Constitutive properties for isotropic energies in ideal nonlinear elasticity for solid materials: Numerical evidences for invertibility and monotonicity in different stress–strain pairs
Restricted accessResearch articleFirst published online March, 2026
Constitutive properties for isotropic energies in ideal nonlinear elasticity for solid materials: Numerical evidences for invertibility and monotonicity in different stress–strain pairs
As a service for the solid mechanics community, we gather in this paper constitutive properties of a collective list of isotropic elastic energies for compressible materials. Of interest to us are the invertibility and monotonicity of certain stress–strain pairs. The calculations are done numerically by our own evaluation algorithm and presented in a yes/no table. Such an overview has been missing up to now. It is intended to expand the table with further energies as time goes on and updates will be found on arxive.
NeffPGhibaIDLankeitJ. The exponentiated Hencky-logarithmic strain energy. Part I: Constitutive issues and rank–one convexity. J Elast2015; 121: 143–234.
2.
MartinRJVossJGhibaID, et al. Monotonicity of isotropic tensor functions on the set of symmetric matrices: Hill’s generalization of the Chandler-Davis-Lewis convexity theorem revised. in preparation.
3.
OgdenRW. Non-linear elastic deformations. Mathematics and its Applications. Chichester: Ellis Horwood, p. 349, 1983.
4.
HillR. On constitutive inequalities for simple materials – I. J Mech Phys Solids1968; 16(4): 229–242.
5.
HillR. On constitutive inequalities for simple materials II. J Mech Phys Solids1968; 16(5): 315–322.
6.
HillR. Constitutive inequalities for isotropic elastic solids under finite strain. Proc R Soc Lond Ser A Math Phys Sci1970; 314(1519): 457–472.
7.
GhibaIDGmeinederFHolthausenS, et al. The Biot stress–right stretch relation for the compressible Neo-Hooke-Ciarlet-Geymonat model and Rivlin’s cube problem. Commun Math Anal Appl2024; 4(1): 112–150.
8.
BendixsonI. Sur les racines d’une équation fondamentale. Acta Math1902; 25: 359–365.
9.
GaoDYNeffPRoventaI, et al. On the convexity of nonlinear elastic energies in the right Cauchy-Green tensor. J Elast2015; 127: 303–308.
10.
EisenträgerSMaurerLJuhreD, et al. Implementation of isotropic hyperelastic material models: a “template” approach. Acta Mech2025; 236: 1899–1934.
11.
MooneyM. A theory of large elastic deformation. J Appl Phys1940; 11(9): 582–592.
12.
RivlinR. Large elastic deformations of isotropic materials. IV Further developments of the general theory. Philos Trans R Soc Lond A Math Phys Eng Sci1948; 241(835): 379–397.
13.
GlügeRKalischJ. Graphical representations of the regions of rank-one-convexity of some strain energies. Technische Mechanik2012; 32: 227–237.
14.
SimoJCPisterKS. Remarks on rate constitutive equations for finite deformation problems: computational implications. Comput Methods Appl Mech Eng1984; 46(2): 201–215.
15.
NeffPEidelBMartinRJ. Geometry of logarithmic strain measures in solid mechanics. Arch Ration Mech Anal2016; 222: 507–572.
16.
NeffPGrabanKSchweickertE, et al. The axiomatic introduction of arbitrary strain tensors by Hans Richter–a commented translation of strain tensor, strain deviator and stress tensor for finite deformations. Math Mech Solids2020; 25(5): 1060–1080.
17.
GrabanKSchweickertEMartinRJ, et al. A commented translation of Hans Richter’s early work “the isotropic law of elasticity”. Math Mech Solids2019; 24(8): 2649–2660.
18.
NeffPMünchIMartinRJ. Rediscovering GF Becker’s early axiomatic deduction of a multiaxial nonlinear stress–strain relation based on logarithmic strain. Math Mech Solids2016; 21(7): 856–911.
19.
FungYC. Inversion of a class of nonlinear stress-strain relationships of biological soft tissues. J Biomech Eng1979; 101(1): 23–27.
20.
BeattyM. Topics in finite elasticity: hyperelasticity of rubber, elastomers, and biological tissues – with examples. Appl Mech Rev1987; 40(12): 1699–1734.
21.
SaravananU. On the adequacy of the existing restrictions on the constitutive relations to ensure reasonable elastic response of compressible bodies. Mech Res Commun2011; 38(2): 123–125.
22.
LurieAI. Non-linear theory of elasticity. Amsterdam: Elsevier, 2012.
23.
GentAN. A new constitutive relation for rubber. Rubber Chem Technol1996; 69(1): 59–61.
24.
WollnerMPHolzapfelGANeffP. In search of constitutive conditions in isotropic hyperelasticity: polyconvexity versus true-stress-true-strain monotonicity. J Mech Phys Solids2026; 209: 106465.
25.
Anssari-BenamA. Large isotropic elastic deformations: on a comprehensive model to correlate the theory and experiments for incompressible rubber-like materials. J Elast2023; 153(2): 219–244.
26.
CarrollMM. A strain energy function for vulcanized rubbers. J Elast2011; 103: 173–187.
27.
GaoYC. Large deformation field near a crack tip in rubber-like material. Theor Appl Fract Mech1997; 26(3): 155–162.
28.
MurnaghanFD. Finite deformations of an elastic solid. Am J Math1937; 59(2): 235–260.
29.
HartmannSNeffP. Polyconvexity of generalized polynomial type hyperelastic strain energy functions for near incompressibility. Int J Solids Struct2003; 40(11): 2767–2791.
30.
KorobeynikovSNLarichkinAYNeffP. Two Types of Compressible Isotropic Neo-Hookean Material Models. Springer Briefs in Applied Sciences and Technology, Springer, Cham, 2026. https://doi.org/10.1007/978-3-032-06050-1
31.
MartinRNeffP. Some remarks on monotonicity of primary matrix functions on the set of symmetric matrices. Arch Appl Mech2015; 85: 1761–1778.
32.
KrawietzA. A comprehensive constitutive inequality in finite elastic strain. Arch Ration Mech Anal1975; 58: 127–149.
33.
OgdenRW. Inequalities associated with the inversion of elastic stress-deformation relations and their implications. Math Proc Camb Philos Soc1977; 81(02): 313–324.
34.
d’AgostinoMVHolthausenSBernardiniD, et al. A constitutive condition for idealized isotropic Cauchy elasticity involving the logarithmic strain. J Elast2025; 157: 23.
35.
NeffPHolthausenSd’AgostinoMV, et al. Hypo-elasticity, Cauchy-elasticity, corotational stability and monotonicity in the logarithmic strain. J Mech Phys Solids2025; 202: 106074.
36.
NeffPHolthausenSKorobeynikovSN, et al. A natural requirement for objective corotational rates–on structure preserving corotational rates. Acta Mech2024; 236: 2657–2689.
37.
JogCSPatilKD. Conditions for the onset of elastic and material instabilities in hyperelastic materials. Arch Appl Mech2013; 83: 1–24.
38.
NeffPHusemannNJHolthausenS, et al. Rate-form equilibrium for an isotropic Cauchy-elastic formulation: Part I: modeling. J Nonlinear Sci2025; 36(1): 8.
39.
BlesgenTHusemannNJHolthausenS, et al. Rate-form equilibrium for an isotropic Cauchy-elastic formulation: part II: local existence of solutions. In preparation, 2026.
40.
NeffPHusemannNJNguetcho TchakoutioAS, et al. The corotational stability postulate: positive incremental Cauchy stress moduli for diagonal, homogeneous deformations in isotropic nonlinear elasticity. Int J NonLin Mech2025; 174: 105033.
41.
NeffPHusemannNJHolthausenS, et al. Truesdell’s Hauptproblem: On constitutive stability in idealized isotropic nonlinear elasticity and a 500 euro challenge. IUTAM Symposium Computational Fracture Mechanics in Multi Field Problems, Physikzentrum Bad Honnef, 8– 13 December 2024, 2024.
42.
BallJM. Convexity conditions and existence theorems in nonlinear elasticity. Arch Ration Mech Anal1977; 63: 337–403.
43.
KnowlesJKSternbergE. On the failure of ellipticity of the equations for finite elastostatic plane strain. Arch Ration Mech Anal1976; 63(4): 321–336.
44.
KnowlesJKSternbergE. On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics. J Elast1978; 8(4): 329–379.
45.
SethBR. Generalized strain measure with applications to physical problems. In: RainerMA (ed.) Second-order effects in elasticity, plasticity and fluid dynamics. Oxford: Pergamon Press, 1961, pp. 162–172.
