We consider three different compressible versions of the conventional incompressible neo-Hookean material model. The different versions are not new and have been used in various model studies. They each give neo-Hookean behavior in an appropriate incompressible limit. The three versions each show some basic differences with respect to each other as regards the qualitative nature of the approach to the neo-Hookean limit. The purpose of this study is to exhibit these differences.
SussmanTBatheKJ. A finite element formulation for nonlinear incompressible elastic and inelastic analysis. Comput Struct1987; 26: 357–409.
2.
WeissJMakerBNGovindjeeS. Finite element implementation of incompressible, transversely isotropic hyperelasticity. Comput Methods Appl Mech Eng1996; 135: 107–128.
3.
LevinsonMBurgessI. A comparison of some simple constitutive relations for slightly compressible rubber-like materials. Int J Mech Sci1971; 13: 563–572.
4.
BoyceMCArrudaEM. Constitutive models of rubber elasticity: a review. Rubber Chem Technol2000; 73: 504–523.
5.
HorganCOMurphyJG. A generalization of Hencky’s strain-energy density to model the large deformations of slightly compressible solid rubbers. Mech Mater2009; 41: 943–950.
6.
RivlinRS. Large elastic deformations of isotropic materials, I: Fundamental concepts. Philos Trans R Soc Lond A1948; 240: 459–490.
7.
KuboR. Large elastic deformation of rubber. J Phys Soc Japan1948; 3: 312–317.
8.
TruesdellCNollW. The Nonlinear Field Theories of Mechanics (Handbuch der Physik, vol. III/3) Berlin: Springer-Verlag, 1965.
9.
RajagopalKRWinemanAS. A constitutive equation for nonlinear solids which undergo deformation induced microstructural changes. Int J Plast1992; 8: 385–395.
10.
QiuGYPenceTJ. Remarks on the behavior of simple directionally reinforced incompressible nonlinearly elastic solids. J Elast1997; 49: 1–30.
11.
Ben-AmarMGorielyA. Growth and instability in elastic tissues. J Mech Phys Solids2005; 53: 2284–2319.
12.
HaritonIDebottonGGasserTC. Stress-modulated collagen fiber remodeling in a human carotid bifurcation. J Theor Biol2007; 248: 460–470.
13.
BallJM. Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Philos Trans R Soc Lond A1982; 306: 557–611.
14.
TsaiHPenceTJKirkinisE. Swelling induced finite strain flexure in a rectangular block of an isotropic elastic material. J Elast2004; 75: 69–89.
15.
GuoZCanerFPengX. On constitutive modelling of porous neo-Hookean composites. J Mech Phys Solids2008; 56: 2338–2357.
16.
RajagopalKRSrinivasaARWinemanAS. On the shear and bending of a degrading polymer beam. Int J Plast2007; 23: 1618–1636.
17.
BaekSValentinAHumphreyJ. Biochemomechanics of cerebral vasospasm and its resolution: II. constitutive relations and model simulations. Ann Biomed Eng2007; 35: 1498–1509.
18.
DemirkoparanHPenceTJ. Swelling of an internally pressurized nonlinearly elastic tube with fiber reinforcing. Int J Solids Struct2007; 44: 4009–4029.
19.
PenceTJTsaiH. Swelling-induced cavitation of elastic spheres. Math Mech Solids2006; 11: 527–551.
20.
BaekSPenceTJ. On swelling induced degradation of fiber reinforced polymers. Int J Eng Sci2009; 47: 1100–1109.
21.
BaekSPenceTJ. On mechanically induced degradation of fiber-reinforced hyperelastic materials. Math Mech Solids2011; 16: 406–434.
22.
BonetJWoodR. Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge: Cambridge University Press, 1997.
23.
PucciESaccomandiG. On universal relations in continuum mechanics. Continuum Mech Thermodyn1997; 9: 61–72.
24.
HorganCOSaccomandiG. Simple torsion of isotropic, hyperelastic, incompressible materials with limiting chain extensibility. J Elast1999; 56: 159–170.
25.
WinemanA. Some results for generalized neo-Hookean elastic materials. Int J Non Linear Mech2005; 40: 271–279.
26.
CurriePK. The attainable region of strain-invariant space for elastic materials. Int J Non Linear Mech2004; 39: 833–842.
27.
CiarletPGGeymonatG. Sur les lois de comportement en élasticité non linéaire compressible. CR Acad Sci Paris Sér II1982; 295: 423–426.
28.
OgdenRW. Non-linear Elastic Deformations. New York: Dover, 1984.
29.
CiarletP. Mathematical Elasticity, vol. I. Amsterdam: Elsevier, 1988.
30.
HolzapfelGA. Nonlinear Solid Mechanics: a Continuum Approach for Engineering. Chichester: Wiley, 2000.
31.
ScottNH. The incremental bulk modulus, Young’s modulus and Poisson’s ratio in nonlinear isotropic elasticity: Physically reasonable response. Math Mech Solids2007; 12: 526–542.
32.
FloryP. Thermodynamic relations for high elastic materials. Trans Faraday Soc1961; 57: 829–938.
33.
OgdenR. Large deformation isotropic elasticity: on the correlation of theory and experiment for compressible rubberlike solids. Proc R Soc Lond Ser A, Math Phys Sci1972; 328: 567–583.
34.
EhlersWEipperG. The simple tension problem at large volumetric strains computed from finite hyperelastic material laws. Acta Mech1998; 130: 17–27.
35.
HorganCOMurphyJG. On the volumetric part of strain-energy functions used in the constitutive modeling of slightly compressible solid rubbers. Int J Solids Struct2009; 46: 3078–3085.
36.
PenceTJ. Distortion of anisotropic hyperelastic solids under pure pressure loading: compressibility, incompressibility and near incompressibility. J Elasticity2014; 2: 251–273.
37.
DollSSchweizerhofK. On the development of volumetric strain energy functions. J Appl Mech2000; 67: 17–21.
38.
HartmannSNeffP. Polyconvexity of generalized polynomial-type hyperelastic strain energy functions for near-incompressibility. Int J Solids Struct2003; 40: 2767–2791.
39.
BlatzPJ. On the thermostatic behavior of elastomers. In: Polymer Networks, Structure and Mechanical Properties. New York: Plenum Press, 1971.
40.
GorbYWaltonJW. Dependence of the frequency spectrum of small amplitude vibrations superimposed on finite deformations of a nonlinear, cylindrical elastic body on residual stress. Int J Eng Sci2010; 48: 1289–1312.
41.
GouKJoshiSWaltonJ. Recovery of material parameters of soft hyperelastic tissue by an inverse spectral technique. Int J Eng Sci2012; 56: 1–16.
42.
BlatzPKoW. Application of finite elastic theory to the deformation of rubbery materials. Trans Soc Rheol1962; 6: 223–251.
43.
BeattyMFStalnakerDO. The Poisson function of finite elasticity. J Appl Mech1986; 53: 807–813.
