The governing equations for plane deformations of isotropic incompressible hyperelastic materials are highly nonlinear, and consequently, very few exact solutions are known. As far as we are aware, the Varga material is the only material in which results of any generality are known. In this paper, we show that for the Varga material, the full governing equations are in fact linearizable.
VargaOH.Stress-strain behaviour of elastic materials. New York: Interscience, 1966.
2.
DickieRASmithTL.Viscoelastic properties of rubber vulcanizates under large deformations in equal biaxial tension, pure shear and simple tension. Trans Soc Rheol1971; 36: 91–110.
3.
HoldenJT.A class of exact solutions for finite plane strain deformations of a particular elastic material. Appl Sci Res1968; 19: 171–181.
4.
HillJM.Buckling of long thick-walled circular cylindrical shells of isotropic incompressible hyperelastic materials under uniform external pressure. J Mech Phys Solids1975; 23: 99–112.
5.
HillJMArrigoDJ.New families of exact solution for finitely deformed incompressible elastic materials. IMA J Appl Math1995; 54: 109–123.
6.
HillJMArrigoDJ.Transformations and equation reductions in finite elasticity I: plane strain deformations. Math Mech Solids1996; 1: 155–175.
7.
HillJMArrigoDJ.Transformations and equation reductions in finite elasticity III: a general integral for plane strain deformations. Math Mech Solids1999; 4: 3–15.
8.
HillJM.Exact integrals and solutions for finite deformations of the incompressible Varga elastic materials. In: FuYCOgdenR (eds) Nonlinear elasticity theory and applications (London mathematical society lecture note series no. 283). Cambridge: Cambridge University Press, 2001, pp. 160–200.
9.
LieS.Klassifikation und integration von gewohnlichen differentialgleichen zwischen , die eine gruppe von transformationen gestatten. Math Ann1888; 32: 213–228.
10.
ArrigoDJ.Symmetry analysis of differential equations: an introduction. Hoboken, NJ: John Wiley & Sons, 2015.
11.
BlumanGKumeiS.Symmetries and differential equations. New York: Springer-Verlag, 1989.
12.
BlumanGAncoSC.Symmetry and integration methods for differential equations. New York: Springer-Verlag, 2002.
13.
ChernihaRSerovMPliukhinO.Nonlinear reaction-diffusion-convection equations: lie and conditional symmetry, exact solutions and their applications. Boca Raton, FL: CRC Press, 2018.
14.
OlverP.Applications of lie groups to differential equations. New York: Springer-Verlag, 1986.
15.
BlumanGKumeiS.When nonlinear differential equations are equivalent to linear differential equations. SIAM J Appl Math1982; 42: 1157–1173.
ArrigoDJHillJM.Transformations and equation reductions in finite elasticity II: plane stress and axially symmetric deformations. Math Mech Solids1996; 1: 172–192.