Analysis of Smart Piezo-Magneto-Thermo-Elastic Composite and Reinforced Plates: Part I – Model Development

D. A. Hadjiloizi ; A.L. Kalamkarov ; Ch. Metti ; A. V. Georgiades

Curved and Layered Structures, Tome 1 (2014), / Harvested from The Polish Digital Mathematics Library

Résumé

A comprehensive micromechanical model for the analysis of a smart composite piezo-magneto-thermoelastic thin plate with rapidly-varying thickness is developed in the present paper. A rigorous three-dimensional formulation is used as the basis of multiscale asymptotic homogenization. The asymptotic homogenization model is developed using static equilibrium equations and the quasi-static approximation of Maxwell’s equations. The work culminates in the derivation of a set of differential equations and associated boundary conditions. These systems of equations are called unit cell problems and their solution yields such coefficients as the effective elastic, piezoelectric, piezomagnetic, dielectric permittivity and others. Among these coefficients, the so-called product coefficients are also determined which are present in the behavior of the macroscopic composite as a result of the interactions and strain transfer between the various phases but can be absent from the constitutive behavior of some individual phases of the composite material. The model is comprehensive enough to allow calculation of such local fields as mechanical stress, electric displacement and magnetic induction. In part II of this work, the theory is illustrated by means of examples pertaining to thin laminated magnetoelectric plates of uniform thickness and wafer-type smart composite plates with piezoelectric and piezomagnetic constituents. The practical importance of the model lies in the fact that it can be successfully employed to tailor the effective properties of a smart composite plate to the requirements of a particular engineering application by changing certain geometric or material parameters. The results of the model constitute an important refinement over previously established work. Finally, it is shown that in the limiting case of a thin elastic plate of uniform thickness the derived model converges to the familiar classical plate model.

Détail

Publié le : 2014-01-01
EUDML-ID : urn:eudml:doc:276956

Bibliographie

[1] Newnham R. E., Skinner D. P., Cross L. E., Connectivity and piezoelectric-pyroelectric composites,Mat. Res. Bull. 13 (1978) 525-536. [Crossref]

[2] Nan C.-W., Bichurin M. I., Dong S., Viehland D. and Srinivasan G., Multiferroic magnetoelectric composites: Historical perspective, status, and future directions, J. Appl. Phys 031101(1) – 031101 (2008) (35) . [Crossref]

[3] Bichurin M., Petrov V., Priya S., Bhalla A., Multiferroic magnetoelectric composites and their applications, Advances in Condensed Matter Physics, Article ID 129794 (2012) 1-3.

[4] Srinivasan G., Magnetoelectric composites, Annual Review of Materials Research, 40 (2010) 153-178. [Crossref]

[5] Bhatra D., Masud Md., De S. K., Chauduri B. K., Large magnetoelectric effect and low-loss high relative permittivity in 0-3 CuO/PVDF composite films exhibiting unusual ferromagnetism at room temperature, J. Phys. D: Appl. Phys. 45 (2012) 485002.

[6] Zhou H.-M., Li C., Xuan L.-M., Wei J., Zhao J.-X. Equivalent circuit method research of resonant magnetoelectric characteristic in magnetoelectric laminate composites using nonlinear magnetostrictive constitutive model, Smart Materials and Structures 20 (2011) 035001. [Crossref]

[7] Ju S., Chae S. H., Choi Y., Lee S., Lee H. W., Ji C.-H., A low frequency vibration energy harvester using magnetoelectric laminate composite, Smart Materials and Structures 22 (2013) 115037. [Crossref]

[8] Oh S. R., Wong T. C., Tan C. W., Yao K., Tay F. E., Fabrication of polymer multilayers on flexible substrates for energy harvesting, Smart Materials and Structures 23 (2014) 015013. [Crossref]

[9] Semenov A. A., Karmanenko S. F., Demidov V. E., Kalinikos B. A., Srinivasan G., Slavin A. N., Mantese J. V., Ferriteferroelectric layered structures for electrically and magnetically tunable microwave resonators, Applied Physics Letters 88 (2006) 033503. [Crossref]

[10] Lottermoser T., Lonkai T., Amann U., Hohlwein D., Ihringer J., Fiebig M., Magnetic phase control by an electric field, Nature 430 (2004) 541-544.

[11] Shen Y., McLaughlin K. L., Gao J., Gray D., Shen L., Wang Y., Li M., Berry D., Li J., Viehland D. AC magnetic dipole localization by a magnetoelectric sensor, Smart Materials and Structures 21 (2012) 065007. [Crossref]

[12] Zhai J., Xing Z., Dong S., Li J., Viehland D., Detection of pico- Tesla magnetic fields using magnetoelectric sensors at room temperature, Applied Physics Letters 88 (2006) 062510. [Crossref]

[13] Harshe G., Doherty J. P., Newnham R. E., Theoretical modeling of 3-0/0-3 magnetoelectric composites, International Journal of Applied Electromagnetics in Materials, 4(2) (1993) 145-159

[14] Harshe G., Doherty J. P., Newnham R. E., Theoretical modeling of multilayer magnetoelectric composites, International Journal of Applied Electromagnetics in Materials, 4(2) (1993) 161- 171 .

[15] Avellaneda M., Harshé G., Magnetoelectric effect in piezoelectric/ magnetostrictive multilayer (2-2) composites, J. Intel.Mat. Syst. Str., 5 (1994) 501-513. [Crossref]

[16] Huang J. H., Kuo W. S., The analysis of piezoelectric/ piezomagnetic compositematerials containing ellipsoidal inclusions, Journal of Applied Physics, 81(3) (1997) 1378-1386 . [Crossref]

[17] Huang J. H., Analytical predictions for themagnetoelectric coupling in piezomagnetic materials reinforced by piezoelectric ellipsoidal inclusions, Physical Review B, 58(1) (1998) 12-15. [Crossref]

[18] Huang J. H., Chiu Y. H.,Liu H. K., Magneto-Electro-Elastic Eshelby tensors for a piezoelectric-piezomagnetic composite reinforced by ellipsoidal inclusions, Journal of Applied Physics, 83(10) (1998) 5364-5370. [Crossref]

[19] Huang J. H., Liu H. K., Dai W. L., The optimized fiber volume fraction for magnetoelectric coupling effect in piezoelectricpiezomagnetic continuous fiber reinforced composites International, Journal of Engineering Science, 38(11) (2000) 1207- 1217. [Crossref]

[20] Bichurin M. I., Petrov V. N., Srinivasan G., Modeling of magnetoelectric effect in ferromagnetic/piezoelectricmultilayer composites, Ferroelectrics, 280 (2002) 165-175. [Crossref]

[21] Bichurin M. I., Petrov V. N., Averkin S. V., Liverts E., Present status of theoretical modeling the magnetoelectric effect in magnetostrictive-piezoelectric nanostructures. Part I: Low frequency electromechanical resonance ranges, J. Appl. Phys., 107(5), (2010) 053904(1)-053904(11). [Crossref]

[22] Soh A. K., Liu J. X., On the constitutive equations of magnetoelectroelastic solids, Journal of Intelligent Materials Systems and Structures, 16 (2005) 597-602.

[23] Bravo-Castillero J., Rodrigues-Ramos R., Mechkour H., Otero J., Sabina F.J., Homogenization of magneto-electro-elastic multilaminated materials, Q J Mechanics Appl Math, 61(3) (2008) 311-332 . [Crossref] | Zbl 1147.74040

[24] Ni Y., Priya S. and Khachaturyan A. G., Modeling of magnetoelectric effect in polycrystalline multiferroic laminates influenced by the orientations of applied electric/magnetic fields, J Appl Phys, 105 (2009) 083914(1)-083914(4). [Crossref]

[25] Akbarzadeh A. H., Babaei M. H., Chen Z. T., The thermoelectromagnetoelastic behavior of a rotating functionally graded piezoelectric cylinder, Smart Mater. Struct., 20 (2011) 065008(1)- 065008(11). [Crossref]

[26] Eshelby J. D., The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proc. R. Soc. Lond. A, 241(1226) (1957) 376-396. | Zbl 0079.39606

[27] Mori T., Tanaka K., Average stress in matrix and average energy of materials with misfitting inclusions, Acta Metallurgica et Materialia, 21 (1973) 571-574. [Crossref]

[28] Kirchner H. O. K. , Alshits V. I., Elastically anisotropic angularly inhomogeneous media II. The Green’s function for piezoelectric, piezomagnetic and magnetoelectric media, Philosophical Magazine A, 74(4) (1996) 861-885. [Crossref]

[29] Pan E., Heyliger R. P., Free vibrations of simply supported and multilayered magneto-electro-elastic plates, Journal of Sound and Vibration, 252(3) (2002) 429-442. [Crossref]

[30] Benveniste Y., Milton G. W., New exact results for the effective electric, elastic, piezoelectric and other properties of composite ellipsoid assemblages, Journal of the Mechanics and Physics of Solids, 51(10) (2003) 1773-1813. | Zbl 1077.74591

[31] Nan C. W., Magnetoelectric effect in composite of piezoelectric and piezomagnetic phases, Physical ReviewB, 50(9) (1994) 6082-6088.

[32] Spyropoulos C. P., Sih G. C. , Song Z. F., Magnetoelectroelastic composite with poling parallel to plane of line crack under out-of-plane deformation, Theoretical and Applied Fracture Mechanics, 40(2) (2003) 281-289. [Crossref]

[33] Tang T., Yu W., Variational Asymptotic homogenization of heterogeneous electromagnetoelastic materials, Int. J. Eng. Sci., 46 (2008) 741-757. [Crossref] | Zbl 1213.74140

[34] Tang T., Yu W., Micromechanical modeling of the multiphysical behavior of smart materials using the variational asymptotic method, SmartMater. Struct., 18(12) (2009) 125026 (1)-125026 (14).

[35] Sunar M., Al-Garni Z., Ali M. H., Kahraman R., Finite Element modeling of thermopiezomagnetic smart structures, AIAA Journal, 40(9) (2002) 1846-1851. [Crossref]

[36] Lee J., Boyd I.V. J.G., Lagoudas D.C., Effective properties of three-phase electro-magneto-elastic composites, Int. J. Eng. Sci., 43 (2005) 790-825. [Crossref] | Zbl 1211.74182

[37] Liu Y. X., Wan J. G., Liu J.-M., Nan C. W., Numerical modeling of magnetoelectric effect in a composite structure, J. Appl. Phys., 94(8) (2003) 5111-5117. [Crossref]

[38] Mininger X., Galopin N., Dennemont Y., Bouillault F., 3D finite element model formagnetoelectric sensors, The European Physical Journal of Applied Physics, 52(2) (2010) 23303(1)- 23303(5).

[39] Sun K. H., Kim Y. Y., Design ofmagnetoelectric multiferroic heterostructures by topology optimization, J. Phys. D: Appl. Phys., 44 (2011) 185003(1)- 185003(8).

[40] Bensoussan A., Lions J. L., Papanicolaou G., Asymptotic analysis for periodic structures, Amsterdam: North-Holland, 1978. | Zbl 0404.35001

[41] Sanchez-Palencia E., Non-Homogeneous media and vibration theory. Lecture Notes in Physics, Berlin: Springer-Verlag, 1980. | Zbl 0432.70002

[42] Bakhvalov N., Panasenko G., Homogenisation: Averaging processes in periodic media, Amsterdam: Kluwer Academic Publishers, 1984. | Zbl 0607.73009

[43] Cioranescu D., Donato P., An Introduction to homogenization ,Oxford: Oxford University Press, 1999. | Zbl 0939.35001

[44] Kalamkarov A. L., Composite and Reinforced Elements of Construction ,New York: Wiley,1992. | Zbl 0809.73006

[45] Kalamkarov A. L., Kolpakov A. G., Analysis, design and optimization of composite structures ,New York: Wiley, 1997. | Zbl 0936.74002

[46] Kalamkarov A. L., Georgiades A. V., Modeling of Smart Composites on Account of Actuation, Thermal Conductivity and Hygroscopic Absorption Composites part B Eng, 33 (2002) 141- 152.

[47] Georgiades A. V., Challagulla K. S., Kalamkarov A. L., Asymptotic homogenization modeling of smart composite generally orthotropic grid-reinforced shells. Part II-Applications, European Journal of Mechanics A-Solids, 29 (2010) 541-556. [Crossref]

[48] Hassan E. M., Kalamkarov A. L., Georgiades A. V., Challagulla K. S., Asymptotic homogenization model for smart 3D gridreinforced composite structures with generally orthotropic constituents, SmartMaterials and Structures, 18(7) art. (2009) 075006.

[49] Saha G. C., Kalamkarov A. L., Georgiades A. V., Micromechanical analysis of effective piezoelastic properties of smart composite sandwich shells made of generally orthotropic materials, Smart Materials and Structures, 16(3) (2007) 866-883. [Crossref]

[50] Guedes J. M. and Kikuchi N., Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods, Comput. Methods Appl. Mech. Engrg., 83 (1990) 143-198. [Crossref] | Zbl 0737.73008

[51] Sevostianov I., Kachanov M., Effect of interphase layers on the overall elastic and conductive properties of matrix composites. Applications to nanosize inclusion, Int. J. Solids Struct., 44 (2007) 1304-1315. [Crossref] | Zbl 1124.74041

[52] Duvaut G., Analyse fonctionnelle et méchanique des milieux continus, Proceedings of the 14th IUTAM Congress (Delft, Holland) (1976) 119-132.

[53] Duvaut G., Metellus A.-M., Homogénéisation d’une plaque mince en flexion de structure périodique et symétrique, C.R. Acad. Sci., Ser. A. 283 (1976) 947-950. | Zbl 0368.73058

[54] Andrianov I. V., Manevich L. I., Shell design using the homogenization method, Uspekhi Mekh, 6 (1983) 3-29.

[55] Andrianov I. V., Lesnichaya V., Manevich L. I., Homogenization methods in the statics and dynamics of ribbed shells (Moscow, Nauka) (1985). | Zbl 0576.73054

[56] Caillerie D Equations de la diffusion stationnaire dans un domaine comportant une distribution périodique d’inclusions aplaties de grande conductivité, C.R. Acad. Sci., Ser. 1 292(1) (1981) 115-118. | Zbl 0468.73149

[57] Caillerie D., Homogénéisation des equation de la diffusion stationnaire dans les domaines cylindrique aplatis, Anal. Numér., 15 (1981) 295-319. | Zbl 0483.35003

[58] Kohn R. V., Vogelius M., A new model for thin plates with rapidly varying thickness, Int. J. of Solids and Struct., 20 (1984) 333-350. | Zbl 0532.73055

[59] Kohn R. V., Vogelius M., A new model for thin plates with rapidly varying thickness, II: A convergence proof, Quart. J. Appl. Math., 43 (1985) 1-22. | Zbl 0565.73046

[60] Challagulla K. S., Georgiades A. V., Kalamkarov A. L., Asymptotic homogenization modeling of smart composite gener ally orthotropic grid-reinforced shells. Part I-Theory, European Journal of Mechanics A-Solids, 29 (2010) 530-540. [Crossref]

[61] Kalamkarov A. L., Kolpakov A. G., A new asymptotic model for a composite piezoelastic plate, International Journal of Solids and Structures, 38 (2001) 6027-6044. | Zbl 1063.74032

[62] Hadjiloizi D. A., Georgiades A. V., Kalamkarov A. L. Dynamic modeling and determination of effective properties of smart composite plates with rapidly varying thickness, International Journal of Engineering Science, 56 (2012) 63-85. [Crossref]

[63] Hadjiloizi D.A., Georgiades, A.V, Kalamkarov, A.L, Jothi S., Micromechanical Model of Piezo-Magneto-Thermo-Elastic Composite Structures: Part I-Theory, European Journal of Mechanics A-Solids, 39, (2013), 298-312. [Crossref]

[64] Hadjiloizi D.A., Georgiades A.V., Kalamkarov A.L, Jothi S., Micromechanical Model of Piezo-Magneto-Thermo-Elastic Composite Structures: Part II-Applications, European Journal ofMechanics A-Solids, 39, (2013), 313-326.

[65] Kalamkarov A. L., Georgiades A. V., Asymptotic homogenization models for smart composite plates with rapidly varying thickness: Part I-Theory, Journal of Multiscale Computational Engineering, 2(1) (2004) 133-148.

[66] Georgiades A.V., Kalamkarov A. L., Asymptotic homogenization models for smart composite plates with rapidly varying thickness: Part II-Applications, Journal ofMultiscale Computational Engineering, 2(1) (2004) 149-174. [Crossref]

[67] Hadjiloizi D.A., Kalamkarov A.L., Metti Ch., Georgiades A.V., Analysis of Piezo-Magneto-Thermo-Elastic Composite and Reinforced Plates: Part II – Applications, Curved and Layered Structures, 1 (2014) 32-58.

[68] Podstrigach Ya. S. and Shvets R.N., Thermoelasticity of Thin Shells, Naukova Dumka Publ., Kiev, 1978.

[69] Podstrigach Ya. S., Lomakin V. A., Kolyano Yu. M., Thermoelasticity of Non-homogeneous Structures, Nauka, Moscow, 1984.

[70] Gibson R. F., Principles of Composite Material Mechanics, McGraw-Hill, New York, 1994.

[71] Kalamkarov A.L. (2014) Asymptotic Homogenization Method and Micromechanical Models for Composite Materials and Thin-Walled Composite Structures, in “Mathematical Methods and Models in Composites,” pp. 1-60, Imperial College Press, London. | Zbl 1302.74136

[72] Kalamkarov A.L. and Challagulla K.S. (2013) Effective Properties of Composite Materials, Reinforced Structures and Smart Composites. Asymptotic Homogenization Approach, in “Effective Properties of Heterogeneous Materials,” Solid Mechanics and Its Applications, Vol. 193, pp. 283-363. Springer, Dordrecht, New York.

[73] Vinson J. R., Sierakowski R. L., The Behavior of Structures Composed of Composite Materials, Kluwer Academic Publishers, Dordrecht, Netherlands, 2002. | Zbl 1029.74001