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A nonlinear strain gradient finite element for microbeams and microframes

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Abstract

The aim of this paper is to develop a total Lagrangian finite element formulation for the geometrically nonlinear deformation analysis of microbeams and microframes. A general form of the first strain gradient elasticity theory of Mindlin is employed to capture size effects at micron scales. Moreover, the von Kármán strain tensor and its spatial gradient are used to include the effects of geometric nonlinearity. Due to the higher-order nature of the gradient-based governing equilibrium equations, the interpolation functions of the extension and bending field variables are chosen to be the standard \(C^1\) and \(C^2\) functions, respectively. Accordingly, a novel two-node strain gradient microbeam element is introduced. The formulation is then extended to include the deformation of microbeams with arbitrary orientation in a plane, which enables us to analyze arbitrary planar microframes composed of several microbeams at different orientations. The full Newton–Raphson scheme is used to solve the resulting nonlinear algebraic equations. Three different examples are analyzed to show the accuracy and performance of the proposed element at linear as well as nonlinear regimes of deformation. It is shown that the new element can successfully capture the so-called size effect as well as geometric nonlinearity at large distortion of microbeams and microframes at micron scales.

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References

  1. Zhao, X., Abdel-Rahman, E.M., Nayfeh, A.H.: A reduced-order model for electrically actuated microplates. J. Micromech. Microeng. 14, 900–906 (2004)

    Article  Google Scholar 

  2. Faris, W., Nayfeh, A.H.: Mechanical response of a capacitive microsensor under thermal load. Commun. Nonlinear Sci. Numer. Simul. 12, 776–783 (2007)

    Article  MATH  Google Scholar 

  3. Singh, M.P.: Application of biolog FF microplate for substrate utilization and metabolite profiling of closely related fungi. J. Microbiol. Methods 77, 102108 (2009)

    Article  Google Scholar 

  4. Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W.: Strain gradient plasticity: theory and experiment. Acta Metall. Mater. 42, 475–487 (1994)

    Article  Google Scholar 

  5. Nix, W.D., Gao, H.: Indentation size effects in crystalline materials: a law for strain gradient plasticity. J. Mech. Phys. Solids 46, 411–425 (1998)

    Article  MATH  Google Scholar 

  6. Fleck, N.A., Hutchinson, J.W.: Strain gradient plasticity. Adv. Appl. Mech. 33, 295–361 (1997)

    Article  MATH  Google Scholar 

  7. Fleck, N.A., Hutchinson, J.W.: A reformulation of strain gradient plasticity. J. Mech. Phys. Solids 49, 2245–2271 (2001)

    Article  MATH  Google Scholar 

  8. Aifantis, E.C.: On the microstructural origin of certain inelastic models. Trans. ASME. J. Eng. Mater. Technol. 106, 326–330 (1984)

    Article  Google Scholar 

  9. Aifantis, E.C.: The physics of plastic deformation. Int. J. Plast. 3, 211–247 (1987)

    Article  MATH  Google Scholar 

  10. Aifantis, K.E., Willis, J.R.: The role of interfaces in enhancing the yield strength of composites and polycrystals. J. Mech. Phys. Solids 53, 1047–1070 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gurtin, M.E., Anand, L.: Thermodynamics applied to gradient theories involving the accumulated plastic strain: The theories of Aifantis and Fleck and Hutchinson and their generalization. J. Mech. Phys. Solids 57, 405–421 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  14. Aifantis, E.C.: On the role of gradients in the localization of deformation and fracture. Int. J. Eng. Sci. 30, 1279–1299 (1992)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  16. 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 

  17. Ru, Q.C., Aifantis, E.C.: A simple approach to solve boundary-value problems in gradient elasticity. Acta Mech. 101, 59–68 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  18. Askes, H., Aifantis, E.C.: Comments on Model and analysis of size-stiffening in nanoporous cellular solids by Wang and Lam [J. Mater. Sci. 44, 985991 (2009)]. J. Mater. Sci. 46, 6158–6161 (2011)

    Article  Google Scholar 

  19. Aifantis, E.C.: On the gradient approach-relation to Eringen’s nonlocal theory. Int. J. Eng. Sci. 49, 1367–1377 (2011)

    Article  MathSciNet  Google Scholar 

  20. Askes, H., Aifantis, E.C.: 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)

    Article  Google Scholar 

  21. Aifantis, E.C.: Exploring the applicability of gradient elasticity to certain micro/nano reliability problems. Microsyst. Technol. 15, 109–115 (2009)

    Article  Google Scholar 

  22. Sun, B., Aifantis, E.C.: Gradient elasticity formulations for micro/nanoshells. J. Nanomater. 2014, 1–4 (2014)

    Google Scholar 

  23. Askes, H., Aifantis, E.C.: Gradient elasticity and flexural wave dispersion in carbon nanotubes. Phys. Rev. 80, 195412 (2009)

    Article  Google Scholar 

  24. Kong, S.L., Zhou, S.J., Nie, Z.F., Wang, K.: Static and dynamic analysis of microbeams based on strain gradient elasticity theory. Int. J. Eng. Sci. 47, 487–498 (2009)

    Article  MATH  Google Scholar 

  25. Wang, B., Zhao, J., Zhou, S.: A micro scale Timoshenko beam model based on strain gradient elasticity theory. Eur. J. Mech. A Solids 29, 591–599 (2010)

    Article  Google Scholar 

  26. Akgöz, B., Civalek, Ö.: Buckling analysis of functionally graded microbeams based on the strain gradient theory. Acta Mech. 224, 2185–2201 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Akgöz, B., Civalek, Ö.: A novel microstructure-dependent shear deformable beam model. Int. J. Mech. Sci. 99, 10–20 (2015)

    Article  MATH  Google Scholar 

  28. Akgöz, B., Civalek, Ö.: Modeling and analysis of micro-sized plates resting on elastic medium using the modified couple stress theory. Meccanica 48, 863–873 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Akgöz, B., Civalek, Ö.: A microstructure-dependent sinusoidal plate model based on the strain gradient elasticity theory. Acta Mech. 226, 2277–2294 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lazopoulos, A.K.: Dynamic response of thin strain gradient elastic beams. Int. J. Mech. Sci. 58, 27–33 (2012)

    Article  Google Scholar 

  31. Artan, R., Batra, R.C.: Free vibrations of a strain gradient beam by the method of initial values. Acta Mech. 223, 2393–2409 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kahrobaiyan, M.H., Asghari, M., Rahaeifard, M., Ahmadian, M.T.: A nonlinear strain gradient beam formulation. Int. J. Eng. Sci. 49, 1256–1267 (2011)

    Article  MathSciNet  Google Scholar 

  33. Asghari, M., Kahrobaiyan, M.H., Nikfar, M., Ahmadian, M.T.: A size-dependent nonlinear Timoshenko microbeam model based on the strain gradient theory. Acta Mech. 223, 1233–1249 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  34. Lazopoulos, A.K., Lazopoulos, K.A., Palassopoulos, G.: Nonlinear bending and buckling for strain gradient elastic beams. Appl. Math. Modell. 38, 253–262 (2014)

    Article  MathSciNet  Google Scholar 

  35. Lazopoulos, A.K.: Non-smooth bending and buckling of a strain gradient elastic beam with non-convex stored energy function. Acta Mech. 225, 825–834 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  36. Rajabi, F., Ramezani, S.: A nonlinear microbeam model based on strain gradient elasticity theory with surface energy. Arch. Appl. Mech. 82, 363–376 (2012)

    Article  MATH  Google Scholar 

  37. Rajabi, F., Ramezani, S.: A nonlinear microbeam model based on strain gradient elasticity theory. Acta Mech. Solida Sin. 26, 21–34 (2013)

    Article  MATH  Google Scholar 

  38. Ramezani, S.: A micro scale geometrically nonlinear Timoshenko beam model based on strain gradient elasticity theory. Int. J. Nonlinear Mech. 47, 863–873 (2012)

    Article  Google Scholar 

  39. Ghayesh, M.H., Amabili, M., Farokhi, H.: Nonlinear forced vibrations of a microbeam based on the strain gradient elasticity theory. Int. J. Eng. Sci. 63, 52–60 (2013)

    Article  MathSciNet  Google Scholar 

  40. Kahrobaiyan, M.H., Asghari, M., Ahmadian, M.T.: Strain gradient beam element. Finite Elem. Anal. Des. 68, 63–75 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  42. Zhang, B., He, Y., Liu, D., Gan, Z., Shen, L.: Non-classical Timoshenko beam element based on the strain gradient elasticity theory. Finite Elem. Anal. Des. 79, 22–39 (2014)

    Article  MathSciNet  Google Scholar 

  43. Pegios, I.P., Papargyri-Beskou, S., Beskos, D.E.: Finite element static and stability analysis of gradient elastic beam structures. Acta Mech. 226, 745–768 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  45. Ramezani, S.: Nonlinear vibration analysis of micro-plates based on strain gradient elasticity theory. Nonlinear Dyn. 73, 1399–1421 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  46. Wriggers, P.: Nonlinear Finite Element Methods. Springer, Berlin (2008)

    MATH  Google Scholar 

  47. Crisfield, M.A.: Nonlinear Finite Element Analysis of Solids and Structures, vol. 1. Wiley, Chichester (1991)

    MATH  Google Scholar 

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

    Article  Google Scholar 

  49. McElhaney, K.W., Valssak, J.J., Nix, W.D.: Determination of indenter tip geometry and indentation contact area for depth sensing indentation experiments. J. Mater. Res. 13, 1300–1306 (1998)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  51. Shrotriya, P., Allameh, S.M., Lou, J., Buchheit, T., Soboyejo, W.O.: On the measurement of the plasticity length scale parameter in LIGA nickel foils. Mech. Mater. 35, 233–243 (2003)

    Article  Google Scholar 

  52. Martínez-Pãneda, E., Niordson, C.F.: On fracture in finite strain gradient plasticity. Int. J. Plast. 80(80), 154–167 (2016)

    Article  Google Scholar 

  53. Oran, C., Kassimali, A.: Large deformations of framed structures under static and dynamic loads. Comput. Struct. 6, 539–547 (1976)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Guilan, Rasht, Iran

    Farzam Dadgar-Rad & Alireza Beheshti

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  1. Farzam Dadgar-Rad
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  2. Alireza Beheshti
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Correspondence to Farzam Dadgar-Rad.

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Dadgar-Rad, F., Beheshti, A. A nonlinear strain gradient finite element for microbeams and microframes. Acta Mech 228, 1941–1964 (2017). https://doi.org/10.1007/s00707-017-1798-3

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  • DOI: https://doi.org/10.1007/s00707-017-1798-3

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