Abstract
In the present work, a non-linear finite element model for the analysis of composite and sandwich plates is proposed. The underlying variational formulation is based on the kinematics of the refined zigzag theory (RZT) as well as on a modified version of the two-field Hellinger–Reissner (HR) functional, where the displacements and the transverse shear stresses are involved as independent field variables. In fact, a consistent expression for each shear stress function is obtained by integrating the local equilibrium equations written in terms of derivated strain components. Regardless of the number of material layers, the proposed four-node plate element, which is labeled HR–RZT, exhibits only seven nodal degrees of freedom. The additionally introduced variables are effectively eliminated on the element level. One key advantage of the HR–RZT formulation is the fact that interlayer-continuity of the transverse shear stresses is automatically obtained without resorting to any post-processing procedures. The same applies to the zero stress conditions at the outer surfaces of the laminate. Further, due to the enhanced kinematics, the layerwise distortion of the laminate’s cross-section can also be accurately captured. The performance of the novel element formulation is studied by several numerical applications, where the solutions of the two-dimensional HR–RZT elements are verified by comparison with fully three-dimensional finite element models.
Similar content being viewed by others
References
Altenbach H, Altenbach J, Kissing W (2018) Mechanics of composite structural elements. Springer, Singapore
Ambartsumian S (1958) On a general theory of anisotropic shells. J Appl Math Mech 22(2):305–319
Auricchio F, Sacco E (1999) A mixed-enhanced finite-element for the analysis of laminated composite plates. Int J Numer Methods Eng 44(10):1481–1504
Auricchio F, Sacco E (2003) Refined first-order shear deformation theory models for composite laminates. J Appl Mech 70(3):381–390
Auricchio F, Balduzzi G, Khoshgoftar M, Rahimi G, Sacco E (2014) Enhanced modeling approach for multilayer anisotropic plates based on dimension reduction method and Hellinger–Reissner principle. Compos Struct 118:622–633
Averill R (1994) Static and dynamic response of moderately thick laminated beams with damage. Compos Eng 4(4):381–395
Brank B, Carrera E (2000) Multilayered shell finite element with interlaminar continuous shear stresses: a refinement of the Reissner–Mindlin formulation. Int J Numer Methods Eng 48(6):843–874
Carrera E (1996) \(C^0\) Reissner–Mindlin multilayered plate elements including zig-zag and interlaminar stress continuity. Int J Numer Methods Eng 39(11):1797–1820
Carrera E (1997) \(c_z^0\) requirements—models for the two dimensional analysis of multilayered structures. Compos Struct 37(3–4):373–383
Carrera E (1998) Evaluation of layerwise mixed theories for laminated plates analysis. AIAA J 36(5):830–839
Carrera E (2000) A priori vs. a posteriori evaluation of transverse stresses in multilayered orthotropic plates. Compos Struct 48(4):245–260
Carrera E, Demasi L (2002) Classical and advanced multilayered plate elements based upon pvd and rmvt. Part 1: derivation of finite element matrices. Int J Numer Methods Eng 55(2):191–231
Carrera E, Demasi L (2002) Classical and advanced multilayered plate elements based upon PVD and RMVT. Part 2: numerical implementations. Int J Numer Methods Eng 55(3):253–291
Carrera E (2002) Theories and finite elements for multilayered, anisotropic, composite plates and shells. Arch Comput Methods Eng 9(2):87–140
Carrera E (2003) Historical review of zig-zag theories for multilayered plates and shells. Appl Mech Rev 56(3):287–308
Carrera E (2003) Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking. Arch Comput Methods Eng 10(3):215–296
Carrera E (2004) On the use of the murakami’s zig-zag function in the modeling of layered plates and shells. Comput Struct 82(7–8):541–554
Di Sciuva M (1986) Bending, vibration and buckling of simply supported thick multilayered orthotropic plates: an evaluation of a new displacement model. J Sound Vib 105(3):425–442
Di Sciuva M (1987) An improved shear-deformation theory for moderately thick multilayered anisotropic shells and plates. J Appl Mech 54(3):589–596
Di Sciuva M (1992) Multilayered anisotropic plate models with continuous interlaminar stresses. Compos Struct 22(3):149–167
Dvorkin E, Bathe KJ (1984) A continuum mechanics based four-node shell element for general non-linear analysis. Eng Comput 1(1):77–88
Eijo A, Oñate E, Oller S (2013) A four-noded quadrilateral element for composite laminated plates/shells using the refined zigzag theory. Int J Numer Methods Eng 95(8):631–660
Gherlone M, Tessler A, Di Sciuva M (2011) \(C^0\) beam elements based on the refined zigzag theory for multilayered composite and sandwich laminates. Compos Struct 93(11):2882–2894
Gruttmann F, Wagner W, Meyer L, Wriggers P (1993) A nonlinear composite shell element with continuous interlaminar shear stresses. Comput Mech 13(3):175–188
Gruttmann F, Wagner W (1994) On the numerical analysis of local effects in composite structures. Compos Struct 29(1):1–12
Gruttmann F, Wagner W, Knust G (2016) A coupled global-local shell model with continuous interlaminar shear stresses. Comput Mech 57(2):237–255
Gruttmann F, Wagner W (2017) Shear correction factors for layered plates and shells. Comput Mech 59(1):129–146
Gruttmann F, Knust G, Wagner W (2017) Theory and numerics of layered shells with variationally embedded interlaminar stresses. Comput Methods Appl Mech Eng 326:713–738
Iurlaro L, Gherlone M, Di Sciuva M, Tessler A (2015) Refined zigzag theory for laminated composite and sandwich plates derived from Reissner’s mixed variational theorem. Compos Struct 133:809–817
Kant T, Manjunatha B (1994) On accurate estimation of transverse stresses in multilayer laminates. Comput Struct 50(3):351–365
Kant T, Swaminathan K (2002) Analytical solutions for the static analysis of laminated composite and sandwich plates based on a higher order refined theory. Compos Struct 56(4):329–344
Klinkel S, Gruttmann F, Wagner W (1999) A continuum based three-dimensional shell element for laminated structures. Comput Struct 71(1):43–62
Klinkel S, Gruttmann F, Wagner W (2006) A robust non-linear solid shell element based on a mixed variational formulation. Comput Methods Appl Mech Eng 195(1):179–201
Malkus D, Hughes T (1978) Mixed finite element methods-reduced and selective integration techniques: a unification of concepts. Comput Methods Appl Mech Eng 15(1):63–81
Murakami H (1986) Laminated composite plate theory with improved in-plane responses. J Appl Mech 53(3):661–666
Oñate E, Eijo A, Oller S (2012) Simple and accurate two-noded beam element for composite laminated beams using a refined zigzag theory. Comput Methods Appl Mech Eng 213:362–382
Reddy J (1984) A simple higher-order theory for laminated composite plates. J Appl Mech 51(4):745–752
Reddy J, Robbins D (1994) Theories and computational models for composite laminates. Appl Mech Rev 47(6):147–169
Reddy J (2003) Mechanics of laminated composite plates and shells: theory and analysis, 2nd edn. CRC Press, Boca Raton
Rolfes R, Rohwer K (1997) Improved transverse shear stresses in composite finite elements based on first order shear deformation theory. Int J Numer Methods Eng 40(1):51–60
Rolfes R, Noor A, Sparr H (1998) Evaluation of transverse thermal stresses in composite plates based on first-order shear deformation theory. Comput Methods Appl Mech Eng 167(3–4):355–368
Schürg M, Wagner W, Gruttmann F (2009) An enhanced fsdt model for the calculation of interlaminar shear stresses in composite plate structures. Comput Mech 44(6):765–776
Taylor R (2019) FEAP-A finite element analysis program. http://projects.ce.berkeley.edu/feap/
Tessler A, Hughes T (1985) A three-node mindlin plate element with improved transverse shear. Comput Methods Appl Mech Eng 50(1):71–101
Tessler A, Di Sciuva M, Gherlone M (2009) A refined zigzag beam theory for composite and sandwich beams. J Compos Mater 43(9):1051–1081
Tessler A, Di Sciuva M, Gherlone M (2010) A consistent refinement of first-order shear deformation theory for laminated composite and sandwich plates using improved zigzag kinematics. J Mech Mater Struct 5(2):341–367
Tessler A (2015) Refined zigzag theory for homogeneous, laminated composite, and sandwich beams derived from Reissner’s mixed variational principle. Meccanica 50(10):2621–2648
Toledano A, Murakami H (1987) A composite plate theory for arbitrary laminate configurations. J Appl Mech 54(1):181–189
Versino D, Gherlone M, Mattone M, Di Sciuva M, Tessler A (2013) \(C^0\) triangular elements based on the refined zigzag theory for multilayer composite and sandwich plates. Compos B Eng 44(1):218–230
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix 1
Components of \({\mathbf {T}}_{\alpha 0}^k\):
Components of \({\widehat{{\mathbf {T}}}}_\alpha ^k(z)\):
with
Components of \({\widetilde{{\mathbf {T}}}}_\alpha \):
Components of \({\overline{{\mathbf {T}}}}_\alpha ^k\):
Appendix 2
Standard FSDT constitutive matrices:
Additional zigzag contributions:
with
and
with
and
with
Rights and permissions
About this article
Cite this article
Köpple, M., Wagner, W. A mixed finite element model with enhanced zigzag kinematics for the non-linear analysis of multilayer plates. Comput Mech 65, 23–40 (2020). https://doi.org/10.1007/s00466-019-01750-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00466-019-01750-y