- -

Multiphysics and Thermodynamic Formulations for Equilibrium and Non-equilibrium Interactions: Non-linear Finite Elements Applied to Multi-coupled Active Materials

RiuNet: Institutional repository of the Polithecnic University of Valencia

Share/Send to

Cited by

Statistics

Multiphysics and Thermodynamic Formulations for Equilibrium and Non-equilibrium Interactions: Non-linear Finite Elements Applied to Multi-coupled Active Materials

Show full item record

Pérez-Aparicio, JL.; Palma, R.; Taylor, R. (2016). Multiphysics and Thermodynamic Formulations for Equilibrium and Non-equilibrium Interactions: Non-linear Finite Elements Applied to Multi-coupled Active Materials. Archives of Computational Methods in Engineering. 23:535-583. doi:10.1007/s11831-015-9149-9

Por favor, use este identificador para citar o enlazar este ítem: http://hdl.handle.net/10251/84597

Files in this item

Item Metadata

Title: Multiphysics and Thermodynamic Formulations for Equilibrium and Non-equilibrium Interactions: Non-linear Finite Elements Applied to Multi-coupled Active Materials
Author:
UPV Unit: Universitat Politècnica de València. Departamento de Mecánica de los Medios Continuos y Teoría de Estructuras - Departament de Mecànica dels Medis Continus i Teoria d'Estructures
Universitat Politècnica de València. Escuela Técnica Superior de Ingeniería del Diseño - Escola Tècnica Superior d'Enginyeria del Disseny
Issued date:
Abstract:
[EN] Combining several theories this paper presents a general multiphysics framework applied to the study of coupled and active materials, considering mechanical, electric, magnetic and thermal fields. The framework is ...[+]
Copyrigths: Reserva de todos los derechos
Source:
Archives of Computational Methods in Engineering. (issn: 1134-3060 ) (eissn: 1886-1784 )
DOI: 10.1007/s11831-015-9149-9
Publisher:
Springer Verlag (Germany)
Publisher version: http://doi.org/10.1007/s11831-015-9149-9
Thanks:
This research was partially supported by grants CSD2008-00037 Canfranc Underground Physics, Polytechnic University of Valencia under programs PAID 02-11-1828 and 05-10-2674. The first author used the grant Generalitat ...[+]
Type: Artículo

References

Abraham M (1910) Sull’elettrodinamica di Minkowski. Rend Circ Mat 30:33–46

Allik H, Hughes TJR (1970) Finite elment method for piezoelectric vibration. Int J Numer Methods Eng 2:151–157

Antonova EE, Looman DC (2005) Finite elements for thermoelectric device analysis in ANSYS. In: International conference on thermoelectrics [+]
Abraham M (1910) Sull’elettrodinamica di Minkowski. Rend Circ Mat 30:33–46

Allik H, Hughes TJR (1970) Finite elment method for piezoelectric vibration. Int J Numer Methods Eng 2:151–157

Antonova EE, Looman DC (2005) Finite elements for thermoelectric device analysis in ANSYS. In: International conference on thermoelectrics

Atulasimha J, Flatau AB (2011) A review of magnetostrictive iron–gallium alloys. Smart Mater Struct 20:1–15

Ballato A (1995) Piezoelectricity: old effect, new thrusts. IEEE Trans Ultrason Ferroelectr Freq Control 42(5):916–926

Baoyuan S, Jiantong W, Jun Z, Min Q (2003) A new model describing physical effects in crystals: the diagrammatic and analytic methods for macro-phenomenological theory. J Mater Process Technol 139:444–447

Bargmann S, Steinmann P (2005) Finite element approaches to non-classical heat conduction in solids. Comput Model Eng Sci 9(2):133–150

Bargmann S, Steinmann P (2006) Theoretical and computational aspects of non-classical thermoelasticity. Comput Methods Appl Mech Eng 196:516–527

Bargmann S, Steinmann P (2008) Modeling and simulation of first and second sound in solids. Int J Solids Struct 45:6067–6073

Barnett SM (2010) Resolution of the Abraham–Minkowski dilemma. Phys Rev Lett 104:070401

Benbouzid MH, Meunier G, Meunier G (1995) Dynamic modelling of giant magnetostriction in Terfenol-D rods by the finite element method. IEEE Trans Magn 31(3):1821–1824

Benbouzid MH, Reyne G, Meunier G (1993) Nonlinear finite element modelling of giant magnetostriction. IEEE Trans Magn 29(6):2467–2469

Benbouzid MH, Reyne G, Meunier G (1995) Finite elment modelling of magnetostrictive devices: investigations for the design of the magnetic circuit. IEEE Trans Magn 31(3):1813–1816

Besbes M, Ren Z, Razek A (1996) Finite element analysis of magneto-mechanical coupled phenomena in magnetostrictive materials. IEEE Trans Magn 32(3):1058–1061

Biot MA (1956) Thermoelasticity and irreversible thermodynamics. J Appl Phys 27(3):240–253

Bisio G, Cartesegna M, Rubatto G (2001) Thermodynamic analysis of elastic systems. Energy Convers Manag 42:799–812

Blun SL (1974) Materials for radiation detection. National Academy of Sciences, Washington

Bonet J, Wood RD (1997) Nonlinear continuum mechanics for finite element analysis. Cambridge University Press, Cambridge

Borovik-Romanov AS (1960) Piezomagnetism in the antiferromagnetic fluorides of cobalt and manganese. Sov Phys 11:786

Bowyer P (2005) The momentum of light in media: the Abraham–Minkowski controversy. http://bit.ly/1M7wyAT

Brauer JR, Ruehl JJ, MacNeal BE, Hirtenfelder F (1995) Finite element analysis of Hall effect and magnetoresistance. IEEE Trans Electron Devices 42(2):328–333

Bustamante R, Dorfmann A, Ogden RW (2009) On electric body forces and Maxwell stresses in nonlinearly electroelastic solids. Int J Eng Sci 47:1131–1141

Callen HB (1948) The application of Onsager’s reciprocal relations to thermoelectric, thermomagnetic, and galvanomagnetic effects. Phys Rev 73(11):1349–1358

Callen HB (1985) Thermodynamics and an introduction to thermostatistics. Wiley, New York

Carter JP, Booker JR (1989) Finite element analysis of coupled thermoelasticity. Comput Struct 31(1):73–80

Cattaneo C (1938) Sulla conduzione del calore. Atti Semin Mat Fis Univ Modena 3:83–1013

Chaplik AV (2000) Some exact solutions for the classical Hall effect in an inhomogeneous magnetic field. JETP Lett 72:503

Chen PJ, Gurtin ME (1968) On a theory of heat conduction involving two temperatures. J Z Angew Math Phys ZAMP 19(4):614–627

Chu LJ, Haus HA, Penfield P (1966) The force density in polarizable and magnetizable fluids. In: Proceedings of the IEEE

Clin Th, Turenne S, Vasilevskiy D, Masut RA (2009) Numerical simulation of the thermomechanical behavior of extruded bismuth telluride alloy module. J Electron Mater 38(7):994–1001

Coleman BD (1964) Thermodynamics of materials with memory. Arch Ration Mech Anal 17:1–46

de Groot SR (1961) Non-equilibrium themodynamics of systems in an electromagnetic field. J Nucl Energy C Plasma Phys 2:188–194

de Groot SR, Mazur P (1984) Non-equilibrium thermodynamics. Dover, Mineola

Debye P (1913) On the theory of anomalous dispersion in the region of long-wave electromagnetic radiation. Verh dtsch phys Ges 15:777–793

del Castillo LF, García-Colín LS (1986) Thermodynamic basis for dielectric relaxation in complex materials. Phys Rev B 33(7):4944–4951

Delves RT (1964) Figure of merit for Ettingshausen cooling. Br J Appl Phys 15:105–106

Dorf RC (1997) The electrical engineering handbook. CRC Press, UK

Earle R, Richards JFC (1956) Theophrastus: on stones. Ohio State University, Columbus

Ebling D, Jaegle M, Bartel M, Jacquot A, Bottner H (2009) Multiphysics simulation of thermoelectric systems for comparison with experimental device performance. J Electron Mater 38(7):1456–1461

El-Karamany AS, Ezzat MA (2011) On the two-temperature Green–Naghdi thermoelasticity theories. J Therm Stress 34:1207–1226

Eringen AC (1980) Mechanics of continua. Robert E Krieger, Malabar

Eringen AC, Maugin GA (1990) Electrodynamics of continua I. Springer, New York

Ersoy Y (1984) A new nonlinear constitutive theory for conducting magnetothermoelastic solids. Int J Eng Sci 22(6):683–705

Ersoy Y (1986) A new nonlinear constitutive theory of electric and heat conductions for magnetoelastothermo-electrical anisotropic solids. Int J Eng Sci 24(6):867–882

Ferrari A, Mittica A (2013) Thermodynamic formulation of the constitutive equations for solids and fluids. Energy Convers Manag 66:77–86

Galushko D, Ermakov N, Karpovski M, Palevski A, Ishay JS, Bergman DJ (2005) Electrical, thermoelectric and thermophysical properties of hornet cuticle. Semicond Sci Technol 20:286–289

Gao JL, Du QG, Zhang XD, Jiang XQ (2011) Thermal stress analysis and structure parameter selection for a Bi2Te3-based thermoelectric module. J Electron Mater 40(5):884–888

Gaudenzi P, Bathe KJ (1995) An iterative finite element procedure for the analysis of piezoelectric continua. J Intell Mater Syst Struct 6:266–273

Gavela D, Pérez-Aparicio JL (1998) Peltier pellet analysis with a coupled, non-linear 3D finite element model. In: 4th European workshop on thermoelectrics

Goudreau GL, Taylor RL (1972) Evaluation of numerical integration methods in elastodynamics. Comput Methods Appl Mech Eng 2:69–97

Griffiths DJ (1999) Introduction to electrodynamics. Prentice-Hall Inc, Upper Saddle River

Gros L, Reyne G, Body C, Meunier G (1998) Strong coupling magneto mechanical methods applied to model heavy magnetostrictive actuators. IEEE Trans Magn 34(5):3150–3153

Gurtin ME, Williams WO (1966) On the Clausius–Duhem inequality. J Z Angew Math Phys ZAMP 17(5):626–633

Hamader VM, Patil TA, Chovan SH (1987) Free vibration response of two-dimensional magneto-electro-elastic laminated plates. Build Mater Sci 9:249–253

Hausler C, Milde G, Balke H, Bahr HA, Gerlach G (2001) 3-D modeling of pyroelectric sensor arrays part I: multiphysics finite-element simulation. IEEE Sens J 8(12):2080–2087

He Y (2004) Heat capacity, thermal conductivity and thermal expansion of barium titanate-based ceramics. Thermochimica 419:135–141

Hernández-Lemus E, Orgaz E (2002) Hysteresis in nonequilibrium steady states: the role of dissipative couplings. Rev Mex Fís 48:38–45

Hinds EA (2009) Momentum exchange between light and a single atom: Abraham or Minkowski? Phys Rev Lett 102:050403

Hirsinger L, Billardon R (1995) Magneto-elastic finite element analysis including magnetic forces and magnetostriction effects. IEEE Trans Magn 31(3):1877–1880

Huang MJ, Chou PK, Lin MC (2008) An investigation of the thermal stresses induced in a thin-film thermoelectric cooler. J Therm Stress 31:438–454

IEEE Standards Board (1988) IEEE standard on piezoelectricity. ANSI/IEEE Std 176-1987. doi: 10.1109/IEEESTD.1988.79638

IEEE Standards Board (1991) IEEE standard on magnetostrictive materials: piezomagnetic nomenclature. IEEE Std 319-1990. doi: 10.1109/IEEESTD.1991.101048

Ioffe Institute (2013) INSb—indium antimonide. Ioffe Institute. www.ioffe.rssi.ru/SVA/NSM/Semicond/InSb/index.html

Jackson JD (1962) Classical electrodynamics. Wiley, New York

Jaegle M (2008) Multiphysics simulation of thermoelectric systems—modeling of Peltier—cooling and thermoelectric generation. In: Proceedings of the COMSOL

Jaegle M, Bartel M, Ebling D, Jacquot A, Bottner H (2008) Multiphysics simulation of thermoelectric systems. In: European conference on thermoelectrics ECT2008

Jiménez JL, Campos I (1996) Advanced electromagnetism: foundations, theory and applications, chapter The balance equations of energy and momentum in classical electrodynamics. World Scientific Publishing, Singapore

Johnstone S (2008) Is there potential for use of the Hall effect in analytical science? Analyst 133:293–296

Jou D, Lebon G (1996) Extended irreversible thermodynamics. Springer, Berlin

Kaltenbacher M, Kaltenbacher B, Hegewald T, Lerch R (2010) Finite element formulation for ferroelectric hysteresis of piezoelectric materials. J Intell Mater Syst Struct 21:773–785

Kaltenbacher M, Meiler M, Ertl M (2009) Physical modeling and numerical computation of magnetostriction. Int J Comput Math Electr Electron Eng 28(4):819–832

Kamlah M, Bohle U (2001) Finite element analysis of piezoceramic components taking into account ferroelectric hysteresis behavior. Int J Solids Struct 38:605–633

Kannan KS, Dasgupta A (1997) A nonlinear Galerkin finite-element theory for modeling magnetostrictive smart structures. Smart Mater Struct 6:341–350

Kiang J, Tong L (2010) Nonlinear magneto-mechanical finite element analysis of Ni–Mn–Ga single crystals. Smart Mater Struct 19:1–17

Kinsler P, Favaro A, McCall MW (2009) Four Poynting theorems. Eur J Phys 30:983–993

Klinckel S, Linnemann K (2008) A phenomenological constitutive model for magnetostrictive materials and ferroelectric ceramics. Proc Appl Math Mech 8:10507–10508

Kosmeier D (2013) Hornets: Gentle Giants! Wikipedia: the free encyclopedia. www.hornissenschutz.de/hornets.htm

Lahmer T (2008) Forward and inverse problems in piezoelectricity. PhD thesis, Universität Erlangen-Nürnberg

Landau LD, Lifshitz EM (1982) Mechanics. Butterworth-Heinemann, Oxford

Landau LD, Lifshitz EM (1984) Electrodynamics of continuous media. Pergamon Press, Oxford

Landis CM (2002) A new finite-element formulation for electromechanical boundary value problems. Int J Numer Methods Eng 55:613–628

Díaz Lantada A (2011) Handbook of active materials for medical devices: advances and applications. CRC Press, Boca Raton

Lebon G, Jou D, Casas-Vázquez J (2008) Understanding non-equilibrium thermodynamics. Springer, Berlin

Linnemann K, Klinkel S (2006) A constitutive model for magnetostrictive materials—theory and finite element implementation. Proc Appl Math Mech 6:393–394

Linnemann K, Klinkel S, Wagner W (2009) A constitutive model for magnetostrictive and piezoelectric materials. Int J Solids Struct 46:1149–1166

Llebot JE, Jou D, Casas-Vázquez J (1983) A thermodynamic approach to heat and electric conduction in solids. Physica 121(A):552–562

Lu X, Hanagud V (2004) Extended irreversible thermodynamics modeling for self-heating and dissipation in piezoelectric ceramics. IEEE Trans Ultrason Ferroelectr Freq Control 51(12):1582–1592

Lubarda VA (2004) On thermodynamic potentials in linear thermoelasticity. Int J Solids Struct 41:7377–7398

Mansuripur M (2012) Trouble with the lorentz law of force: incompatibility with special relativity and momentum conservation. Phys Rev Lett 108:193901

Maruszewski B, Lebon G (1986) An extended irreversible thermodynamic description of electrothermoelastic semiconductors. Int J Eng Sci 24(4):583–593

McMeeking RM, Landis CM (2005) Electrostatic forces and stored energy for deformable dielectric materials. J Appl Mech 72:581–590

McMeeking RM, Landis CM, Jimenez MA (2007) A principle of virtual work for combined electrostatic and mechanical loading of materials. Int J Non Linear Mech 42:831–838

MELCOR (2000) Thermoelectric handbook. Melcor, a unit of Laird Technologies. http://www.lairdtech.com

Minkowski H (1908) Nachr. ges. wiss. Gottingen 53

Naranjo B, Gimzewski JK, Putterman S (2005) Observation of nuclear fusion driven by a pyroelectric crystal. Nature 28(434):1115–1117

Nédélec JC (1980) Mixed finite elements in $${R}^3$$ R 3 . Numer Math 35:314–345

Nettleton RE, Sobolev SL (1995) Applications of extended thermodynamics to chemical, rheological, and transport processes: a special survey part I. approaches and scalar rate processes. J Non-Equilib Thermodyn 20:205–229

Nettleton RE, Sobolev SL (1995) Applications of extended thermodynamics to chemical, rheological, and transport processes: a special survey part II. vector transport processes, shear relaxation and rheology. J Non-Equilib Thermodyn 20:297–331

Nettleton RE, Sobolev SL (1996) Applications of extended thermodynamics to chemical, rheological, and transport processes: a special survey part III. wave phenomena. J Non-Equilib Thermodyn 21:1–16

Newmark N (1959) A method of computation for structural dynamics. ASCE J Eng Mech 85:67–94

Newnham RE (2005) Properties of materials: anisotropy, symmetry, structure. Oxford University Press, Oxford

Nour AE, Abd-Alla N, Maugin GA (1990) Nonlinear equations for thermoelastic magnetizable conductors. Int J Eng Sci 27(7):589–603

Nowacki A (1962) International series of monographs in aeronautics and astronautics. Pergamon Press, Oxford

Okumura H, Hasegawa Y, Nakamura H, Yamaguchi S (1999) A computational model of thermoelectric and thermomagnetic semiconductors. In: 18th international conference on thermoelectrics

Okumura H, Yamaguchi S, Nakamura H, Ikeda K, Sawada K (1998) Numerical computation of thermoelectric and thermomagnetic effects. In: 17th international conference on thermoelectrics

Oliver X, Agelet C (2000) Continuum mechanics for engineers. Edicions UPC, Barcelona. http://hdl.handle.net/2099.3/36197

Shankar K, Kondaiah P, Ganesan N (2013) Pyroelectric and pyromagnetic effects on multiphase magneto-electro-elastic cylindrical shells for axisymmetric temperature. Smart Mater Struct 22(2):025007

Palma R, Pérez-Aparicio JL, Bravo R (2013) Study of hysteretic thermoelectric behavior in photovoltaic materials using the finite element method, extended thermodynamics and inverse problems. Energy Convers Manag 65:557–563

Palma R, Pérez-Aparicio JL, Taylor RL (2012) Non-linear finite element formulation applied to thermoelectric materials under hyperbolic heat conduction model. Comput Method Appl Mech Eng 213–216:93–103

Palma R, Rus G, Gallego R (2009) Probabilistic inverse problem and system uncertainties for damage detection in piezoelectrics. Mech Mater 41:1000–1016

Pérez-Aparicio JL, Gavela D (1998) 3D, non-linear coupled, finite element model of thermoelectricity. In: 4th European workshop on thermoelectrics

Pérez-Aparicio JL, Palma R, Taylor RL (2012) Finite element analysis and material sensitivity of Peltier thermoelectric cells coolers. Int J Heat Mass Transf 55:1363–1374

Pérez-Aparicio JL, Sosa H (2004) A continuum three-dimensional, fully coupled, dynamic, non-linear finite element formulation for magnetostrictive materials. Smart Mater Struct 13:493–502

Perez-Aparicio JL, Sosa H, Palma R (2007) Numerical investigations of field-defect interactions in piezoelectric ceramics. Int J Solids Struct 44:4892–4908

Pérez-Aparicio JL, Taylor RL, Gavela D (2007) Finite element analysis of nonlinear fully coupled thermoelectric materials. Comput Mech 40:35–45

Qi H, Fang D, Yao Z (1997) FEM analysis of electro-mechanical coupling effect of piezoelectric materials. Comput Mater Sci 8:283–290

Pérez-Aparicio JL, Palma R, Abouali-Sánchez S (2014) Complete finite element method analysis of galvanomagnetic and thermomagnetic effects. Appl Therm Eng (submitted)

Perez-Aparicio JL, Palma R, Moreno-Navarro P (2014) Elasto-thermoelectric non-linear, fully coupled, and dynamic finite element analysis of pulsed thermoelectrics. Appl Therm Eng (submitted)

Ramírez F, Heyliger PR, Pan E (2006) Free vibration response of two-dimensional magneto-electro-elastic laminated plates. J Sound Vib 292:626–644

Reitz JR, Milford FJ (1960) Foundations of electromagnetic theory. Addison-Wesley, Boston

Reng Z, Ionescu B, Besbes M, Razek A (1995) Calculation of mechanical deformation of magnetic materials in electromagnetic devices. IEEE Trans Magn 31(3):1873–1876

Restuccia L (2010) On a thermodynamic theory for magnetic relaxation phenomena due to n microscopic phenomena described by n internal variables. J Non-Equilib Thermodyn 35:379–413

Restuccia L, Kluitenberg GA (1988) On generalizations of the Debye equation for dielectric relaxation. Phys A 154:157–182

Restuccia L, Kluitenberg GA (1992) On the heat dissipation function for dielectric relaxation phenomena in anisotropic media. Int J Eng Sci 30(3):305–315

Riffat SB, Ma X (2003) Thermoelectrics: a review of present and potential applications. Appl Therm Eng 23:913–935

Rinaldi C, Brenner H (2002) Body versus surface forces in continuum mechanics: is the Maxwell stress tensor a physically objective Cauchy stress? Phys Rev E 65:036615

Rowe DM (ed) (1995) CRC handbook of thermoelectrics. CRC Press, UK

Rus G, Palma R, Pérez-Aparicio JL (2009) Optimal measurement setup for damage detection in piezoelectric plates. Int J Eng Sci 47:554–572

Rus G, Palma R, Pérez-Aparicio JL (2012) Experimental design of dynamic model-based damage identification in piezoelectric ceramics. Mech Syst Signal Process 26:268–293

Sadiku MNO (2001) Numerical techniques in electromagnetics. CRC Press LLC, Boca Raton

Semenov AS, Kessler H, Liskowsky A, Balke H (2006) On a vector potential formulation for 3D electromechanical finite element analysis. Commun Numer Methods Eng 22:357–375

Serra E, Bonaldi M (2008) A finite element formulation for thermoelastic damping analysis. Int J Numer Methods Eng 78(6):671–691

Several. Wikipedia. Wikipedia: The Free Encyclopedia, Several

Soh AK, Liu JX (2005) On the constitutive equations of magnetoelectroelastic solids. J Intell Mater Syst Struct 16:597–602

Stefanescu DM (2011) Handbook of force transducers: principles and components. Springer, Berlin

Tamma KK, Namburu RR (1992) An effective finite element modeling/analysis approach for dynamic thermoelasticity due to second sound effects. Comput Mech 9:73–84

Tang T, Yu W (2009) Micromechanical modeling of the multiphysical behavior of smart materials using the variational asymptotic method. Smart Mater Struct 18:1–14

Taylor RL (2010) FEAP a finite element analysis program: user manual. University of California, Berkeley. http://www.ce.berkeley.edu/feap

Thurston RN (1994) Warren p. Mason (1900–1986) physicist, engineer, inventor, author, teacher. IEEE Trans Ultrason Ferroelectr Freq Control 41(4):425–434

Tian X, Shen Y, Chen C, He T (2006) A direct finite element method study of generalized thermoelastic problems. Int J Solids Struct 43:2050–2063

Tinder RF (2008) Tensor properties of solids: phenomenological development of the tensor properties of crystals. Morgan and Claypool, San Rafael

Truesdell C (1968) Thermodynamics for beginners, in irreversible aspects of continuum mechanics. Springer, Berlin

Tzou HS, Ye R (1996) Pyroelectric and thermal strain effects of piezoelectric (PVDF and PZT) devices. Mech Syst Signal Process 10(4):459–469

Walser R (1972) Application of pyromagnetic phenomena to radiation detection. IEEE Trans Magn 8(3):619

Woodbridge K, Ertl ME (1978) Pulsed Ettingshausen cooling in bismuth. J Phys F Met Phys 8(9):1941–1945

Yan R, Wang B, Yang Q, Liu F, Cao S, Huang W (2004) A numerical model of displacement for giant magnetostrictive actuator. IEEE Trans Magn 14(2):1914–1917

Yoo B, Hirano M, Hirata K (2008) Fully coupled electro-magneto-mechanical analysis method of magnetostrictive actuator using 3D finite element method. In: Proceedings of the 2008 international conference on electrical machines

Youssef HM (2006) Theory of two-temperature-generalized thermoelasticity. IMA J Appl Math 71:383–390

Youssef HM (2011) Theory of two-temperature-generalized thermoelasticity without energy dissipation. J Therm Stress 34:138–146

Yu N, Imatani S, Inoue T (2006) Hyperbolic thermoelastic analysis due to pulsed heat input by numerical solution. JSME Int J Ser A 49(2):180–187

Zeng X, Rajapakse RKND (2004) Effects of remanent field on an elliptical flaw and a crack in a poled piezoelectric ceramic. Comput Mater Sci 30:433–440

Zhou L, Tang DW, Araki N (2006) Coupled finite element analysis of generalized thermoelasticity in semi-infinite medium. JSME Int J Ser A 49(2):195–200

Zienkiewicz OC, Taylor RL, Zhu JZ (2013) The finite element method: the basis, 7th edn. Elsevier Butterworth-Heinemann, Amsterdam

[-]

This item appears in the following Collection(s)

Show full item record