Skip to main content
SpringerLink
Log in

Axisymmetric couple stress elasticity and its finite element formulation with penalty terms

  • Original
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

Length scale dependent deformation in polymers has been reported in the literature in different experiments at the micron and submicron length scales. Such length scale dependent deformation behavior can be described using higher order gradient theories. A numerical approach for axisymmetric problems is presented here where a couple stress elasticity theory is employed. For the numerical formulation, a penalty finite element approach is proposed and implemented with C 0 axisymmetric elements. In this approach, rotations are introduced as nodal variables independent of nodal displacements, and the penalty term is used to minimize the difference in rotations determined from nodal displacements and nodal rotations. Numerical simulations are performed on different examples to assess the performance of the suggested approach. In particular, a circular cylinder with spherical inclusions and the interaction of the inclusion with the boundary are studied. It is found that with the length scale parameter the distance with which the free boundary affects the stress state of the inclusion increases as well.

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. Cheng Y.T., Cheng C.M.: Scaling, dimensional analysis, and indentation measurements. Mater. Sci. Eng. R 44, 91–149 (2004)

    Article  Google Scholar 

  2. Fleck N.A., Muller G.M., Ashby M.F., Hutchinson J.W.: Strain gradient plasticity: theory and experiments. Acta Metall. Mater. 42, 457–487 (1994)

    Google Scholar 

  3. Stolken J.S., Evans A.G.: A microbend test method for measuring the plasticity length scale. Acta Mater. 46, 5109–5115 (1998)

    Article  Google Scholar 

  4. Nakamura Y., Yamaguchi M., Okubo M., Matsumoto T.: Effects of particle size on mechanical and impact properties of epoxy resin filled with spherical silica. J. Appl. Polym. Sci. 45, 1281–1289 (1992)

    Article  Google Scholar 

  5. Tjernlund J.A., Gamstedt E.K., Gudmundson P.: Length-scale effects on damage development in tensile loading of glass-sphere filled epoxy. Int. J. Solids Struct. 43, 7337–7357 (2006)

    Article  MATH  Google Scholar 

  6. Han C.-S., Nikolov S.: Indentation size effects of polymers and related rotation gradients. J. Mater. Res. 22, 1662–1672 (2007)

    Article  Google Scholar 

  7. Han C.-S.: Influence of the molecular structure on indentation size effect in polymers. Mater. Sci. Eng. A 527, 619–624 (2010)

    Article  Google Scholar 

  8. Alisafaei F., Han C.-S., Sanei S.H.R.: On the time and indentation depth dependence of hardness, dissipation and stiffness in polydimethylsiloxane. Polym. Test. 32, 1220–1228 (2013)

    Article  Google Scholar 

  9. Tatiraju R.V.S., Han C.-S.: Rate dependence of indentation size effects in silicone filled rubber. J. Mech. Mater. Struct. 5, 277–288 (2010)

    Article  Google Scholar 

  10. Lam D.C.C., Yang F., Chong A.C.M., Wang J., Tong P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51, 1477–1508 (2003)

    Article  MATH  Google Scholar 

  11. Shu J.Y., King W.E., Fleck N.A.: Finite elements for materials with strain gradient effects. Int. J. Numer. Methods Eng. 44, 373–391 (1999)

    Article  MATH  Google Scholar 

  12. Han C.-S., Ma A., Roters F., Raabe D.: A Finite Element approach with patch projection for strain gradient plasticity formulations. Int. J. Plast. 23, 690–710 (2007)

    Article  MATH  Google Scholar 

  13. Lee M.G., Han C.-S.: An explicit finite element approach with patch projection technique for strain gradient plasticity formulations. Comput. Mech. 49, 171–183 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Eringen A.C., Suhubi E.S.: Nonlinear theory of simple micro-elastic solids—I. Int. J. Eng. Sci. 2, 189+ (1964)

    Article  MathSciNet  Google Scholar 

  15. Mindlin R.D., Tiersten H.F.: Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11, 415–448 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  16. Toupin R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11, 385+ (1962)

    Article  MathSciNet  Google Scholar 

  17. Koiter W.T.: Couple stresses in the theory of elasticity I and II. Proc. K Ned. Akad. Wet. B 67, 17–44 (1964)

    MATH  Google Scholar 

  18. Yang F., Chong A.C.M., Lam D.C.C., Tong P.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39, 2731–2743 (2002)

    Article  MATH  Google Scholar 

  19. Asghari M., Kahrobaiyan M.H., Rahaeifard M.: Investigation of the size effects in Timoshenko beams based on the couple stress theory. Arch. Appl. Mech. 81, 863–874 (2011)

    Article  MATH  Google Scholar 

  20. Akgoz B., Civalek O.: Analysis of micro-sized beams for various boundary conditions based on the strain gradient elasticity theory. Arch. Appl. Mech. 82, 423–443 (2012)

    Article  Google Scholar 

  21. Zhou, S.-S., Gao, X.-L.: A nonclassical model for circular Mindlin plates based on a modified couple stress theory. ASME J. Appl. Mech. 81(5), 051014–051014-8 (2014). doi:10.1115/1.4026274

  22. Wang Y.-G., Lin W.-H., Zhou C.-L.: Nonlinear bending of the size-dependent circular microplates based on the modified coupe stress theory. Arch. Appl. Mech. 84, 391–400 (2014)

    Article  MATH  Google Scholar 

  23. Mousavi, S.M., Paavola, J.: Analysis of plate in second strain gradient elasticity. Arch. Appl. Mech. (2014). doi:10.1007/s00419-014-0871-9

  24. Akgoz B., Civalek O.: Free vibration analysis for single-layered graphene sheets in an elastic matrix via modified couple stress theory. Mater. Des. 42, 164–171 (2012)

    Article  Google Scholar 

  25. Wang J., Lam D.C.C.: Nanostiffening in polymeric nanocomposites. Comput. Mater. Contin. 17, 215–232 (2010)

    Google Scholar 

  26. Anderson W.B., Lakes R.S.: Size effects due to Cosserat elasticity and surface damage in closed-cell polymethacrylimide foam. J. Mater. Sci. 29, 6413–6419 (1994)

    Article  Google Scholar 

  27. Askes H., Bennett T., Aifantis E.C.: A new formulation and C0-implementationof dynamically consistent gradient elasticity. Int. J. Numer. Methods Eng. 72, 111–126 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  28. Petera J., Pittman J.F.T.: Isoparametric Hermite elements. Int. J. Numer. Methods Eng. 37, 3489–3519 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  29. Akarapu S., Zbib H.M.: Numerical analysis of plane cracks in strain-gradient elastic materials. Int. J. Fract. 141, 403–430 (2006)

    Article  MATH  Google Scholar 

  30. Fischer P., Klassen M., Mergheim J., Steinmann P., Müller R.: Isogeometric analysis of 2D gradient elasticity. Comput. Mech. 47, 325–334 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  31. Amanatidou E., Aravas N.: Mixed finite element formulations of strain-gradient elasticity problems. Comput. Methods Appl. Mech. Eng. 191, 1723–1751 (2002)

    Article  MATH  Google Scholar 

  32. Askes H., Gutierrez M.A.: Implicit gradient elasticity. Int. J. Numer. Methods Eng. 67, 400–416 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  33. Zervos A.: Finite element for elasticity with microstructure and gradient elasticity. Int. J. Numer. Methods Eng. 73, 564–595 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  34. Engel G., Garikipati K., Hughes T.J.R., Larson M.G., Mazzei L., Taylor R.L.: Continuous/discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Comput. Methods Appl. Mech. Eng. 191, 3669–3750 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  35. Wood R.D.: Finite element analysis of plane couple-stress problems using first order stress functions. Int. J. Numer. Methods Eng. 26(2), 489–509 (1988)

    Article  MATH  Google Scholar 

  36. Garg N., Han C.-S.: A penalty finite element approach for couple stress elasticity. Comput. Mech. 52, 709–720 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  37. Zhao J., Chen W., Ji B.: A weak continuity condition of FEM for axisymmetric couple stress theory and an 18-DOF triangular axisymmetric element. Finite Elem. Anal. Des. 46, 632–644 (2010)

    Article  MathSciNet  Google Scholar 

  38. Nikolov, S., Han, C.-S., Raabe, D.: On the origin of size effects in small-strain elasticity of solid polymers. Int. J. Solids Struct. 44, 1582–1592. Corrigendum in Int. J. Solids Struct. 44, 7713–7713 (2007)

  39. Kennedy T.C., Kim J.B.: Dynamic stress-concentrations in micropolar elastic-materials. Comput. Struct. 45, 53–60 (1992)

    Article  MATH  Google Scholar 

  40. Pothier A., Rencis J.J.: 3-Dimensional finite-element formulation for microelastic solids. Comput. Struct. 51, 1–21 (1994)

    Article  MATH  Google Scholar 

  41. Providas E., Kattis M.A.: Finite element method in plane Cosserat elasticity. Comput. Struct. 80, 2059–2069 (2002)

    Article  Google Scholar 

  42. Li L., Xie S.: Finite element method for linear micropolar elasticity and numerical study of some scale effects phenomena in MEMS. Int. J. Mech. Sci. 46, 1571–1587 (2004)

    Article  MATH  Google Scholar 

  43. Huang F.Y., Yan B.H., Yan J.L., Yang D.U.: Bending analysis of micropolar elastic beam using a 3-D finite element method. Int. J. Eng. Sci. 38, 275–286 (2000)

    Article  MATH  Google Scholar 

  44. Gomez J., Basaran C.: Computational implementation of Cosserat continuum. Int. J. Mater. Prod. Tech. 34, 3–36 (2009)

    Article  Google Scholar 

  45. Lawson, C.L., Hanson, R.J.: Bounds for the condition number of a triangular matrix. In: Solving Least Square Problems, Society for Industrial and Applied Mathematics, Philadelphia, pp 28–35 (1995)

  46. Lemeire F.: Bounds for condition numbers of triangular and trapezoid matrices. BIT 15, 58–64 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  47. O’Connell P.A., McKenna G.B.: The stiffening of ultrathin polymer films in the rubbery regime: the relative contributions of membrane stress and surface tension. J. Polym. Sci. Part B Polym. Phys. 47, 2441–2448 (2009)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering (Dept. # 3295), University of Wyoming, Laramie, WY, 82071, USA

    Nitin Garg & Chung-Souk Han

Authors
  1. Nitin Garg
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chung-Souk Han
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chung-Souk Han.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Garg, N., Han, CS. Axisymmetric couple stress elasticity and its finite element formulation with penalty terms. Arch Appl Mech 85, 587–600 (2015). https://doi.org/10.1007/s00419-014-0932-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-014-0932-0

Keywords

Search

Navigation