The exponentiated Hencky-logarithmic strain energy. Part II: Coercivity, planar polyconvexity and existence of minimizers

Abstract

We consider a family of isotropic volumetric–isochoric decoupled strain energies

$$F \mapsto W_{\rm eH}(F):=\widehat{W}_{\rm eH}(U):=\left\{\begin{array}{lll}\frac{\mu}{k}\,e^{k\,\|{\rm dev}_n{\rm log} {U}\|^2}+\frac{\kappa}{2\hat{k}}\,e^{\hat{k}\,[{\rm tr}({\rm log} U)]^2}&\text{if}& \det\, F > 0,\\+\infty &\text{if} &\det F\leq 0,\end{array}\right.$$

based on the Hencky-logarithmic (true, natural) strain tensor log U, where μ > 0 is the infinitesimal shear modulus, \({\kappa=\frac{2\mu+3\lambda}{3} > 0}\) is the infinitesimal bulk modulus with \({\lambda}\) the first Lamé constant, \({k,\hat{k}}\) are dimensionless parameters, \({F=\nabla \varphi}\) is the gradient of deformation, \({U=\sqrt{F^T F}}\) is the right stretch tensor and \({{\rm dev}_n{\rm log} {U} ={\rm log} {U}-\frac{1}{n}{\rm tr}({\rm log} {U})\cdot{1\!\!1}}\) is the deviatoric part (the projection onto the traceless tensors) of the strain tensor log U. For small elastic strains, the energies reduce to first order to the classical quadratic Hencky energy

$$\begin{array}{ll}F\mapsto W{_{\rm H}}(F):=\widehat{W}_{_{\rm H}}(U)&:={\mu}\,\|{\rm dev}_n{\rm log} U\|^2+\frac{\kappa}{2}\,[{\rm tr}({\rm log} U)]^2,\end{array}$$

which is known to be not rank-one convex. The main result in this paper is that in plane elastostatics the energies of the family \({W_{_{\rm eH}}}\) are polyconvex for \({k\geq \frac{1}{3},\,\widehat{k}\geq \frac{1}{8}}\) , extending a previous finding on its rank-one convexity. Our method uses a judicious application of Steigmann’s polyconvexity criteria based on the representation of the energy in terms of the principal invariants of the stretch tensor U. These energies also satisfy suitable growth and coercivity conditions. We formulate the equilibrium equations, and we prove the existence of minimizers by the direct methods of the calculus of variations.

This is a preview of subscription content, access via your institution.

References

  1. 1

    Ball J.M.: Constitutive inequalities and existence theorems in nonlinear elastostatics. In: Knops, R.J. (eds) Herriot Watt Symposion: Nonlinear Analysis and Mechanics., volume 1, pp. 187–238. Pitman, London (1977)

  2. 2

    Ball J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rat. Mech. Anal. 63, 337–403 (1977)

    Article  MATH  Google Scholar 

  3. 3

    Ball J.M. et al.: Some open problems in elasticity. In: Newton, P. (eds) Geometry, Mechanics, and Dynamics, pp. 3–59. Springer, New-York (2002)

  4. 4

    Balzani D., Neff P., Schröder J., Holzapfel G.A.: A polyconvex framework for soft biological tissues. Adjustment to experimental data. Int. J. Solids Struct. 43(20), 6052–6070 (2006)

    Article  MATH  Google Scholar 

  5. 5

    Bîrsan M., Neff P., Lankeit J.: Sum of squared logarithms: an inequality relating positive definite matrices and their matrix logarithm. J. Inequal. Appl. 2013(1), 168 (2013)

    Article  Google Scholar 

  6. 6

    Borwein J.M., Vanderwerff J.D.: Convex Functions. Constructions, Characterizations and Counterexamples. Cambridge University Press, Cambridge (2010)

    Google Scholar 

  7. 7

    Bruhns O.T., Xiao H., Mayers A.: Constitutive inequalities for an isotropic elastic strain energy function based on Hencky’s logarithmic strain tensor. Proc. R. Soc. Lond. A 457, 2207–2226 (2001)

    Article  MATH  Google Scholar 

  8. 8

    Bruhns O.T., Xiao H., Mayers A.: Finite bending of a rectangular block of an elastic Hencky material. J. Elast. 66(3), 237–256 (2002)

    Article  MATH  Google Scholar 

  9. 9

    Busemann H., Ewald G., Shephard G.C.: Convex bodies and convexity on Grassmann cones. I–IV. Math. Ann. 151, 1–14 (1963)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10

    Chen Y.C.: Stability and bifurcation of homogeneous deformations of a compressible elastic body under pressure load. Math. Mech. Solids 1(1), 57–72 (1996)

    MathSciNet  MATH  Google Scholar 

  11. 11

    Dacorogna B.: Direct Methods in the Calculus of Variations, volume 78 of Applied Mathematical Sciences, 2 edn. Springer, Berlin (2008)

    Google Scholar 

  12. 12

    Dacorogna B., Douchet J., Gangbo W., Rappaz J.: Some examples of rank-one convex functions in dimension two. Proc. R. Soc. Edinb. Sect. A Math. 114(1–2), 135–150 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13

    Dacorogna B., Koshigoe H.: On the different notions of convexity for rotationally invariant functions. Ann. Fac. Sci. Toulouse 2, 163–184 (1993)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14

    Dacorogna B., Marcellini P.: Implicit Partial Differential Equations. Birkhäuser, Boston (1999)

    Google Scholar 

  15. 15

    Dacorogna B., Maréchal P. et al.: A note on spectrally defined polyconvex functions. In: Carozza, M. (eds) Proceedings of the Workshop “New Developments in the calculus of variations”, pp. 27–54. Edizioni Scientifiche Italiane, Napoli (2006)

  16. 16

    Davis C.: All convex invariant functions of hermitian matrices. Arch. Math. 8, 276–278 (1957)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17

    Hartmann S., Neff P.: Polyconvexity of generalized polynomial type hyperelastic strain energy functions for near incompressibility. Int. J. Solids Struct. 40(11), 2767–2791 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18

    Henann D.L., Anand L.: A large strain isotropic elasticity model based on molecular dynamics simulations of a metallic glass. J. Elast. 104(1–2), 281–302 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19

    Hencky H.: Über die Form des Elastizitätsgesetzes bei ideal elastischen Stoffen. Z. Techn. Physik 9, 215–220, http://www.uni--due.de/imperia/md/content/mathematik/ag_neff/hencky1928(see also the technical translation NASA TT–21602), (1928)

  20. 20

    Hencky H.: Das Superpositionsgesetz eines endlich deformierten relaxationsfähigen elastischen Kontinuums und seine Bedeutung für eine exakte Ableitung der Gleichungen für die zähe Flüssigkeit in der Eulerschen Form. Ann. der Physik, 2, 617–630, http://www.uni--due.de/imperia/md/content/mathematik/ag_neff/hencky_superposition1929, (1929)

  21. 21

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

  22. 22

    Hencky H.: The law of elasticity for isotropic and quasi-isotropic substances by finite deformations. J. Rheol. 2, 169–176, http://www.uni-due.de/imperia/md/content/mathematik/ag_neff/henckyjrheology31, (1931)

  23. 23

    Lankeit J., Neff P., Nakatsukasa Y.: The minimization of matrix logarithms: On a fundamental property of the unitary polar factor. Linear Alg. Appl. 449(0), 28–42 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24

    Lewis A.S.: Convex analysis on the Hermitian matrices. SIAM J. Optim. 6(1), 164–177 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25

    Lewis A.S.: The mathematics of eigenvalue optimization. Math. Program. 97(1–2 (B)), 155–176 (2003)

    MathSciNet  MATH  Google Scholar 

  26. 26

    Lewis, A.S., Overton M.L.: Eigenvalue optimization. In: Acta Numerica, Vol. 5, pp. 149–190. Cambridge University Press, Cambridge (1996)

  27. 27

    Mielke A.: Necessary and suffcient conditions for polyconvexity of isotropic functions. J. Conv. Anal. 12(2), 291–314 (2005)

    MathSciNet  MATH  Google Scholar 

  28. 28

    Neff P.: Mathematische Analyse multiplikativer Viskoplastizität. Ph.D. thesis, Technische Universität Darmstadt. Shaker Verlag, ISBN:3-8265-7560-1, http://www.uni-due.de/~hm0014/Download_files/cism_convexity08, Aachen, (2000)

  29. 29

    Neff P., Eidel B., Osterbrink F., Martin R.: A Riemannian approach to strain measures in nonlinear elasticity. C. R. Acad. Sci. 342, 254–257 (2014)

    Google Scholar 

  30. 30

    Neff, P., Eidel, B., Martin R.: Geometry, solid mechanics and logarithmic strain measures. The Hencky energy is the squared geodesic distance of the deformation gradient to SO(n) in any left-invariant, right-O(n)-invariant Riemannian metric on GL(n). in preparation (2015)

  31. 31

    Neff, P., Ghiba, I.D.: The exponentiated Hencky-logarithmic strain energy. Part III: Coupling with idealized isotropic finite strain plasticity. Preprint arXiv:1409.7555, special issue in honour of D.J. Steigmann, Cont. Mech. Thermodyn. (to appear, 2015)

  32. 32

    Neff, P., Ghiba, I.D., Lankeit J.: The exponentiated Hencky-logarithmic strain energy. Part I: Constitutive issues and rank–one convexity. J. Elasticity (to appear, 2015)

  33. 33

    Neff P., Nakatsukasa Y., Fischle A.: A logarithmic minimization property of the unitary polar factor in the spectral norm and the Frobenius matrix norm. SIAM J. Matrix Anal. 35, 1132–1154 (2014)

    MathSciNet  Article  Google Scholar 

  34. 34

    Ogden R.W.: Non-linear Elastic Deformations. Mathematics and Its Applications, 1st edn. Ellis Horwood, Chichester (1983)

    Google Scholar 

  35. 35

    Pipkin A.C.: Convexity conditions for strain-dependent energy functions for membranes. Arch. Rat. Mech. Anal. 121(4), 361–376 (1993)

    MathSciNet  Article  Google Scholar 

  36. 36

    Rosakis P.: Characterization of convex isotropic functions. J. Elast. 49, 257–267 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  37. 37

    Rosakis P., Simpson H.: On the relation between polyconvexity and rank-one convexity in nonlinear elasticity. J. Elast. 37, 113–137 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  38. 38

    Schröder J., Neff P.: Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions. Int. J. Solids Struct. 40(2), 401–445 (2003)

    Article  MATH  Google Scholar 

  39. 39

    Schröder J., Neff P., Balzani D.: A variational approach for materially stable anisotropic hyperelasticity. Int. J. Solids Struct. 42(15), 4352–4371 (2005)

    Article  MATH  Google Scholar 

  40. 40

    Schröder J., Neff P., Ebbing V.: Anisotropic polyconvex energies on the basis of crystallographic motivated structural tensors. J. Mech. Phys. Solids 56(12), 3486–3506 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  41. 41

    Šilhavý M.: The Mechanics and Thermomechanics of Continuous Media. Springer, Berlin (1997)

    Google Scholar 

  42. 42

    Šilhavý M. et al.: Convexity conditions for rotationally invariant functions in two dimensions. In: Sequeira, (eds) Applied Nonlinear Analysis, Kluwer Academic Publisher, New-York (1999)

  43. 43

    Šilhavý M.: On isotropic rank one convex functions. Proc. R. Soc. Edinb. 129, 1081–1105 (1999)

    Article  MATH  Google Scholar 

  44. 44

    Šilhavý M.: Rank 1 convex hulls of isotropic functions in dimension 2 by 2. Math. Bohemica 126(2), 521–529 (2001)

    MATH  Google Scholar 

  45. 45

    Šilhavý M.: Monotonicity of rotationally invariant convex and rank 1 convex functions. Proc. R. Soc. Edinb. 132(2), 419–435 (2002)

    Article  MATH  Google Scholar 

  46. 46

    Šilhavý M.: An O(n) invariant rank 1 convex function that is not polyconvex. Theoret. Appl. Mech. 28, 325–336 (2002)

    MathSciNet  Google Scholar 

  47. 47

    Šilhavý M.: On SO(n)-invariant rank 1 convex functions. J. Elast. 71, 235–246 (2003)

    Article  MATH  Google Scholar 

  48. 48

    Steigmann D.: Tension-field theory. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 429(1876), 141–173 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  49. 49

    Steigmann D.: Frame-invariant polyconvex strain-energy functions for some anisotropic solids. Math. Mech. Solids 8(5), 497–506 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  50. 50

    Steigmann D.: On isotropic, frame-invariant, polyconvex strain-energy functions. Q. J. Mech. Appl. Math. 56(4), 483–491 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  51. 51

    Templet G.J., Steigmann D.J.: On the theory of diffusion and swelling in finitely deforming elastomers. Math. Mech. Complex Syst. 1, 105–128 (2013)

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Ionel-Dumitrel Ghiba.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Mathematics Subject Classification

  • 74B20
  • 74G64
  • 35Q74
  • 35E10

Keywords

  • Finite isotropic elasticity
  • Logarithmic strain
  • Polyconvexity
  • Existence of minimizers
  • Plane elastostatics
  • Coercivity