Abstract
The development of constitutive equations formulated in the resultant nonlinear shell theory is presented. The specific features of the present shell theory are drilling rotation naturally included in the formulation and asymmetric measures of strains and stress resultants. The special attention in the chapter is given to recent achievements: progressive failure analysis of laminated shells and elastoplastic constitutive relation for shells made of functionally graded material (FGM). The modified Hashin criterion is used to estimate failure initiation in laminates and stiffness degradation approach in the last ply failure computations. The numerical results obtained for axially compressed C-shaped column are compared with experimental load-deflection curve. The Cosserat plane stress assumption, Tamura-Tomota-Ozawa (TTO) model and improved method of shear correction factor calculation are applied in the elastoplastic constitutive relation for FGM shell. The proposed formulation is tested in numerical examples: rectangular compressed plate and channel section clamped beam. The influence of TTO model parameters and Cosserat characteristic length is investigated.
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References
Reissner, E.: Linear and nonlinear theory of shells. In: Fung, Y.C., Sechler, E.E. (eds.) Thin Shell Structures, pp. 29–44. Prentice-Hall, Englewood Cliffs (1974)
Libai, A., Simmonds, J.G.: The Nonlinear Theory of Elastic Shells. Cambridge University Press, Cambridge (1998)
Chróścielewski, J., Makowski, J., Pietraszkiewicz, W.: Statyka i Dynamika Powłok Wielopłatowych: Nieliniowa teoria i metoda elementów skończonych (Statics and Dynamics of Multifold Shells: Nonlinear Theory and Finite Element Method). Wydawnictwo IPPT PAN, Warszawa (2004)
Altenbach, J., Altenbach, H., Eremeyev, V.A.: On generalized Cosserat-type theories of plates and shells: a short review and bibliography. Arch. Appl. Mech. 80, 73–92 (2010). https://doi.org/10.1007/s00419-009-0365-3
Neff, P.: A geometrically exact Cosserat shell-model including size effects, avoiding degeneracy in the thin shell limit. Part I: Formal dimensional reduction for elastic plates and existence of minimizers for positive Cosserat couple modulus. Contin. Mech. Thermodyn. 16, 577–628 (2004). https://doi.org/10.1007/s00161-004-0182-4
Chróścielewski, J., Makowski, J., Stumpf, H.: Genuinely resultant shell finite elements accounting for geometric and material non-linearity. Int. J. Numer. Methods Eng. 35, 63–94 (1992). https://doi.org/10.1002/nme.1620350105
Pietraszkiewicz, W., Konopińska, V.: Drilling couples and refined constitutive equations in the resultant geometrically non-linear theory of elastic shells. Int. J. Solids Struct. 51, 2133–2143 (2014). https://doi.org/10.1016/j.ijsolstr.2014.02.022
Burzyński, S., Chróścielewski, J., Daszkiewicz, K., Pietraszkiewicz, W., Sabik, A., Sobczyk, B., Witkowski, W.: On constitutive relations in the resultant non-linear theory of shells. In: Kołakowski, Z., Mania, R.J. (eds.) Statics, Dynamics and Stability of Structures. Selected Problems of Solid Mechanics, pp. 298–318. Lodz University of Technology, Lodz (2016)
Makowski, J., Stumpf, H.: Finite strains and rotations in shells. In: Pietraszkiewicz, W. (ed.) Finite Rotations in Structural Mechanics. Lecture Notes in Engineering, vol. 19, pp. 175–194. Springer, Berlin (1986)
Eremeyev, V.A., Pietraszkiewicz, W.: Local symmetry group in the general theory of elastic shells. J. Elast. 85, 125–152 (2006). https://doi.org/10.1007/s10659-006-9075-z
Altenbach, H., Eremeyev, V.A.: On the linear theory of micropolar plates. ZAMM Zeitschrift fur Angew. Math. und Mech. 89, 242–256 (2009). https://doi.org/10.1002/zamm.200800207
Chróscielewski, J., Witkowski, W.: On some constitutive equations for micropolar plates. ZAMM Zeitschrift fur Angew. Math. und Mech. 90, 53–64 (2010). https://doi.org/10.1002/zamm.200900366
Chróścielewski, J., Witkowski, W.: FEM analysis of Cosserat plates and shells based on some constitutive relations. ZAMM Zeitschrift fur Angew. Math. und Mech. 91, 400–412 (2011). https://doi.org/10.1002/zamm.201000090
Burzyński, S., Chróścielewski, J., Witkowski, W.: Geometrically nonlinear FEM analysis of 6-parameter resultant shell theory based on 2-D Cosserat constitutive model. ZAMM - J. Appl. Math. Mech. Zeitschrift für Angew. Math. und Mech. 96, 191–204 (2016). https://doi.org/10.1002/zamm.201400092
Burzyński, S., Chróścielewski, J., Witkowski, W.: Elastoplastic material law in 6-parameter nonlinear shell theory. In: Pietraszkiewicz, W., Górski, J. (eds.) 10th Jubilee Conference on Shell Structures—Theory and Applications (SSTA), pp. 377–380. CRC Press, London (2014)
Burzyński, S., Chróścielewski, J., Witkowski, W.: Elastoplastic law of Cosserat type in shell theory with drilling rotation. Math. Mech. Solids. 20, 790–805 (2015). https://doi.org/10.1177/1081286514554351
Daszkiewicz, K., Chróścielewski, J., Witkowski, W.: Geometrically nonlinear analysis of functionally graded shells based on 2-D Cosserat constitutive model. Eng. Trans. 62, 109–130 (2014)
Burzyński, S., Chróścielewski, J., Daszkiewicz, K., Witkowski, W.: Geometrically nonlinear FEM analysis of FGM shells based on neutral physical surface approach in 6-parameter shell theory. Compos. Part B Eng. 107, 203–213 (2016). https://doi.org/10.1016/j.compositesb.2016.09.015
Tamura, I., Tomota, Y., Ozawa, M.: Strength and ductility of Iron-Nickel-Carbon alloys composed of austenite and martensite with various strength. In: 3rd International Conference on Strength of Metals and Alloys, pp. 611–615. Institute of Metal and Iron, Cambridge (1973)
Burzyński, S., Chróścielewski, J., Daszkiewicz, K., Witkowski, W.: Elastoplastic nonlinear FEM analysis of FGM shells of Cosserat type. Compos. Part B Eng. 154, 478–491 (2018). https://doi.org/10.1016/j.compositesb.2018.07.055
Chróścielewski, J., Sabik, A., Sobczyk, B., Witkowski, W.: 2-D constitutive equations for orthotropic Cosserat type laminated shells in finite element analysis. Compos. Part B Eng. 165, 335–353 (2019). https://doi.org/10.1016/j.compositesb.2018.11.101
Chróścielewski, J., Kreja, I., Sabik, A., Witkowski, W.: Modeling of composite shells in 6-parameter nonlinear theory with drilling degree of freedom. Mech. Adv. Mater. Struct. 18, 403–419 (2011). https://doi.org/10.1080/15376494.2010.524972
Chróścielewski, J., Sabik, A., Sobczyk, B., Witkowski, W.: Nonlinear FEM 2D failure onset prediction of composite shells based on 6-parameter shell theory. Thin-Walled Struct. 105, 207–219 (2016). https://doi.org/10.1016/j.tws.2016.03.024
Sobczyk, B.: FEM analysis of composite materials failure in nonlinear six field shell theory. Doctoral Thesis (2016)
Sabik, A.: Progressive failure analysis of laminates in the framework of 6-field non-linear shell theory. Compos. Struct. 200, 195–203 (2018). https://doi.org/10.1016/j.compstruct.2018.05.069
Debski, H., Teter, A.: Effect of load eccentricity on the buckling and post-buckling states of short laminated Z-columns. Compos. Struct. 210, 134–144 (2019). https://doi.org/10.1016/j.compstruct.2018.11.044
Kim, Y.J.: State of the practice of FRP composites in highway bridges. Eng. Struct. 179, 1–8 (2019). https://doi.org/10.1016/j.engstruct.2018.10.067
Siwowski, T., Kulpa, M., Rajchel, M., Poneta, P.: Design, manufacturing and structural testing of all-composite FRP bridge girder. Compos. Struct. 206, 814–827 (2018). https://doi.org/10.1016/j.compstruct.2018.08.048
Birman, V., Kardomateas, G.A.: Review of current trends in research and applications of sandwich structures. Compos. Part B Eng. 142, 221–240 (2018). https://doi.org/10.1016/j.compositesb.2018.01.027
Amaro, A.M., Pinto, M.I.M., Reis, P.N.B., Neto, M.A., Lopes, S.M.R.: Structural integrity of glass/epoxy composites embedded in cement or geopolymer mortars. Compos. Struct. 206, 509–516 (2018). https://doi.org/10.1016/j.compstruct.2018.08.060
Zhang, X., Shi, Y., Li, Z.-X.: Experimental study on the tensile behavior of unidirectional and plain weave CFRP laminates under different strain rates. Compos. Part B Eng. 164, 524–536 (2019). https://doi.org/10.1016/j.compositesb.2019.01.067
Zhang, Z., He, M., Liu, A., Singh, H.K., Ramakrishnan, K.R., Hui, D., Shankar, K., Morozov, E.V.: Vibration-based assessment of delaminations in FRP composite plates. Compos. Part B Eng. 144, 254–266 (2018). https://doi.org/10.1016/j.compositesb.2018.03.003
Gliszczynski, A., Kubiak, T., Borkowski, L.: Experimental investigation of pre-damaged thin-walled channel section column subjected to compression. Compos. Part B Eng. 147, 56–68 (2018). https://doi.org/10.1016/j.compositesb.2018.04.022
Altaee, M., Cunningham, L.S., Gillie, M.: Practical application of CFRP strengthening to steel floor beams with web openings: a numerical investigation. J. Constr. Steel Res. 155, 395–408 (2019). https://doi.org/10.1016/j.jcsr.2019.01.006
Chróścielewski, J., Miśkiewicz, M., Pyrzowski, Ł., Rucka, M., Sobczyk, B., Wilde, K.: Modal properties identification of a novel sandwich footbridge—comparison of measured dynamic response and FEA. Compos. Part B Eng. 151, 245–255 (2018). https://doi.org/10.1016/j.compositesb.2018.06.016
Reddy, J.N.: Mechanics of Laminated Composite Plates and Shells, Theory and Analysis, 2nd edn. CRC Press, Boca Raton, London, New York, Washington.C (2004)
Kaw, A.: Mechanics of Composite Materials, 2nd edn. Taylor & Francis Group, Boca Raton, London, New York (2006)
Davila, C.G., Camanho, P.P., Rose, C.A.: Failure criteria for FRP laminates. J. Compos. Mater. 39, 323–345 (2005). https://doi.org/10.1177/0021998305046452
Hinton, M., Kaddour, A., Soden, P.: A further assessment of the predictive capabilities of current failure theories for composite laminates: comparison with experimental evidence. Compos. Sci. Technol. 64, 549–588 (2004). https://doi.org/10.1016/S0266-3538(03)00227-6
Kaddour, A.S., Hinton, M.J., Soden, P.D.: A comparison of the predictive capabilities of current failure theories for composite laminates: additional contributions. Compos. Sci. Technol. 64, 449–476 (2004). https://doi.org/10.1016/S0266-3538(03)00226-4
Soden, P., Kaddour, A., Hinton, M.: Recommendations for designers and researchers resulting from the world-wide failure exercise. Compos. Sci. Technol. 64, 589–604 (2004). https://doi.org/10.1016/S0266-3538(03)00228-8
Puck, A., Schürmann, H.: Failure analysis of FRP laminates by means of physically based phenomenological models. Compos. Sci. Technol. 62, 1633–1662 (2002). https://doi.org/10.1016/S0266-3538(01)00208-1
Reddy, Y.S.N., Dakshina Moorthy, C.M., Reddy, J.N.: Non-linear progressive failure analysis of laminated composite plates. Int. J. Non. Linear. Mech. 30, 629–649 (1995). https://doi.org/10.1016/0020-7462(94)00041-8
Xie, D., Biggers, S.B.: Postbuckling analysis with progressive damage modeling in tailored laminated plates and shells with a cutout. Compos. Struct. 59, 199–216 (2003). https://doi.org/10.1016/S0263-8223(02)00233-7
Ambur, D.R., Jaunky, N., Hilburger, M., Dávila, C.G.: Progressive failure analyses of compression-loaded composite curved panels with and without cutouts. Compos. Struct. 65, 143–155 (2004). https://doi.org/10.1016/S0263-8223(03)00184-3
Bai, J.B., Shenoi, R.A., Yun, X.Y., Xiong, J.J.: Progressive damage modelling of hybrid RTM-made composite Π-joint under four-point flexure using mixed failure criteria. Compos. Struct. 159, 327–334 (2017). https://doi.org/10.1016/j.compstruct.2016.09.083
Matzenmiller, A., Lubliner, J., Taylor, R.L.: A constitutive model for anisotropic damage in fiber-composites. Mech. Mater. 20, 125–152 (1995). https://doi.org/10.1016/0167-6636(94)00053-0
Lee, C.S., Kim, J.H., Kim, S.K., Ryu, D.M., Lee, J.M.: Initial and progressive failure analyses for composite laminates using Puck failure criterion and damage-coupled finite element method. Compos. Struct. 121, 406–419 (2015). https://doi.org/10.1016/j.compstruct.2014.11.011
Lopes, C.S., Camanho, P.P., Gürdal, Z., Tatting, B.F.: Progressive failure analysis of tow-placed, variable-stiffness composite panels. Int. J. Solids Struct. 44, 8493–8516 (2007). https://doi.org/10.1016/j.ijsolstr.2007.06.029
Gliszczynski, A., Kubiak, T.: Progressive failure analysis of thin-walled composite columns subjected to uniaxial compression. Compos. Struct. 169, 52–61 (2017). https://doi.org/10.1016/j.compstruct.2016.10.029
Sabik, A.: Direct shear stress vs strain relation for fiber reinforced composites. Compos. Part B Eng. 139, 24–30 (2018). https://doi.org/10.1016/j.compositesb.2017.11.057
Shen, M., Bever, M.B.: Gradients in polymeric materials. J. Mater. Sci. 7, 741–746 (1972). https://doi.org/10.1007/BF00549902
Jha, D.K., Kant, T., Singh, R.K.: A critical review of recent research on functionally graded plates. Compos. Struct. 96, 833–849 (2013). https://doi.org/10.1016/j.compstruct.2012.09.001
Swaminathan, K., Naveenkumar, D.T., Zenkour, A.M., Carrera, E.: Stress, vibration and buckling analyses of FGM plates—a state- of-the-art review. Compos. Struct. 120, 10–31 (2015). https://doi.org/10.1016/j.compstruct.2014.09.070
Williamson, R.L., Rabin, B.H., Drake, J.T.: Finite element analysis of thermal residual stresses at graded ceramic-metal interfaces. Part I. Model description and geometrical effects. J. Appl. Phys. 74, 1310–1320 (1993). https://doi.org/10.1063/1.354910
Drake, J.T., Williamson, R.L., Rabin, B.H.: Finite element analysis of thermal residual stresses at graded ceramic-metal interfaces. Part II. Interface optimization for residual stress reduction. J. Appl. Phys. 74, 1321–1326 (1993). https://doi.org/10.1063/1.354911
Jin, Z.H., Paulino, G.H., Dodds, R.H.: Cohesive fracture modeling of elastic-plastic crack growth in functionally graded materials. Eng. Fract. Mech. 70, 1885–1912 (2003). https://doi.org/10.1016/S0013-7944(03)00130-9
Baghani, M., Fereidoonnezhad, B.: Limit analysis of FGM circular plates subjected to arbitrary rotational symmetric loads using von-Mises yield criterion. Acta Mech. 224, 1601–1608 (2013). https://doi.org/10.1007/s00707-013-0828-z
Gunes, R., Aydin, M., Kemal Apalak, M., Reddy, J.N.: Experimental and numerical investigations of low velocity impact on functionally graded circular plates. Compos. Part B Eng. 59, 21–32 (2014). https://doi.org/10.1016/j.compositesb.2013.11.022
Xu, G., Huang, H., Chen, B., Chen, F.: Buckling and postbuckling of elastoplastic FGM plates under inplane loads. Compos. Struct. 176, 225–233 (2017). https://doi.org/10.1016/j.compstruct.2017.04.061
Kleiber, M., Taczała, M., Buczkowski, R.: Elasto-plastic response of thick plates built in functionally graded material using the third order plate theory. In: Advances in Computational Plasticity, pp. 185–199 (2018)
Huang, H., Han, Q.: Elastoplastic buckling of axially loaded functionally graded material cylindrical shells. Compos. Struct. 117, 135–142 (2014). https://doi.org/10.1016/j.compstruct.2014.06.018
Zhang, Y., Huang, H., Han, Q.: Buckling of elastoplastic functionally graded cylindrical shells under combined compression and pressure. Compos. Part B Eng. 69, 120–126 (2015). https://doi.org/10.1016/j.compositesb.2014.09.024
Kalali, A.T., Hassani, B., Hadidi-Moud, S.: Elastic-plastic analysis of pressure vessels and rotating disks made of functionally graded materials using the isogeometric approach. J. Theor. Appl. Mech. 113 (2016). https://doi.org/10.15632/jtam-pl.54.1.113
Akis, T.: Elastoplastic analysis of functionally graded spherical pressure vessels. Comput. Mater. Sci. 46, 545–554 (2009). https://doi.org/10.1016/j.commatsci.2009.04.017
Jrad, H., Mars, J., Wali, M., Dammak, F.: Geometrically nonlinear analysis of elastoplastic behavior of functionally graded shells. Eng. Comput. (2018). https://doi.org/10.1007/s00366-018-0633-3
Mathew, T.V., Natarajan, S., Martínez-Pañeda, E.: Size effects in elastic-plastic functionally graded materials. Compos. Struct. 204, 43–51 (2018). https://doi.org/10.1016/j.compstruct.2018.07.048
Jeong, J., Ramezani, H., Münch, I., Neff, P.: A numerical study for linear isotropic Cosserat elasticity with conformally invariant curvature. ZAMM Zeitschrift fur Angew. Math. und Mech. 89, 552–569 (2009). https://doi.org/10.1002/zamm.200800218
Fischmeister, H., Karlsson, B.: Plastizitatseigenschaften Grob-Zweiphasiger Werkstoffe. Zeitschrift für Met. 68, 311–327 (1977)
Nguyen, T.K., Sab, K., Bonnet, G.: First-order shear deformation plate models for functionally graded materials. Compos. Struct. 83, 25–36 (2008). https://doi.org/10.1016/j.compstruct.2007.03.004
Singha, M.K., Prakash, T., Ganapathi, M.: Finite element analysis of functionally graded plates under transverse load. Finite Elem. Anal. Des. 47, 453–460 (2011). https://doi.org/10.1016/j.finel.2010.12.001
Daszkiewicz, K.: A family of hybrid mixed elements in 6-parameter shell theory, geometrically nonlinear analysis of functionally graded shells. Doctoral Thesis (in Polish) (2017)
de Borst, R.: Simulation of strain localization: a reappraisal of the Cosserat continuum. Eng. Comput. 8, 317–332 (1991)
de Souza Neto, E.A., Peric, D., Owen, D.R.: Computational Methods for Plasticity: Theory and Applications (2009)
Simo, J.C., Hughes, T.J.R.: Computational Inelasticity. Springer New York, Inc. (1998)
Eberlein, R., Wriggers, P.: Finite element concepts for finite elastoplastic strains and isotropic stress response in shells: theoretical and computational analysis. Comput. Methods Appl. Mech. Eng. 171, 243–279 (1999). https://doi.org/10.1016/S0045-7825(98)00212-6
Tan, X.G., Vu-Quoc, L.: Efficient and accurate multilayer solid-shell element: non-linear materials at finite strain. Int. J. Numer. Methods Eng. 63, 2124–2170 (2005). https://doi.org/10.1002/nme.1360
Abaqus 6.14-2 User Manual. Dassault Systemes Simulia Corp., Providence, RI, USA (2014)
Acknowledgements
The research reported in this paper was supported by the National Science Centre, Poland with the grant UMO-2015/17/B/ST8/02190. Parallel solver for CAM elements is developed on the basis of HSL, a collection of Fortran codes for large-scale scientific computation. http://www.hsl.rl.ac.uk. Abaqus calculations were carried out at the Academic Computer Centre in Gdańsk.
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Burzyński, S., Chróścielewski, J., Daszkiewicz, K., Sabik, A., Sobczyk, B., Witkowski, W. (2019). Recent Achievements in Constitutive Equations of Laminates and Functionally Graded Structures Formulated in the Resultant Nonlinear Shell Theory. In: Altenbach, H., Chróścielewski, J., Eremeyev, V., Wiśniewski, K. (eds) Recent Developments in the Theory of Shells . Advanced Structured Materials, vol 110. Springer, Cham. https://doi.org/10.1007/978-3-030-17747-8_11
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