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Original Articles

Finite element solution of vibrations and buckling of laminated thin plates in hygro-thermal environment based on strain gradient theory

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Pages 4383-4396 | Received 10 May 2022, Accepted 20 Jun 2022, Published online: 27 Jul 2022

References

  • G. Autuori, F. Cluni, V. Gusella, and P. Pucci, Mathematical models for nonlocal elastic composite materials, Adv. Nonlinear Anal., vol. 6, no. 4, pp. 355–382, 2017. DOI: 10.1515/anona-2016-0186.
  • Y. Gholami, R. Ansari, and R. Gholami, Three-dimensional nonlinear primary resonance of functionally graded rectangular small-scale plates based on strain gradeint elasticity theory, Thin-Walled Struct., vol. 150, p. 106681, 2020. DOI: 10.1016/j.tws.2020.106681.
  • P. Trovalusci and G. Augusti, A continuum model with microstructure for materials with flaws and inclusions, J. Phys. IV France, vol. 08, no. PR8, pp. Pr8-383–Pr8–390, 1998. DOI: 10.1051/jp4:1998847.
  • P. Trovalusci, D. Capecchi, and G. Ruta, Genesis of the multiscale approach for materials with microstructure, Arch. Appl. Mech., vol. 79, no. 11, pp. 981–997, 2009. DOI: 10.1007/s00419-008-0269-7.
  • A. Pagani and E. Carrera, Unified one-dimensional finite element for the analysis of hyperelastic soft materials and structures, Mech. Adv. Mater. Struct., pp. 1–14, 2021. DOI: 10.1080/15376494.2021.2013585.
  • R. Barretta, S. Ali Faghidian, F. M. de Sciarra, and F. P. Pinnola, Timoshenko nonlocal strain gradient nanobeams: Variational consistency, exact solutions and carbon nanotube young moduli, Mech. Adv. Mater. Struct., vol. 28, no. 15, pp. 1523–1536, 2021. DOI: 10.1080/15376494.2019.1683660.
  • B. Wang, J. Liu, A. K. Soh, and N. Liang, Exact strain gradient modelling of prestressed nonlocal diatomic lattice metamaterials, Mech. Adv. Mater. Struct., pp. 1–17, 2022. DOI: 10.1080/15376494.2022.2062629.
  • A. Caporale, H. Darban, and R. Luciano, Exact closed-form solutions for nonlocal beams with loading discontinuities, Mech. Adv. Mater. Struct., vol. 29, no. 5, pp. 694–704, 2022. DOI: 10.1080/15376494.2020.1787565.
  • M. Pelliciari and A. M. Tarantino, Equilibrium paths of a three-bar truss in finite elasticity with an application to graphene, Math. Mech. Solids, vol. 25, no. 3, pp. 705–726, 2020. DOI: 10.1177/1081286519887470.
  • M. Pelliciari and A. M. Tarantino, Equilibrium and stability of anisotropic hyperelastic graphene membranes, J. Elast., vol. 144, no. 2, pp. 169–195, 2021a. DOI: 10.1007/s10659-021-09837-5.
  • M. Pelliciari and A. M. Tarantino, A nonlinear molecular mechanics model for graphene subjected to large in-plane deformations, Int. J. Eng. Sci., vol. 167, p. 103527, 2021b. DOI: 10.1016/j.ijengsci.2021.103527.
  • M. Pelliciari, D. P. Pasca, A. Aloisio, and A. M. Tarantino, Size effect in single layer graphene sheets and transition from molecular mechanics to continuum theory, Int. J. Mech. Sci., vol. 214, p. 106895, 2022. DOI: 10.1016/j.ijmecsci.2021.106895.
  • G. Sciarra and S. Vidoli, Asymptotic fracture modes in strain-gradient elasticity: Size effects and characteristic lengths for isotropic materials, J. Elast., vol. 113, no. 1, pp. 27–53, 2013. DOI: 10.1007/s10659-012-9409-y.
  • D. T. Le, J.-J. Marigo, C. Maurini, and S. Vidoli, Strain-gradient vs damage-gradient regularizations of softening damage models, Comput. Methods Appl. Mech. Eng., vol. 340, pp. 424–450, 2018. DOI: 10.1016/j.cma.2018.06.013.
  • M. A. Roudbari, T. D. Jorshari, C. Lü, R. Ansari, A. Z. Kouzani, and M. Amabili, A review of size-dependent continuum mechanics models for micro-and nano-structures, Thin-Walled Struct., vol. 170, p. 108562, 2022. DOI: 10.1016/j.tws.2021.108562.
  • F. P. Pinnola, M. S. Vaccaro, R. Barretta, and F. M. de Sciarra, Finite element method for stress-driven nonlocal beams, Eng. Anal. Bound. Elem., vol. 134, pp. 22–34, 2022. DOI: 10.1016/j.enganabound.2021.09.009.
  • R. Barretta, R. Luciano, and F. Marotti de Sciarra, A fully gradient model for Euler-Bernoulli nanobeams, Math. Probl. Eng., vol. 2015, pp. 1–8, 2015. DOI: 10.1155/2015/495095.
  • R. Barretta, S. Ali Faghidian, R. Luciano, C. Medaglia, and R. Penna, Stress-driven two-phase integral elasticity for torsion of nano-beams, Compos. B: Eng., vol. 145, pp. 62–69, 2018. DOI: 10.1016/j.compositesb.2018.02.020.
  • A. Apuzzo, R. Barretta, S. Faghidian, R. Luciano, and F. Marotti de Sciarra, Nonlocal strain gradient exact solutions for functionally graded inflected nano-beams, Compos. B: Eng., vol. 164, pp. 667–674, 2019. DOI: 10.1016/j.compositesb.2018.12.112.
  • R. Barretta, F. Fabbrocino, R. Luciano, F. M. de Sciarra, and G. Ruta, Buckling loads of nano-beams in stress-driven nonlocal elasticity, Mech. Adv. Mater. Struct., vol. 27, no. 11, pp. 869–875, 2020. DOI: 10.1080/15376494.2018.1501523.
  • A. C. Eringen, A unified theory of thermomechanical materials, Int. J. Eng. Sci., vol. 4, no. 2, pp. 179–202, 1966. DOI: 10.1016/0020-7225(66)90022-X.
  • A. C. Eringen, Nonlocal polar elastic continua, Int. J. Eng. Sci., vol. 10, no. 1, pp. 1–16, 1972. DOI: 10.1016/0020-7225(72)90070-5.
  • A. C. Eringen and D. G. B. Edelen, On nonlocal elasticity, Int. J. Eng. Sci., vol. 10, no. 3, pp. 233–248, 1972. DOI: 10.1016/0020-7225(72)90039-0.
  • S. Altan and E. C. Aifantis, On the structure of the mode III crack-tip in gradient elasticity, Scr. Metall. Mater., vol. 26, no. 2, pp. 319–324, 1992. DOI: 10.1016/0956-716X(92)90194-J.
  • B. S. Altan and E. C. Aifantis, On some aspects in the special theory of gradient elasticity, J. Mech. Behav. Mater., vol. 8, no. 3, pp. 231–282, 1997. DOI: 10.1515/JMBM.1997.8.3.231.
  • R. D. Mindlin, Microstructure in linear elasticity, Arch. Rational Mech. Anal., vol. 16, no. 1, pp. 51–78, 1964. DOI: 10.1007/BF00248490.
  • C. Q. Ru and E. C. Aifantis, A simple approach to solve boundary-value problems in gradient elasticity, Acta Mech., vol. 101, no. 1–4, pp. 59–68, 1993. DOI: 10.1007/BF01175597.
  • E. C. Aifantis, Update on a class of gradient theories, Mech. Mater., vol. 35, no. 3–6, pp. 259–280, 2003. DOI: 10.1016/S0167-6636(02)00278-8.
  • H. Askes and E. C. Aifantis, Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results, Int. J. Solids Struct., vol. 48, no. 13, pp. 1962–1990, 2011. DOI: 10.1016/j.ijsolstr.2011.03.006.
  • J. Kim and J. N. Reddy, A general third-order theory of functionally graded plates with modified couple stress effect and the von kármán nonlinearity: Theory and finite element analysis, Acta Mech., vol. 226, no. 9, pp. 2973–2998, 2015. DOI: 10.1007/s00707-015-1370-y.
  • A. Ashoori and M. J. Mahmoodi, A nonlinear thick plate formulation based on the modified strain gradient theory, Mech. Adv. Mater. Struct., vol. 25, no. 10, pp. 813–819, 2018. DOI: 10.1080/15376494.2017.1308588.
  • K. K. Żur, M. Arefi, J. Kim, and J. N. Reddy, Free vibration and buckling analyses of magneto-electro-elastic FGM nanoplates based on nonlocal modified higher-order sinusoidal shear deformation theory, Compos. B: Eng., vol. 182, p. 107601, 2020. DOI: 10.1016/j.compositesb.2019.107601.
  • P. Trovalusci, Molecular Approaches for Multifield Continua: Origins and Current Developments, Springer, Vienna, pp. 211–278, 2014.
  • N. Fantuzzi, P. Trovalusci, and S. Dharasura, Mechanical behavior of anisotropic composite materials as micropolar continua, Front. Mater., vol. 6, p. 59, 2019. DOI: 10.3389/fmats.2019.00059.
  • M. Tuna and P. Trovalusci, Stress distribution around an elliptic hole in a plate with “implicit” and “explicit” non-local models, Compos. Struct., vol. 256, p. 113003, 2021. DOI: 10.1016/j.compstruct.2020.113003.
  • E. Carrera and V. Zozulya, Carrera unified formulation (CUF) for the micropolar plates and shells. I. higher order theory, Mech. Adv. Mater. Struct., vol. 29, no. 6, pp. 773–795, 2022. DOI: 10.1080/15376494.2020.1793241.
  • R. Luciano and J. R. Willis, Non-local constitutive response of a random laminate subjected to configuration-dependent body force, J. Mech. Phys. Solids, vol. 49, no. 2, pp. 431–444, 2001. DOI: 10.1016/S0022-5096(00)00031-4.
  • A. Apuzzo, R. Barretta, F. Fabbrocino, S. A. Faghidian, R. Luciano, and F. Marotti de Sciarra, Axial and torsional free vibrations of elastic nano-beams by stress-driven two-phase elasticity, J. Appl. Comput. Mech., vol. 5, pp. 402–413, 2019.
  • M. Pelliciari and A. M. Tarantino, Equilibrium paths for von Mises trusses in finite elasticity, J Elast., vol. 138, no. 2, pp. 145–168, 2020. DOI: 10.1007/s10659-019-09731-1.
  • L. Lanzoni and A. M. Tarantino, The bending of beams in finite elasticity, J. Elast., vol. 139, no. 1, pp. 91–121, 2020. DOI: 10.1007/s10659-019-09746-8.
  • L. Lanzoni and A. M. Tarantino, Bending of nanobeams in finite elasticity, Int. J. Mech. Sci., vol. 202–203, p. 106500, 2021. DOI: 10.1016/j.ijmecsci.2021.106500.
  • B. Babu and B. Patel, On the finite element formulation for second-order strain gradient nonlocal beam theories, Mech. Adv. Mater. Struct., vol. 26, no. 15, pp. 1316–1332, 2019. DOI: 10.1080/15376494.2018.1432807.
  • G. Zhang, C. Zheng, C. Mi, and X.-L. Gao, A microstructure-dependent Kirchhoff plate model based on a reformulated strain gradient elasticity theory, Mech. Adv. Mater. Struct., vol. 29, no. 17, pp. 2521–2523, 2022. DOI: 10.1080/15376494.2020.1870054.
  • F. Cornacchia, N. Fantuzzi, R. Luciano, and R. Penna, Solution for cross- and angle-ply laminated Kirchhoff nano plates in bending using strain gradient theory, Compos. B: Eng., vol. 173, p. 107006, 2019. DOI: 10.1016/j.compositesb.2019.107006.
  • J. N. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, 2nd ed., CRC Press, Florida, USA, 2004.
  • M. Bacciocchi and A. M. Tarantino, Analytical solutions for vibrations and buckling analysis of laminated composite nanoplates based on third-order theory and strain gradient approach, Compos. Struct., vol. 272, p. 114083, 2021a. DOI: 10.1016/j.compstruct.2021.114083.
  • M. Bacciocchi and A. M. Tarantino, Third-order theory for the bending analysis of laminated thin and thick plates including the strain gradient effect, Materials, vol. 14, no. 7, pp. 1771, 2021b. DOI: 10.3390/ma14071771.
  • J. Niiranen, J. Kiendl, A. H. Niemi, and A. Reali, Isogeometric analysis for sixth-order boundary value problems of gradient-elastic Kirchhoff plates, Comput. Methods Appl. Mech. Eng., vol. 316, pp. 328–348, 2017. DOI: 10.1016/j.cma.2016.07.008.
  • J. Niiranen and A. H. Niemi, Variational formulations and general boundary conditions for sixth-order boundary value problems of gradient-elastic Kirchhoff plates, Eur. J. Mech.-A/Solids, vol. 61, pp. 164–179, 2017. DOI: 10.1016/j.euromechsol.2016.09.001.
  • M. Bacciocchi, N. Fantuzzi, and A. J. M. Ferreira, Conforming and nonconforming laminated finite element Kirchhoff nanoplates in bending using strain gradient theory, Comput. Struct., vol. 239, p. 106322, 2020. DOI: 10.1016/j.compstruc.2020.106322.
  • M. Bacciocchi, N. Fantuzzi, R. Luciano, and A. M. Tarantino, Linear eigenvalue analysis of laminated thin plates including the strain gradient effect by means of conforming and nonconforming rectangular finite elements, Comput. Struct., vol. 257, p. 106676, 2021. DOI: 10.1016/j.compstruc.2021.106676.
  • J. N. Reddy, Introduction to the Finite Element Method 4th Edition, McGraw-Hill Education, New York, USA, 2018.
  • J. Zhao, W. Chen, and S. Lo, A refined nonconforming quadrilateral element for couple stress/strain gradient elasticity, Int. J. Numer. Methods Eng., vol. 85, no. 3, pp. 269–288, 2011. DOI: 10.1002/nme.2962.
  • M. Bacciocchi, N. Fantuzzi, and A. J. M. Ferreira, Static finite element analysis of thin laminated strain gradient nanoplates in hygro-thermal environment, Contin. Mech. Thermodyn., vol. 33, no. 4, pp. 969–992, 2021. DOI: 10.1007/s00161-020-00940-x.
  • O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method: Solid and Fluid Mechanics, Dynamics and Non-Linearity, vol. 2, McGraw-Hill, New York, USA, 1989.
  • A. Carini, F. Genna, and ODe Donato, Introduzione al Metodo Degli Elementi Finiti, Progetto Leonardo, Esculapio, Bologna, Italy, 1996.
  • P. G. Ciarlet, The Finite Element Method for Elliptic Problems, SIAM, Philadelphia, USA, 2002.
  • F. K. Bogner, R. L. Fox, and L. A. Schmit, The generation of inter-element-compatible stiffness and mass matrices by the use of interpolation formulae. In: Matrix Methods in Structural Mechanics Proceedings, Wright-Patterson Air Force Base, USA, 1965. p. 397.
  • A. Adini and R. W. Clough, Analysis of Plate Bending by the Finite Element Method, University of California, USA, 1960.
  • R. J. Melosh, Basis for derivation of matrices for the direct stiffness method, AIAA J., vol. 1, no. 7, pp. 1631–1637, 1963. DOI: 10.2514/3.1869.
  • J. E. Walz, R. E. Fulton, and N. J. Cyrus, Accuracy and convergence of finite element approximations, Technical Report, National Aeronautics and Space Administration Hampton Langley Research Center, Virginia, USA, 1968.
  • B. Babu and B. Patel, A new computationally efficient finite element formulation for nanoplates using second-order strain gradient Kirchhoff’s plate theory, Compos. B: Eng., vol. 168, pp. 302–311, 2019. DOI: 10.1016/j.compositesb.2018.12.066.
  • E. Carrera, F. Miglioretti, and M. Petrolo, Accuracy of refined finite elements for laminated plate analysis, Compos. Struct., vol. 93, no. 5, pp. 1311–1327, 2011. DOI: 10.1016/j.compstruct.2010.11.007.
  • E. Carrera, A. Pagani, and R. Augello, Large deflection of composite beams by finite elements with node-dependent kinematics, Comput. Mech., vol. 69, no. 6, pp. 1481–1500, 2022. DOI: 10.1007/s00466-022-02151-4.
  • A. Pagani, R. Augello, and E. Carrera, Numerical simulation of deployable ultra-thin composite shell structures for space applications and comparison with experiments, Mech. Adv. Mater. Struct., pp. 1–13, 2022. DOI: 10.1080/15376494.2022.2037173.
  • B. Babu and B. Patel, Analytical solution for strain gradient elastic Kirchhoff rectangular plates under transverse static loading, Eur. J. Mech.-A/Solids, vol. 73, pp. 101–111, 2019. DOI: 10.1016/j.euromechsol.2018.07.007.
  • K. Lazopoulos, On the gradient strain elasticity theory of plates, Eur. J. Mech.-A/Solids, vol. 23, no. 5, pp. 843–852, 2004. DOI: 10.1016/j.euromechsol.2004.04.005.
  • K. Lazopoulos, On bending of strain gradient elastic micro-plates, Mech. Res. Commun., vol. 36, no. 7, pp. 777–783, 2009. DOI: 10.1016/j.mechrescom.2009.05.005.
  • S. Papargyri-Beskou and D. E. Beskos, Static, stability and dynamic analysis of gradient elastic flexural Kirchhoff plates, Arch. Appl. Mech., vol. 78, no. 8, pp. 625–635, 2008. DOI: 10.1007/s00419-007-0166-5.
  • S. Papargyri-Beskou, A. Giannakopoulos, and D. Beskos, Variational analysis of gradient elastic flexural plates under static loading, Int. J. Solids Struct., vol. 47, no. 20, pp. 2755–2766, 2010. DOI: 10.1016/j.ijsolstr.2010.06.003.
  • F. Cornacchia, F. Fabbrocino, N. Fantuzzi, R. Luciano, and R. Penna, Analytical solution of cross-and angle-ply nano plates with strain gradient theory for linear vibrations and buckling, Mech. Adv. Mater. Struct., vol. 28, no. 12, pp. 1201–1215, 2021. DOI: 10.1080/15376494.2019.1655613.
  • G. Tocci Monaco, N. Fantuzzi, F. Fabbrocino, and R. Luciano, Semi-analytical static analysis of nonlocal strain gradient laminated composite nanoplates in hygrothermal environment, J Braz. Soc. Mech. Sci. Eng., vol. 43, no. 5, pp. 1–20, 2021a. DOI: 10.1007/s40430-021-02992-9.
  • G. Tocci Monaco, N. Fantuzzi, F. Fabbrocino, and R. Luciano, Hygro-thermal vibrations and buckling of laminated nanoplates via nonlocal strain gradient theory, Compos. Struct., vol. 262, p. 113337, 2021b. DOI: 10.1016/j.compstruct.2020.113337.
  • G. Tocci Monaco, N. Fantuzzi, F. Fabbrocino, and R. Luciano, Critical temperatures for vibrations and buckling of magneto-electro-elastic nonlocal strain gradient plates, Nanomaterials, vol. 11, no. 1, pp. 87, 2021c. DOI: 10.3390/nano11010087.
  • G. Tocci Monaco, N. Fantuzzi, F. Fabbrocino, and R. Luciano, Trigonometric solution for the bending analysis of magneto-electro-elastic strain gradient nonlocal nanoplates in hygro-thermal environment, Mathematics, vol. 9, no. 5, p. 567, 2021d. DOI: 10.3390/math9050567.
  • S. Brischetto, Hygrothermal loading effects in bending analysis of multilayered composite plates, Comput. Model. Eng. Sci. (CMES), vol. 88, pp. 367–417, 2012.
  • S. Brischetto and E. Carrera, Coupled thermo-electro-mechanical analysis of smart plates embedding composite and piezoelectric layers, J. Therm. Stresses, vol. 35, no. 9, pp. 766–804, 2012. DOI: 10.1080/01495739.2012.689232.
  • S. Brischetto and E. Carrera, Static analysis of multilayered smart shells subjected to mechanical, thermal and electrical loads, Meccanica, vol. 48, no. 5, pp. 1263–1287, 2013. DOI: 10.1007/s11012-012-9666-7.
  • S. Brischetto, R. Leetsch, E. Carrera, T. Wallmersperger, and B. Kröplin, Thermo-mechanical bending of functionally graded plates, J. Therm. Stresses, vol. 31, no. 3, pp. 286–308, 2008. DOI: 10.1080/01495730701876775.
  • H. Matsunaga, Thermal buckling of cross-ply laminated composite and sandwich plates according to a global higher-order deformation theory, Compos. Struct., vol. 68, no. 4, pp. 439–454, 2005. DOI: 10.1016/j.compstruct.2004.04.010.
  • H. Matsunaga, Thermal buckling of angle-ply laminated composite and sandwich plates according to a global higher-order deformation theory, Compos. Struct., vol. 72, no. 2, pp. 177–192, 2006. DOI: 10.1016/j.compstruct.2004.11.016.
  • L. Chu, G. Dui, and Y. Zheng, Thermally induced nonlinear dynamic analysis of temperature-dependent functionally graded flexoelectric nanobeams based on nonlocal simplified strain gradient elasticity theory, Eur. J. Mech.-A/Solids, vol. 82, p. 103999, 2020. DOI: 10.1016/j.euromechsol.2020.103999.
  • F. Hildebrand, E. Reissner, and G. Thomas, Notes on the foundations of the theory of small displacements of orthotropic shells, Technical Report, 1949.
  • K. H. Lo, R. M. Christensen, and E. M. Wu, A high-order theory of plate deformation-part 1: Homogeneous plates, J. Appl. Mech., vol. 44, no. 4, pp. 663–668, 1977a. DOI: 10.1115/1.3424154.
  • K. H. Lo, R. M. Christensen, and E. M. Wu, A high-order theory of plate deformation-part 2: Laminated plates, J. Appl. Mech., vol. 44, no. 4, pp. 669–676, 1977b. DOI: 10.1115/1.3424155.

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