58
2 Size-Dependent Theories of Beams, Plates and Shells
References
1. Fu, Y., Zhang, J.: Electromechanical dynamic buckling phenomenon in symmetric electric
fields actuated microbeams considering material damping. Acta Mech. 212, 29–42 (2010)
2. Moghimi, Z.M., Ahmadian, M.T.: Static pull-in analysis of electrostatically actuated
microbeams using homotopy perturbation method. Appl. Math. Model. 34, 1032–1046 (2010)
3. Jia, X.L., Yang, J., Kitipornchai, S.: Pull-in instability of geometrically nonlinear microswitches under electrostatic and Casimir forces. Acta Mech. 218, 161–174 (2011)
4. Stolken, J.S., Evans, A.G.: Microbend test method for measuring the plasticity length scale.
Acta Mater. 46(14), 5109–5115 (1998)
5. Nix, W.D.: Mechanical properties of thin films. Metall. Trans. A 20, 2217–2245 (1989)
6. Ma, Q., Clarke, D.R.: Size dependent hardness of silver single crystals. J. Mater. Res. 10(4),
853–863 (1995)
7. Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W.: Strain gradient plasticity: theory
and experiments. Acta Metall. Mater. 42, 475–487 (1994)
8. Chong, A.C.M., Lam, D.C.C.: Strain gradient plasticity effect in indentation hardness of
polymers. Mater. Res. 14(10), 4103–4110 (1999)
9. Dell’Isola, F., Andreaus, U., Placidi, L.: At the origins and in the vanguard of peridynamics,
non-local and higher-gradient continuum mechanics: An underestimated and still topical
contribution of Gabrio Piola. Math. Mech. Sol. 20, 887–928 (2015)
10. Dell’Isola, F., Della Corte, A., Esposito, R., Russo, L.: Some cases of unrecognized transmission of scientific knowledge: from antiquity to Gabrio Piola’s peridynamics and generalized
continuum theories. In: Altenbach, H., Forest, S. (eds.) Generalized Continua as Models for
Classical and Advanced Materials, pp. 77–128. Springer, Berlin (2016)
11. Cosserat, E., Cosserat, F.: Théorie des Corps Déformables. Hermann et Fils, Paris (1909)
12. Toupin, R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11, 385–414
(1962)
13. Mindlin, R.D., Tiersten, H.F.: Effects of couple-stresses in linear elasticity. Arch. Ration.
Mech. Anal. 11, 415–448 (1962)
14. Koiter, W.T.: Couple-stresses in the theory of elasticity, I and II. Proc. K. Ned. Akad. Wet. B
67, 17–44 (1964)
15. Yang, F., Chong, M., Lam, D.C.C., Tong, P.: Couple stress based strain gradient theory for
elasticity. Int. J. Solids Struct. 39, 2731–2743 (2002)
16. Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964)
17. Mindlin, R.D.: Second gradient of strain and surface-tension in linear elasticity. Int. J. Sol.
Struct. 1, 417–38 (1965)
18. Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J., Tong, P.: Experiments and theory in strain
gradient elasticity. J. Mech. Phys. Soli 51, 1477–1508 (2003)
19. Eringen, A.C.: Simple microfluids. Int. J. Eng. Sci. 2, 205–217 (1964)
20. Eringen, A.C.: Linear theory of micropolar elasticity. J. Math. Mech. 15, 909–923 (1966)
21. Eringen, A.C.: Micropolar fluids with stretch. Int. J. Eng. Sci. 7, 115–127 (1969)
22. Altenbach, H., Eremeyev, V.A.: Strain rate tensors and constitutive equations of inelastic
micropolar materials. Int. J. Plast. 63, 3–17 (2014)
23. Ansari, R., Bazdid-Vahdati, M., Shakouri, A., Norouzzadeh, A., Rouhi, H.: Micromorphic
prism element. Math. Mech. Sol. 22(6), 17–29 (2016)
24. Ansari, R., Bazdid-Vahdati, M., Shakouri, A., Norouzzadeh, A., Rouhi, H.: Micromorphic
first-order shear deformable plate element. Mecca. 51, 1797–809 (2016)
25. Ansari, R., Shakouri, A.H., Bazdid-Vahdati, M., Norouzzadeh, A., Rouhi, H.: A nonclassical
finite element approach for the nonlinear analysis of micropolar plates. J. Comput. Nonlin.
Dyn. 12, 011019–12 (2016)
26. Chen, Y., Lee, J.D.: Determining material constants in micromorphic theory through phonon
dispersion relations. Int. J. Eng. Sci. 41, 871–886 (2003)
27. Eringen, A.C.: Microcontinuum Field Theories: I Foundations and Solids. Springer, Berlin
(1999)
2 Size-Dependent Theories of Beams, Plates and Shells
References
1. Fu, Y., Zhang, J.: Electromechanical dynamic buckling phenomenon in symmetric electric
fields actuated microbeams considering material damping. Acta Mech. 212, 29–42 (2010)
2. Moghimi, Z.M., Ahmadian, M.T.: Static pull-in analysis of electrostatically actuated
microbeams using homotopy perturbation method. Appl. Math. Model. 34, 1032–1046 (2010)
3. Jia, X.L., Yang, J., Kitipornchai, S.: Pull-in instability of geometrically nonlinear microswitches under electrostatic and Casimir forces. Acta Mech. 218, 161–174 (2011)
4. Stolken, J.S., Evans, A.G.: Microbend test method for measuring the plasticity length scale.
Acta Mater. 46(14), 5109–5115 (1998)
5. Nix, W.D.: Mechanical properties of thin films. Metall. Trans. A 20, 2217–2245 (1989)
6. Ma, Q., Clarke, D.R.: Size dependent hardness of silver single crystals. J. Mater. Res. 10(4),
853–863 (1995)
7. Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W.: Strain gradient plasticity: theory
and experiments. Acta Metall. Mater. 42, 475–487 (1994)
8. Chong, A.C.M., Lam, D.C.C.: Strain gradient plasticity effect in indentation hardness of
polymers. Mater. Res. 14(10), 4103–4110 (1999)
9. Dell’Isola, F., Andreaus, U., Placidi, L.: At the origins and in the vanguard of peridynamics,
non-local and higher-gradient continuum mechanics: An underestimated and still topical
contribution of Gabrio Piola. Math. Mech. Sol. 20, 887–928 (2015)
10. Dell’Isola, F., Della Corte, A., Esposito, R., Russo, L.: Some cases of unrecognized transmission of scientific knowledge: from antiquity to Gabrio Piola’s peridynamics and generalized
continuum theories. In: Altenbach, H., Forest, S. (eds.) Generalized Continua as Models for
Classical and Advanced Materials, pp. 77–128. Springer, Berlin (2016)
11. Cosserat, E., Cosserat, F.: Théorie des Corps Déformables. Hermann et Fils, Paris (1909)
12. Toupin, R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11, 385–414
(1962)
13. Mindlin, R.D., Tiersten, H.F.: Effects of couple-stresses in linear elasticity. Arch. Ration.
Mech. Anal. 11, 415–448 (1962)
14. Koiter, W.T.: Couple-stresses in the theory of elasticity, I and II. Proc. K. Ned. Akad. Wet. B
67, 17–44 (1964)
15. Yang, F., Chong, M., Lam, D.C.C., Tong, P.: Couple stress based strain gradient theory for
elasticity. Int. J. Solids Struct. 39, 2731–2743 (2002)
16. Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964)
17. Mindlin, R.D.: Second gradient of strain and surface-tension in linear elasticity. Int. J. Sol.
Struct. 1, 417–38 (1965)
18. Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J., Tong, P.: Experiments and theory in strain
gradient elasticity. J. Mech. Phys. Soli 51, 1477–1508 (2003)
19. Eringen, A.C.: Simple microfluids. Int. J. Eng. Sci. 2, 205–217 (1964)
20. Eringen, A.C.: Linear theory of micropolar elasticity. J. Math. Mech. 15, 909–923 (1966)
21. Eringen, A.C.: Micropolar fluids with stretch. Int. J. Eng. Sci. 7, 115–127 (1969)
22. Altenbach, H., Eremeyev, V.A.: Strain rate tensors and constitutive equations of inelastic
micropolar materials. Int. J. Plast. 63, 3–17 (2014)
23. Ansari, R., Bazdid-Vahdati, M., Shakouri, A., Norouzzadeh, A., Rouhi, H.: Micromorphic
prism element. Math. Mech. Sol. 22(6), 17–29 (2016)
24. Ansari, R., Bazdid-Vahdati, M., Shakouri, A., Norouzzadeh, A., Rouhi, H.: Micromorphic
first-order shear deformable plate element. Mecca. 51, 1797–809 (2016)
25. Ansari, R., Shakouri, A.H., Bazdid-Vahdati, M., Norouzzadeh, A., Rouhi, H.: A nonclassical
finite element approach for the nonlinear analysis of micropolar plates. J. Comput. Nonlin.
Dyn. 12, 011019–12 (2016)
26. Chen, Y., Lee, J.D.: Determining material constants in micromorphic theory through phonon
dispersion relations. Int. J. Eng. Sci. 41, 871–886 (2003)
27. Eringen, A.C.: Microcontinuum Field Theories: I Foundations and Solids. Springer, Berlin
(1999)