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Recent Achievements in Constitutive Equations of Laminates and Functionally Graded Structures Formulated in the Resultant Nonlinear Shell Theory

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Book cover Recent Developments in the Theory of Shells

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 110))

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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

  1. 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)

    Google Scholar 

  2. Libai, A., Simmonds, J.G.: The Nonlinear Theory of Elastic Shells. Cambridge University Press, Cambridge (1998)

    Book  Google Scholar 

  3. 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)

    Google Scholar 

  4. 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

    Article  Google Scholar 

  5. 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

    Article  Google Scholar 

  6. 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

    Article  Google Scholar 

  7. 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

    Article  Google Scholar 

  8. 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)

    Google Scholar 

  9. 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)

    Chapter  Google Scholar 

  10. 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

    Article  Google Scholar 

  11. 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

    Article  Google Scholar 

  12. 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

    Article  Google Scholar 

  13. 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

    Article  Google Scholar 

  14. 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

    Article  Google Scholar 

  15. 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)

    Google Scholar 

  16. 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

    Article  Google Scholar 

  17. 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)

    Google Scholar 

  18. 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

    Article  Google Scholar 

  19. 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)

    Google Scholar 

  20. 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

    Article  Google Scholar 

  21. 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

    Article  Google Scholar 

  22. 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

    Article  Google Scholar 

  23. 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

    Article  Google Scholar 

  24. Sobczyk, B.: FEM analysis of composite materials failure in nonlinear six field shell theory. Doctoral Thesis (2016)

    Google Scholar 

  25. 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

    Article  Google Scholar 

  26. 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

    Article  Google Scholar 

  27. 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

    Article  Google Scholar 

  28. 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

    Article  Google Scholar 

  29. 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

    Article  CAS  Google Scholar 

  30. 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

    Article  Google Scholar 

  31. 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

    Article  CAS  Google Scholar 

  32. 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

    Article  CAS  Google Scholar 

  33. 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

    Article  CAS  Google Scholar 

  34. 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

    Article  Google Scholar 

  35. 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

    Article  Google Scholar 

  36. 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)

    Google Scholar 

  37. Kaw, A.: Mechanics of Composite Materials, 2nd edn. Taylor & Francis Group, Boca Raton, London, New York (2006)

    Google Scholar 

  38. 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

    Article  CAS  Google Scholar 

  39. 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

    Article  CAS  Google Scholar 

  40. 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

    Article  CAS  Google Scholar 

  41. 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

    Article  Google Scholar 

  42. 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

    Article  Google Scholar 

  43. 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

    Article  Google Scholar 

  44. 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

    Article  Google Scholar 

  45. 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

    Article  Google Scholar 

  46. 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

    Article  Google Scholar 

  47. 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

    Article  Google Scholar 

  48. 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

    Article  Google Scholar 

  49. 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

    Article  Google Scholar 

  50. 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

    Article  Google Scholar 

  51. 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

    Article  CAS  Google Scholar 

  52. Shen, M., Bever, M.B.: Gradients in polymeric materials. J. Mater. Sci. 7, 741–746 (1972). https://doi.org/10.1007/BF00549902

    Article  CAS  Google Scholar 

  53. 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

    Article  Google Scholar 

  54. 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

    Article  Google Scholar 

  55. 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

    Article  CAS  Google Scholar 

  56. 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

    Article  CAS  Google Scholar 

  57. 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

    Article  Google Scholar 

  58. 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

    Article  Google Scholar 

  59. 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

    Article  Google Scholar 

  60. 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

    Article  Google Scholar 

  61. 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)

    Google Scholar 

  62. 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

    Article  Google Scholar 

  63. 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

    Article  Google Scholar 

  64. 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

  65. 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

    Article  CAS  Google Scholar 

  66. 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

    Article  Google Scholar 

  67. 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

    Article  Google Scholar 

  68. 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

    Article  Google Scholar 

  69. Fischmeister, H., Karlsson, B.: Plastizitatseigenschaften Grob-Zweiphasiger Werkstoffe. Zeitschrift für Met. 68, 311–327 (1977)

    CAS  Google Scholar 

  70. 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

    Article  Google Scholar 

  71. 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

    Article  Google Scholar 

  72. 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)

    Google Scholar 

  73. de Borst, R.: Simulation of strain localization: a reappraisal of the Cosserat continuum. Eng. Comput. 8, 317–332 (1991)

    Article  Google Scholar 

  74. de Souza Neto, E.A., Peric, D., Owen, D.R.: Computational Methods for Plasticity: Theory and Applications (2009)

    Google Scholar 

  75. Simo, J.C., Hughes, T.J.R.: Computational Inelasticity. Springer New York, Inc. (1998)

    Google Scholar 

  76. 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

    Article  Google Scholar 

  77. 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

    Article  Google Scholar 

  78. Abaqus 6.14-2 User Manual. Dassault Systemes Simulia Corp., Providence, RI, USA (2014)

    Google Scholar 

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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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Authors and Affiliations

  1. Faculty of Civil and Environmental Engineering, Department of Mechanics of Materials and Structures, Gdansk University of Technology, Narutowicza 11/12, 80-233, Gdańsk, Poland

    Stanisław Burzyński, Jacek Chróścielewski, Karol Daszkiewicz, Agnieszka Sabik, Bartosz Sobczyk & Wojciech Witkowski

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  1. Stanisław Burzyński
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  2. Jacek Chróścielewski
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Correspondence to Karol Daszkiewicz .

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Editors and Affiliations

  1. Faculty of Mechanical Engineering, Otto-von-Guericke-Universität Magdeburg, Magdeburg, Germany

    Holm Altenbach

  2. Faculty of Civil and Environmental Engineering, Gdańsk University of Technology, Gdańsk, Poland

    Jacek Chróścielewski

  3. Faculty of Civil and Environmental Engineering, Gdańsk University of Technology, Gdańsk, Poland

    Victor A. Eremeyev

  4. Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, Poland

    Krzysztof Wiśniewski

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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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