Skip to main content
SpringerLink
Log in
Book cover

R for Finite Element Analyses of Size-dependent Microscale Structures pp 13–45Cite as

Bending of Microstructure-Dependent MicroBeams and Finite Element Implementations with R

  • Chapter
  • First Online:

  • 477 Accesses

Part of the book series: SpringerBriefs in Applied Sciences and Technology ((BRIEFSCOMPUTAT))

Abstract

The bending response of microstructure-dependent miniature beam-like structures is examined within the theoretical framework of the modified couple stress theory. Predicated on a planar assumed displacement field, and adopting the small-strain, linearly elastic and Timoshenko’s shear deformable beam theory assumptions, the equilibrium equations governing the bending response of these structures are established via the variational method. Finite element solutions of the model are sought. The finite element solutions are implemented in the R programming language and then employed to illustrate the influence of size-effect on the bending response of microscale beams.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    In the words of AEH Love in Theoretical Mechanics, such models facilitate “the description of motion of material bodies”.

  2. 2.

    The words size-effect and small-scale effect are used interchangeably in this text. In the context of this book, they are used to describe the dependency of the response of small-scale structures on their size.

  3. 3.

    A scalar quantity typically measured in energy per volume.

  4. 4.

    In two-dimensional approximations, the field variable (such as displacement or stress field) depends on two coordinates, and the boundary conditions are imposed on a line. Conversely, in one-dimensional approximations, the field variable is a function of one coordinate and the boundary conditions are imposed on points.

  5. 5.

    The displacement-strain relations are obtained with the assumption that the axial, shear and curvature strains along with the rotation of the beam are small in comparison to unity.

  6. 6.

    We have replaced \(\mu\), originally in Eq. (2.10), by G.

  7. 7.

    This is to remedy the fact that the shear stress has been taken to be independent of z, and hence uniform across the thickness.

  8. 8.

    The total work done is also a scalar quantity like the strain energy.

  9. 9.

    From the knowledge that \(\gamma = \left( { - \phi + \frac{dw}{d\xi }} \right) = c.\)

  10. 10.

    This is true for static problems. For dynamic problems, more elements are necessary to obtain accurate results.

  11. 11.

    Please note that page 140 of the reference contains other results, here we have only done the computation for the case of \(L = 12\,h,\,l = 1/3h\).

  12. 12.

    As expected the value of the slope at node 2 is approximately zero.

  13. 13.

    Please note that page 140 of the reference contains other results, here we have only done the computation for the case of \(L = 12\,h,\,l = 1/3h\).

References

  1. J.F. Doyle, Static and Dynamic Analysis of Structures: With an Emphasis on Mechanics and Computer Matrix Methods (Springer, Netherlands, 1991)

    Book  Google Scholar 

  2. M.H. Kahrobaiyan, M. Asghari, M.T. Ahmadian, A strain gradient Timoshenko beam element: Application to MEMS. Acta Mech. 226, 505–525 (2015)

    Article  MathSciNet  Google Scholar 

  3. R.A. Coutu Jr., P.E. Kladitis, L. Starman, J.R. Reid, A comparison of micro-switch analytic, finite element, and experimental results. Sens. Actuators, A 115, 252–258 (2004)

    Article  Google Scholar 

  4. V. Kaajakari, Practical MEMS: Small Gear Pub. (2009)

    Google Scholar 

  5. S. Khakalo, V. Balobanov, J. Niiranen. Modelling size-dependent bending, buckling and vibrations of 2D triangular lattices by strain gradient elasticity models: Applications to sandwich beams and auxetics. Int. J. Eng. Sci. 127, 33–52 (2018)

    Google Scholar 

  6. B.R. Goncalves, A. Karttunen, J. Romanoff, J. Reddy, Buckling and free vibration of shear-flexible sandwich beams using a couple-stress-based finite element. Compos. Struct. 165, 233–241 (2017)

    Article  Google Scholar 

  7. A.T. Karttunen, J. Romanoff, J. Reddy, Exact microstructure-dependent Timoshenko beam element. Int. J. Mech. Sci. 111, 35–42 (2016)

    Article  Google Scholar 

  8. C.L. Dym, I.H. Shames, Solid mechanics: A variational approach, Augmented edn. (Springer, New York, 2013)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. S. Wulfinghoff, E. Bayerschen, T. Böhlke, Micromechanical simulation of the Hall-Petch effect with a crystal gradient theory including a grain boundary yield criterion. PAMM 13, 15–18 (2013)

    Article  Google Scholar 

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

    Google Scholar 

  12. R. Lakes, Size effects and micromechanics of a porous solid. J. Mater. Sci. 18, 2572–2580 (1983)

    Google Scholar 

  13. P.R. Onck, E.W. Andrews, L.J. Gibson, Size effects in ductile cellular solids. Part I: modeling. Int. J. Mech. Sci. 43, 681–699 (2001)

    Google Scholar 

  14. P. Giovine, L. Margheriti, M.P. Speciale, On wave propagation in porous media with strain gradient effects. Comput. Math. Appl. 55, 307–318 (2008)

    Google Scholar 

  15. A. Kelly, Precipitation hardening (Pergamon Press, 1963)

    Google Scholar 

  16. R. Ebeling, M.F. Ashby, Dispersion hardening of copper single crystals. Philos. Mag. 13, 805–834 (1966)

    Google Scholar 

  17. M.F. Ashby, The deformation of plastically non-homogeneous materials. Philos. Mag. 21, 399–424 (1970)

    Google Scholar 

  18. J.W. Hutchinson, Plasticity at the micron scale. Int. J. Solids Struct. 37, 225–238 (2000)

    Article  MathSciNet  Google Scholar 

  19. D.C.C. Lam, A.C.M. Chong, Indentation model and strain gradient plasticity law for glassy polymers. J. Mater. Res. 14, 3784–3788 (1999)

    Article  Google Scholar 

  20. H. Askes, E.C. Aifantis, Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results. Int. J. Solids Struct. 48, 1962–1990 (2011)

    Google Scholar 

  21. J. Dyszlewicz, Micropolar Theory of Elasticity (Springer, Heidelberg, 2004)

    Book  Google Scholar 

  22. H.-T. Thai, T.P. Vo, T.-K. Nguyen, S.-E. Kim, A review of continuum mechanics models for size-dependent analysis of beams and plates. Compos. Struct. 177, 196–219 (2017)

    Article  Google Scholar 

  23. S. Khakalo, J. Niiranen, Form II of Mindlin’s second strain gradient theory of elasticity with a simplification: For materials and structures from nano- to macro-scales. Eur. J. Mech. A/Solids 71, 292–319 (2018)

    Google Scholar 

  24. W. Voigt, Theoretische studien über die elasticitätsverhältnisse der krystalle: Königliche Gesellschaft der Wissenschaften zu Göttingen (1887)

    Google Scholar 

  25. E. Cosserat, F. Cosserat, Théorie des corps déformables (1909)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  27. R.A. Toupin, Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11, 385–414 (1962)

    Article  MathSciNet  Google Scholar 

  28. W. Koiter, Couple-stress in the theory of elasticity, in Proceedings of the K. Ned. Akad. Wet (1964), pp. 17–44

    Google Scholar 

  29. A.C. Eringen, A unified theory of thermomechanical materials. Int. J. Eng. Sci. 4, 179–202 (1966)

    Article  MathSciNet  Google Scholar 

  30. A.C. Eringen, E.S. Suhubi, Nonlinear theory of simple micro-elastic solids—I. Int. J. Eng. Sci. 2, 189–203 (1964)

    Google Scholar 

  31. R.D. Mindlin, Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964)

    Article  MathSciNet  Google Scholar 

  32. R.D. Mindlin, N.N. Eshel, On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4, 109–124 (1968)

    Google Scholar 

  33. R.D. Mindlin, Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1, 417–438 (1965)

    Google Scholar 

  34. K.B. Mustapha, D. Ruan, Size-dependent axial dynamics of magnetically-sensitive strain gradient microbars with end attachments. Int. J. Mech. Sci. 94–95, 96–110 (2015)

    Google Scholar 

  35. K.B. Mustapha, B.T. Wong, Torsional frequency analyses of microtubules with end attachments. ZAMM-J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik 96, 824–842 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

  38. J.N. Reddy, Microstructure-dependent couple stress theories of functionally graded beams. J. Mech. Phys. Solids 59, 2382–2399 (2011)

    Article  MathSciNet  Google Scholar 

  39. H.M. Ma, X.L. Gao, J.N. Reddy, A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J. Mech. Phys. Solids 56, 3379–3391 (2008)

    Article  MathSciNet  Google Scholar 

  40. A.C.M. Chong, F. Yang, D.C.C. Lam, P. Tong, Torsion and bending of micron-scaled structures. J. Mater. Res. 16, 1052–1058 (2001)

    Google Scholar 

  41. A.H. Nayfeh, P.F. Pai, Linear and Nonlinear Structural Mechanics (Wiley, 2008)

    Google Scholar 

  42. J.R. Barber, Elasticity (Springer, Heidelberg, 2002)

    Google Scholar 

  43. E.B. Magrab, Vibrations of Elastic Systems: With Applications to MEMS and NEMS, vol 184 (Springer Science & Business Media, 2012)

    Google Scholar 

  44. M.H. Kahrobaiyan, M. Asghari, M.T. Ahmadian, A Timoshenko beam element based on the modified couple stress theory. Int. J. Mech. Sci. 79, 75–83 (2014)

    Google Scholar 

  45. I. Babuska, B.A. Szabo, I.N. Katz, The p-version of the finite element method. SIAM J. Numer. Anal. 18, 515–545 (1981)

    Article  MathSciNet  Google Scholar 

  46. T. Apel, J.M. Melenk, Interpolation and quasi-Interpolation in h- and hp-version finite element spaces. Encyclopedia of Computational Mechanics, 2nd edn. (2017), pp. 1–33

    Google Scholar 

  47. A.M. Dehrouyeh-Semnani, A. Bahrami, On size-dependent Timoshenko beam element based on modified couple stress theory. Int. J. Eng. Sci. 107, 134–148 (2016)

    Article  MathSciNet  Google Scholar 

  48. A. Arbind, J. Reddy, Nonlinear analysis of functionally graded microstructure-dependent beams. Compos. Struct. 98, 272–281 (2013)

    Article  Google Scholar 

  49. C. Liebold, W.H. Müller, Comparison of gradient elasticity models for the bending of micromaterials. Comput. Mater. Sci. 116, 52–61 (2016)

    Article  Google Scholar 

  50. Z. Li, Y. He, J. Lei, S. Guo, D. Liu, L. Wang, A standard experimental method for determining the material length scale based on modified couple stress theory. Int. J. Mech. Sci. 141, 198–205 (2018)

    Article  Google Scholar 

  51. R. Narayanaswami, H. Adelman, Inclusion of transverse shear deformation in finite element displacement formulations. AIAA J. 12, 1613–1614 (1974)

    Article  Google Scholar 

  52. A. Öchsner, Computational Statics and Dynamics: An Introduction Based on the Finite Element Method (Springer, Singapore, 2016)

    Google Scholar 

  53. T.B. Jones, N.G. Nenadic, Electromechanics and MEMS (Cambridge University Press, 2013)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical, Materials and Manufacturing Engineering, University of Nottingham Malaysia Campus, Semenyih, Selangor, Malaysia

    Khameel B. Mustapha

Authors
  1. Khameel B. Mustapha
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2019 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Mustapha, K.B. (2019). Bending of Microstructure-Dependent MicroBeams and Finite Element Implementations with R. In: R for Finite Element Analyses of Size-dependent Microscale Structures. SpringerBriefs in Applied Sciences and Technology(). Springer, Singapore. https://doi.org/10.1007/978-981-13-7014-4_2

Download citation

  • DOI: https://doi.org/10.1007/978-981-13-7014-4_2

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-13-7013-7

  • Online ISBN: 978-981-13-7014-4

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation