Skip to main content
SpringerLink
Log in

Cosserat point element (CPE) for finite deformation of orthotropic elastic materials

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

An eight node brick Cosserat point element (CPE) has been developed for the numerical solution of three-dimensional problems of hyperelastic nonlinear orthotropic elastic materials. In the Cosserat approach, a strain energy function for the CPE is proposed which satisfies restrictions due to a nonlinear form of the patch test. Part of the strain energy of the CPE is characterized by a three-dimensional strain energy function that depends on physically based nonlinear orthotropic invariants. Special attention has been focused on developing closed form expressions for constitutive coefficients in another part of the strain energy that characterizes the response to inhomogeneous deformations appropriate for orthotropic material response. A number of example problems are presented which demonstrate that the CPE is a robust user friendly element for finite deformations of orthotropic elastic materials, which does not exhibit unphysical locking or hourglassing for thin structures or nearly incompressible materials.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. ABAQUS, Version 6.9EF1. ABAQUS, Providence

  2. Belytschko T, Bindemann LP (1986) Assumed strain stabilization of the 4-node quadrilateral with 1-point quadrature for nonlinear problems. Comput Methods Appl Mech Eng 54: 279–301

    Article  MATH  Google Scholar 

  3. Belytschko T, Bindemann LP (1993) Assumed strain stabilization of the eight node hexahedral element. Comput Methods Appl Mech Eng 105(2): 225–260

    Article  MATH  Google Scholar 

  4. Belytschko T, Ong JS-J, Liu WK, Kennedy JM (1984) Hourglass control in linear and nonlinear problems. Comput Methods Appl Mech Eng 43: 251–276

    Article  MATH  Google Scholar 

  5. Bonet J, Bhargava P (1995) A uniform deformation gradient hexahedron element with artificial hourglass control. Int J Numer Methods Eng 38(16): 2809–2828

    Article  MATH  Google Scholar 

  6. César de Sá JMA, Areias PMA, Jorge RMN (2001) Quadrilateral elements for the solution of elasto-plastic finite strain problems. Int J Numer Methods Eng 51(8): 883–917

    Article  MATH  Google Scholar 

  7. Cowin SC, Van Buskirk WC (1986) Thermodynamics restrictions on the elastic constants of bone. J Biomech 19(1): 85–87

    Article  Google Scholar 

  8. Crisfield MA, Moita GF, Lyons LPR, Jelenic G (1995) Enhanced lower-order element formulations for large strains. Comput Mech 17(1–25): 62–73

    Article  MATH  Google Scholar 

  9. Fredriksson M, Ottosen NS (2007) Accurate eight-node hexahedral element. Int J Numer Methods Eng 72(6): 631–657

    Article  MathSciNet  MATH  Google Scholar 

  10. Green AE, Adkins JE (1960) Large elastic deformations and non-linear continuum mechanics. Clarendon Press, Oxford

    MATH  Google Scholar 

  11. Hutter R, Hora P, Niederer P (2000) Total hourglass control for hyperelastic materials. Comput Methods Appl Mech Eng 189(3): 991–1010

    Article  MathSciNet  MATH  Google Scholar 

  12. Jabareen M, Rubin MB (2007) Hyperelasticity and physical shear buckling of a block predicted by the Cosserat point element compared with inelasticity and hourglassing predicted by other element formulations. Comput Mech 40(3): 447–459

    Article  MathSciNet  MATH  Google Scholar 

  13. Jabareen M, Rubin MB (2007) An improved 3-D Cosserat brick element for irregular shaped elements. Comput Mech 40: 979–1004

    Article  MathSciNet  MATH  Google Scholar 

  14. Jabareen M, Rubin MB (2008) A generalized Cosserat point element (CPE) for isotropic nonlinear elastic materials including irregular 3-D brick and thin structural elements. J Mech Mater Struct 3(8): 1465–1498

    Article  Google Scholar 

  15. Jabareen M, Rubin MB (2008) Modified torsion coefficients for a 3-D brick Cosserat point element. Comput Mech 41(4): 517–525

    Article  MATH  Google Scholar 

  16. Jaquotte OP, Oden JT (1986) An accurate and efficient a posteriori control of hourglass instabilities in underintegrated linear and nonlinear elasticity. Comput Methods Appl Mech Eng 55: 105–128

    Article  Google Scholar 

  17. Kasper E, Taylor RL (2000) A mixed-enhanced strain method. Part I: geometrically linear problems. Comput Struct 75: 237–250

    Article  Google Scholar 

  18. Kasper E, Taylor RL (2000) A mixed-enhanced strain method. Part II: geometrically nonlinear problems. Comput Struct 75: 251–260

    Article  Google Scholar 

  19. Klepach D, Rubin MB (2006) Influence of membrane stresses on postbuckling of rectangular plates using a nonlinear elastic 3-D Cosserat brick element. Comput Mech 39(6): 729–740

    Article  Google Scholar 

  20. Lekhnitskii SG (1963) Theory of elasticity of an anisotropic elastic body. Holden-Day, San Francisco

    MATH  Google Scholar 

  21. Loehnert S, Boerner EFI, Rubin MB, Wriggers P (2005) Response of a nonlinear elastic general Cosserat brick element in simulations typically exhibiting locking and hourglassing. Comput Mech 36(4): 255–265

    Article  MATH  Google Scholar 

  22. Maple, Version 13. Maplesoft Math Software, Waterloo

  23. Mueller-Hoeppe DS, Loehnert S, Wriggers P (2009) A finite deformation brick element with inhomogeneous mode enhancement. Int J Numer Methods Eng 78: 1164–1187

    Article  MathSciNet  MATH  Google Scholar 

  24. Nadler B, Rubin MB (2003) A new 3-D finite element for nonlinear elasticity using the theory of a Cosserat point. Int J Solids Struct 40: 4585–4614

    Article  MATH  Google Scholar 

  25. Pian THH (1964) Derivation of element stiffness matrices by assumed stress distributions. AIAA J 2(7): 1333–1336

    Article  Google Scholar 

  26. Pian THH, Sumihara K (1984) Rational approach for assumed stress finite elements. Int J Numer Methods Eng 20: 1685–1695

    Article  MATH  Google Scholar 

  27. Reese S, Wriggers P (1996) Finite element calculation of the stability behaviour of hyperelastic solids with the enhanced strain methods. Z Angew Math Mech 76(S5): 415–416

    MATH  Google Scholar 

  28. Reese S, Wriggers P (2000) A stabilization technique to avoid hourglassing in finite elasticity. Int J Numer Methods Eng 48: 79–110

    Article  MATH  Google Scholar 

  29. Reese S, Wriggers P, Reddy BD (2000) A new locking-free brick element technique for large deformation problems in elasticity. Comput Struct 75: 291–304

    Article  MathSciNet  Google Scholar 

  30. Rubin MB (1985) On the theory of a Cosserat point and its application to the numerical solution of continuum problems. ASME J Appl Mech 52: 368–372

    Article  MATH  Google Scholar 

  31. Rubin MB (1985) On the numerical solution of one-dimensional continuum problems using the theory of a Cosserat point. ASME J Appl Mech 52: 373–378

    Article  MATH  Google Scholar 

  32. Rubin MB (1995) Numerical solution of two- and three-dimensional thermomechanical problems using the theory of a Cosserat point, theoretical, experimental, and numerical contributions to the mechanics of fluids and solids: a collection of papers in honor of Paul M. Naghdi, edited by J. Casey and M. J. Crochet, Birkäuser, Basel. Spec Issue Z Angew Math Phys 46:308–334

  33. Rubin MB (1996) Restrictions on nonlinear constitutive equations for elastic rods. J Elast 44: 9–36

    Article  MATH  Google Scholar 

  34. Rubin MB (2000) Cosserat theories: shells, rods and points., solid mechanics and its applications. Kluwer, Dordrecht

    Google Scholar 

  35. Rubin MB, Jabareen M (2008) Physically based invariants for nonlinear elastic orthotropic solids. J Elast 90: 1–18

    Article  MathSciNet  MATH  Google Scholar 

  36. Rubin MB, Jabareen M (2010) Further developments of physically based invariants for nonlinear elastic orthotropic solids. Accepted for publication in J. Elast 103:289–294

    Google Scholar 

  37. Simo JC, Armero F (1992) Geometrically nonlinear enhanced strain mixed methods and the method of incompatible modes. Int J Numer Methods Eng 33: 1413–1449

    Article  MathSciNet  MATH  Google Scholar 

  38. Simo JC, Rifai MS (1990) A class of mixed assumed strain methods and the method of incompatible modes. Int J Numer Methods Eng 29(8): 1595–1638

    Article  MathSciNet  MATH  Google Scholar 

  39. Simo JC, Armero F, Taylor RL (1993) Improved versions of assumed enhanced strain tri-linear elements for 3D finite deformation problems. Comput Methods Appl Mech Eng 110(3–4): 359–386

    Article  MathSciNet  MATH  Google Scholar 

  40. Sussmann T, Bathe KJ (1987) A finite element formulation for nonlinear incompressible elastic and inelastic analysis. Comput Struct 26(1–2): 357–409

    Article  Google Scholar 

  41. Wriggers P, Korelc J (1996) On enhanced strain methods for small and finite deformations of solids. Comput Mech 18: 413–428

    Article  MATH  Google Scholar 

  42. Wriggers P, Reese S (1996) A note on enhanced strain methods for large deformations. Comput Methods Appl Mech Eng 135: 201–209

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Civil and Environmental Engineering, Technion—Israel Institute of Technology, 32000, Haifa, Israel

    M. Jabareen

  2. Faculty of Mechanical Engineering, Technion—Israel Institute of Technology, 32000, Haifa, Israel

    L. Sharipova & M. B. Rubin

Authors
  1. M. Jabareen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. Sharipova
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. B. Rubin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Jabareen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jabareen, M., Sharipova, L. & Rubin, M.B. Cosserat point element (CPE) for finite deformation of orthotropic elastic materials. Comput Mech 49, 525–544 (2012). https://doi.org/10.1007/s00466-011-0654-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-011-0654-x

Keywords

Search

Navigation