Abstract
This article deals with multi-operator boundary value problems in viscoelasticity, in particular problems concerning composite bodies which, being inhomogeneous as a whole, are composed of different homogenous viscoelastic materials. It is shown that by application of the Volterra principle, the solution of the viscoelastic problem for a piecewise-homogeneous body can be readily reduced to a purely mathematical problem of involving the construction linear operators which, in most cases, are very complicated functions of several independent operators characterizing the viscoelastic properties of the structural elements. The solutions to the specified class of problems are found using two effective methods: (1) a version of the approximate method of quasi-constant operators using partial approximations and (2) the method of constructing a solution as a series expansion in powers of the Il’yushin hereditary operators. This article presents a detailed description of the methods and gives several examples of their numerical evaluation.
Similar content being viewed by others
References
Rabotnov YuN (1977) Elements of hereditary mechanics of solids. Nauka, Moscow (in Russian)
Il’yushin AA (1968) Approximation method for calculation of structures in the framework of the linear theory of thermoviscoelasticity. Mech Polym 2: 210–221
Il’yushin AA, Pobedrya BE (1970) Fundamentals of the mathematical theory of thermoviscoelasticity. Nauka, Moscow (in Russian)
Pobedrya BE (1984) Mechanics of composite materials. Moscow State University Publishers, Moscow (in Russian)
Pobedrya BE (1981) Numerical methods in the theory of elasticity and plasticity. Moscow State University Publishers, Moscow (in Russian)
Maltsev LE (1984) Extension of the Il’yushin approximation method to a general case of anisotropic body. Mech Compos Mater 3: 417–425
Adamov AA, Matveenko VP, Trufanov NA, Shardakov IN (2003) Methods of applied viscoelasticity. Ural Branch of the Russian Academy of Sciences, Ekaterinburg (in Russian)
Shepery RA(1974) Viscoelastic behavior of composite materials. In: Broutman LJ, Krock RH (eds), Composite materials, vol 2. Sendeckyj GP (ed), Mechanics of composite materials. Academic Press, New York/London
Kaminskiy AA, Selivanov MF (2003) On one method of solving boundary value problems of the linear viscoelasticity theory for anisotropic composites. Int Appl Mech 39(11): 70–80
Sullivan JL (1995) Universal aspects of composite viscoelastic behavior. Polym Compos 16(1): 3–9
Svetashkov AA (2001) Evaluation of effective characteristics of inhomogeneous viscoelastic bodies. Comput Technol 6(1): 52–64
Malikov VG, Shashkov YuM (2002) The stress–strain state of a multi-layer composite cylinder in case of viscoelastic behavior of the material. Prob Mech Eng Reliab Mach 2: 42–45
Malyi VI (1980) Quasiconstant operators in the theory of viscoelasticity of unaged materials. Mech Solids 1: 77–86
Matveenko VP, Troyanovskiy IE, Tsaplina GS (1996) Construction of solutions to problems of the elasticity theory as power series in elastic constants and their application to viscoelasticity. J Appl Math Mech 60(4): 651–659
Vanin GA, Nguyen DD (1996) The creep of orthogonal reinforced spherofibre composites. Mech Compos Mater 32(6): 539–543
Golub VB, Pavlyuk YaV, Fernati PV (2007) Non-stationary creep of linear viscoelastic materials under uniaxial tension and compression conditions. Theor Appl Mech 43: 40–49
Lahellec N, Suquet P (2007) Effective behavior of linear viscoelastic composites: a time-integration approach. Int J Solids Struct 44(2): 507–529
Li D, Hu GK (2007) Effective viscoelastic behavior of particulate polymer composites at finite concentration. J Appl Math Mech 28(3): 297–307
Korchevskaya EA (2008) Optimal design of “sandwich” type laminated composite shells containing viscoelastic layers. Bull Vitebsk Univ 3: 115–120
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Matveenko, V., Trufanov, N. Multi-operator boundary value problems of viscoelasticity of piecewise-homogeneous bodies. J Eng Math 78, 119–129 (2013). https://doi.org/10.1007/s10665-011-9496-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10665-011-9496-y