A contact problem for a viscoelastic plate and an elastic beam

Under consideration is the problem of contact of a viscoelastic plate with an elastic beam. To characterize the viscoelastic deformation of the plate, the hereditary integrals are used. The differential formulation of the problem with the conditions in the form of a system of equalities and inequalities in the domain of possible contact is presented, and its equivalence to a variational inequality is proved. The unique solvability of the problem is proved as well as the existence of the time derivative of the solution. A limit problem is also considered as the bending rigidity of the plate tends to infinity.     

Quasistatic Frictional Problems for Elastic and Viscoelastic Materials

We consider two quasistatic problems which describe the frictional contact between a deformable body and an obstacle, the so-called foundation. In the first problem the body is assumed to have a viscoelastic behavior, while in the other it is assumed to be elastic. The frictional contact is modeled by a general velocity dependent dissipation functional. We derive weak formulations for the models and prove existence and uniqueness results. The proofs are based on the theory of evolution variational inequalities and fixed-point arguments. We also prove that the solution of the viscoelastic problem converges to the solution of the corresponding elastic problem, as the viscosity tensor converges to zero. Finally, we describe a number of concrete contact and friction conditions to which our results apply.     

On the accumulation of eigenvalues of operator pencils connected with the problem of vibrations in a viscoelastic rod

In this paper, we study the problem of the boundary accumulation of a discrete spectrum, which is essential for a boundary-value problem of fourth order arising in the theory of small transverse vibrations in an inhomogeneous viscoelastic rod (a Kelvin—Voigt body). We establish conditions for such an accumulation and its asymptotics, which are expressed in terms of the coefficients defining the problem posed by the differential expression. The results obtained are illustrated by numerical computation data.     

On the problem of a thin rigid inclusion embedded in a Maxwell material

We consider a plane viscoelastic body, composed of Maxwell material, with a crack and a thin rigid inclusion. The statement of the problem includes boundary conditions in the form of inequalities, together with an integral condition describing the equilibrium conditions of the inclusion. An equivalent variational statement is provided and used to prove the uniqueness of the problem’s solution. The analysis is carried out in respect of perfect and non-perfect bonding of the rigid inclusion. Additional smoothness properties of the solutions, namely the existence of the time derivative, are also established.     

On the three-dimensional undulation instability of a rectangular viscoelastic composite plate in biaxial compression

Within the frame work of the second version of small precritical deformation in the three-dimensional linearized theory of stability of deformable bodies (TDLTSDB), the undulation instability problem for a simply supported rectangular plate made of a viscoelastic composite material is investigated in biaxial compression in the plate plane. The corresponding boundary-value problem is solved by employing the Laplace transformation and the principle of correspondence. For comparison and estimation of the accuracy of results given by the TDLTSDB, the same problem is also solved by using various approximate plate theories. The viscoelasticity properties of the plate material are described by the Rabotnov fractional-exponential operator. The numerical results and their discussion are presented for the case where the plate is made of a multilayered viscoelastic composite material. In particular, the variation range of problem parameters is established for which it is necessary to investigate the undulation instability of the viscoelastic composite plate by using the TDLTSDB. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 45, No. 1, pp. 93–108, January–February, 2009.     

Blow-up of solutions for a viscoelastic equation with nonlinear damping

The initial boundary value problem for a viscoelastic equation with nonlinear damping in a bounded domain is considered. By modifying the method, which is put forward by Li, Tasi and Vitillaro, we sententiously proved that, under certain conditions, any solution blows up in finite time. The estimates of the life-span of solutions are also given. We generalize some earlier results concerning this equation.      

Existence of a solution to a dynamic unilateral contact problem for a cracked viscoelastic body

In this paper we study a dynamic unilateral contact problem with friction for a cracked viscoelastic body. The viscoelastic model is characterized by Kelvin–Voigt's law and a nonlocal friction law is investigated here. The existence of a solution to the problem is obtained by using a penalty method. Several estimates are obtained on the solution to the penalized problem, which enable us to pass to the limit by using compactness results. To cite this article: M. Cocou, G. Scarella, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Large deformations of thermorheologically simple bodies

The relation between the true stresses and true strains associated with the simple finite deformation of a viscoelastic body is modified for the case in which the properties of the material depend on temperature and the body is located in an inhomogeneous nonstationary temperature field. This modification is based on the time-temperature equivalence principle. The validity of this principle for the case in question is confirmed by the experimental data of Andrews, Hofman-Bang, and Tobolsky on the true stress relaxation at large deformations of a polyisobutylene specimen. The modified relations are used to solve the problem of the large deformations of a heated cylinder, made of a viscoelastic material whose properties depend on temperature, under the action of internal and external pressures that are functions of time. The case of polyisobutylene is analyzed as an example of a specific viscoelastic material. The analogous problem in the presence of a reinforcing cylindrical shell is also considered. In the latter problem the solution of the solving nonlinear integrodifferential equation is unique and is obtained in the form of an infinite convergent series.Mekhanika Polimerov, Vol. 3, No. 3, pp. 436–447, 1967     

Similarity representation of wave propagation in a nonlinear viscoelastic rod on a group theoretic basis

Similarity representation of wave propagation in a nonlinear viscoelastic rod, subjected to a velocity impact is constructed. By the use of a multiparameter dimensional group of transformations, under which the system of basic equations and auxiliary conditions are invariant, similarity transformations are obtained which are used to construct the similarity representation in the form of an ordinary nonlinear boundary value problem of the original system, consisting of a set of simultaneous nonlinear partial differential equations and the association boundary and initial conditions.     

On Solvability of Some Initial-Boundary Problems for Mathematical Models of the Motion of Nonlinearly Viscous and Viscoelastic Fluids

In the paper, the settings of initial-boundary and initial value problems arising in a number of models of movement of nonlinearly viscous or viscoelastic incompressible fluid are considered, and existence theorems for these problems are presented. In particular, the settings of initial-boundary value problems appearing in the regularized model of the movement of viscoelastic fluid with Jeffris constitutive relation are described. The theorems for the existence of weak and strong solutions for these problems in bounded domains are given. The initial value problem for a nonlinearly viscous fluid on the whole space is considered. The estimates on the right-hand side and initial conditions under which there exist local and global solutions of this problem are presented. The modification of Litvinov's model for laminar and turbulent flows with a memory is described. The existence theorem for weak solutions of initial-boundary value problem appearing in this model is given.     

Averaging of acoustic equation for partially perforated viscoelastic material with channels filled by a liquid

Acoustic equations for combined media consisting of partially perforated viscoelastic material and viscous incompressible liquid filling pores are considered. An averaged model is constructed for the model under consideration, and boundary conditions connecting equations of the obtained averaged model on the boundary between solid viscoelastic material and porous viscoelastic material filled by a viscous incompressible liquid are found. The convergence of limit problems to the solution of corresponding averaged problem with respect to the norm of the space L ² is proved.     

Analysis of a dynamic unilateral contact problem for a cracked viscoelastic body

This work deals with the mathematical analysis of a dynamic unilateral contact problem with friction for a cracked viscoelastic body. We consider here a Kelvin-Voigt viscoelastic material and a nonlocal friction law. To prove the existence of a solution to the unilateral problem with friction, an auxiliary penalized problem is studied. Several estimates on the penalized solutions are given, which enable us to pass to the limit by using compactness results.     

Thermal convection in a simple fluid with fading memory

The problem of thermal convection in a viscoelastic fluid with fading memory is studied. After describing what is meant by a fluid with fading memory, we establish existence and uniqueness for the linearized thermal convection written in an arbitrary bounded domain of the three-dimensional space. Finally, we derive conditions on the Rayleigh number, which guarantee the exponential decay in the linearized stability problem.     

Analysis of a dynamic unilateral contact problem for a cracked viscoelastic body

This work deals with the mathematical analysis of a dynamic unilateral contact problem with friction for a cracked viscoelastic body. We consider here a Kelvin-Voigt viscoelastic material and a nonlocal friction law. To prove the existence of a solution to the unilateral problem with friction, an auxiliary penalized problem is studied. Several estimates on the penalized solutions are given, which enable us to pass to the limit by using compactness results. Received: February 16, 2005     

Local Buckling around an Interfacial Crack in a Viscoelastic Sandwich Plate

Buckling around an interfacial crack in a clamped sandwich plate is studied. The layers of the plate are assumed to be linearly viscoelastic, isotropic, and homogeneous. The investigations are carried out within the framework of a piecewise homogeneous body model with the use of a three-dimensional linearized theory of stability. The corresponding boundary value problems are solved numerically by employing the FEM and the Laplace transform. The calculated critical times are presented for various problem parameters. In this case, the upper and lower layers are assumed to be viscoelastic and are described by Rabotnov operators, whereas the midlayer is regarded as purely elastic. The influence of rheological parameters on the critical time is also analyzed.     

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.     

Nonlinear evolution of the travelling waves at the surface of a thin viscoelastic falling film

The problem of hydrodynamic instability of a thin condensate viscoelastic liquid film flowing down on the outer surface of an axially moving vertical cylinder is investigated. In order to improve the accuracy of numerical results, the viscoelastic and heat transfer parameters have been included into the governing equations. Also, the analytical solutions are obtained by utilizing the long-wave perturbation method. The influence of some physical parameters is discussed in both linear and nonlinear steps of the problem. It has been revealed that the stability of the film flow is weakened when the radius of cylinder and the temperature difference are reduced. Moreover, it is found that the increment of down-moving motion of the cylinder can enhance the flow stability. Further, the thin film flow can be destabilized by the viscoelastic property. The results show that both supercritical stability and subcritical instability can take place within the film flow system given appropriate conditions. Moreover, the absence of Reynolds number leads to an obvious difference in the behavior of some physical parameters.     

A linearized theory of thermoviscoelasticity

A generalized linearized theory of thermoviscoelasticity, including the effect of heat formation, is presented. The linearized equations of motion, of state, and for the energy are given together with the linearized boundary conditions for large initial deformations. Attention is drawn to the fact that the equations which have been derived can be used for the solution of problems concerning the stability of viscoelastic bodies, the propagation of waves in viscoelastic materials which are subjected to deformation, and problems concerning the stress-deformed state of viscoelastic elements. The problem of the propagation of plane waves in viscoelastic materials which are subjected to deformation is considered as an example.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 2, pp. 214–221, March–April, 1972.     

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.     

一类非线性积分偏微分方程的初值问题

讨论初值问题 整体经典解的存在性。该问题来源于粘弹性力学。在关于已知函数的一些正则性假设和p'(s)≥c₁>0,|q'(s)|≤const,λ(0)<0,λ'(0)<λ2(0)的条件下,通过能量估计,证明了该问题整体经典解的存在性。