Classical invariants, despite most of them having unclear physical interpretation and not having experimental advantages, have been extensively used in modeling nonlinear magneto-elastic materials. In this paper, a new set of spectral invariants, which have some advantages over classical invariants, is proposed to model the behavior of transversely isotropic nonlinear magneto-elastic bodies. The novel spectral invariant formulation, which is shown to be more general, is used to analytically solve some simple magneto-mechanical boundary value problems. With the aid of the proposed spectral invariants it is possible to study, in a much simpler manner, the effect of different types of deformations on the response of the magneto-elastic material.
BustamanteRDorfmannAOgdenRW. On variational formulations in nonlinear magnetoelastostatics. Math Mech Solids2008; 13: 725–745.
4.
BustamanteRDorfmannAOgdenRW. Numerical solution of finite geometry boundary value problems in nonlinear magnetoelasticity. Int J Solids Struct2011; 48: 874–883.
5.
ChatzigeorgiouGJaviliASteinmannP. Unified magnetomechanical homogenization framework with application to magnetorheological elastomers. Math Mech Solids2014; 19: 193–211.
6.
DanasKKankanalaSVTriantafyllidisN. Experiments and modelling of iron-particled-filled magnetorheological elastomers. J Mech Phys Solids2012; 60(1): 120–138.
OgdenRWSteigmannDJ. Mechanics and electrodynamics of magneto- and electro-elastic materials (CISM Courses And Lectures Series, vol. 527). Wien: Springer, 2010.
14.
Ponte-CastañedaPGalipeauE. Homogenization-based constitutive models for magnetorheological elastomers at finite strain. J Mech Phys Solids2011; 59: 194–215.
15.
PucciESaccomandiG. On the controllable states of elastic dielectric and magnetoelastic solids. Int J Eng Sci1993; 31: 251–256.
16.
SalasEBustamanteR. Numerical solution of some boundary value problems in nonlinear magneto-elasticity. J Intel Mat Syst Str2015; 26(2): 156–171.
17.
SaxenaPHossainHSteinmannP. Nonlinear magneto-viscoelasticity of transversally isotropic magneto-active polymers. Proc R Soc A2014; 470: 20140082.
18.
SaxenaPHossainHSteinmannP. A theory of finite deformation magneto-viscoelasticity. Int J Solids Struct2014; 50: 3886–3897.
19.
SteigmannDJ. Equilibrium theory for magnetic elastomers and magnetoelastic membranes. Int J Nonlinear Mech2004; 39: 1193–1216.
20.
VuDKSteinmannP. Material and spatial motion problems in nonlinear electro- and magneto-elastostatics. Math Mech Solids2010; 15: 239–257.
21.
BednarekS. The giant magnetostriction in ferromagnetic composites within an elastomer matrix. Appl Phys A-Mater1999; 68: 63–67.
22.
BellanCBossisG. Field dependence of viscoelastic properties of MR elastomers. Int J Mod Phys B2002; 16: 2447–2453.
23.
BicaI. The influence of the magnetic field on the elastic properties of anisotropic magnetorheological elastomers. J Ind Eng Chem2012; 18: 1666–1669.
24.
BoczkowskaAAwietjanSF. Smart composites of urethane elastomers with carbonyl iron. J Mater Sci2009; 44: 4104–4111.
25.
BoczkowskaAAwietjanSF. Microstructure and properties of magnetorheological elastomers. In: BoczkowskaA, (ed.) Advanced elastomers-technology, properties and applications. Croatia: InTech, DOI: 10.5772/2784, 2012, 147–180.
26.
BöseH. Viscoelastic properties of silicone-based magnetorheological elastomers. Int J Mod Phys B2007; 21: 4790–4797.
27.
DengHGongX. Application of magnetorheological elastomer to vibration absorber. Commun Nonlinear Sci2008; 13: 1938–1947.
28.
GhafoorianfarNWangXGordaninejadF. On the sensing of magnetorheological elastomers. In: LynchJP. (eds.) Sensors and smart structures technologies for civil, mechanical, and aerospace systems (SPIE Proceedings, vol. 8692). SPIE Press, 2013, paper 869214.
29.
GinderJMClarkSMSchlotterWFNicholsME. Magnetostrictive phenomena in magnetorheological elastomers. Int J Mod Phys B2002; 16: 2412–2418.
30.
GordaninejadFWangWMysoreP. Behavior of thick magnetorheological elastomers. J Intel Mat Syst Str2012; 23: 1033–1039.
31.
JollyMRCarlsonJDMuñozBC. A model of the behaviour of magnetorheological materials. Smart Mater Struct1996; 5: 607–614.
GinderJMNicholsMEElieLDClarkSM. Controllable stiffness components based on magnetorheological elastomers. In: Wereley, NM (ed.) Smart structures and materials: smart structures and integrated systems (SPIE Proceedings, vol. 3985). SPIE Press, 2000, 418–425.
45.
GinderJMSchlotterWFNicholsME. Magnetorheological elastomers in tunable vibration absorbers. In: InmanDJ, (ed.) Smart structures and materials: damping and isolation (SPIE Proceedingsvol. 4331). SPIE Press, 2001, 103–110.
46.
KashimaSMiyasakaFHirataK. Novel soft actuator using magnetorheological elastomer. IEEE T Magn2012; 48: 1649–1652.
47.
LiWZhangX. Research and applications of MR elastomers. Recent Pat Mech Eng2008; 1: 161–166.
48.
YalcintasMDaiH. Vibration suppression capabilities of magnetorheological materials based adaptive structures. Smart Mater Struct2004; 13: 1–11.
49.
ZhuJTXuZDGuoYQ. Magnetoviscoelasticity parametric model of a MR elastomer vibration device. Smart Mater Struct2012; 21: 075034.
50.
VargaZFilipcseiGZrngiM. Smart composites with controlled anisotropy. Polymer2005; 47: 7779–7787.
51.
HossainMSaxenaPSteinmannP. Modelling the mechanical aspects of the curing process of magneto-sensitive elastomeric materials. Int J Solids Struct2015; 58: 257–269.
52.
HossainMChatzigeorgiouGMeraghniFSteinmannP. A multi-scale approach to model the curing process in magneto-sensitive polymeric materials. Int J Solids Struct2015; 69–70: 34–44.
53.
BustamanteRMerodioJ. Constitutive structure in coupled non-linear electro-elasticity: invariants descriptions and constitutive restrictions. Int J Nonlinear Mech2011; 46: 1315–1323.
54.
SpencerAJM. Theory of invariants. In: EringenAC, (ed.) Continuum physics, vol. 1. New York: Academic Press, 1971, 239–353.
55.
ZhengQS. Theory of representations for tensor functions. A unified invariant approach to constitutive equations. Appl Mech Rev1994; 47: 545–587.
56.
ShariffMHBM. Nonlinear transversely isotropic elastic solids: an alternative representation. Q J Mech Appl Math2008; 61: 129–149.
57.
ShariffMHBM. Physical invariants for nonlinear orthotropic solids. Int J Solids Struct2011; 48: 1906–1914.
58.
ShariffMHBM. Nonlinear orthotropic elasticity: only six invariants are independent. J Elasticity2013; 110: 237–241.
59.
ShariffMHBMBustamanteR. On the independence of strain invariants of two preferred direction nonlinear elasticity. Int J Eng Sci2015; 97: 18-25.
60.
BustamanteRShariffMHBM. A principal axis formulation for nonlinear magnetoelastic deformations: isotropic bodies. Eur J Mech A-Solids2015; 50: 17–27.
61.
OgdenRW. Non-linear Elastic Deformations. New York: Dover, 1997.
62.
TruesdellCAToupinR. The classical field theories. In: Handbuch der physik, vol. III/1. Berlin: Springer, 1960.
63.
KovetzA. Electromagnetic TheoryOxford: University Press, 2000.
64.
HossainMSteinmannP. More hyperelastic models for rubber-like materials: consistent tangent operator and comparative study. J Mech Behav Mater2013; 22(1-2): 27–50.
65.
TierstenHF. Coupled magnetomechanical equations for magnetically saturated insulators. J Math Phys1964; 5: 1298–1318.
