A refined assumed strain finite element model for statics and dynamics of laminated plates

Continuum Mechanics and Thermodynamics - Tập 32 - Trang 665-692 - 2018
Antonio Cazzani1, Mauro Serra1, Flavio Stochino1, Emilio Turco2
1Department of Civil and Environmental Engineering and Architecture (DICAAR), University of Cagliari, Cagliari, Italy
2Department of Architecture, Design and Urban Planning (DADU), University of Sassari, Alghero, Italy

Tóm tắt

This work discusses some alternate models of a mixed assumed strain finite element which has been developed for laminated plates. After a brief theoretical review about this kind of plates and their possible finite element formulation, specifically devised for predicting the mechanical behavior of such structures, we discuss four possible assumptions for strains generating four kinds of mixed assumed strain finite elements. Several numerical tests performed on the aforementioned finite elements are thoroughly discussed in order to sketch some guidelines which can be useful when dealing with laminated plate problems.

Tài liệu tham khảo

Reddy, J.N.: Mechanics of Laminated Composite Plates and Shells—Theory and Analysis, 2nd edn. CRC Press, Boca Raton (2004) Qatu, M.S.: Vibration of Laminated Shells and Plates, 1st edn. Academic Press, Oxford (2004) Yang, P.C., Norris, C.H., Stavsky, Y.: Elastic wave propagation in heterogeneous plates. Int. J. Solids Struct. 2, 665–684 (1966) Whitney, J.M., Pagano, N.J.: Shear deformation in heterogeneous anisotropic plates. J. Appl. Mech. ASME 37, 1031–1036 (1970) Rolfes, R., Rohwer, K.: Improved transverse shear stresses in composite finite elements based on first order shear deformation theory. Int. J. Numer. Methods Eng. 40, 51–60 (1997) Rolfes, R., Rohwer, K., Ballerstaedt, M.: Efficient linear transverse normal stress analysis of layered composite plates. Comput. Struct. 68, 643–652 (1998) Yu, W., Hodges, D.H., Volovoi, V.V.: Asymptotically accurate 3-D recovery from Reissner-like composite plate finite elements. Comput. Struct. 81, 439–454 (2003) Naumenko, K., Eremeyev, V.A.: A layer-wise theory for laminated glass and photovoltaic panels. Compos. Struct. 112, 283–291 (2014) Naumenko, K., Eremeyev, V.A.: A layer-wise theory of shallow shells with thin soft core for laminated glass and photovoltaic applications. Compos. Struct. 178, 434–446 (2017) Altenbach, H., Eremeyev, V.A., Naumenko, K.: On the use of the first order shear deformation plate theory for the analysis of three-layer plates with thin soft core layer. J. Appl. Math. Mech. Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) 95(10), 1004–1011 (2015) Chepiga, V.E.: Refined theory of multilayered shells. Sov. Appl. Mech. 12, 1127–1130 (1976). English translation from Prikladnaia Mekhanika, vol. 12, pp. 45–49 Lo, K.H., Christensen, R.M., Wu, E.M.: A higher-order theory of plate deformation: part 2, laminated plates. J. Appl. Mech. ASME 44, 669–676 (1977) Reddy, J.N.: A simple higher-order theory for laminated composite plates. J. Appl. Mech. ASME 51, 745–752 (1984) Pandya, B.N., Kant, T.: Flexure analysis of laminated composites using refined higher-order \(C^0\) plate bending elements. Comput. Methods Appl. Mech. Eng. 66, 173–198 (1988) Yoda, T., Atluri, S.N.: Post-buckling analysis of stiffened laminated composite panels, using a higher-order shear deformation theory. Comput. Mech. 9, 390–404 (1992) Yong, Y.-K., Cho, Y.: Higher-order, partial hybrid stress, finite element formulation for laminated plate and shell analysis. Comput. Struct. 57, 817–827 (1995) Gaudenzi, P., Mannini, A., Carbonaro, R.: Multi-layer higher order finite elements for the analysis of free-edge stresses in composite laminates. Int. J. Numer. Methods Eng. 41, 851–873 (1998) Poniatovskii, V.V.: Theory for plates of medium thickness. PMM 26, 478–486 (1962). English translation from Prikladnaia Matematika i Mekhanika, vol. 26, pp. 335–341 Cicala, P.: Consistent approximations in shell theory. J. Eng. Mech. Div. ASCE 88, 45–74 (1962) Reddy, J.N.: A generalization of two-dimensional theories of laminated composite plates. Commun. Appl. Numer. Methods 3, 173–180 (1987) Di Sciuva, M.: An improved shear deformation theory for moderately thick multi-layered anisotropic shells and plates. J. Appl. Mech. ASME 54, 589–596 (1987) Robbins, D.H., Reddy, J.N.: Modeling of thick composites using a layerwise laminate theory. Int. J. Numer. Methods Eng. 36, 655–677 (1993) Bisegna, P., Sacco, E.: A layer-wise laminate theory rationally deduced from the three-dimensional elasticity. J. Appl. Mech. ASME 64, 538–545 (1997) Pagano, N.J.: Exact solutions for rectangular bidirectional composites and sandwich plates. J. Compos. Mater. 4, 20–34 (1970) Pagano, N.J., Hatfield, S.J.: Elastic behavior of multilayered bidirectional composites. AIAA J. 10, 931–933 (1972) Liou, W., Sun, C.T.: A three-dimensional hybrid stress isoparametric element for the analysis of laminated composite plates. Comput. Struct. 25, 241–249 (1987) Cen, S., Long, Y., Yao, Z.: A new hybrid-enhanced displacement-based element for the analysis of laminated composite plates. Comput. Struct. 80, 819–833 (2002) Whitney, J.M.: Shear correction factors for orthotropic laminates under static load. J. Appl. Mech. ASME 40, 302–304 (1973) Vlachoutsis, S.: Shear correction factors for plates and shells. Int. J. Numer. Methods Eng. 33, 1537–1552 (1992) Savoia, M., Laudiero, F., Tralli, A.: A two-dimensional theory for the analysis of laminated plates. Comput. Mech. 14, 38–51 (1994) Noor, A.K., Burton, W.S.: Assessment of computational models for multilayered anisotropic plates. Compos. Struct. 14, 223–265 (1990) Auricchio, F., Sacco, E.: Refined first-order shear deformation theory models for composite laminates. J. Appl. Mech. ASME 70, 381–390 (2003) Qi, Y., Knight, N.F.: A refined first-order shear-deformation theory and its justification by plane strain bending problem of laminated plates. Comput. Struct. 33, 49–64 (1996) Mau, S.T., Tong, P., Pian, T.H.H.: Finite element solutions for laminated plates. J. Compos. Mater. 6, 304–311 (1972) Spilker, R.L., Orringer, O., Witmer, E.A.: Use of the hybrid-stress finite-element model for the static and dynamic analysis of composite plates and shell. Technical Report ASRL TR 181–2, MIT (1976) Spilker, R.L., Munir, N.I.: A hybrid-stress quadratic serendipity displacement Mindlin plate bending element. Comput. Struct. 12, 11–21 (1980) Spilker, R.L.: Hybrid-stress eight-node element for thin and thick multilayered laminated plates. Int. J. Numer. Methods Eng. 18, 801–828 (1982) Cazzani, A., Rizzi, N.L., Stochino, F., Turco, E.: Modal analysis of laminates by a mixed assumed-strain finite element model. Math. Mech. Solids 23(1), 99–119 (2018) Garusi, E., Cazzani, A., Tralli, A.: An unsymmetric stress formulation for Reissner–Mindlin plates: a simple and locking-free hybrid rectangular element. Int. J. Comput. Eng. Sci. 5, 589–618 (2004) Cazzani, A., Garusi, E., Tralli, A., Atluri, S.N.: A four-node hybrid assumed-strain finite element for laminated composite plates. Comput. Mater. Continua 2, 23–38 (2005) Bathe, K.-J.: Finite Element Procedures. Prentice-Hall, Upper Saddle River (1996) Malvern, L.E.: Introduction to the Mechanics of a Continuous Medium. Prentice Hall, Englewood Cliffs (1969) Cazzani, A., Atluri, S.N.: Four-noded mixed finite elements, using unsymmetric stresses, for linear analysis of membranes. Comput. Mech. 11, 229–251 (1993) Cook, R.D., Malkus, D.S., Plesha, M.E.: Concept and Applications of Finite Element Analysis. Wiley, New York (1989) Clough, R.W., Penzien, J.: Dynamics of Structures. McGraw-Hill, New York (1975) Savoia, M., Laudiero, F., Tralli, A.: A refined theory for laminated beams: part I—a new high order approach. Meccanica 28(1), 39–51 (1993) Savoia, M., Tralli, A., Laudiero, F.: A refined theory for laminated beams: part II—an iterative variational approach. Meccanica 28(3), 217–225 (1993) Pilkey, W.D.: Formulas for Stress, Strain, and Structural Matrices, 2nd edn. Wiley, New York (2005) Qatu, M.S., Leissa, A.W.: Vibration of Continuous Systems, 1st edn. McGraw-Hill, New York (2011) Bilotta, A., Formica, G., Turco, E.: Performance of a high-continuity finite element in three-dimensional elasticity. Int. J. Numer. Methods Biomed. Eng. 26, 1155–1175 (2010) Greco, L., Cuomo, M.: B-Spline interpolation of Kirchhoff-Love space rods. Comput. Methods Appl. Mech. Eng. 256, 251–269 (2013) Greco, L., Cuomo, M.: An implicit \(G^1\) multi patch B-spline interpolation for Kirchhoff-Love space rod. Comput. Methods Appl. Mech. Eng. 269, 173–197 (2014) Cazzani, A., Malagù, M., Turco, E.: Isogeometric analysis of plane curved beams. Math. Mech. Solids 21(5), 562–577 (2016) Cazzani, A., Malagù, M., Turco, E.: Isogeometric analysis: a powerful numerical tool for the elastic analysis of historical masonry arches. Contin. Mech. Thermodyn. 28(1), 139–156 (2016) Cazzani, A., Malagù, M., Turco, E., Stochino, F.: Constitutive models for strongly curved beams in the frame of isogeometric analysis. Math. Mech. Solids 21(2), 182–209 (2016) Greco, L., Cuomo, M.: An isogeometric implicit \(\text{ G }^1\) mixed finite element for Kirchhoff space rods. Comput. Methods Appl. Mech. Eng. 298, 325–349 (2016) Cazzani, A., Stochino, F., Turco, E.: An analytical assessment of finite elements and isogeometric analysis of the whole spectrum of Timoshenko beams. J. Appl. Math. Mech. Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) 96(10), 1220–1244 (2016) Altenbach, H., Eremeyev, V.A.: On the linear theory of micropolar plates. J. Appl. Math. Mech. Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) 89(4), 242–256 (2009) Altenbach, H., Eremeyev, V.A.: On the shell theory on the nanoscale with surface stresses. Int. J. Eng. Sci. 49(12), 1294–1301 (2011) Šilhavỳ, M.: A direct approach to nonlinear shells with application to surface-substrate interactions. Math. Mech. Complex Syst. 1(2), 211–232 (2013) Turco, E.: Tools for the numerical solution of inverse problems in structural mechanics: review and research perspectives. Eur. J. Environ. Civ. Eng. 21(5), 509–554 (2017) 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. Solids 20(8), 887–928 (2015) 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, Cham (2016) Placidi, L., Barchiesi, E.: Energy approach to brittle fracture in strain-gradient modelling. Proc. R. Soc. A Math. Phys. Eng. Sci. 474(20170878), 1–19 (2018) Rizzi, N., Varano, V., Gabriele, S.: Initial postbuckling behavior of thin-walled frames under mode interaction. Thin Walled Struct. 68, 124–134 (2013) Gabriele, S., Rizzi, N., Varano, V.: A 1D higher gradient model derived from Koiter’s shell theory. Math. Mech. Solids 21(6), 737–746 (2016) Cazzani, A., Wagner, N., Ruge, P., Stochino, F.: Continuous transition between traveling mass and traveling oscillator using mixed variables. Int. J. Non-Linear Mech. 66, 82–95 (2015) Acito, M., Stochino, F., Tattoni, S.: Structural response and reliability analysis of RC beam subjected to explosive loading. Appl. Mech. Mater. 82, 434–439 (2011) Stochino, F.: RC beams under blast load: reliability and sensitivity analysis. Eng. Fail. Anal. 66, 544–565 (2016) dell’Isola, F., Steigmann, D., Della Corte, A.: Synthesis of fibrous complex structures: designing microstructure to deliver targeted macroscale response. Appl. Mech. Rev. 67(6), 060804 (2015) Giorgio, I., Grygoruk, R., dell’Isola, F., Steigmann, D.J.: Pattern formation in the three-dimensional deformations of fibered sheets. Mech. Res. Commun. 69, 164–171 (2015) dell’Isola, F., Giorgio, I., Pawlikowski, M., Rizzi, N.L.: Large deformations of planar extensible beams and pantographic lattices: heuristic homogenisation, experimental and numerical examples of equilibrium. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 472(2185), 1–23 (2016) Scerrato, D., Zhurba Eremeeva, I.A., Lekszycki, T., Rizzi, N.L.: On the effect of shear stiffness on the plane deformation of linear second gradient pantographic sheets. J. Appl. Math. Mech. Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) 96(11), 1268–1279 (2016) Scerrato, D., Giorgio, I., Rizzi, N.L.: Three-dimensional instabilities of pantographic sheets with parabolic lattices: numerical investigations. Zeitschrift für Angewandte Mathematik und Physik (ZAMP) 67(53), 1–19 (2016) Giorgio, I., Rizzi, N.L., Turco, E.: Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis. Proc. R. Soc. A Math. Phys. Eng. Sci. 473(20170636), 1–21 (2017) Misra, A., Poorsolhjouy, P.: Granular micromechanics based micromorphic model predicts frequency band gaps. Contin. Mech. Thermodyn. 28(1), 215–234 (2016) Giorgio, I., Galantucci, L., Della Corte, A., Del Vescovo, D.: Piezo-electromechanical smart materials with distributed arrays of piezoelectric transducers: current and upcoming applications. Int. J. Appl. Electromagn. Mech. 47(4), 1051–1084 (2015) Giorgio, I., Culla, A., Del Vescovo, D.: Multimode vibration control using several piezoelectric transducers shunted with a multiterminal network. Arch. Appl. Mech. 79, 859–879 (2009) Enakoutsa, K., Della Corte, A., Giorgio, I.: A model for elastic flexoelectric materials including strain gradient effects. Math. Mech. Solids 21(2), 242–254 (2016) Maurini, C., dell’Isola, F., Pouget, J.: On models of layered piezoelectric beams for passive vibration control. J. Phys. IV 115, 307–316 (2004) Aymerich, F., Serra, M.: An ant colony optimization algorithm for stacking sequence design of composite laminates. Comput. Model. Eng. Sci. 13(1), 49–65 (2006) Aymerich, F., Serra, M.: Optimization of laminate stacking sequence for maximum buckling load using the ant colony optimization (ACO) metaheuristic. Compos. Part A Appl. Sci. Manuf. 39(2), 262–272 (2008) Banichuk, N.V., Ivanova, S.Y., Ragnedda, F., Serra, M.: Multiobjective approach for optimal design of layered plates against penetration of strikers. Mech. Based Des. Struct. Mach. 41(2), 189–201 (2013) Altenbach, H., Eremeyev, V.A.: Direct approach-based analysis of plates composed of functionally graded materials. Arch. Appl. Mech. 78(10), 775–794 (2008) Altenbach, H., Eremeyev, V.A.: Eigen-vibrations of plates made of functionally graded material. Comput. Mater. Continua 9(2), 153–178 (2009) Altenbach, H., Eremeyev, V.A.: Analysis of the viscoelastic behavior of plates made of functionally graded materials. J. Appl. Math. Mech. Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) 88(5), 332–341 (2008) Altenbach, H., Eremeyev, V.A.: On the bending of viscoelastic plates made of polymer foams. Acta Mech. 204(3–4), 137 (2009)