Henky's logarithmic strain and dual stress-strain and strain-stress relations in isotropic finite hyperelasticity

International Journal of Solids and Structures

Volumn 40, Issue 6, 2003, Pages 1455-1463

  Xiao, Heng ^a   Chen, Liang Sen ^b  

^a RUHR UNIVERSITY BOCHUM   (Germany)

^b NANCHANG UNIVERSITY   (China)

Author keywords

Duality; Finite strain; Hyperelasticity; Strain stress relation; Stress strain relation

Indexed keywords

ELASTOPLASTICITY; STRAIN; STRESS ANALYSIS;

STRAIN-STRESS RELATIONSHIP;

ELASTICITY;

STRESS-STRAIN ANALYSIS;

EID: 0037372380     PISSN: 00207683     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0020-7683(02)00653-4     Document Type: Article

References (44)

1. Anand, L., 1979. On H. Henky's approximate strain-energy function for moderate deformations. J. Appl. Mech. 46, 78-82.
2. Anand, L., 1986. Moderate deformations in extension-torsion of incompressible isotropic elastic materials. J. Mech. Phys. Solids 34, 293-304.
3. Bažant, Z.P., 1998. Easy-to-compute tensors with symmetric inverse approximating Hencky finite strain and its rate. J. Engng. Mater. Techn. 120, 131-136.
4. Blume, J.A., 1992. On the form of the inverted stress-strain law for isotropic hyperelastic solids. Int. J. Non-linear Mech. 27, 413-421.
5. de Boer, R., 1967. Die elastisch-plastische Biegung eines Plattenstreifens aus inkompressiblem Werkstoff bei endlichen Formänderungen. Ing.-Arch. 36, 145-154.
6. de Boer, R., Bruhns, O.T., 1969. Zur Berechnung der Eigenspannungen bei einem durch endliche Biegung verformten inkompressiblen Plattenstreifen. Acta Mech., 146-159.
7. Bonet, J., Wood, R.D., 1997. Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press, Cambridge.
8. Bruhns, O.T., 1970. Die Berücksichtigung einer isotropen Werkstoffverfestigung bei der elastisch-plastischen Blechbiegung mit endlichen Formänderungen. Ing.-Arch. 39, 63-72.
9. Bruhns, O.T., 1971. Elastoplastische Scheibenbiegung bei endlichen Formänderungen. Z. Angew. Math. Mech. 51, T101-T103.
10. Bruhns, O.T., Thermann, K., 1969. Elastisch-plastische Biegung eines Plattenstreifens bei endlichen Formänderungen. Ing.-Arch. 38, 141-152.
11. Bruhns, O.T., Xiao, H., Meyers, A., 1999. Self-consistent Eulerian rate type elastoplasticity models based upon the logarithmic stress rate. Int. J. Plasticity 15, 479-520.
12. Bruhns, O.T., Xiao, H., Meyers, A., 2001a. The Hencky model of elasticity: a study on Poynting effect and stress response in torsion of tubes and rods. Arch. Mech. 52, 489-509.
13. Bruhns, O.T., Xiao, H., Meyers, A., 2001b. A self-consistent Eulerian rate type model for finite deformation elastoplasticity with isotropic damage. Int. J. Solids Struct. 38, 657-683.
14. Chiskis, A., Parnes, R., 2000. Linear stress-strain relations in nonlinear elasticity. Acta Mech. 146, 109-113.
15. Curnier, A., Rakotomanana, L., 1991. Generalized strain and stress measures: critical survey and new results. Engng. Trans. 39, 461-538.
16. Eterovic, A.L., Bathe, K.J., 1990. A hyperelastic-based large strain elasto-plastic constitutive formulation with combined isotropic-kinematic hardening using the logarithmic stress and strain measures. Int. J. Num. Meth. Engng. 30, 1099-1115.
17. Fitzjerald, J.E., 1980. A tensorial Hencky measure of strain and strain rate for finite deformations. J. Appl. Phys. 51, 5111-5115.
18. Freed, A.D., 1995. Natural strain. J. Engng. Mater. Techn. 117, 379-385.
19. Gurtin, M.E., 1981. An Introduction to Continuum Mechanics. Academic Press, New York.
20. Hencky, H., 1928. Über die Form des Elastizitätsgesetzes bei ideal elastischen Stoffen. Z. Techn. Phys. 9, 215-220, 457.
21. Hencky, H., 1931. The law of elasticity for isotropic and quasi-isotropic substances by finite deformations. J. Rheology 2, 169-176.
22. Hencky, H., 1933. The elastic behaviour of vulcanized rubber. Rubber Chem. Techn. 6, 217-224.
23. Hill, R., 1968. Constitutive inequalities for simple materials. J. Mech. Phys. Solids 16, 229-242, 315-322.
24. Hill, R., 1970. Constitutive inequalities for isotropic elastic solids under finite strain. Proc. R. Soc. Lond. A 326, 131-147.
25. Hill, R., 1978. Aspect of invariance in solid mechanics. Adv. Appl. Mech. 18, 1-75.
26. Hoger, A., 1987. The stress conjugate to logarithmic strain. Int. J. Solids Struct. 23, 1645-1656.
27. Kollmann, F.G., Sansour, C., 1997. Viscoplastic shells: theory and numerical analysis. Arch. Mech. 49, 477-511.
28. Miehe, C., 1998. A formulation of finite elastoplasticity based on dual co- and contravariant eigenvector triads normalized with respect to a plastic metric. Comp. Meth. Appl. Mech. Engng. 159, 223-260.
29. Miehe, C., Stein, E., Wagner, W., 1994. Associative multiplicative elasto-plasticity: formulation and aspects of the numerical implementation including stability analysis. Comput. Struct. 52, 969-978.
30. Ogden, R.W., 1970. Compressible isotropic elastic solids under finite strain-constitutive inequalities. Quart. J. Mech. Appl. Math. 23, 457-468.
31. Ogden, R.W., 1984. Non-Linear Elastic Deformations. Ellis Horwood, Chichester.
32. Raniecki, B., Nguyen, H.V., 1984. Isotropic elasto-plastic solids at finite strain and arbitrary pressure. Arch. Mech. 36, 687-704.
33. Sansour, C., 2001. On the dual variable of the logarithmic strain tensor, the dual variable of the Cauchy stress tensor, and related issues. Int. J. Solids Struct. 38, 9221-9232.
34. Schieck, B., Stumpf, H., 1995. The appropriate corotational rate, exact formula for the plastic spin and constitutive model for finite elastoplasticity. Int. J. Solids Struct. 32, 3643-3667.
35. Šilhaý M., 1997. The Mechanics and Thermodynamics of Continuous Media. Springer, Berlin.
36. Stören, S., Rice, J.R., 1975. Localized necking in thin sheets. J. Mech. Phys. Solids 23, 421-441.
37. Stumpf, H., Schieck, B., 1994. Theory and analysis of shells undergoing finite elastic-plastic strains and rotations. Acta Mech. 106, 1-21.
38. Truesdell, C., Noll, W., 1965. The non-linear field theories of mechanics. In: Flügge, S. (Ed.), Handbuch der Physik, vol. III/3. Springer, Berlin.
39. Weber, G., Anand, L., 1990. Finite deformation constitutive equations and a time integration procedure for isotropic, hyperelastic-viscoplastic solids. Comp. Meth. Appl. Mech. Engng. 79, 173-202.
40. Xiao, H., Bruhns, O.T., Meyers, A., 1997a. Hypo-elasticity model based upon the logarithmic stress rate. J. Elasticity 47, 51-68.
41. Xiao, H., Bruhns, O.T., Meyers, A., 1997b. Logarithmic strain, logarithmic spin and logarithmic rate. Acta Mech. 124, 89-105.
42. Xiao, H., Bruhns, O.T., Meyers, A., 1998. Strain rates and material spins. J. Elasticity 52, 1-41.
43. Xiao, H., Bruhns, O.T., Meyers, A., 1999. Existence and uniqueness of the integrable-exactly hypoelastic equation τ@̊, =(D)I+2GD and its significance to finite inelasticity. Acta Mech. 138, 31-50.
44. Xiao, H., Bruhns, O.T., Meyers, A., 2000. The choice of objective rates in finite elastoplasticity: general results on the uniqueness of the logarithmic rate. Proc. R. Soc. Lond. A 456, 1865-1882.