Polyconvexity is an important mathematical condition imposed on a strain energy function. In particular, it is sufficient for the ellipticity of the constitutive equation and for the material stability and becomes especially crucial in the context of nonlinear elasticity. In combination with another condition referred to as coercivity, polyconvexity ensures existence of the global minimizer of the total elastic energy which implies a solution of a boundary value problem. While a great variety of polyconvex energies are known for isotropic and have recently been proposed for anisotropic elastic solids, there are, to the best of our knowledge, no results on polyconvexity for electro- and magneto-elastic materials. In the present paper, we extend the notion of polyconvexity to the coupled electro- and magneto-elastic response and formulate polyconvex free energy functions for electro- and magneto-sensitive elastomers. In analogy to the purely elastic response, these free energy functions will ensure the positive features of the constitutive equations mentioned above, although a strict mathematical proof of this fact should be supplied later. The proposed model is able to describe the electro- and magnetostriction and demonstrates good agreement with the corresponding experimental data.
VuDKSteinmannPPossartG. Numerical modelling of non-linear electroelasticity. Int J Numer Meth Eng2006; 70(6): 685–704.
19.
OtténioMDestradeMOgdenRW. Incremental magnetoelastic deformations with applications to surface instability. J Elast2008; 90: 19–42.
20.
DestradeMOgdenRW. On magneto-acoustic waves in finitely deformed elastic solids. Math Mech Solid2011; 16: 594–604.
21.
OgdenRW. Incremental elastic motions superimposed on afinite deformation in the presence of an electromagnetic field. Int J Non-Lin Mech2009; 44: 570–580.
GinderJMClarkSMSchlotterWF. Magnetostrictive phenomena in magnetorheological elastomers. Int J Mod Phys2002; 16: 2412–2418.
24.
BustamanteRMerodioJ. Constitutive structure in coupled non-linear electro-elasticity: Invariant descriptions and constitutive restrictions. Int J Non-Lin Mech2011; 46: 1315–1323.
25.
GalipeauEPonte CastañedaP. A finite strain constitutive model for magnetorheological elastomers: Magnetic torques and fiber rotations. J Mech Phys Solid2013; 61: 1065–1090.
26.
GalipeauEPonte CastañedaP. Giant field-induced strains in magnetoactive elastomer composites. Proc R Soc Lond A2013; 469: 20130385.
27.
Ponte CastañedaPGalipeauE. Homogenization-based constitutive models for magnetorheological elastomers at finite strain. J Mech Phys Solid2011; 59: 194–215.
28.
DanasKKankanalaSVTriantafyllidisN. Experiments and modeling of iron-particle-filled magnetorheological elastomers. J Mech Phys Solid2012; 60: 120–138.
29.
MorreyCB. Quasi-convexity and the lower semicontinuity of multiple integrals. Pacif J Math1952; 2: 25–53.
30.
AcerbiEFuscoN. Semicontinuity problems in the calculus of variations. Arch Rat Mech Anal1984; 86: 125–145.
31.
BallJMMuratF. W1,p -quasiconvexity and variational problems for multiple integrals. J Func Anal1984; 58: 225–253.
32.
BallJM. Convexity conditions and existence theorems in non-linear elasticity. Arch Rat Mech Anal1977; 63: 337–403.
DacorognaB. Direct methods in the calculus of variations (Applied Mathematical Sciences, vol. 78). Berlin: Springer, 1989.
35.
HartmannSNeffP. Polyconvexity of generalized polynomial-type hyperelastic strain energy functions for near-incompressibility. Int J Solid Struct2003; 40: 2767–2791.
36.
RaoultA. Non-polyconvexity of the stored energy function of a St. Venant-Kirchhoff material. Apl Mat1986; 6: 417–419.
37.
SchröderJNeffP. Invariant formulation of hyperelastic transverse isotropy. Int J Solid Struct2003; 40: 401–445.
38.
SteigmannDJ. Frame-invariant polyconvex strain-energy functions for some anisotropic solids. Math Mech Solid2003; 8: 497–506.
39.
BalzaniDNeffPSchröderJ. A polyconvex framework for soft biological tissues: Adjustment to experimental data. Int J Solid Struct2006; 43: 6052–6070.
40.
KambouchevNFernandezJRadovitzkyR. A polyconvex model for materials with cubic symmetry. Modell Simul Mater Sci Eng2007; 15: 451–467.
41.
SchröderJNeffPEbbingV. Anisotropic polyconvex energies on the basis of crystallographic motivated structural tensors. J Mech Phys Solid2008; 56: 3486–3506.
42.
HelfensteinJJabareenMMazzaE. On non-physical response in models for fiber-reinforced hyperelastic materials. Int J Solid Struct2010; 47(16): 2056–2061.
43.
ItskovMAkselN. A class of orthotropic and transversely isotropic hyperelastic constitutive models based on a polyconvex strain energy function. Int J Solid Struct2004; 41: 3833–3848.
44.
EhretAEItskovM. A polyconvex hyperelastic model for fiber-reinforced materials in application to soft tissues. J Mater Sci2007; 42: 8853–8863.
45.
EhretAEItskovM. Modeling of anisotropic softening phenomena: Application to soft biological tissues. Int J Plast2009; 25: 901.
46.
ItskovMEhretAEMavrilasD. A polyconvex anisotropic strain-energy function for soft collagenous tissues. Biomech Model Mechanobio2006; 5: 17–26.
47.
DorfmannLOgdenRW. Nonlinear theory of electroelastic and magnetoelastic interactions. New York, NY: Springer, 2014.
48.
SmithGF. On isotropic functions of symmetric tensors, skew-symmetric tensors and vectors. Int J Eng Sci1971; 9: 899–916.
49.
BoehlerJP. On irreducible representations for isotropic scalar functions. Zeitsch Angew Math Mech1977; 57: 323–327.
50.
PennisiSTrovatoM. On the irreducibility of Professor G.F. Smith’s representations for isotropic functions. Int J Eng Sci1987; 25(8): 1059–1065.
51.
BednarekS. The giant magnetostriction in ferromagnetic composites within an elastomer matrix. Appl Phys A: Mater Sci Process1999; 68(1): 63–67.
