References
289
76. Grigolyuk, E.I., Chulkov, P.P.: Stability and Oscillation of Sandwich Shells. Mashinostroenie,
Moscow (1973) (in Russian)
77. Krysko, V.A., Koch, M.I., Krysko, A.V., Zhigalov, M.V.: Chaotic phase synchronization of
vibration multilayer beam structures. J. Appl. Mech. Tech. Phys. 53(3), 451–459 (2012)
78. Awrejcewicz, J., Krysko, A.V., Zhigalov, M.V., Saltykova, O.A., Krysko, V.A.: Chaotic vibrations in flexible multilayered Bernoulli-Euler and Timoshenko type beams. Lat. Am. J. Solids
Struct. 5(4), 319–363 (2008)
79. Alshorbagy, A.E., Eltaher, M.A., Mahmoud, F.F.: Free vibration characteristics of a functionally graded beam by finite element method. Appl. Math. Model. 35(1), 412–425 (2011)
80. Li, L., Xiaobai, L., Yujin, H.: Free vibration analysis of nonlocal strain gradient beams made
of functionally graded material. Int. J. Eng. Sci. 102, 77–92 (2016)
81. Nazemnezhad, R., Hosseini-Hashemi, S.: Nonlocal nonlinear free vibration of functionally
graded nanobeams. Compos. Struct. 110, 192–199 (2014)
82. Rahimi, G.H., Davoodinik, A.R.: Thermal behavior analysis of the functionally graded Timoshenko’s beam. IUST Int. J. Eng. Sci. 19(5–1), 105–113 (2008)
83. Li, Y., Meguid, S.A., Fu, Y., Xu, D.: Nonlinear analysis of thermally and electrically actuated
functionally graded material microbeam. Proc. R. Soc. A 470, 20130473 (2014)
84. Bao, G., Wang, L.: Multiple cracking in functionally graded ceramic/metal coatings. Int. J.
Solids Struct. 32, 2853–2871 (1995)
85. Lee, Y.D., Erdogan, F.: Residual/thermal stress in FGM and laminated thermal barrier coatings.
Int. J. Fract. 69, 145–165 (1995)
86. Chung, Y.L., Chi, S.H.: The residual stress of functionally graded materials. J. Chin. Inst. Civ.
Hydraul. Eng. 13, 1–9 (2001)
87. Shafiei, N., Mousavi, A., Ghadiri, M.: On size-dependent nonlinear vibration of porous and
imperfect functionally graded tapered microbeams. Int. J. Eng. Sci. 106, 42–56 (2016)
88. Ansari, R., Gholami, R., Sahmani, S.: Free vibration analysis of size-dependent functionally
graded microbeams based on the strain gradient Timoshenko beam theory. Compos. Struct.
94, 221–228 (2011)
89. Vel, S.S., Batra, R.C.: Exact solution for thermoelastic deformations of functionally graded
thick rectangular plates. AIAA J. 40, 1421–1433 (2002)
90. Hatta, H., Taya, M.: Effective thermal conductivity of a misoriented short fiber composite. J.
Appl. Phys. 58(7), 2478–2486 (1985)
91. Rosen, B.W., Hashin, Z.: Effective thermal expansion coefficients and specific heats of composite materials. Int. J. Eng. Sci. 8, 157–173 (1970)
92. Hill, R.: A self-consistent mechanics of composite materials. J. Mech. Phys. Solids 13(2),
213–222 (1965)
93. Hashin, Z.: Assessment of the self consistent scheme approximation: conductivity of particulate composites. J. Compos. Mater. 2, 284–300 (1968)
94. Azadi, M.: A finite element method (FEM) free and forced lateral vibration analysis of beams
made of functionally graded materials (FGMs). J. Mech. Sci. Technol. 25(1), 69–80 (2011)
95. Shen, H.: Functionally Graded Materials, Non-linear Analysis of Plates and Shells. CRC
Press, New York (2009)
96. Young-Wann, K.: Temperature dependent vibration analysis of functionally graded rectangular plates. J. Sound Vib. 284, 531–549 (2005)
97. Zhong, Z., Yu, T.: Analytical solution of a cantilever functionally graded beam. Compos. Sci.
Technol. 67, 481–488 (2007)
98. Ying, J., Li, C.F., Chen, W.Q.: Two-dimensional elasticity solutions for functionally graded
beams resting on elastic foundations. Compos. Struct. 84, 209–219 (2008)
99. Li, X.F.: A unified approach for analyzing static and dynamic behaviors of functionally graded
Timoshenko and Euler-Bernoulli beams. J. Sound Vib. 318, 1210–1229 (2008)
100. Kadoli, R., Akhtar, K., Ganesan, N.: Static analysis of functionally graded beams using higher
order shear deformation theory. Appl. Math. Model. 32, 2509–2525 (2008)
101. Arbind, A., Reddy, J.N.: Nonlinear analysis of functionally graded microstructure-dependent
beams. Compos. Struct. 98, 272–281 (2013)
289
76. Grigolyuk, E.I., Chulkov, P.P.: Stability and Oscillation of Sandwich Shells. Mashinostroenie,
Moscow (1973) (in Russian)
77. Krysko, V.A., Koch, M.I., Krysko, A.V., Zhigalov, M.V.: Chaotic phase synchronization of
vibration multilayer beam structures. J. Appl. Mech. Tech. Phys. 53(3), 451–459 (2012)
78. Awrejcewicz, J., Krysko, A.V., Zhigalov, M.V., Saltykova, O.A., Krysko, V.A.: Chaotic vibrations in flexible multilayered Bernoulli-Euler and Timoshenko type beams. Lat. Am. J. Solids
Struct. 5(4), 319–363 (2008)
79. Alshorbagy, A.E., Eltaher, M.A., Mahmoud, F.F.: Free vibration characteristics of a functionally graded beam by finite element method. Appl. Math. Model. 35(1), 412–425 (2011)
80. Li, L., Xiaobai, L., Yujin, H.: Free vibration analysis of nonlocal strain gradient beams made
of functionally graded material. Int. J. Eng. Sci. 102, 77–92 (2016)
81. Nazemnezhad, R., Hosseini-Hashemi, S.: Nonlocal nonlinear free vibration of functionally
graded nanobeams. Compos. Struct. 110, 192–199 (2014)
82. Rahimi, G.H., Davoodinik, A.R.: Thermal behavior analysis of the functionally graded Timoshenko’s beam. IUST Int. J. Eng. Sci. 19(5–1), 105–113 (2008)
83. Li, Y., Meguid, S.A., Fu, Y., Xu, D.: Nonlinear analysis of thermally and electrically actuated
functionally graded material microbeam. Proc. R. Soc. A 470, 20130473 (2014)
84. Bao, G., Wang, L.: Multiple cracking in functionally graded ceramic/metal coatings. Int. J.
Solids Struct. 32, 2853–2871 (1995)
85. Lee, Y.D., Erdogan, F.: Residual/thermal stress in FGM and laminated thermal barrier coatings.
Int. J. Fract. 69, 145–165 (1995)
86. Chung, Y.L., Chi, S.H.: The residual stress of functionally graded materials. J. Chin. Inst. Civ.
Hydraul. Eng. 13, 1–9 (2001)
87. Shafiei, N., Mousavi, A., Ghadiri, M.: On size-dependent nonlinear vibration of porous and
imperfect functionally graded tapered microbeams. Int. J. Eng. Sci. 106, 42–56 (2016)
88. Ansari, R., Gholami, R., Sahmani, S.: Free vibration analysis of size-dependent functionally
graded microbeams based on the strain gradient Timoshenko beam theory. Compos. Struct.
94, 221–228 (2011)
89. Vel, S.S., Batra, R.C.: Exact solution for thermoelastic deformations of functionally graded
thick rectangular plates. AIAA J. 40, 1421–1433 (2002)
90. Hatta, H., Taya, M.: Effective thermal conductivity of a misoriented short fiber composite. J.
Appl. Phys. 58(7), 2478–2486 (1985)
91. Rosen, B.W., Hashin, Z.: Effective thermal expansion coefficients and specific heats of composite materials. Int. J. Eng. Sci. 8, 157–173 (1970)
92. Hill, R.: A self-consistent mechanics of composite materials. J. Mech. Phys. Solids 13(2),
213–222 (1965)
93. Hashin, Z.: Assessment of the self consistent scheme approximation: conductivity of particulate composites. J. Compos. Mater. 2, 284–300 (1968)
94. Azadi, M.: A finite element method (FEM) free and forced lateral vibration analysis of beams
made of functionally graded materials (FGMs). J. Mech. Sci. Technol. 25(1), 69–80 (2011)
95. Shen, H.: Functionally Graded Materials, Non-linear Analysis of Plates and Shells. CRC
Press, New York (2009)
96. Young-Wann, K.: Temperature dependent vibration analysis of functionally graded rectangular plates. J. Sound Vib. 284, 531–549 (2005)
97. Zhong, Z., Yu, T.: Analytical solution of a cantilever functionally graded beam. Compos. Sci.
Technol. 67, 481–488 (2007)
98. Ying, J., Li, C.F., Chen, W.Q.: Two-dimensional elasticity solutions for functionally graded
beams resting on elastic foundations. Compos. Struct. 84, 209–219 (2008)
99. Li, X.F.: A unified approach for analyzing static and dynamic behaviors of functionally graded
Timoshenko and Euler-Bernoulli beams. J. Sound Vib. 318, 1210–1229 (2008)
100. Kadoli, R., Akhtar, K., Ganesan, N.: Static analysis of functionally graded beams using higher
order shear deformation theory. Appl. Math. Model. 32, 2509–2525 (2008)
101. Arbind, A., Reddy, J.N.: Nonlinear analysis of functionally graded microstructure-dependent
beams. Compos. Struct. 98, 272–281 (2013)