Document Type : Article

**Authors**

Faculty of Civil Engineering, Babol Noshirvani University of Technology, Babol, Iran.

**Abstract**

In this paper to include small scale effect, the augmented Love displacement potential functions (DPF) are developed for isotropic micro or Nano scales medium based on couple stress theory. By substituting the new DPF in equilibrium equations, governing equations are simplified to two linear partial differential equations of sixth and fourth order. Then the governing differential equations are solved for simply supported rectangular plate using the separation of variable method with satisfying exact boundary conditions without any simplification assumptions. Displacements, bending and torsional moments of rectangular plate are obtained for different length scale parameters, aspect and Poisson’s ratios. The obtained results are compared with other studies which show excellent agreement between them.

**Keywords**

- Micromechanics
- couple stress theory
- length scale parameter
- rectangular plate
- displacement potential function

**Main Subjects**

References:

1. Sadd, M. H., Elasticity Theory, Applications, and Numerics, Elsevier, 2nd edition (2009).

2. Asghari, M. "Geometrically nonlinear micro-plate formulation based", International Journal of Engineering Science, 51, pp. 292-309 (2012).

3. McFarland, A.W. and Colton, J.S. "Role of material microstructure in plate stiffness with relevance to micro cantilever sensors", Journal of Micromechanics and Microengineering, 15, pp. 1060-1067 (2005).

4. Papargyri-Beskou, S. and Beskos, D.E. "Static, stability and dynamic analysis of gradient elastic flexural Kirchhoff plates", Archive of Applied Mechanics, 78, pp. 625-635 (2008).

5. Nowacki, W. "The linear theory of micropolar elasticity", International Center for Mechanical Sciences, 151, pp. 1-43 (1974).

6. Wang, X. and Lee, J.D. "Micromorphic theory: a gateway to Nano world", International Journal of Smart and Nano Materials, 1, pp. 115-135 (2010).

7. Toupin, R.A. "Elastic materials with couple-stresses", Archive for Rational Mechanics and Analysis, 11(1), pp. 385-414 (1962).

8. Mindlin, R. and Tiersten, H.F. "Effects of couplestresses in linear elasticity", Archive for Rational Mechanics and Analysis, 11, pp. 415-488 (1962).

9. Koiter, W.T. "Couple-stresses in the theory of elasticity", I and II Proc. Koninklijke Nederlandse Akademie Van Weteschappen - Proceedings Series B - Physical Sciences, 67, pp. 17-44 (1964).

10. Nowacki, W., Theory of Asymmetric Elasticity, Pergamon Press, Oxford (1986).

11. Asghari, M., Kahrobaiyan, M.H., Rahaeifard, M., and Ahmadian, M.T. "Investigation of the size effects in Timoshenko beams based on the couple stress theory", Archive of Applied Mechanics, 81, pp. 863-874 (2011).

12. Kong, S., Zhou, S., Nie, Z., and Wang, K. "The sizedependent natural frequency of Bernoulli-Euler microbeams", International Journal of Engineering Science, 46, pp. 427-437 (2008).

13. Yang, F., Chong, A., Lam, D., and Tong, P. "Couple stress based strain gradient theory for elasticity", International Journal of Solids and Structures, 39, pp. 2731-2743 (2002).

14. Park, S.K. and Gao, X.L. "Bernoullie Euler beam model based on a modified couple stress theory", Journal of Micromechanics and Microengineering, 16, pp. 2355-2359 (2006).

15. Ma, H., Gao, X.-L., and Reddy, J. "A microstructuredependent Timoshenko beam model based on a modified couple stress theory", Journal of the Mechanics and Physics of Solids, 56, pp. 3379-3391 (2008).

16. Ke, L. and Wang, Y. "Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory", Composite Structures, 93, pp. 342-350 (2011).

17. Tsiatas, G. "A new Kirchhoff plate model based on a modified couple stress theory", International Journal of Solids and Structures, 46, pp. 2757-2764 (2009).

18. Setoodeh, A.R. and Rezaei, M. "An explicit solution for the size-dependent large amplitude transverse vibration of thin functionally graded micro-plates", Scientia Iranica, 25(2), pp. 799-812 (2018).

19. Karimzadeh, A., Ahmadian, M.T., and Rahaeifard, M. "Effect of size dependency on in-plane vibration of circular micro-ring", Scientia Iranica, 24(4), pp. 1996- 2008 (2017).

20. Rahaeifard, M. and Mojahedi, M. "Size-dependent dynamic behavior of electrostatically actuated microaccelerometers under mechanical shock", International Journal of Structural Stability and Dynamics, 1750042 (2016).

21. Mojahedi, M. "Size dependent dynamic behavior of electrostatically actuated microbridges", International Journal of Engineering Science, 111, pp. 74-85 (2017).

22. Akbas, S.D. "Free vibration of edge cracked functionally graded microscale beams based on the modified couple stress theory", International Journal of Structural Stability and Dynamics, 1750033 (2016).

23. Mohammad-Abadi, M. and Daneshmehr, A. "Modified couple stress theory applied to dynamic analysis of composite laminated beams by considering different beam theories", International Journal of Engineering Science, 87, pp. 83-102 (2015).

24. Ansari, R., and Gholami, R. "Size-dependent buckling and postbuckling analyses of first-order shear deformable magneto-electro-thermo elastic nanoplates based on the nonlocal elasticity theory", International Journal of Structural Stability and Dynamics, 1750014 (2016).

25. Arani, A. and Jafari, G. "Nonlinear vibration analysis of laminated composite Mindlin micro/nanoplates resting on orthotropic Pasternak medium using DQM", Applied Mathematics and Mechanics, 36(8), pp. 1033-1044 (2015).

26. Batra, R., Porfiri, M., and Spinello, D. "Reducedorder models for microelectromechanical rectangular and circular plates incorporating the Casimir force", International Journal of Solids and Structures, 45(11), pp. 3558-3583 (2008).

27. Li, Y. and Pan, E. "Static bending and free vibration of a functionally graded piezoelectric microplate based on the modified couple-stress theory", International Journal of Engineering Science, 97, pp. 40-59 (2015).

28. Changizi, A., Stiharu, I., Olbrechts, B., and Raskin, J.- P. "Extraction method for the residual stress in multilayer microplates under large deflection based on static deflection analysis", Journal of Microelectromechanical Systems, 24(4), pp. 1150-1163 (2015).

29. Fathalilou, M., Sadeghi, M., and Rezazadeh, G. "Micro-inertia effects on the dynamic characteristics of micro-beams considering the couple stress theory", Mechanics Research Communications, 60, pp. 74-80 (2014).

30. Hadjesfandiari, A.R. and Dargush, G.F. "Fundamental solutions for isotropic size-dependent couple stress elasticity", International Journal of Solids and Structures, 50, pp. 1253-1265 (2013).

31. Hadjesfandiari, A.R. and Dargush, G.F. "Couple stress theory for solids", International Journal of Solids and Structures, 48, pp. 2496-2510 (2011).

32. Love, A.E.H.. A Treatise on the Mathematical Theory of Elasticity, 4th Ed., Dover, New York, MR0010851 (6:79e) (1944).

33. Tran-Cong, T. "On the completeness and uniqueness of Papkovich-Neuber and the non-axisymmetric Boussinesq, Love and Burgatti solutions in general cylindrical coordinates", Journal of Elasticity, 36, pp. 227-255 (1995).

34. Pak, R.Y.S. and Eskandari-Ghadi, M. "On the completeness of a method of potentials in elastodynamics", Quart. Appl. Math., 65(4), pp. 789-797 (2007).

35. Eskandari-Ghadi, M. and Pak, R.Y.S. "Elastodynamics and elastostatics by a unified method of potentials for x3-convex domains", Journal of Elasticity, 92, pp. 187-194 (2008).

36. Eskandari-Ghadi, M. "A complete solution of the wave equations for transversely isotropic media", Journal of Elasticity, 81, pp. 1-19 (2005).

37. Wang, M.Z. and Wang, W. "Completeness and nonuniqueness of general solutions of transversely isotropic elasticity", Int. J. Solids Structure, 32(3/4), pp. 501-513 (1995).

38. Moslemi, A., Navayi Neya, B., and Vaseghi Amiri, J. "3-D elasticity buckling solution for simply supported thick rectangular plates using displacement potential functions", Applied Mathematical Modelling, 40, pp. 5717-5730 (2016).

39. Nematzadeh, M., Eskandari-Ghadi, M., and Navayi Neya, B. "An analytical solution for transversely isotropic simply supported thick rectangular plates using displacement potential functions", Strain Analysis, 46, pp. 121-142 (2010).

40. Yakhkeshi, F. and Navayi Neya, B. "Governing equations of micro-scale plate in terms of displacement potential functions", 8th National Congress on Civil Engineering, Babol Noshirvani University of Technology, Babol, Iran (7-8 May, 2014).

41. Apostol, T.M., Mathematical Analysis, A Modern Approach to Advanced Calculus, Addison-Wesley Publishing Co., London (1957).

42. Szilard, R., Theories and Applications of Plate Analysis, ISBN 0-471-42989-9 (2004).

43. Timoshenko, S.P. and Krieger, W., Theory of Plates and Shells, Second Ed., McGraw-Hill Book Company (1959).

2. Asghari, M. "Geometrically nonlinear micro-plate formulation based", International Journal of Engineering Science, 51, pp. 292-309 (2012).

3. McFarland, A.W. and Colton, J.S. "Role of material microstructure in plate stiffness with relevance to micro cantilever sensors", Journal of Micromechanics and Microengineering, 15, pp. 1060-1067 (2005).

4. Papargyri-Beskou, S. and Beskos, D.E. "Static, stability and dynamic analysis of gradient elastic flexural Kirchhoff plates", Archive of Applied Mechanics, 78, pp. 625-635 (2008).

5. Nowacki, W. "The linear theory of micropolar elasticity", International Center for Mechanical Sciences, 151, pp. 1-43 (1974).

6. Wang, X. and Lee, J.D. "Micromorphic theory: a gateway to Nano world", International Journal of Smart and Nano Materials, 1, pp. 115-135 (2010).

7. Toupin, R.A. "Elastic materials with couple-stresses", Archive for Rational Mechanics and Analysis, 11(1), pp. 385-414 (1962).

8. Mindlin, R. and Tiersten, H.F. "Effects of couplestresses in linear elasticity", Archive for Rational Mechanics and Analysis, 11, pp. 415-488 (1962).

9. Koiter, W.T. "Couple-stresses in the theory of elasticity", I and II Proc. Koninklijke Nederlandse Akademie Van Weteschappen - Proceedings Series B - Physical Sciences, 67, pp. 17-44 (1964).

10. Nowacki, W., Theory of Asymmetric Elasticity, Pergamon Press, Oxford (1986).

11. Asghari, M., Kahrobaiyan, M.H., Rahaeifard, M., and Ahmadian, M.T. "Investigation of the size effects in Timoshenko beams based on the couple stress theory", Archive of Applied Mechanics, 81, pp. 863-874 (2011).

12. Kong, S., Zhou, S., Nie, Z., and Wang, K. "The sizedependent natural frequency of Bernoulli-Euler microbeams", International Journal of Engineering Science, 46, pp. 427-437 (2008).

13. Yang, F., Chong, A., Lam, D., and Tong, P. "Couple stress based strain gradient theory for elasticity", International Journal of Solids and Structures, 39, pp. 2731-2743 (2002).

14. Park, S.K. and Gao, X.L. "Bernoullie Euler beam model based on a modified couple stress theory", Journal of Micromechanics and Microengineering, 16, pp. 2355-2359 (2006).

15. Ma, H., Gao, X.-L., and Reddy, J. "A microstructuredependent Timoshenko beam model based on a modified couple stress theory", Journal of the Mechanics and Physics of Solids, 56, pp. 3379-3391 (2008).

16. Ke, L. and Wang, Y. "Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory", Composite Structures, 93, pp. 342-350 (2011).

17. Tsiatas, G. "A new Kirchhoff plate model based on a modified couple stress theory", International Journal of Solids and Structures, 46, pp. 2757-2764 (2009).

18. Setoodeh, A.R. and Rezaei, M. "An explicit solution for the size-dependent large amplitude transverse vibration of thin functionally graded micro-plates", Scientia Iranica, 25(2), pp. 799-812 (2018).

19. Karimzadeh, A., Ahmadian, M.T., and Rahaeifard, M. "Effect of size dependency on in-plane vibration of circular micro-ring", Scientia Iranica, 24(4), pp. 1996- 2008 (2017).

20. Rahaeifard, M. and Mojahedi, M. "Size-dependent dynamic behavior of electrostatically actuated microaccelerometers under mechanical shock", International Journal of Structural Stability and Dynamics, 1750042 (2016).

21. Mojahedi, M. "Size dependent dynamic behavior of electrostatically actuated microbridges", International Journal of Engineering Science, 111, pp. 74-85 (2017).

22. Akbas, S.D. "Free vibration of edge cracked functionally graded microscale beams based on the modified couple stress theory", International Journal of Structural Stability and Dynamics, 1750033 (2016).

23. Mohammad-Abadi, M. and Daneshmehr, A. "Modified couple stress theory applied to dynamic analysis of composite laminated beams by considering different beam theories", International Journal of Engineering Science, 87, pp. 83-102 (2015).

24. Ansari, R., and Gholami, R. "Size-dependent buckling and postbuckling analyses of first-order shear deformable magneto-electro-thermo elastic nanoplates based on the nonlocal elasticity theory", International Journal of Structural Stability and Dynamics, 1750014 (2016).

25. Arani, A. and Jafari, G. "Nonlinear vibration analysis of laminated composite Mindlin micro/nanoplates resting on orthotropic Pasternak medium using DQM", Applied Mathematics and Mechanics, 36(8), pp. 1033-1044 (2015).

26. Batra, R., Porfiri, M., and Spinello, D. "Reducedorder models for microelectromechanical rectangular and circular plates incorporating the Casimir force", International Journal of Solids and Structures, 45(11), pp. 3558-3583 (2008).

27. Li, Y. and Pan, E. "Static bending and free vibration of a functionally graded piezoelectric microplate based on the modified couple-stress theory", International Journal of Engineering Science, 97, pp. 40-59 (2015).

28. Changizi, A., Stiharu, I., Olbrechts, B., and Raskin, J.- P. "Extraction method for the residual stress in multilayer microplates under large deflection based on static deflection analysis", Journal of Microelectromechanical Systems, 24(4), pp. 1150-1163 (2015).

29. Fathalilou, M., Sadeghi, M., and Rezazadeh, G. "Micro-inertia effects on the dynamic characteristics of micro-beams considering the couple stress theory", Mechanics Research Communications, 60, pp. 74-80 (2014).

30. Hadjesfandiari, A.R. and Dargush, G.F. "Fundamental solutions for isotropic size-dependent couple stress elasticity", International Journal of Solids and Structures, 50, pp. 1253-1265 (2013).

31. Hadjesfandiari, A.R. and Dargush, G.F. "Couple stress theory for solids", International Journal of Solids and Structures, 48, pp. 2496-2510 (2011).

32. Love, A.E.H.. A Treatise on the Mathematical Theory of Elasticity, 4th Ed., Dover, New York, MR0010851 (6:79e) (1944).

33. Tran-Cong, T. "On the completeness and uniqueness of Papkovich-Neuber and the non-axisymmetric Boussinesq, Love and Burgatti solutions in general cylindrical coordinates", Journal of Elasticity, 36, pp. 227-255 (1995).

34. Pak, R.Y.S. and Eskandari-Ghadi, M. "On the completeness of a method of potentials in elastodynamics", Quart. Appl. Math., 65(4), pp. 789-797 (2007).

35. Eskandari-Ghadi, M. and Pak, R.Y.S. "Elastodynamics and elastostatics by a unified method of potentials for x3-convex domains", Journal of Elasticity, 92, pp. 187-194 (2008).

36. Eskandari-Ghadi, M. "A complete solution of the wave equations for transversely isotropic media", Journal of Elasticity, 81, pp. 1-19 (2005).

37. Wang, M.Z. and Wang, W. "Completeness and nonuniqueness of general solutions of transversely isotropic elasticity", Int. J. Solids Structure, 32(3/4), pp. 501-513 (1995).

38. Moslemi, A., Navayi Neya, B., and Vaseghi Amiri, J. "3-D elasticity buckling solution for simply supported thick rectangular plates using displacement potential functions", Applied Mathematical Modelling, 40, pp. 5717-5730 (2016).

39. Nematzadeh, M., Eskandari-Ghadi, M., and Navayi Neya, B. "An analytical solution for transversely isotropic simply supported thick rectangular plates using displacement potential functions", Strain Analysis, 46, pp. 121-142 (2010).

40. Yakhkeshi, F. and Navayi Neya, B. "Governing equations of micro-scale plate in terms of displacement potential functions", 8th National Congress on Civil Engineering, Babol Noshirvani University of Technology, Babol, Iran (7-8 May, 2014).

41. Apostol, T.M., Mathematical Analysis, A Modern Approach to Advanced Calculus, Addison-Wesley Publishing Co., London (1957).

42. Szilard, R., Theories and Applications of Plate Analysis, ISBN 0-471-42989-9 (2004).

43. Timoshenko, S.P. and Krieger, W., Theory of Plates and Shells, Second Ed., McGraw-Hill Book Company (1959).

Transactions on Nanotechnology (F)

May and June 2019Pages 1997-2006