Large Deformation Hermitian Finite Element Coupled Thermoelasticity Analysis of Wave Propagation and Reflection in a Finite Domain

Subject Areas : Mechanics of Solids

M Mirparizi ¹ , M Shariyat ² , A.R Fotuhi ³

1 - Department of Mechanical Engineering, Yazd University, Yazd, Iran
2 - Faculty of Mechanical Engineering, K.N. Toosi University of Technology, Tehran, Iran
3 - Department of Mechanical Engineering, Yazd University, Yazd, Iran

Received: 2021-08-01 Accepted : 2021-10-07 Published : 2021-12-30

Keywords: Wave propagation and reflection, Finite strain, Coupled thermoelasticity, Surface traction, Thermal Shock,

Abstract :

In the present paper, a finite element nonlinear coupled thermoelasticity formulation is presented for analysis of the wave propagation, reflection, and mixing phenomena in the finite length isotropic solids. The governing equations are derived based on the second Piola-Kirchhoff stress and the full form of Green’s strain-displacement tensors to account for the large deformations and finite strains. In contrast to the available researches, the assumption of very small temperature changes compared to the reference temperature is released in the present research. Galerkin’s method, a weak formulation, and cubic elements are employed to obtain the time-dependent non-linear finite element governing equations. The proposed solution procedure to the resulting highly nonlinear and time-dependent governing equations employs an updating algorithm and Newmark’s numerical time integration method. The wave propagation and reflection phenomena are investigated for both the mechanical and thermal shocks and time variations of distributions of the resulting displacements, temperature rises, and stresses are illustrated graphically and discussed comprehensively. Furthermore, the effects of the non-linear terms are discussed comprehensively. Results reveal that in the non-linear analysis, no fixed speed of wave propagation can be defined.

References:

[1] Melan E., Parkus H., 1953, Wärmespannungen in Platten, Wärmespannungen, Springer.
[2] Nowacki W., 1970, Problems of thermoelasticity, Progress in Aerospace Sciences 10: 1-63.
[3] Nowacki W., 1986, Thermoelasticity, Polish Scientific Publishers, Warsaw, and Pergamon Press, Oxford.
[4] Nowacki W., 1975, Dynamic Problems of Thermoelasticity, Springer Science & Business Media.
[5] Danilovskaya V.I., 1950, Thermal stresses in an elastic half-Space arising after a sudden heating of its boundary, Pelageya Yakovlevna Kochina 14(3): 316-318.
[6] Manoach E., Ribeiro P., 2004, Coupled, thermoelastic, large amplitude vibrations of Timoshenko beams, International Journal of Mechanical Sciences 46(11): 1589-1606.
[7] Vujošević L., Lubarda V., 2002, Finite-strain thermoelasticity based on multiplicative decomposition of deformation gradient, Theoretical and Applied Mechanics 28-29: 379-399.
[8] Abo-El-Nour N., Ghaleb A., Maugin G., 1994, Harmonic wave generation in nonlinear thermoelasticity, International Journal of Engineering Science 32(7): 1103-1116.
[9] Slemrod M., 1981, Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional non-linear thermoelasticity, Archive for Rational Mechanics and Analysis 76(2): 97-133.
[10] Rawy E., Iskandar L., Ghaleb A., 1998, Numerical solution for a nonlinear, one-dimensional problem of thermoelasticity, Journal of Computational and Applied Mathematics 100(1): 53-76.
[11] Sweilam N., 2007, Harmonic wave generation in non linear thermoelasticity by variational iteration method and Adomian's method, Journal of Computational and Applied Mathematics 207(1): 64-72.
[12] Rogovoi A., Stolbova O., 2008, Evolutionary model of finite-strain thermoelasticity, Journal of Applied Mechanics and Technical Physics 49(3): 500-509.
[13] Kocakaplan S., Tassoulas J.L., 2019, Wave propagation in initially-stressed elastic rods, Journal of Sound and Vibration 443: 293-309
[14] Marzani A., Viola E., Bartoli I., Scalea F.L., Rizzo P., 2008, A semi-analytical finite element formulation for modeling stress wave propagation in axisymmetric damped waveguides, Journal of Sound and Vibration 318: 488-505.
[15] Mirparizi M., Fotuhi A.R., Shariyat M., 2020, Nonlinear coupled thermoelastic analysis of thermal wave propagation in a functionally graded finite solid undergoing finite strain, Journal of Thermal Analysis and Calorimetry 139: 2309-2320.
[16] Shariyat M., Khaghani M., Lavasani S.M.H., 2010, Nonlinear thermoelasticity, vibration, and stress wave propagation analyses of thick FGM cylinders with temperature-dependent material properties, European Journal of Mechanics - A/Solids 29(3): 378-391.
[17] Shariyat M., 2009, A rapidly convergent nonlinear transfinite element procedure for transient thermoelastic analysis of temperature-dependent functionally graded cylinders, Journal of Solid Mechanics 1(4):313-327.
[18] Shariyat M., 2012, Nonlinear transient stress and wave propagation analyses of the FGM thick cylinders, employing a unified generalized thermoelasticity theory, International Journal of Mechanical Sciences 65(1):24-37.
[19] Parkus H., 1976, Thermoelasticity, Springer, New York.
[20] Eslami M.R., Hetnarski R.B., Ignaczak J., Noda N., Sumi N., Tanigawa Y., 2013, Theory of Elasticity and Thermal Stresses, Springer.
[21] Hetnarski R.B., Eslami M.R., 2009, Thermal Stresses, Springer.
[22] Eslami M.R., 2014, Finite Elements Methods in Mechanics, Springer International Publishing Switzerland.
[23] Shariyat M., Behzad H., Shaterzadeh A.R., 2018, 3D thermomechanical buckling analysis of perforated annular sector plates with multiaxial material heterogeneities based on curved B-spline elements, Composite Structures 188: 89-103.
[24] Shariyat M., Niknami A., 2016. Impact analysis of strain-rate-dependent composite plates with SMA wires in thermal environments: Proposing refined coupled thermoelasticity, constitutive, and contact models, Composite Structures 136: 191-203.
[25] Niknami A., Shariyat M., 2016, Refined constitutive, bridging, and contact laws for including effects of the impact-induced temperature rise in impact responses of composite plates with embedded SMA wires, Thin-Walled Structures 106: 166-178.
[26] Shariyat M., Ghafourinam M., 2019, Hygrothermomechanical creep and stress redistribution analysis of thick-walled FGM spheres with temperature and moisture dependent material properties and inelastic radius changes, International Journal of Pressure Vessels and Piping 169: 94-114.
[27] Shariyat M., Jahanshahi S., Rahimi H., 2019, Nonlinear Hermitian generalized hygrothermoelastic stress and wave propagation analyses of thick FGM spheres exhibiting temperature, moisture, and strain-rate material dependencies, Composite Structures 229: 111364.
[28] Shariyat M., Ghaznavi A., Hosseini S.H., 2020, On inefficiency of the shape memory alloys in dynamically loaded sandwich plates with structural damping: New 3D zigzag-viscoelasticity theory and asymmetric transformations, Thin–Walled Structures 155: 106879.
[29] Shariyat M., Lavasani S.M.H., Khaghani M., 2010, Nonlinear transient thermal stress and elastic wave propagation analyses of thick temperature-dependent FGM cylinders, using a second-order point-collocation method, Applied Mathematical Modelling 34: 898-918.
[30] Youssef M., Lehaibi E., 2007, State-space approach of two-temperature generalized thermoelasticity, International Journal of Solids and Structures 44(5): 1550-1562.
[31] Shariyat M., 2009, A nonlinear Hermitian transfinite element method for transient behavior analysis of hollow functionally graded cylinders with temperature-dependent materials under thermo-mechanical loads, International Journal of Pressures Vessels and Piping 86: 280-289.
[32] Azadi M., Shariyat M., 2010, Nonlinear transient transfinite element thermal analysis of thick-walled FGM cylinders with temperature-dependent material properties, Meccanica 45(3): 305-18.
[33] Alipour M.M., Shariyat M., 2018, Analytical layerwise stress and deformation analysis of laminated composite plates with arbitrary shapes of interfacial imperfections and discontinuous lateral deflections, Composite Structures 200: 88-102.
[34] Shariyat M., Eslami M.R., 1996, Isoparametric finite-element thermoelasto-plastic creep analysis of shells of revolution, International Journal of Pressures Vessels and Piping 68(3): 249-59.
[35] Ting E., Chen H., 1982, A unified numerical approach for thermal stress waves, Computers & Structures 15(2): 165-175.
[36] Lee T.W., Sim W.J., 1992, Efficient time-domain finite element analysis for dynamic coupled thermoelasticity, Computers & Structures 45(4): 785-93.
[37] Stailess Steel, Table of Technical Properties, Materials and Applications Series, 2007, http://www.euro-inox.org.
[38] Gould P.L., 1998, Analysis of Shells and Plates, Prentice Hall.