Skip to main content
SpringerLink
Log in

Exact solution for thermal–mechanical post-buckling of functionally graded micro-beams

  • Original Paper
  • Published:
CEAS Aeronautical Journal Aims and scope Submit manuscript

Abstract

In this paper, buckling and post-buckling analysis of functionally graded (FG) micro-beams subjected to the thermal gradient are presented. The FG micro-beam is embedded on an elastic substrate medium modeled by Winkler-Pasternak foundation. To capture the size effect, a modified couple stress theory is applied. Based on the minimum potential energy principle, and using Timoshenko beam hypothesis and von Kármán strains, governing differential equations are derived. Both Fourier series solution and exact analytical method are presented for solving the system of coupled differential equations. Because Fourier series solution cannot satisfy all boundary conditions, it is not able to predict the post-buckling response of FG micro-beams, correctly. Moreover, in numerical result section, the effects of length-scale parameter, power-law index, transverse and axial load, temperature gradient, and elastic foundation constants on the post-buckling behavior of simply supported FG micro-beams are investigated. Obtained responses demonstrated that the buckling of the FG micro-beams are significantly sensitive to the variation of mentioned parameters.

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
Fig. 13

Similar content being viewed by others

Abbreviations

U :

Strain energy

\(\sigma\) :

Cauchy stress

m :

Couple stress

\(\epsilon\) :

Strain

\(\chi\) :

Curvature

u :

Displacement

\(k_{w}\) :

Winkler constant

\(\Psi\) :

Rotation angle

\(q\left( x \right)\) :

Transverse loading

b :

Beam width

\(M_{11}\) :

Equivalent bending moment

\(N_{T}\) :

Thermal gradient force

\(\left( {EC} \right)_{eq}\) :

Flexural-axial equivalent stiffness

\(\left( {\beta A} \right)_{eq}\) :

Size effect equivalent stiffness

n :

Fourier Series mode number

\(\theta\) :

Rotation

\(\delta\) :

Kronecker delta

\(\ell\) :

Length scale parameter

\(\lambda ,\mu\) :

Lame constants

E :

Modulus of elasticity

ν :

Poisson ratio

\(k_{p}\) :

Pasternak constant

F :

Reaction of elastic foundation

\(N_{a}\) :

Axial load

\(N_{11}\) :

Equivalent axial force

\(M_{12}^{m}\) :

Equivalent couple stress moment

\(M_{T}\) :

Thermal gradient moment

\(\left( {EI} \right)_{{{\text{eq}}}}\) :

Flexural equivalent stiffness

W :

Potential energy

\(N_{a}^{cr}\) :

Critical axial load

G :

Shear modulus

n :

Power-law index

\(\Lambda\) :

Material property

h :

Beam thickness

m :

Metal

c :

Ceramic

w :

Transverse deflection

x :

Horizontal axis

T :

Temperature

\(V_{13}\) :

Equivalent Shear force

\(\alpha_{T}\) :

Coefficient of thermal expansion

\(\left( {EA} \right)_{eq}\) :

Axial equivalent stiffness

\(\left( {GA} \right)_{eq}\) :

Shear equivalent stiffness

\(W_{n} , G_{n} , Q_{n}\) :

Fourier series coefficients

\(k_{s}\) :

Shear correction factor

References

  1. Ma, W., Huang, T., Guo, S., Yang, C., Ding, Y., Hu, C.: Atomic force microscope study of the aging/rejuvenating effect on asphalt morphology and adhesion performance. Constr. Build. Mater. 205, 642–655 (2019)

    Article  Google Scholar 

  2. Fan, R., Luo, Y., Li, L., Wu, Q., Ren, Z., Peng, B.: Large-range fiber microsphere micro-displacement sensor. Opt. Fiber Technol. 48, 173–178 (2019)

    Article  Google Scholar 

  3. Li, M., et al.: Recent advances on photocatalytic fuel cell for environmental applications. The marriage of photocatalysis and fuel cells. Sci. Total Environ. 668, 966–978 (2019)

    Article  Google Scholar 

  4. Michael, A., Kwok, C.Y.: Piezoelectric micro-lens actuator. Sens. Actuators A Phys. 236, 116–129 (2015)

    Article  Google Scholar 

  5. Chan, Y.J., Huang, J.-W.: Multiple-point vibration testing with micro-electromechanical accelerometers and micro-controller unit. Mechatronics 44, 84–93 (2017)

    Article  Google Scholar 

  6. Sung, M., Shin, K., Moon, W.: A micro-machined hydrophone employing a piezoelectric body combined on the gate of a field-effect transistor. Sens. Actuators A Phys. 237, 155–166 (2016)

    Article  Google Scholar 

  7. Kabel, J., et al.: Micro-mechanical evaluation of SiC-SiC composite interphase properties and debond mechanisms. Compos. Part B Eng. 131, 173–183 (2017)

    Article  Google Scholar 

  8. Ding, J., et al.: Micro-mechanism of the effect of grain size and temperature on the mechanical properties of polycrystalline TiAl. Comput. Mater. Sci. 158, 76–87 (2019)

    Article  Google Scholar 

  9. Eringen, A.C.: Linear theory of micropolar elasticity. J. Math. Mech. 15, 909–923 (1966)

    MathSciNet  MATH  Google Scholar 

  10. Mindlin, R.D., Eshel, N.N.: On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4(1), 109–124 (1968)

    Article  Google Scholar 

  11. Mindlin, R.D.: Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1(4), 417–438 (1965)

    Article  Google Scholar 

  12. Mindlin, R.D., Tiersten, H.F.: Effects of couple-stresses in linear elasticity. Arch Ration Mech Anal 11(1), 415–448 (1962)

    Article  MathSciNet  Google Scholar 

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

  14. Koiter, W.T.: Couple-stress in the theory of elasticity. Proc. K. Ned. Akad. Wet 67, 17–44 (1964)

    MathSciNet  MATH  Google Scholar 

  15. Toupin, R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11(1), 385–414 (1962)

    Article  MathSciNet  Google Scholar 

  16. 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(10), 2731–2743 (2002)

    MATH  Google Scholar 

  17. Winkler, E.: Die Lehre von der Elasticitaet und Festigkeit: mit besonderer Rücksicht auf ihre Anwendung in der Technik für polytechnische Schulen, Bauakademien, Ingenieue, Maschinenbauer, Architecten, etc, vol. 1. Dominicus (1867)

  18. Pasternak, P.L.: On a new method of analysis of an elastic foundation by means of two foundation constants (in Russian), Gosudarstvennoe Izdatelstvo Literaturi po Stroitelstvu I Arkhitekture, USSR, Moscow, USSR (1954)

  19. Asghari, M., Ahmadian, M.T., Kahrobaiyan, M.H., Rahaeifard, M.: On the size-dependent behavior of functionally graded micro-beams. Mater. Des. 31(5), 2324–2329 (2010)

    Article  Google Scholar 

  20. Mohammadi-Alasti, B., Rezazadeh, G., Borgheei, A.-M., Minaei, S., Habibifar, R.: On the mechanical behavior of a functionally graded micro-beam subjected to a thermal moment and nonlinear electrostatic pressure. Compos. Struct. 93(6), 1516–1525 (2011)

    Article  Google Scholar 

  21. Rezaiee-Pajand, M., Rajabzadeh-Safaei, N.: Nonlocal static analysis of a functionally graded material curved nanobeam. Mech. Adv. Mater. Struct. 25(7), 539–547 (2018)

    Article  Google Scholar 

  22. Azimi, M., Mirjavadi, S.S., Shafiei, N., Hamouda, A.M.S., Davari, E.: Vibration of rotating functionally graded Timoshenko nano-beams with nonlinear thermal distribution. Mech. Adv. Mater. Struct. 25(6), 467–480 (2018)

    Article  Google Scholar 

  23. Ebrahimi, F., Barati, M.R.: A modified nonlocal couple stress-based beam model for vibration analysis of higher-order FG nanobeams. Mech. Adv. Mater. Struct. 25(13), 1121–1132 (2018)

    Article  Google Scholar 

  24. Habibi, B., Beni, Y.T., Mehralian, F.: Free vibration of magneto-electro-elastic nanobeams based on modified couple stress theory in thermal environment. Mech. Adv. Mater. Struct. 26(7), 601–613 (2019)

    Article  Google Scholar 

  25. Babaei, A., Rahmani, A.: On dynamic-vibration analysis of temperature-dependent Timoshenko microbeam possessing mutable nonclassical length scale parameter. Mech. Adv. Mater. Struct. (2018). https://doi.org/10.1080/15376494.2018.1516252

    Article  Google Scholar 

  26. Rezaiee-Pajand, M., Mokhtari, M.: Size dependent buckling analysis of nano sandwich beams by two schemes. Mech. Adv. Mater. Struct. 27, 975–990 (2019)

    Article  Google Scholar 

  27. Sahmani, S., Aghdam, M.M., Rabczuk, T.: Nonlinear bending of functionally graded porous micro/nano-beams reinforced with graphene platelets based upon nonlocal strain gradient theory. Compos. Struct. 186, 68–78 (2018)

    Article  Google Scholar 

  28. Bhattacharya, S., Das, D.: Free vibration analysis of bidirectional-functionally graded and double-tapered rotating micro-beam in thermal environment using modified couple stress theory. Compos. Struct. 215, 471–492 (2019)

    Article  Google Scholar 

  29. Reddy, J.N.: Microstructure-dependent couple stress theories of functionally graded beams. J. Mech. Phys. Solids 59(11), 2382–2399 (2011)

    Article  MathSciNet  Google Scholar 

  30. Fallah, A., Aghdam, M.M.: Nonlinear free vibration and post-buckling analysis of functionally graded beams on nonlinear elastic foundation. Eur. J. Mech. A Solids 30(4), 571–583 (2011)

    Article  Google Scholar 

  31. Lei, J., He, Y., Guo, S., Li, Z., Liu, D.: Thermal buckling and vibration of functionally graded sinusoidal microbeams incorporating nonlinear temperature distribution using DQM. J. Therm. Stress 40(6), 665–689 (2015)

    Article  Google Scholar 

  32. Ma, H.M., Gao, X.L., Reddy, J.N.: A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J. Mech. Phys. Solids 56(12), 3379–3391 (2008)

    Article  MathSciNet  Google Scholar 

  33. Komijani, M., Esfahani, S.E., Reddy, J.N., Liu, Y.P., Eslami, M.R.: Nonlinear thermal stability and vibration of pre/post-buckled temperature-and microstructure-dependent functionally graded beams resting on elastic foundation. Compos. Struct. 112, 292–307 (2014)

    Article  Google Scholar 

  34. Das, D.: Nonlinear forced vibration analysis of higher order shear-deformable functionally graded microbeam resting on nonlinear elastic foundation based on modified couple stress theory. Proc. Inst. Mech. Eng. L 223(9), 1773–1790 (2019)

    Google Scholar 

  35. Dehrouyeh-Semnani, A.M., Nikkhah-Bahrami, M.: The influence of size-dependent shear deformation on mechanical behavior of microstructures-dependent beam based on modified couple stress theory. Compos. Struct. 123, 325–336 (2015)

    Article  Google Scholar 

  36. Kiani, Y., Eslami, M.R.: Thermal buckling analysis of functionally graded material beams. Int. J. Mech. Mater. Des. 6(3), 229–238 (2010)

    Article  Google Scholar 

  37. Al-Basyouni, K.S., Tounsi, A., Mahmoud, S.R.: Size dependent bending and vibration analysis of functionally graded micro beams based on modified couple stress theory and neutral surface position. Compos. Struct. 125, 621–630 (2015)

    Article  Google Scholar 

  38. Ma, L.S., Lee, D.W.: A further discussion of nonlinear mechanical behavior for FGM beams under in-plane thermal loading. Compos. Struct. 93(2), 831–842 (2011)

    Article  Google Scholar 

  39. Alizada, A.N., Sofiyev, A.H.: Modified Young’s moduli of nano-materials taking into account the scale effects and vacancies. Meccanica 46(5), 915–920 (2011)

    Article  MathSciNet  Google Scholar 

  40. Najafov, A.M., Sofiyev, A.H., Hui, D., Karaca, Z., Kalpakci, V., Ozcelik, M.: Stability of EG cylindrical shells with shear stresses on a Pasternak foundation. Steel Compos. Struct. 17(4), 453–470 (2014)

    Article  Google Scholar 

  41. Asghari, M., Kahrobaiyan, M.H., Ahmadian, M.T.: A nonlinear Timoshenko beam formulation based on the modified couple stress theory. Int J Eng Sci 48(12), 1749–1761 (2010)

    Article  MathSciNet  Google Scholar 

Download references

Funding

This study was not funded by any company.

Author information

Authors and Affiliations

  1. Civil Engineering Department, Ferdowsi University of Mashhad, Mashhad, Iran

    Mohammad Rezaiee-Pajand & Farhad Kamali

Authors
  1. Mohammad Rezaiee-Pajand
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Farhad Kamali
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohammad Rezaiee-Pajand.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

Appendices

Appendix 1

$$A_{1} y^{\left( 6 \right)} \left( x \right) + A_{2} y^{\left( 4 \right)} \left( x \right) + A_{3} y^{^{\prime\prime}} \left( x \right) + A_{4} y\left( x \right) = f\left( x \right)$$
$$y\left( x \right) = y_{h} \left( x \right) + y_{p} \left( x \right)$$
$$y_{h} \left( x \right) = C_{1} e^{{m_{1} x}} + C_{2} e^{{m_{2} x}} + C_{3} e^{{m_{3} x}} + C_{4} e^{{m_{4} x}} + C_{5} e^{{m_{5} x}} + C_{6} e^{{m_{6} x}}$$
$$\begin{gathered} m_{1} = - m_{2} = \surd ( - \frac{{A_{2} }}{{3A_{1} }} + \frac{{2^{1/3} A_{2}^{2} }}{{3A_{1} \left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} + \sqrt {4\left( { - A_{2}^{2} + 3A_{1} A_{3} } \right)^{3} + \left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} } \right)^{2} } } \right)^{1/3} }} \hfill \\ - \frac{{2^{1/3} A_{3} }}{{\left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} + \sqrt {4\left( { - A_{2}^{2} + 3A_{1} A_{3} } \right)^{3} + \left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} } \right)^{2} } } \right)^{1/3} }} \hfill \\ + \frac{{\left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} + \sqrt {4\left( { - A_{2}^{2} + 3A_{1} A_{3} } \right)^{3} + \left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} } \right)^{2} } } \right)^{1/3} }}{{3 \times 2^{1/3} A_{1} }}) \hfill \\ \end{gathered}$$
$$\begin{gathered} m_{3} = - m_{4} \hfill \\ = \surd ( - \frac{{A_{2} }}{{3A_{1} }} \hfill \\ - \frac{{A_{2}^{2} }}{{3 \times 2^{2/3} A_{1} \left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} + \sqrt {4\left( { - A_{2}^{2} + 3A_{1} A_{3} } \right)^{3} + \left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} } \right)^{2} } } \right)^{1/3} }} \hfill \\ + \frac{{{\text{i}}A_{2}^{2} }}{{2^{2/3} \sqrt 3 A_{1} \left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} + \sqrt {4\left( { - A_{2}^{2} + 3A_{1} A_{3} } \right)^{3} + \left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} } \right)^{2} } } \right)^{1/3} }} \hfill \\ + \frac{{A_{3} }}{{2^{2/3} \left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} + \sqrt {4\left( { - A_{2}^{2} + 3A_{1} A_{3} } \right)^{3} + \left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} } \right)^{2} } } \right)^{1/3} }} \hfill \\ - \frac{{{\text{i}}\sqrt 3 A_{3} }}{{2^{2/3} \left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} + \sqrt {4\left( { - A_{2}^{2} + 3A_{1} A_{3} } \right)^{3} + \left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} } \right)^{2} } } \right)^{1/3} }} \hfill \\ - \frac{{\left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} + \sqrt {4\left( { - A_{2}^{2} + 3A_{1} A_{3} } \right)^{3} + \left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} } \right)^{2} } } \right)^{1/3} }}{{6 \times 2^{1/3} A_{1} }} \hfill \\ - \frac{{{\text{i}}\left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} + \sqrt {4\left( { - A_{2}^{2} + 3A_{1} A_{3} } \right)^{3} + \left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} } \right)^{2} } } \right)^{1/3} }}{{2 \times 2^{1/3} \sqrt 3 A_{1} }}) \hfill \\ \end{gathered}$$
$$\begin{gathered} m_{5} = - m_{6} \hfill \\ = \surd ( - \frac{{A_{2} }}{{3A_{1} }} \hfill \\ - \frac{{A_{2}^{2} }}{{3 \times 2^{2/3} A_{1} \left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} + \sqrt {4\left( { - A_{2}^{2} + 3A_{1} A_{3} } \right)^{3} + \left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} } \right)^{2} } } \right)^{1/3} }} \hfill \\ - \frac{{{\text{i}}A_{2}^{2} }}{{2^{2/3} \sqrt 3 A_{1} \left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} + \sqrt {4\left( { - A_{2}^{2} + 3A_{1} A_{3} } \right)^{3} + \left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} } \right)^{2} } } \right)^{1/3} }} \hfill \\ + \frac{{A_{3} }}{{2^{2/3} \left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} + \sqrt {4\left( { - A_{2}^{2} + 3A_{1} A_{3} } \right)^{3} + \left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} } \right)^{2} } } \right)^{1/3} }} \hfill \\ + \frac{{{\text{i}}\sqrt 3 A_{3} }}{{2^{2/3} \left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} + \sqrt {4\left( { - A_{2}^{2} + 3A_{1} A_{3} } \right)^{3} + \left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} } \right)^{2} } } \right)^{1/3} }} \hfill \\ - \frac{{\left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} + \sqrt {4\left( { - A_{2}^{2} + 3A_{1} A_{3} } \right)^{3} + \left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} } \right)^{2} } } \right)^{1/3} }}{{6 \times 2^{1/3} A_{1} }} \hfill \\ + \frac{{{\text{i}}\left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} + \sqrt {4\left( { - A_{2}^{2} + 3A_{1} A_{3} } \right)^{3} + \left( { - 2A_{2}^{3} + 9A_{1} A_{2} A_{3} - 27A_{1}^{2} A_{4} } \right)^{2} } } \right)^{1/3} }}{{2 \times 2^{1/3} \sqrt 3 A_{1} }}) \hfill \\ \end{gathered}$$

Appendix 2

$$A_{1j} = \frac{1}{{m_{j}^{3} }}{ }$$
$$A_{2j} = \frac{{e^{{m_{j} L}} }}{{m_{j}^{3} }}$$
$$A_{3j} = \frac{1}{{m_{j} }} - \frac{{k_{w} m_{j} + \left( {k_{p} - N_{a} } \right)m_{j}^{3} - \frac{{\left( {\beta A} \right)_{eq} m_{j}^{5} }}{2}}}{{\left( {EI} \right)_{eq} - \frac{{\left( {EC} \right)_{eq}^{2} }}{{\left( {EA} \right)_{eq} }} + \frac{{\left( {\beta A} \right)_{eq} }}{2}}}$$
$$A_{4j} = \frac{{{\text{e}}^{{Lm_{j} }} }}{{m_{j} }} - \frac{{{\text{e}}^{{Lm_{j} }} \left( {2k_{w} m_{j}^{2} + 2\left( {N_{a} - k_{p} } \right)m_{j}^{4} + \left( {\beta A} \right)_{eq} m_{j}^{6} } \right)}}{{2m_{j} \left[ {\left( {EI} \right)_{eq} - \frac{{\left( {EC} \right)_{eq}^{2} }}{{\left( {EA} \right)_{eq} }} + \frac{{\left( {\beta A} \right)_{eq} }}{2}} \right]}}$$
$$A_{5j} = \frac{{m_{j}^{2} \left[ {4\left( {EC} \right)_{eq}^{2} - \left( {EA} \right)_{eq} \left( {4\left( {EI} \right)_{eq} + \left( {\beta A} \right)_{eq} } \right)} \right]\left[ {2k_{w} + 2\left( {N_{a} - k_{p} } \right)m_{j}^{2} + \left( {\beta A} \right)_{eq} m_{j}^{4} \left] { - \left( {\beta A} \right)_{eq} } \right[\left( {EA} \right)_{eq} \left( {2\left( {EI} \right)_{eq} + \left( {\beta A} \right)_{eq} } \right) - 2\left( {EC} \right)_{eq}^{2} } \right]}}{{4m_{j} \left[ {2\left( {EC} \right)_{eq}^{2} - \left( {EA} \right)_{eq} \left( {2\left( {EI} \right)_{eq} + \left( {\beta A} \right)_{eq} } \right)} \right]}}$$
$$A_{6j} = \frac{{{\text{e}}^{{Lm_{j} }} \left[\left( {\beta A} \right)_{eq} {\left( {2\left( {EC} \right)_{eq}^{2} - \left( {EA} \right)_{eq} \left( {2\left( {EI} \right)_{eq} + \left( {\beta A} \right)_{eq} } \right)} \right) + m_{j}^{2} \left( {4\left( {EC} \right)_{eq}^{2} - \left( {EA} \right)_{eq} \left( {4\left( {EI} \right)_{eq} + \left( {\beta A} \right)_{eq} } \right)} \right)\left( {2k_{w} + 2\left( {N_{a} - k_{p} } \right)m_{j}^{2} + \left( {\beta A} \right)_{eq} m_{j}^{4} } \right)} \right]}}{{4\left( {2\left( {EC} \right)_{eq}^{2} - \left( {EA} \right)_{eq} \left( {2\left( {EI} \right)_{eq} + \left( {\beta A} \right)_{eq} } \right)} \right)m_{j} }}$$

where \(j=1, 2, 3, 4, 5, 6\)

$$r_{1} = r_{3} = r_{5} = 0$$
$$r_{2} = q\left[ {\frac{{L^{4} \left( {N_{a} - k_{p} } \right)}}{{24k_{w}^{2} }} - \frac{{L^{6} }}{{720k_{w} }}} \right]$$
$$r_{4} = q\left[ {\frac{{L^{2} }}{{2\left( {EI} \right)_{eq} - \frac{{2\left( {EC} \right)_{eq}^{2} }}{{\left( {EA} \right)_{eq} }} + \left( {\beta A} \right)_{eq} }} + \frac{{(N_{a} - k_{p} )L^{2} }}{{2k_{w}^{2} }} - \frac{{L^{4} }}{{24k_{w} }}} \right]$$
$$r_{6} = - \frac{{qL^{2} }}{96}\left[ {24 + \frac{{48\left( {\left( {EA} \right)_{eq} \left( {EI} \right)_{eq} - \left( {EC} \right)_{eq}^{2} } \right)}}{{\left( {EA} \right)_{eq} \left( {2\left( {EI} \right)_{eq} + \left( {\beta A} \right)_{eq} } \right) - 2\left( {EC} \right)_{eq}^{2} }} + \frac{{\left( {\beta A} \right)_{eq} \left( {12k_{p} + L^{2} k_{w} - 12N_{a} } \right)}}{{k_{w}^{2} }}} \right]$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rezaiee-Pajand, M., Kamali, F. Exact solution for thermal–mechanical post-buckling of functionally graded micro-beams. CEAS Aeronaut J 12, 85–100 (2021). https://doi.org/10.1007/s13272-020-00480-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13272-020-00480-9

Keywords

Search

Navigation