Skip to main content
SpringerLink
Log in

A comparative study of modified strain gradient theory and modified couple stress theory for gold microbeams

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

A Correction to this article was published on 25 July 2019

This article has been updated

Abstract

Microbeams are common structures encountered in micro- and nano-electromechanical systems. Their mechanical response cannot be modelled by local theories of continuum mechanics due to size effect, which becomes more pronounced as the structural length scale approaches the microstructural length scale. The size effect can be circumvented by higher-order continuum theories. In this study, Euler–Bernoulli microbeams are analysed with modified strain gradient theory (MSGT) and modified couple stress theory (MCST). The weak forms for the numerical implementation are obtained by using variational methods. Then, the set of algebraic equations for the finite element method are derived. As a novel aspect, the performance of MSGT and MCST is compared and the length scale parameters of these theories are identified for gold microbeams from the existing experimental results in the literature. With the help of the identified parameters, the cut-off point for the applicability of the classical beam theories for gold microbeams is assessed. The study suggests use of higher-order theories for the state-of-the-art gold microbeam structures having thickness \(t<30\,\upmu {\hbox {m}}\).

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Change history

  • 25 July 2019

    This is an erratum to the previously published version due to systematical errors in figures. Only the legends and axes names of the Figs.

  • 25 July 2019

    This is an erratum to the previously published version due to systematical errors in figures. Only the legends and axes names of the Figs.

References

  1. Aero, E., Kuvshinskii, E.: Fundamental equations of the theory of elastic media with rotationally interacting particles. Fizika Tverdogo Tela 2, 1399–1409 (1960)

    MathSciNet  Google Scholar 

  2. Akgöz, 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 

  3. Akgöz, B., Civalek, Ömer: Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams. Int. J. Eng. Sci. 49, 1268–1280 (2011)

    Article  MathSciNet  Google Scholar 

  4. Akgöz, B., Civalek, Ömer: A size-dependent shear deformation beam model based on the strain gradient elasticity theory. Int. J. Eng. Sci. 70, 1–14 (2013)

    Article  MathSciNet  Google Scholar 

  5. Alper, S., Akin, T.: A single-crystal silicon symmetrical and decoupled MEMS gyroscope on an insulating substrate. J. Microelectromech. Syst. 14, 707–717 (2005)

    Article  Google Scholar 

  6. Asghari, M., Kahrobaiyan, M., Ahmadian, M.: A nonlinear Timoshenko beam formulation based on the modified couple stress theory. Int. J. Eng. Sci. 48, 1749–1761 (2010)

    Article  MathSciNet  Google Scholar 

  7. 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  Google Scholar 

  8. 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 

  9. Baghani, M.: Analytical study on size-dependent static pull-in voltage of microcantilevers using the modified couple stress theory. Int. J. Eng. Sci. 54, 99–105 (2012)

    Article  Google Scholar 

  10. Berry, C., Wang, N., Hashemi, M., Unlu, M., Jarrahi, M.: Significant performance enhancement in photoconductive terahertz optoelectronics by incorporating plasmonic contact electrodes. Nat. Commun. 4, 1622 (2013)

    Article  Google Scholar 

  11. Cauchy, A.: Mémoire sur les systèmes isotropes de points matériels, Oeuvres complètes. 1re Série – Tome II, 351–386 (1850a)

  12. Cauchy, A.: Mémoire sur les vibrations d’un double système de molécules et de l’éther continu dans un corps cristallisé., 1re Série – Tome II, 338–350 (1850b)

  13. Challamel, N., Wang, C.M.: The small length scale effect for a non-local cantilever beam: a paradox solved. Nanotechnology 19, 345703 (2008)

    Article  Google Scholar 

  14. Chasiotis, I., Knauss, W.G.: A new microtensile tester for the study of MEMS materials with the aid of atomic force microscopy. Exp. Mech. 42, 51–57 (2002)

    Article  Google Scholar 

  15. Cook, R., Malkus, D., Plesha, M., Witt, R.: Concepts and Applications of Finite Element Analysis. Wiley, Hoboken (2002)

    Google Scholar 

  16. Cosserat, E., Cosserat, F.: Théorie des corps déformables. Hermann Archives, Hermann (1909)

    MATH  Google Scholar 

  17. Ericksen, J.L., Truesdell, C.: Exact theory of stress and strain in rods and shells. Arch. Ration. Mech. Anal. 1, 295–323 (1957)

    Article  MathSciNet  Google Scholar 

  18. Eringen, A.: Simple microfluids. Int. J. Eng. Sci. 2, 205–217 (1964)

    Article  MathSciNet  Google Scholar 

  19. Eringen, A.: Theory of micropolar continua. In: Proceedings of the ninth midwestern mechanics conference, Wisconsin (1965)

  20. Eringen, A.: Theory of micropolar elasticity. Fracture 1, 621–729 (1968)

    MATH  Google Scholar 

  21. Eringen, A.: Nonlocal polar elastic continua. Int. J. Eng. Sci. 10, 1–16 (1972)

    Article  MathSciNet  Google Scholar 

  22. Eringen, A., Edelen, D.: On nonlocal elasticity. Int. J. Eng. Sci. 10, 233–248 (1972)

    Article  MathSciNet  Google Scholar 

  23. Eringen, A.C. (ed.): Nonlocal Continuum Field Theories. Springer, New York (2004)

    MATH  Google Scholar 

  24. Espinosa, H., Prorok, B., Fischer, M.: A methodology for determining mechanical properties of freestanding thin films and MEMS materials. J. Mech. Phys. Solids 51, 47–67 (2003)

    Article  Google Scholar 

  25. Fang, X.Q., Zhu, C.S.: Size-dependent nonlinear vibration of nonhomogeneous shell embedded with a piezoelectric layer based on surface/interface theory. Compos. Struct. 160, 1191–1197 (2017)

    Article  Google Scholar 

  26. Farokhi, H., Ghayesh, M.H., Amabili, M.: Nonlinear dynamics of a geometrically imperfect microbeam based on the modified couple stress theory. Int. J. Eng. Sci. 68, 11–23 (2013)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  28. Fleck, N.A., Hutchinson, J.W.: A phenomenological theory for strain gradient effects in plasticity. J. Mech. Phys. Solids 41, 1825–1857 (1993)

    Article  MathSciNet  Google Scholar 

  29. Fleck, N.A., Hutchinson, J.W.: Strain gradient plasticity. Adv. Appl. Mech. 33, 296–358 (1997)

    MATH  Google Scholar 

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

    Article  Google Scholar 

  31. 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 

  32. Gianola, D., Sharpe, W.: Techniques for testing thin films in tension. Exp. Tech. 28, 23–27 (2004)

    Article  Google Scholar 

  33. Greer, J., Nix, W.: Size dependence of mechanical properties of gold at the sub-micron scale. Appl. Phys. A 80, 1625–1629 (2005)

    Article  Google Scholar 

  34. Greer, J.R., Oliver, W.C., Nix, W.D.: Size dependence of mechanical properties of gold at the micron scale in the absence of strain gradients. Acta Mater. 53, 1821–1830 (2005)

    Article  Google Scholar 

  35. Grioli, G.: Elasticità asimmetrica. Annali di Matematica Pura ed Applicata, Series 4(50), 389–417 (1960)

    Article  Google Scholar 

  36. Gudlavalleti, S.: Mechanical Testing of Solid Materials at the Micro-Scale. Ph.D. thesis, Massachusetts Institute of Technology (2001)

  37. 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  Google Scholar 

  38. Jiang, W., Cho, M.K., Wu, F. (eds.): Attitude Angular Measurement System Based on MEMS Accelerometer. SPIE-Intl Soc Optical Eng, Bellingham (2014)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  40. Kahrobaiyan, M., Asghari, M., Rahaeifard, M., Ahmadian, M.: Investigation of the size-dependent dynamic characteristics of atomic force microscope microcantilevers based on the modified couple stress theory. Int. J. Eng. Sci. 48, 1985–1994 (2010)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  42. Kahrobaiyan, M., Rahaeifard, M., Ahmadian, M.: A size-dependent yield criterion. Int. J. Eng. Sci. 74, 151–161 (2014)

    Article  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  44. Kong, S., Zhou, S., Nie, Z., Wang, K.: The size-dependent natural frequency of Bernoulli–Euler micro-beams. Int. J. Eng. Sci. 46, 427–437 (2008)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  46. Kulah, H., Najafi, K.: Energy scavenging from low-frequency vibrations by using frequency up-conversion for wireless sensor applications. IEEE Sensors J. 8, 261–268 (2008)

    Article  Google Scholar 

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

    Article  Google Scholar 

  48. Lee, H.L., Chang, W.J.: Sensitivity of v-shaped atomic force microscope cantilevers based on a modified couple stress theory. Microelectron. Eng. 88, 3214–3218 (2011)

    Article  Google Scholar 

  49. 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  Google Scholar 

  50. Lim, C.: Equilibrium and static deflection for bending of a nonlocal nanobeam. Adv. Vib. Eng. 8, 277–300 (2009)

    Google Scholar 

  51. Ma, H., Gao, X.L., Reddy, J.N.: 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 

  52. Ma, Q., Clarke, D.R.: Size dependent hardness of silver single crystals. J. Mater. Res. 10, 853–863 (1995)

    Article  Google Scholar 

  53. McElhaney, K.W., Vlassak, 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 

  54. Miller, D.C., Herrmann, C.F., Maier, H.J., George, S.M., Stoldt, C.R., Gall, K.: Thermo-mechanical evolution of multilayer thin films: Part i. mechanical behavior of au/cr/si microcantilevers. Thin Solid Films 515, 3208–3223 (2007)

    Article  Google Scholar 

  55. Mindlin, R.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16 (1964)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  59. Mitcheson, P., Yeatman, E., Rao, G., Holmes, A., Green, T.: Energy harvesting from human and machine motion for wireless electronic devices. Proc. IEEE 96, 1457–1486 (2008)

    Article  Google Scholar 

  60. Mohammad-Abadi, M., Daneshmehr, A.: Size dependent buckling analysis of microbeams based on modified couple stress theory with high order theories and general boundary conditions. Int. J. Eng. Sci. 74, 1–14 (2014)

    Article  MathSciNet  Google Scholar 

  61. Nowacki, W.: Theory of Micropolar Elasticity. Springer, New York (1970)

    Book  Google Scholar 

  62. Park, H.S., Gall, K., Zimmerman, J.A.: Deformation of FCC nanowires by twinning and slip. J. Mech. Phys. Solids 54, 1862–1881 (2006)

    Article  Google Scholar 

  63. Park, S.K., Gao, X.L.: Bernoulli–Euler beam model based on a modified couple stress theory. J. Micromech. Microeng. 16, 2355–2359 (2006)

    Article  Google Scholar 

  64. Phadikar, J., Pradhan, S.: Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates. Comput. Mater. Sci. 49, 492–499 (2010)

    Article  Google Scholar 

  65. Poole, W., Ashby, M., Fleck, N.: Micro-hardness of annealed and work-hardened copper polycrystals. Scripta Mater. 34, 559–564 (1996)

    Article  Google Scholar 

  66. Pradhan, S.: Nonlocal finite element analysis and small scale effects of CNTs with timoshenko beam theory. Finite Elem. Anal. Des. 50, 8–20 (2012)

    Article  Google Scholar 

  67. Rahaeifard, M., Ahmadian, M., Firoozbakhsh, K.: A strain gradient based yield criterion. Int. J. Eng. Sci. 77, 45–54 (2014)

    Article  Google Scholar 

  68. Ramezani, S., Naghdabadi, R., Sohrabpour, S.: Analysis of micropolar elastic beams. Eur. J. Mech. A. Solids 28, 202–208 (2009)

    Article  Google Scholar 

  69. Rebeiz, G.M.: RF MEMS. Wiley-Blackwell, Hoboken (2003)

    Book  Google Scholar 

  70. Reddy, J.: Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci. 45, 288–307 (2007)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  72. Roque, C., Ferreira, A., Reddy, J.: Analysis of timoshenko nanobeams with a nonlocal formulation and meshless method. Int. J. Eng. Sci. 49, 976–984 (2011)

    Article  Google Scholar 

  73. Shaat, M., Mohamed, S.: Nonlinear-electrostatic analysis of micro-actuated beams based on couple stress and surface elasticity theories. Int. J. Mech. Sci. 84, 208–217 (2014)

    Article  Google Scholar 

  74. Park, S.K., Gao, X.-L.: Variational formulation of a modified couple stress theory and its application to a simple shear problem. Zeitschrift für angewandte Mathematik und Physik ZAMP 59, 904–907 (2008)

    Article  MathSciNet  Google Scholar 

  75. Stelmashenko, N., Walls, M., Brown, L., Milman, Y.: Microindentations on w and mo oriented single crystals: an STM study. Acta Metall. Mater. 41, 2855–2865 (1993)

    Article  Google Scholar 

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

    Article  Google Scholar 

  77. Taati, E., Najafabadi, M.M., Tabrizi, H.B.: Size-dependent generalized thermoelasticity model for timoshenko microbeams. Acta Mech. 225, 1823–1842 (2013)

    Article  MathSciNet  Google Scholar 

  78. Toupin, R.: Elastic materials with couple stress. Arch. Ration. Mech. Anal 11, 385–414 (1962)

    Article  Google Scholar 

  79. Triantafyllou, A., Giannakopoulos, A.: Structural analysis using a dipolar elastic timoshenko beam. Eur. J. Mech. A. Solids 39, 218–228 (2013)

    Article  MathSciNet  Google Scholar 

  80. Unlu, M., Hashemi, M., Berry, C.W., Li, S., Yang, S.H., Jarrahi, M.: Switchable scattering meta-surfaces for broadband terahertz modulation. Nat. Sci. Rep. 4, 5708 (2014)

    Article  Google Scholar 

  81. Vatankhah, R., Kahrobaiyan, M., Alasty, A., Ahmadian, M.: Nonlinear forced vibration of strain gradient microbeams. Appl. Math. Model. 37, 8363–8382 (2013)

    Article  MathSciNet  Google Scholar 

  82. Voigt, W.: Theoretische studien über die elasticitätsverhältnisse der krystalle. i. ableitung der grundgleichungen aus der annahme mit polarität begabter moleküle, Abhandlungen der Mathematischen Classe der Königlichen Gesellschaft der Wissenschaften zu Göttingen 24, 3–52 (1887)

  83. 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 

  84. Wang, C.M., Kitipornchai, S., Lim, C.W., Eisenberger, M.: Beam bending solutions based on nonlocal timoshenko beam theory. J. Eng. Mech. 134, 475–481 (2008)

    Article  Google Scholar 

  85. Wang, C.M., Zhang, Y.Y., Ramesh, S.S., Kitipornchai, S.: Buckling analysis of micro- and nano-rods/tubes based on nonlocal timoshenko beam theory. J. Phys. D Appl. Phys. 39, 3904–3909 (2006)

    Article  Google Scholar 

  86. Wang, L., Liang, C., Prorok, B.: A comparison of testing methods in assessing the elastic properties of sputter-deposited gold films. Thin Solid Films 515, 7911–7918 (2007)

    Article  Google Scholar 

  87. Wang, Y., Bokor, J.: Ultra-high-resolution monolithic thermal bubble inkjet print head. J. Micro/Nanolith. MEMS MOEMS 6, 043009 (2007)

    Article  Google Scholar 

  88. Weihs, T.P., Hong, S., Bravman, J.C., Nix, W.D.: Mechanical deflection of cantilever microbeams: a new technique for testing the mechanical properties of thin films. J. Mater. Res. 3, 931–942 (1988)

    Article  Google Scholar 

  89. Wriggers, P.: Nonlinear Finite Element Methods. Springer, New York (2008)

    MATH  Google Scholar 

  90. Gao, X.-L., Park, S.K.: Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem. Int. J. Solids Struct. 44, 7486–7499 (2007)

    Article  Google Scholar 

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

    Article  Google Scholar 

  92. 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 

  93. Zhang, Y.Y., Wang, C.M., Challamel, N.: Bending, buckling, and vibration of micro/nanobeams by hybrid nonlocal beam model. J. Eng. Mech. 136, 562–574 (2010)

    Article  Google Scholar 

  94. Zhao, J., Zhou, S., Wang, B., Wang, X.: Nonlinear microbeam model based on strain gradient theory. Appl. Math. Model. 36, 2674–2686 (2012)

    Article  MathSciNet  Google Scholar 

  95. Zhu, C.S., Fang, X.Q., Liu, J.X.: Surface energy effect on buckling behavior of the functionally graded nano-shell covered with piezoelectric nano-layers under torque. Int. J. Mech. Sci. 133, 662–673 (2017)

    Article  Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge the financial support from Tubitak under grant number 116M258.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Middle East Technical University, Dumlupınar Bulvarı 1, 06800, Ankara, Turkey

    Murat Kandaz & Hüsnü Dal

Authors
  1. Murat Kandaz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hüsnü Dal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hüsnü Dal.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix A: Tensor operators

The tensor operators used in Section 2 are summarized in Table 2.

Table 2 Description of tensor operators used in Section 2, i.e. (\(\cdot \)), (), (:), (\(\otimes \)), \((\bullet )^{{\mathop {T}\limits ^{}}}\), and \((\bullet )^{{\mathop {T}\limits ^{13}}}\) in compact and indicial notation
Full size table

Appendix B: Shape functions for MSGT

The elements of the shape function \({\varvec{N}}\) for MSGT are as given below. \(N_4\), \(N_5\), and \(N_6\) are symmetric with \(N_1\), \(N_2\), and \(N_3\), respectively, as shown in Fig. 4.

$$\begin{aligned} N_1=\,&[12\text {cosh}(q(s-1))-12\text {cosh}(qs)+12\text {cosh}(q) + 6q^2s^2 - 4q^2s^3 + 8q^2\text {cosh}(q) \nonumber \\&- q^3\text {sinh}(q) + 6q\text {sinh}(qs) \nonumber \\&- 6q^2s - 18q\text {sinh}(q) + 4q^2 + 6q\text {sinh}(q(s-1)) + 12qs\text {sinh}(q) - 6q^2s\text {cosh}(q) - 6q^2s^2\text {cosh}(q) \nonumber \\&+ 4q^2s^3\text {cosh}(q) + 3q^3s^2\text {sinh}(q) - 2q^3s^3\text {sinh}(q) - 12]/D\nonumber \\ N_2=&-[12\text {sinh}(q(s-1))-12\text {sinh}(qs)+12\text {sinh}(q) - 2q^3s^3+4q^2\text {sinh}(q)-12qs+4q^2\text {sinh}(q(s-1))\nonumber \\&+2q^3s-12q\text {cosh}(q)+12q\text {cosh}(q(s-1))+2q^2\text {sinh}(qs)+12qs\text {cosh}(q)\nonumber \\&+4q^3s\text {cosh}(q)-12q^2s\text {sinh}(q)\nonumber \\&-q^4s\text {sinh}(q)-6q^3s^2\text {cosh}(q)+2q^3s^3\text {cosh}(q)+6q^2s^2\text {sinh}(q)\nonumber \\&+2q^4s^2\text {sinh}(q)-q^4s^3\text {sinh}(q)]L/(-qD)\nonumber \\ N_3=&-[6\text {sinh}(q(s-1))-2q-6\text {sinh}(qs)+6\text {sinh}(q)+q^3s^2-q^3s^3\nonumber \\&+q^2\text {sinh}(q)+6qs+q^2\text {sinh}(q(s-1))\nonumber \\&+2q\text {cosh}(qs)-12qs^2+4qs^3-4q\text {cosh}(q)+4q\text {cosh}(q(s-1))-6qs\text {cosh}(q)\nonumber \\&+12qs^2\text {cosh}(q)-4qs^3\text {cosh}(q)\nonumber \\&-q^3s\text {cosh}(q)+4q^2s\text {sinh}(q)+2q^3s^2\text {cosh}(q)-q^3s^3\text {cosh}(q)\nonumber \\&-9q^2s^2\text {sinh}(q)+4q^2s^3\text {sinh}(q)]L^2/(-qD)\nonumber \\ N_4=&-[12\text {cosh}(q(s-1))-12\text {cosh}(qs)-12\text {cosh}(q) + 6q^2s^2 - 4q^2s^3 + 6q\text {sinh}(qs) \nonumber \\&- 6q^2s - 6q\text {sinh}(q) + 6q\text {sinh}(q(s-1)) + 12qs\text {sinh}(q) - 6q^2s\text {cosh}(q) - 6q^2s^2\text {cosh}(q) \nonumber \\&+ 4q^2s^3\text {cosh}(q) + 3q^3s^2\text {sinh}(q) - 2q^3s^3\text {sinh}(q) +12]/D\nonumber \\ N_5=&-[12q-12\text {sinh}(q(s-1))+12\text {sinh}(qs)-12\text {sinh}(q) +6q^3s^2- 2q^3s^3+2q^2\text {sinh}(q)-12qs\nonumber \\&+2q^2\text {sinh}(q(s-1))-12q\text {cosh}(qs)-4q^3s+4q^2\text {sinh}(qs)+12qs\text {cosh}(q)-2q^3s\text {cosh}(q)\nonumber \\&+2q^3s^3\text {cosh}(q)-6q^2s^2\text {sinh}(q)+q^4s^2\text {sinh}(q)-q^4s^3\text {sinh}(q)]L/(-qD)\nonumber \\ N_6=&-[6\text {sinh}(q(s-1))-4q-6\text {sinh}(qs)+6\text {sinh}(q)-2q^3s^2+q^3s^3+6qs+4q\text {cosh}(qs)\nonumber \\&+4qs^3+q^3s-2q\text {cosh}(q)+2q\text {cosh}(q(s-1))-q^2\text {sinh}(qs)-6qs\text {cosh}(q)+4qs^3\text {cosh}(q)\nonumber \\&+2q^2s\text {sinh}(q)-q^3s^2\text {cosh}(q)+q^3s^3\text {cosh}(q)+3q^2s^2\text {sinh}(q)+4q^2s^3\text {sinh}(q)]L^2/(-qD), \end{aligned}$$
(61)

where

$$\begin{aligned} q=L/\sqrt{c_1/c_2}, ~ s=x/L, ~ D=24\text {cosh}(q)+8q^2\text {cosh}(q)-q^3\text {sinh}(q)-24q\text {sinh}(q)+4q^2-24 \end{aligned}$$
(62)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kandaz, M., Dal, H. A comparative study of modified strain gradient theory and modified couple stress theory for gold microbeams. Arch Appl Mech 88, 2051–2070 (2018). https://doi.org/10.1007/s00419-018-1436-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-018-1436-0

Keywords

Search

Navigation