A dynamic thermoviscoelastic contact problem with the second sound effect

被引:18
作者
Berti, Alessia [1 ]
Copetti, Maria I. M. [2 ]
Fernandez, Jose R. [3 ]
Naso, Maria Grazia [4 ]
机构
[1] Univ E Campus, I-22060 Novedrate, CO, Italy
[2] Univ Fed Santa Maria, Lab Anal Numer & Astrofis, Dept Matemat, BR-97119900 Santa Maria, RS, Brazil
[3] Univ Vigo, Dept Matemat Aplicada 1, Escola Enxeneria Telecomunicac, Vigo 36310, Spain
[4] Univ Brescia, Dipartimento Ingn Civile Architettura Terr Ambien, I-25133 Brescia, Italy
关键词
Dynamic thermoviscoelastic contact; Second sound; Energy decay; Finite element approximation; A priori error estimates; Numerical simulations; NUMERICAL-ANALYSIS; EXPONENTIAL STABILITY; NONLINEAR THERMOELASTICITY; ASYMPTOTIC-BEHAVIOR; BEAM; VIBRATIONS; EXISTENCE; DAMAGE; NOISE; BODY;
D O I
10.1016/j.jmaa.2014.07.049
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
This paper deals with a contact problem describing the mechanical and thermal evolution of a damped extensible thermoviscoelastic beam under the Cattaneo law, relating the heat flux to the gradient of the temperature. The beam is rigidly clamped at its left end whereas the right end of the beam moves vertically between reactive stops like a nonlinear spring. Existence and uniqueness of the solution is proved, as well as the exponential decay of the related energy. Then, fully discrete approximations are introduced by using the classical finite element method and the implicit Euler scheme to approximate the spatial variable and to discretize the time derivatives, respectively. An a priori error estimates result is proved, from which the linear convergence of the algorithm is deduced. The case where the two stops are rigid is also studied from the point of view of the existence and longtime behavior of the solutions. Finally, some numerical simulations are presented to demonstrate the accuracy of the approximation and the behavior of the solution. (C) 2014 Elsevier Inc. All rights reserved.
引用
收藏
页码:1163 / 1195
页数:33
相关论文
共 52 条
[1]   VIBRATIONS OF A NONLINEAR DYNAMIC BEAM BETWEEN TWO STOPS [J].
Andrews, K. T. ;
M'Bengue, M. F. ;
Shillor, M. .
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B, 2009, 12 (01) :23-38
[2]   Numerical analysis of dynamic thermoviscoelastic contact with damage of a rod [J].
Andrews, KT ;
Fernández, JR ;
Shillor, M .
IMA JOURNAL OF APPLIED MATHEMATICS, 2005, 70 (06) :768-795
[3]   A thermoviscoelastic beam with a tip body [J].
Andrews, KT ;
Fernández, JR ;
Shillor, M .
COMPUTATIONAL MECHANICS, 2004, 33 (03) :225-234
[4]  
[Anonymous], 2002, CBMS-NSF Regional Conference Series in Applied Mathematics
[5]  
[Anonymous], 1986, Ann Mat Pura Appl, DOI [DOI 10.1007/BF01762360, DOI 10.1007/BF01762360.MR916688]
[6]   Analysis of dynamic nonlinear thermoviscoelastic beam problems [J].
Berti, A. ;
Copetti, M. I. M. ;
Fernandez, J. R. ;
Naso, M. G. .
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 2014, 95 :774-795
[7]  
Berti A., 2014, Z ANGEW MATH PHYS, V66, P1
[8]   VIBRATIONS OF A DAMPED EXTENSIBLE BEAM BETWEEN TWO STOPS [J].
Berti, Alessia ;
Naso, Maria Grazia .
EVOLUTION EQUATIONS AND CONTROL THEORY, 2013, 2 (01) :35-54
[9]   UNILATERAL DYNAMIC CONTACT OF TWO VISCOELASTIC BEAMS [J].
Berti, Alessia ;
Naso, Maria Grazia .
QUARTERLY OF APPLIED MATHEMATICS, 2011, 69 (03) :477-507
[10]  
Bonfanti G, 2009, SER ADV MATH APPL SC, V82, P123