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Valery P. Smyshlyaev - One of the best experts on this subject based on the ideXlab platform.
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Homogenization of a thermo-chemo-viscoelastic Kelvin-Voigt Model
Journal of Mathematical Physics, 2013Co-Authors: Andrey Amosov, Grigory Panasenko, Ilya Kostin, Valery P. SmyshlyaevAbstract:The paper is devoted to a Model for the procedure of formation of a composite material constituted of solid fibers and of a solidifying matrix. The solidification process for the matrix depends on the temperature and on the degree of cure, which are used for the Modeling of the mechanical properties of the matrix. Namely, the mechanical properties are described by Kelvin-Voigt viscoelastic equation with rapidly oscillating periodic coefficients depending on the temperature and the degree of cure. The latter are in turn solutions of a thermo-chemical problem with rapidly varying coefficients. We prove an error estimate for approximation of the viscoelastic problem by the same equation but with the coefficients depending on solution to the homogenized thermo-chemical problem. This estimate, in combination with our recent estimates for the viscoelastic (with time-dependent coefficients) and thermo-chemical homogenization problems, generates the overall error bound for the asymptotic solution to the full coup...
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MEMORY EFFECT IN HOMOGENIZATION OF A VISCOELASTIC KELVIN–VOIGT Model WITH TIME-DEPENDENT COEFFICIENTS
Mathematical Models and Methods in Applied Sciences, 2009Co-Authors: Zouhair Abdessamad, Grigory Panasenko, Ilya Kostin, Valery P. SmyshlyaevAbstract:This paper is motivated by Modeling the procedure of formation of a composite material constituted of solid fibers and of a solidifying matrix. The solidification process for the matrix depends on the temperature and on the reticulation rate which thereby influence the mechanical properties of the matrix. The mechanical properties are described by a viscoelastic medium equation of Kelvin-Voigt type with rapidly oscillating periodic coefficients depending on the temperature and the reticulation rate. That is Modeled as an initial boundary value problem with time-dependent elasticity and viscosity tensors to account for the solidification, and the mechanical and/or thermal forcing. First we prove the existence and uniqueness of the solution for the problem and obtain a priori estimates. Then we derive the homogenized problem, characterize its coefficients including explicit memory terms, and prove that it admits a unique solution. Finally, we prove error bounds for the asymptotic solution, and establish some related regularity properties of the homogenized solution.
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Homogenization of thermo-viscoelastic Kelvin–Voigt Model
Comptes Rendus Mécanique, 2007Co-Authors: Zouhair Abdessamad, Grigory Panasenko, Ilya Kostin, Valery P. SmyshlyaevAbstract:We consider an e-periodic composite material, e≪1, constituted of periodic fibres surrounded by a polymer matrix, solidifying under a heating process. The mechanical behaviour of the material is described by the Kelvin–Voigt visco-elasticity equation with rapidly oscillating space and time dependent coefficients. This time dependence is caused by the dependence of the state of the material on the temperature, that is a solution of a thermo-chemical Model studied earlier. The existence and uniqueness of a solution of the Kelvin–Voigt visco-elasticity Model are proved, the homogenized Model is obtained and the existence and uniqueness of its solution are studied. The estimates for the difference between the solution of the original problem and the homogenized one are obtained. To cite this article: Z. Abdessamad et al., C. R. Mecanique 335 (2007).
Kumbakonam R. Rajagopal - One of the best experts on this subject based on the ideXlab platform.
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Implicit constitutive relations for describing the response of visco-elastic bodies
International Journal of Non-Linear Mechanics, 2020Co-Authors: Roger Bustamante, Kumbakonam R. Rajagopal, O. Orellana, R. MenesesAbstract:Abstract A constitutive relation is proposed for viscoelastic bodies that is a generalization of the classic Kelvin-Voigt Model, wherein the left Cauchy Green tensor, the symmetric part of the velocity gradient, and the Cauchy stress tensor are implicitly related. The Model developed includes several Models that are being used in the literature to describe the elastic and viscoelastic response of bodies. In this paper, we study special homogeneous deformations of a slab within the context of the implicit viscoelastic Model.
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On the states of stress and strain adjacent to a crack in a strain-limiting viscoelastic body.
Mathematics and mechanics of solids : MMS, 2017Co-Authors: Hiromichi Itou, Victor A. Kovtunenko, Kumbakonam R. RajagopalAbstract:The viscoelastic Kelvin–Voigt Model is considered within the context of quasi-static deformations and generalized with respect to a nonlinear constitutive response within the framework of limiting ...
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On Kelvin-Voigt Model and its generalizations
Evolution Equations and Control Theory, 2012Co-Authors: Miroslav Bulíček, Josef Málek, Kumbakonam R. RajagopalAbstract:We consider a generalization of the Kelvin-Voigt Model where the elastic part of the Cauchy stress depends non-linearly on the linearized strain and the dissipative part of the Cauchy stress is a nonlinear function of the symmetric part of the velocity gradient. The assumption that the Cauchy stress depends non-linearly on the linearized strain can be justified if one starts with the assumption that the kinematical quantity, the left Cauchy-Green stretch tensor, is a nonlinear function of the Cauchy stress, and linearizes under the assumption that the displacement gradient is small. Long-time and large data existence, uniqueness and regularity properties of weak solution to such a generalized Kelvin-Voigt Model are established for the non-homogeneous mixed boundary value problem. The main novelty with regard to the mathematical analysis consists in including nonlinear (non-quadratic) dissipation in the problem.
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A note on a reappraisal and generalization of the Kelvin–Voigt Model
Mechanics Research Communications, 2009Co-Authors: Kumbakonam R. RajagopalAbstract:Abstract In this short note we show that expressing the strain and the symmetric part of the velocity gradient as functions of the stress instead of providing constitutive relations for the stress, as is the norm, allows one to obtain a far wider class of Models to describe the response of viscoelastic bodies than those that are possible within the classical framework. Such an approach is used to describe the Kelvin–Voigt solid and several of its generalizations. A further generalization that recognizes that a certain class of viscoelastic solids, such as the Kelvin–Voigt solid, can be thought of a mixture of a dissipative fluid and an elastic solid, allows one to develop Models wherein implicit relationships can be provided for the dissipative response and the elastic response.
K Rajabi - One of the best experts on this subject based on the ideXlab platform.
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application of the generalized hooke s law for viscoelastic materials ghvms in nanoscale mass sensing applications of viscoelastic nanoplates a theoretical study
European Journal of Mechanics A-solids, 2018Co-Authors: K Rajabi, S H HosseinihashemiAbstract:Abstract Reviewing the literature reveals that in all previous research works related to the damped vibration analysis of nanoplates except (Rajabi and Hosseini-Hashemi, 2017a), the material damping of nanoplates has been represented by the Kelvin-Voigt Model without any reasonable justification. The Kelvin-Voigt Model has no instantaneous elasticity in creep and also shows unrealistic behavior in relaxation. Due to these drawbacks, the Kelvin-Voigt Model fails to capture time domain characteristics of viscoelastic solid materials correctly. On the other hand, the Zener Model can predict both creep and relaxation functions of a viscoelastic solid material well in the time domain. In the present paper based on the combination of generalized Hooke's law for viscoelastic materials (GHVMs) and the nonlocal elasticity theory, a general 2-D theory of nonlocal viscoelasticity is obtained. A nanoscale mass-sensor is proposed based on the damped vibration analysis of a viscoelastic orthotropic Kirchhoff-Love nanoplate. The material damping of the nanoplate is represented by the Zener Model for illustration purposes. For simply supported boundary conditions, analytical expression is obtained for the eigenfrequencies of the sensor.
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on the application of viscoelastic orthotropic double nanoplates systems as nanoscale mass sensors via the generalized hooke s law for viscoelastic materials and eringen s nonlocal elasticity theory
Composite Structures, 2017Co-Authors: K Rajabi, S H HosseinihashemiAbstract:Abstract Reviewing the literature reveals that in all previous research works related to the damped vibration analysis of nanoplates, the material damping of nanoplates has been represented by Kelvin-Voigt Model without any reasonable justification. The Kelvin-Voigt Model has no instantaneous elasticity in creep and also shows unrealistic behavior in relaxation. Due to these drawbacks, the Kelvin-Voigt Model fails to capture time domain characteristics of viscoelastic solid materials correctly. On the other hand, the Zener Model can predict both creep and relaxation functions of a viscoelastic solid material well in the time domain. The generalized Hooke’s law for viscoelastic materials (GHLVMs) bridges the differential form of linear viscoelasticity and the integral form of linear viscoelasticity. In the present study based on the combination of GHLVMs and the nonlocal elasticity theory, a general 2-D theory of nonlocal viscoelasticity is obtained. A nanoscale mass-sensor is proposed based on the damped frequency analysis of a viscoelastic orthotropic double-nanoplates system (VODNS). The material damping of the nanoplates is represented by the Zener Model. It has been assumed that the nanoplates obey the Kirchhoff-Love plate hypotheses. Detailed parametric study is presented.
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On the application of viscoelastic orthotropic double-nanoplates systems as nanoscale mass-sensors via the generalized Hooke’s law for viscoelastic materials and Eringen’s nonlocal elasticity theory
Composite Structures, 2017Co-Authors: K Rajabi, S.h. Hosseini-hashemiAbstract:Abstract Reviewing the literature reveals that in all previous research works related to the damped vibration analysis of nanoplates, the material damping of nanoplates has been represented by Kelvin-Voigt Model without any reasonable justification. The Kelvin-Voigt Model has no instantaneous elasticity in creep and also shows unrealistic behavior in relaxation. Due to these drawbacks, the Kelvin-Voigt Model fails to capture time domain characteristics of viscoelastic solid materials correctly. On the other hand, the Zener Model can predict both creep and relaxation functions of a viscoelastic solid material well in the time domain. The generalized Hooke’s law for viscoelastic materials (GHLVMs) bridges the differential form of linear viscoelasticity and the integral form of linear viscoelasticity. In the present study based on the combination of GHLVMs and the nonlocal elasticity theory, a general 2-D theory of nonlocal viscoelasticity is obtained. A nanoscale mass-sensor is proposed based on the damped frequency analysis of a viscoelastic orthotropic double-nanoplates system (VODNS). The material damping of the nanoplates is represented by the Zener Model. It has been assumed that the nanoplates obey the Kirchhoff-Love plate hypotheses. Detailed parametric study is presented.
Zouhair Abdessamad - One of the best experts on this subject based on the ideXlab platform.
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MEMORY EFFECT IN HOMOGENIZATION OF A VISCOELASTIC KELVIN–VOIGT Model WITH TIME-DEPENDENT COEFFICIENTS
Mathematical Models and Methods in Applied Sciences, 2009Co-Authors: Zouhair Abdessamad, Grigory Panasenko, Ilya Kostin, Valery P. SmyshlyaevAbstract:This paper is motivated by Modeling the procedure of formation of a composite material constituted of solid fibers and of a solidifying matrix. The solidification process for the matrix depends on the temperature and on the reticulation rate which thereby influence the mechanical properties of the matrix. The mechanical properties are described by a viscoelastic medium equation of Kelvin-Voigt type with rapidly oscillating periodic coefficients depending on the temperature and the reticulation rate. That is Modeled as an initial boundary value problem with time-dependent elasticity and viscosity tensors to account for the solidification, and the mechanical and/or thermal forcing. First we prove the existence and uniqueness of the solution for the problem and obtain a priori estimates. Then we derive the homogenized problem, characterize its coefficients including explicit memory terms, and prove that it admits a unique solution. Finally, we prove error bounds for the asymptotic solution, and establish some related regularity properties of the homogenized solution.
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Homogenization of thermo-viscoelastic Kelvin–Voigt Model
Comptes Rendus Mécanique, 2007Co-Authors: Zouhair Abdessamad, Grigory Panasenko, Ilya Kostin, Valery P. SmyshlyaevAbstract:We consider an e-periodic composite material, e≪1, constituted of periodic fibres surrounded by a polymer matrix, solidifying under a heating process. The mechanical behaviour of the material is described by the Kelvin–Voigt visco-elasticity equation with rapidly oscillating space and time dependent coefficients. This time dependence is caused by the dependence of the state of the material on the temperature, that is a solution of a thermo-chemical Model studied earlier. The existence and uniqueness of a solution of the Kelvin–Voigt visco-elasticity Model are proved, the homogenized Model is obtained and the existence and uniqueness of its solution are studied. The estimates for the difference between the solution of the original problem and the homogenized one are obtained. To cite this article: Z. Abdessamad et al., C. R. Mecanique 335 (2007).
S.h. Hosseini-hashemi - One of the best experts on this subject based on the ideXlab platform.
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On the application of viscoelastic orthotropic double-nanoplates systems as nanoscale mass-sensors via the generalized Hooke’s law for viscoelastic materials and Eringen’s nonlocal elasticity theory
Composite Structures, 2017Co-Authors: K Rajabi, S.h. Hosseini-hashemiAbstract:Abstract Reviewing the literature reveals that in all previous research works related to the damped vibration analysis of nanoplates, the material damping of nanoplates has been represented by Kelvin-Voigt Model without any reasonable justification. The Kelvin-Voigt Model has no instantaneous elasticity in creep and also shows unrealistic behavior in relaxation. Due to these drawbacks, the Kelvin-Voigt Model fails to capture time domain characteristics of viscoelastic solid materials correctly. On the other hand, the Zener Model can predict both creep and relaxation functions of a viscoelastic solid material well in the time domain. The generalized Hooke’s law for viscoelastic materials (GHLVMs) bridges the differential form of linear viscoelasticity and the integral form of linear viscoelasticity. In the present study based on the combination of GHLVMs and the nonlocal elasticity theory, a general 2-D theory of nonlocal viscoelasticity is obtained. A nanoscale mass-sensor is proposed based on the damped frequency analysis of a viscoelastic orthotropic double-nanoplates system (VODNS). The material damping of the nanoplates is represented by the Zener Model. It has been assumed that the nanoplates obey the Kirchhoff-Love plate hypotheses. Detailed parametric study is presented.
