# A finite element implementation of the isotropic exponentiated Hencky-logarithmic model and simulation of the eversion of elastic tubes

## Abstract

We investigate a finite element formulation of the exponentiated Hencky-logarithmic model whose strain energy function is given by

\begin{aligned} W_\mathrm {eH}(\varvec{F}) = \dfrac{\mu }{k}\, e^k \left||\text{ dev }_n \log \varvec{U}\right||^2} + \dfrac{\kappa }{2 \hat{k}}\, e^\hat{k} [\text{ tr } (\log \varvec{U})]^2 }, \end{aligned

where $$\mu >0$$ is the (infinitesimal) shear modulus, $$\kappa >0$$ is the (infinitesimal) bulk modulus, k and $$\hat{k}$$ are additional dimensionless material parameters, $$\varvec{U}=\sqrt{\varvec{F}^T\varvec{F}}$$ is the right stretch tensor corresponding to the deformation gradient $$\varvec{F}$$, $$\log$$ denotes the principal matrix logarithm on the set of positive definite symmetric matrices, $$\text{ dev }_n \varvec{X} = \varvec{X}-\frac{\text{ tr } \varvec{X}}{n}\varvec{1}$$ and $$||\varvec{X} ||= \sqrt{\text{ tr }\varvec{X}^T\varvec{X}}$$ are the deviatoric part and the Frobenius matrix norm of an $$n\times n$$-matrix $$\varvec{X}$$, respectively, and $$\text{ tr }$$ denotes the trace operator. To do so, the equivalent different forms of the constitutive equation are recast in terms of the principal logarithmic stretches by use of the spectral decomposition together with the undergoing properties. We show the capability of our approach with a number of relevant examples, including the challenging “eversion of elastic tubes” problem.

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1. 1.

Note that, although every objective and isotropic energy function can be expressed in terms of the (material) logarithmic strain tensor $$\log \varvec{U}$$ (or the spatial strain tensor $$\log \varvec{V}$$), not every such energy can be expressed in terms of the logarithmic strain measures alone .

2. 2.

The invertibility of the mapping $$\varvec{B}\mapsto \varvec{\sigma }(\varvec{B})$$ also holds for the volumetric-isochorically decoupled representation of the neo-Hooke and Mooney–Rivlin energies, which are in use for slightly compressible materials like rubber.

3. 3.

The first use of the definition $$\nu = -\frac{(\log V)_{22}}{(\log V)_{11}}$$ is due to the famous German scientist Röntgen .

4. 4.

Truesdell remarks on the logarithmic strain that “[b]ecause of the difficulty of calculating the off-diagonal components of [$$\log \varvec{B}$$] in terms of the displacement gradient, Hencky’s theory is hard to use except in trivial cases”. [47, p. 202, (49.4)]

5. 5.

An everted rubber tube (shown in Fig. 15) was described by Truesdell as follows: “We see that the everted piece is a little longer than the other [identical, non-everted tube]. [...] With the naked eye we can see that the wall of the everted piece is a little thinner than it was originally. If we consider the part of the tube that lies a distance from the ends greater than one-fifth of the diameter, we can say that the everted piece, like the undeformed one, is very nearly a right-circular cylinder. We can idealize what we have seen by saying that an infinitely long, elastic, right-circular cylinder can be turned inside out so as to form another right-circular cylinder, having different radii”.

6. 6.

Note that for isotropic materials, the Kirchhoff stress can be obtained directly from the elastic energy as

\begin{aligned} \varvec{\tau } = \frac{\partial W}{\partial \log \varvec{V}} = \frac{\partial W}{\partial \varvec{V}}\cdot \varvec{V}, \end{aligned}

7. 7.

The restriction of the three-dimensional exponentiated Hencky energy to planar strain is not polyconvex, while (56) is. The difference stems from the definition of the two-dimensional isochoric stretches $$\overline{\lambda }_i = (\det \varvec{U})^{-1/2}\,\lambda _i$$.

8. 8.

The compressible hyperelastic models considered in  require the elastic energy potential W to be of the Valanis–Landel form 

\begin{aligned} W(\varvec{F}) = \sum _{i=1}^3 w(\lambda _i) , \end{aligned}
(66)

with a function $$w:[0,\infty )\rightarrow \mathbb {R}$$. In the past, energy functions of this type have been successfully applied in the incompressible case [19, 49], where they are in good agreement with experimental results. However, in the compressible case, an energy function of the form (66) without an additional volumetric energy term always implies zero lateral contraction for uniaxial stresses.

9. 9.

In the (unmodified) compressible case, the Abaqus updated Lagrangian implementation is not energy consistent, since the used Jaumann-rate of the Cauchy stress

\begin{aligned} \overset{\triangle }{\varvec{\sigma }} = \dot{\varvec{\sigma }} + \varvec{\sigma }\cdot \varvec{W} - \varvec{W}\cdot \varvec{\sigma } \end{aligned}

with $$\varvec{W}=\text{ skew }(\varvec{L}) = \frac{1}{2}(\varvec{L}-\varvec{L}^T)$$ is not energy consistent with the Cauchy stress. The principle of virtual work must be implemented correctly for any choice of stress and objective stress-rate.

## References

1. 1.

Anand L (1979) On H. Hencky’s approximate strain energy function for moderate deformations. J Appl Mech 46:78–82

2. 2.

Armero F (2004) Elastoplastic and viscoplastic deformations in solids and structures. In: Stein E, de Borst R, Hughes T (eds) Encyclopedia of computational mechanics, vol. 2: solids and structures. Wiley, London, pp 227–266

3. 3.

Bažant ZP, Vorel J (2012) Objective stress rates in finite strain of inelastic solid and their energy consistency. Technical report, McCormick School of Engineering and Applied Science, Northwestern University

4. 4.

Bažant ZP, Gattu M, Vorel J (2012) Work conjugacy error in commercial finite-element codes: its magnitude and how to compensate for it. Proc R Soc Lond A Math Phys Eng Sci 468:3047–3058

5. 5.

Chen YC, Haughton D (1997) Existence of exact solutions for the eversion of elastic cylinders. J Elast 49(1):79–88

6. 6.

Ehlers W, Eipper G (1998) The simple tension problem at large volumetric strains computed from finite hyperelastic material laws. Acta Mech 130(1):17–27

7. 7.

Flory PJ (1961) Thermodynamic relations for high elastic materials. Trans Faraday Soc 57:829–838

8. 8.

Gent AN (1996) A new constitutive relation for rubber. Rubber Chem Technol 69:59–61

9. 9.

Gent AN, Rivlin RS (1952) Experiments on the mechanics of rubber I: eversion of a tube. Proc Phys Soc B 65:118–121

10. 10.

Ghiba ID, Neff P, Martin RJ (2015) An ellipticity domain for the distortional Hencky logarithmic strain energy. Proc R Soc Lond A Math Phys Sci 471:2184 10.1098/rspa.2015.0510

11. 11.

Ghiba ID, Neff P, Šilhavỳ M (2015) The exponentiated Hencky-logarithmic strain energy. Improvement of planar polyconvexity. Int J Non-Linear Mech 71:48–51. https://doi.org/10.1016/j.ijnonlinmec.2015.01.009

12. 12.

Hencky H (1928) Über die Form des Elastizitätsgesetzes bei ideal elastischen Stoffen. Zeitschrift für technische Physik 9:215–220 . www.uni-due.de/imperia/md/content/mathematik/ag_neff/hencky1928.pdf

13. 13.

Hencky H (1929) Welche Umstände bedingen die Verfestigung bei der bildsamen Verformung von festen isotropen Körpern? Zeitschrift für Physik 55:145–155. www.uni-due.de/imperia/md/content/mathematik/ag_neff/hencky1929.pdf

14. 14.

Hencky H (1933) The elastic behavior of vulcanized rubber. Rubber Chem Technol 6(2):217–224 . https://www.uni-due.de/imperia/md/content/mathematik/ag_neff/hencky_vulcanized_rubber.pdf

15. 15.

Holzapfel GA (2000) Nonlinear solid mechanics. A continuum approach for engineering. Wiley, Chichester

16. 16.

Hughes TJR (1987) The finite element method. Prentice-Hall, Englewood-Cliffs

17. 17.

Ji W, Waas AM, Bažant ZP (2013) On the importance of work-conjugacy and objective stress rates in finite deformation incremental finite element analysis. J Appl Mech 80(4):041–124

18. 18.

Jog CS, Patil KD (2013) Conditions for the onset of elastic and material instabilities in hyperelastic materials. Arch Appl Mech 83(5):661–684. https://doi.org/10.1007/s00419-012-0711-8

19. 19.

Jones DF, Treloar LRG (1975) The properties of rubber in pure homogeneous strain. J Phys D Appl Phys 8(11):1285

20. 20.

Liang X, Tao F, Cai S (2016) Creasing of an everted elastomer tube. Soft Matter 12(37):7726–7730

21. 21.

Martin RJ, Neff P (2016) Minimal geodesics on $${\rm GL}\mathit{(n)}$$ for left-invariant, right-$${\rm O}(n)$$-invariantriemannian metrics. J Geom Mech 8(3):323–357 . arXiv:1409.7849

22. 22.

Miehe C (1994) Aspects of the formulation and finite element implementation of large strain isotropic elasticity. Int J Numer Methods Eng 37(12):1981–2004

23. 23.

Mihai LA, Neff P (2017) Hyperelastic bodies under homogeneous Cauchy stress induced by non-homogeneous finite deformations. Int J Non-Linear Mech 89:93–100. arXiv:1608.05040

24. 24.

Mihai LA, Neff P (2017) Hyperelastic bodies under homogeneous Cauchy stress induced by three-dimensional non-homogeneous deformations. Math Mech Solids. arXiv:1611.01772

25. 25.

Montella G, Govindjee S, Neff P (2016) The exponentiated Hencky strain energy in modeling tire derived material for moderately large deformations. J Eng Mater Technol 138(3):031,008–1–031,008–12

26. 26.

Nedjar B (2002) Frameworks for finite strain viscoelastic-plasticity based on multiplicative decompositions. Part II: computational aspects. Comput Methods Appl Mech Eng 191:1563–1593

27. 27.

Nedjar B (2007) An anisotropic viscoelastic fibre-matrix model at finite strains: continuum formulation and computational aspects. Comput Methods Appl Mech Eng 196:1745–1756

28. 28.

Nedjar B (2011) On a continuum thermodynamics formulation and computational aspects of finite growth in soft tissues. Int J Numer Methods Biomed Eng 27:1850–1866

29. 29.

Nedjar B (2016) On constitutive models of finite elasticity with possible zero apparent Poisson’s ratio. Int J Solids Struct 91:72–77

30. 30.

Nedjar B (2017) A coupled BEM-FEM method for finite strain magneto-elastic boundary-value problems. Comput Mech 59:795–807

31. 31.

32. 32.

Neff P, Eidel B, Martin RJ (2014) The axiomatic deduction of the quadratic Hencky strain energy by Heinrich Hencky. arXiv:1402.4027

33. 33.

Neff P, Eidel B, Martin RJ (2016) Geometry of logarithmic strain measures in solid mechanics. Arch Ration Mech Anal 222(2):507–572. https://doi.org/10.1007/s00205-016-1007-x. arXiv:1505.02203

34. 34.

Neff P, Ghiba ID (2016) The exponentiated Hencky-logarithmic strain energy. Part III: coupling with idealized isotropic finite strain plasticity. Contin Mech Thermodyn 28(1):477–487. https://doi.org/10.1007/s00161-015-0449-y

35. 35.

Neff P, Ghiba ID (2016) Loss of ellipticity for non-coaxial plastic deformations in additive logarithmic finite strain plasticity. Int J Non-Linear Mech 81:122–128. arXiv:1410.2819

36. 36.

Neff P, Ghiba ID, Lankeit J (2015) The exponentiated Hencky-logarithmic strain energy. Part I: constitutive issues and rank-one convexity. J Elast 121(2):143–234. https://doi.org/10.1007/s10659-015-9524-7

37. 37.

Neff P, Lankeit J, Ghiba ID, Martin RJ, Steigmann DJ (2015) The exponentiated Hencky-logarithmic strain energy. Part II: coercivity, planar polyconvexity and existence of minimizers. Z Angew Math Phys 66(4):1671–1693. https://doi.org/10.1007/s00033-015-0495-0

38. 38.

Neff P, Mihai LA (2017) Injectivity of the Cauchy-stress tensor along rank-one connected lines under strict rank-one convexity condition. J Elast 127(2):309–315. arXiv:1608.05247

39. 39.

Ogden RW (1997) Non-linear elastic deformations. Dover, New York

40. 40.

Plešek J, Kruisová A (2006) Formulation, validation and numerical procedures for Hencky’s elasticity model. Comput Struct 84:1141–1150

41. 41.

Reese S, Govindjee S (1998) A theory of finite viscoelasticity and numerical aspects. Int J Solids Struct 35(26–27):3455–3482

42. 42.

Richter H (1948) Das isotrope Elastizitätsgesetz. Z Angew Math Mech 28(7/8):205–209. https://www.uni-due.de/imperia/md/content/mathematik/ag_neff/richter_isotrop_log.pdf

43. 43.

Röntgen WC (1876) Über das Verhältniss der Quercontraction zur Längendilatation bei Kautschuk. Ann Phys 235(12):601–616

44. 44.

Schröder J, von Hoegen M, Neff P (2017) The exponentiated Hencky energy: Anisotropic extension and biomechanical applications. to appear in Comput Mech. arXiv:1702.00394

45. 45.

Simo JC (1998) Numerical analysis and simulation of plasticity. In: Ciarlet P, Lions J (eds) Handbook of numerical analysis, vol VI. North-Holland, Amsterdam, pp 183–499

46. 46.

Simo JC, Hughes TJR (1998) Computational inelasticity. Springer, New York

47. 47.

Truesdell C (1952) Mechanical foundations of elasticity and fluid dynamics. J Ration Mech Anal 1:125–300

48. 48.

Truesdell C (1978) Some challenges offered to analysis by rational thermomechanics: three lectures for the international symposium on continuum mechanics and partial differential equations. N-Holl Math Stud 30:495–603

49. 49.

Valanis KC, Landel RF (1967) The strain-energy function of a hyperelastic material in terms of the extension ratios. J Appl Phys 38(7):2997–3002

50. 50.

Vallée C (1978) Lois de comportement élastique isotropes en grandes déformations. Int J Eng Sci 16(7):451–457

51. 51.

Wegner JL, Haddow JB (2009) Elements of continuum mechanics and thermodynamics. Cambridge University Press, Cambridge

52. 52.

Wriggers P (2008) Nonlinear finite element methods. Springer, Berlin, Heidelberg

53. 53.

Zienkiewicz OC, Taylor RL (2000) The finite element method, vol 1, 5th edn. Butterworth-Heinemann, Oxford

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