstable. An expression is derived for the critical time.Institute of the Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 6, pp. 1130–1133, November–December, 1972.     

On fracture of viscoelastic bodies

On the basis of a thermodynamic variational principle, a criterion for quasistatic crack growth in a viscoelastic solid is deduced: the change in the sum of the scattering and the rate of decrease of elastic energy equals the increment in surface dissipation. The appropriate invariant contour integral is found. The theory proposed is suitable for cracks of any shape for any loading path on viscoelastic solids. Examples are considered, including crack growth with localized viscous dissipation.     

Some geometric properties of metric ultraproducts of finite simple groups

In this article we prove some previously announced results about metric ultraproducts of finite simple groups. We show that any non-discrete metric ultraproduct of alternating or special linear groups is a geodesic metric space. For more general non-discrete metric ultraproducts of finite simple groups, we are able to establish path-connectedness. As expected, these global properties reflect asymptotic properties of various families of finite simple groups.     

Some enumeration problems for matrices over a finite field

The probability that the product of l square matrices of size n over a finite field with q elements will be nilpotent is shown to be 1-[(qⁿ-1)/qⁿ]^l. Analogous, if less elegant, results are obtained for the product to be idempotent.     

Some equilibrium finite element methods for two-dimensional elasticity problems

Summary We consider some equilibrium finite element methods for two-dimensional elasticity problems. The stresses and the displacements are approximated by using piecewise linear functions. We establishL ₂-estimates of orderO(h ²) for both stresses and displacements.     

Some inverse problems for a viscoelastic medium with a physically non-linear inclusion

A plane finite viscoelastic domain with a physically non-linear inclusion of arbitrary form is considered. The problem of finding those loads which, acting on the outer boundary of the domain, are such that they produce a specified uniform stress—strain state in the inclusion, is solved. Examples, in particular, of the optimal deformation and fracture of the inclusion under creep conditions, are considered.     

Modeling of viscoelastic contact-impact problems

An accurate finite element (FE) model for analyzing the response of viscoelastic structure under low-velocity impact is presented. Generalized standard linear solid (Wiechert) model is adopted to simulate the internal damping of the structure, because its capability of describing both creep and relaxation phenomena adequately. Newmark time integration scheme is proposed to transfer the problem into a static one for each time increment. The incremental convex programming method is modified to accommodate viscoelastic dynamic-contact problems. The Lagrange multiplier technique is selected to incorporate the contact condition. One, two and three-dimensional finite element model is presented to compare between the elastic and viscoelastic materials.     

Error estimators and the adaptive finite element method on large strain deformation problems

We generalize the well‐known residual‐based error estimator in linear elasticity to the case of non‐linear deformation problems based on large strain and demonstrate its use in adaptive mesh control. Copyright © 2009 John Wiley & Sons, Ltd.