References
- B. Zhu and Y. Cai, A strain rate-dependent enhanced continuum model for elastic-plastic impact response of metal-ceramic functionally graded composites, Int. J. Impact Eng., vol. 133, pp. 103340, 2019. DOI: https://doi.org/10.1016/j.ijimpeng.2019.103340.
- S. Kumar, K.M. Reddy, A. Kumar, and G.R. Devi, Development and characterization of polymer–ceramic continuous fiber reinforced functionally graded composites for aerospace application, Aerosp. Sci. Technol., vol. 26, no. 1, pp. 185–191, 2013. DOI: https://doi.org/10.1016/j.ast.2012.04.002.
- J. Jang and S. Han, Mechanical properties of glass-fibre mat/pmma functionally gradient composite, Compos. Part A. Appl. Sci. Manuf., vol. 30, no. 9, pp. 1045–1053, 1999. DOI: https://doi.org/10.1016/S1359-835X(99)00021-4.
- W. Pompe, et al., Functionally graded materials for biomedical applications, Mater. Sci. Eng. A., vol. 362, no. 1-2, pp. 40–60, 2003. DOI: https://doi.org/10.1016/S0921-5093(03)00580-X.
- D. Mahmoud and M.A. Elbestawi, Lattice structures and functionally graded materials applications in additive manufacturing of orthopedic implants: a review, J. Manuf. Mater. Process., vol. 1, no. 2, pp. 13, 2017. DOI: https://doi.org/10.3390/jmmp1020013.
- R. Vaßen, H. Kassner, A. Stuke, D.E. Mack, M.O.D. Jarligo, and D. Stöver, Functionally graded thermal barrier coatings with improved reflectivity and high-temperature capability, Mater. Sci. Forum., vol. 631-632, pp. 73–78, 2009. DOI: https://doi.org/10.4028/www.scientific.net/MSF.631-632.73.
- E. Müller, Č. Drašar, J. Schilz, and W.A. Kaysser, Functionally graded materials for sensor and energy applications, Mater. Sci. Eng. A., vol. 362, no. 1-2, pp. 17–39, 2003. DOI: https://doi.org/10.1016/S0921-5093(03)00581-1.
- B. Bartczak, D. Gierczycka-Zbrożek, Z. Gronostajski, S. Polak, and A. Tobota, The use of thin-walled sections for energy absorbing components: a review, Arch. Civil Mech. Eng., vol. 10, no. 4, pp. 5–19, 2010. DOI: https://doi.org/10.1016/S1644-9665(12)60027-2.
- W. Abramowicz, Thin-walled structures as impact energy absorbers, Thin-Wall. Struct., vol. 41, no. 2-3, pp. 91–107, 2003. DOI: https://doi.org/10.1016/S0263-8231(02)00082-4.
- A.L. Kalamkarov, G. Saha, and A. Georgiades, General micromechanical modeling of smart composite shells with application to smart honeycomb sandwich structures, Compos. Struct., vol. 79, no. 1, pp. 18–33, 2007. DOI: https://doi.org/10.1016/j.compstruct.2005.11.026.
- V. Vasiliev, V. Barynin, and A. Rasin, Anisogrid lattice structures–survey of development and application, Compos. Struct., vol. 54, no. 2-3, pp. 361–370, 2001. DOI: https://doi.org/10.1016/S0263-8223(01)00111-8.
- E. Wyart, D. Coulon, T. Pardoen, J.-F. Remacle, and F. Lani, Application of the substructured finite element/extended finite element method (s-fe/xfe) to the analysis of cracks in aircraft thin walled structures, Eng. Fract. Mech., vol. 76, no. 1, pp. 44–58, 2009. DOI: https://doi.org/10.1016/j.engfracmech.2008.04.025.
- K. Naumenko and V.A. Eremeyev, A layer-wise theory of shallow shells with thin soft core for laminated glass and photovoltaic applications, Compos. Struct., vol. 178, pp. 434–446, 2017. DOI: https://doi.org/10.1016/j.compstruct.2017.07.007.
- P.M. Areias, J. Song, and T. Belytschko, Analysis of fracture in thin shells by overlapping paired elements, Comput. Methods Appl. Mech. Eng., vol. 195, no. 41-43, pp. 5343–5360, 2006. DOI: https://doi.org/10.1016/j.cma.2005.10.024.
- X. Zhuang, R. Huang, H. Zhu, H. Askes, and K. Mathisen, A new and simple locking-free triangular thick plate element using independent shear degrees of freedom, Finite Elem. Anal. Des., vol. 75, pp. 1–7, 2013. DOI: https://doi.org/10.1016/j.finel.2013.06.005.
- E. Carrera, M. Cinefra, M. Petrolo, and E. Zappino, Finite Element Analysis of Structures through Unified Formulation, John Wiley & Sons, 2014.
- M.B. Dehkordi, S. Khalili, and E. Carrera, Non-linear transient dynamic analysis of sandwich plate with composite face-sheets embedded with shape memory alloy wires and flexible core-based on the mixed LW (layer-wise)/ESL (equivalent single layer) models, Compos. Part B. Eng., vol. 87, pp. 59–74, 2016. DOI: https://doi.org/10.1016/j.compositesb.2015.10.008.
- E. Carrera, Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking, Arch. Comput. Methods Eng., vol. 10, no. 3, pp. 215–296, 2003. DOI: https://doi.org/10.1007/BF02736224.
- J. Reinoso, M. Paggi, and C. Linder, Phase field modeling of brittle fracture for enhanced assumed strain shells at large deformations: formulation and finite element implementation, Comput. Mech., vol. 59, no. 6, pp. 981–1001, 2017. DOI: https://doi.org/10.1007/s00466-017-1386-3.
- J. Reinoso, G. Catalanotti, A. Blázquez, P. Areias, P. Camanho, and F. París, A consistent anisotropic damage model for laminated fiber-reinforced composites using the 3d-version of the puck failure criterion, Int. J. Solids Struct., vol. 126-127, pp. 37–53, 2017. DOI: https://doi.org/10.1016/j.ijsolstr.2017.07.023.
- K. Rah, W. Van Paepegem, A.-M. Habraken, J. Degrieck, R.A. de Sousa, and R.A. Valente, Optimal low-order fully integrated solid-shell elements, Comput. Mech., vol. 51, no. 3, pp. 309–326, 2013. DOI: https://doi.org/10.1007/s00466-012-0726-6.
- J. Simo, F. Armero, and R. Taylor, Improved versions of assumed enhanced strain tri-linear elements for 3d finite deformation problems, Comput. Methods Appl. Mech. Eng., vol. 110, no. 3-4, pp. 359–386, 1993. DOI: https://doi.org/10.1016/0045-7825(93)90215-J.
- M. Bischoff, and E. Ramm, Shear deformable shell elements for large strains and rotations, Int. J. Numer. Methods. Eng., vol. 40, no. 23, pp. 4427–4449, 1997. DOI: https://doi.org/10.1002/(SICI)1097-0207(19971215)40:23<4427::AID-NME268>3.0.CO;2-9.
- L. Vu-Quoc and X. Tan, Optimal solid shells for non-linear analyses of multilayer composites. I. Statics, Comput. Methods Appl. Mech. Eng., vol. 192, no. 9-10, pp. 975–1016, 2003. DOI: https://doi.org/10.1016/S0045-7825(02)00435-8.
- S. Reese, A large deformation solid-shell concept based on reduced integration with hourglass stabilization, Int. J. Numer. Methods Eng., vol. 69, no. 8, pp. 1671–1716, 2007. DOI: https://doi.org/10.1002/nme.1827.
- O. Zienkiewicz, R. Taylor, and J. Too, Reduced integration technique in general analysis of plates and shells, Int. J. Numer. Methods Eng., vol. 3, no. 2, pp. 275–290, 1971. DOI: https://doi.org/10.1002/nme.1620030211.
- J. Reinoso and A. Blázquez, Application and finite element implementation of 7-parameter shell element for geometrically nonlinear analysis of layered CFRP composites, Compos. Struct., vol. 139, pp. 263–276, 2016. DOI: https://doi.org/10.1016/j.compstruct.2015.12.009.
- F. Tornabene and E. Viola, Static analysis of functionally graded doubly-curved shells and panels of revolution, Meccanica., vol. 48, no. 4, pp. 901–930, 2013. DOI: https://doi.org/10.1007/s11012-012-9643-1.
- E. Carrera, S. Brischetto, M. Cinefra, and M. Soave, Effects of thickness stretching in functionally graded plates and shells, Compos. Part B. Eng., vol. 42, no. 2, pp. 123–133, 2011. DOI: https://doi.org/10.1016/j.compositesb.2010.10.005.
- J. Reinoso, M. Paggi, P. Areias, and A. Blázquez, Surface-based and solid shell formulations of the 7-parameter shell model for layered CFRP and functionally graded power-based composite structures, Mech. Adv. Mater. Struct., vol. 26, no. 15, pp. 1271–1289, 2019. DOI: https://doi.org/10.1080/15376494.2018.1432802.
- T. Van Do, D.H. Doan, N.D. Duc, and T.Q. Bui, Phase-field thermal buckling analysis for cracked functionally graded composite plates considering neutral surface, Compos. Struct., vol. 182, pp. 542–548, 2017. DOI: https://doi.org/10.1016/j.compstruct.2017.09.059.
- R. Arciniega and J. Reddy, Large deformation analysis of functionally graded shells, Int. J. Solids Struct., vol. 44, no. 6, pp. 2036–2052, 2007. DOI: https://doi.org/10.1016/j.ijsolstr.2006.08.035.
- H. Bayesteh and S. Mohammadi, XFEM fracture analysis of shells: the effect of crack tip enrichments, Comput. Mater. Sci., vol. 50, no. 10, pp. 2793–2813, 2011. DOI: https://doi.org/10.1016/j.commatsci.2011.04.034.
- P.M. Areias and T. Belytschko, Non-linear analysis of shells with arbitrary evolving cracks using XFEM, Int. J. Numer. Methods Eng., vol. 62, no. 3, pp. 384–415, 2005. DOI: https://doi.org/10.1002/nme.1192.
- P.D. Zavattieri, Modeling of crack propagation in thin-walled structures using a cohesive model for shell elements, J. Appl. Mech., vol. 73, no. 6, pp. 948–958, 2006. DOI: https://doi.org/10.1115/1.2173286.
- I. Scheider and W. Brocks, Cohesive elements for thin-walled structures, Comput. Mater. Sci., vol. 37, no. 1-2, pp. 101–109, 2006. DOI: https://doi.org/10.1016/j.commatsci.2005.12.042.
- C. G. Dávila, P. P. Camanho, and A. Turon. Cohesive elements for shells. NASA TP Technical Reports, 2007, núm. 214869, 2007. https://dugi-doc.udg.edu/handle/10256/8243
- M. Paggi and P. Wriggers, Stiffness and strength of hierarchical polycrystalline materials with imperfect interfaces, J. Mech. Phys. Solids., vol. 60, no. 4, pp. 557–572, 2012. DOI: https://doi.org/10.1016/j.jmps.2012.01.009.
- J. Reinoso and M. Paggi, A consistent interface element formulation for geometrical and material nonlinearities, Comput. Mech., vol. 54, no. 6, pp. 1569–1581, 2014. DOI: https://doi.org/10.1007/s00466-014-1077-2.
- B. Bourdin, G.A. Francfort, and J.-J. Marigo, Numerical experiments in revisited brittle fracture, J. Mech. Phys. Solids., vol. 48, no. 4, pp. 797–826, 2000. DOI: https://doi.org/10.1016/S0022-5096(99)00028-9.
- T.-T. Nguyen, et al., On the choice of parameters in the phase field method for simulating crack initiation with experimental validation, Int. J. Fract., vol. 197, no. 2, pp. 213–226, 2016., DOI: https://doi.org/10.1007/s10704-016-0082-1.
- E. Tanné, T. Li, B. Bourdin, J.-J. Marigo, and C. Maurini, Crack nucleation in variational phase-field models of brittle fracture, J. Mech. Phys. Solids., vol. 110, pp. 80–99, 2018. DOI: https://doi.org/10.1016/j.jmps.2017.09.006.
- C. Miehe, F. Welschinger, and M. Hofacker, Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations, Int. J. Numer. Methods Eng., vol. 83, no. 10, pp. 1273–1311, 2010. DOI: https://doi.org/10.1002/nme.2861.
- T.K. Mandal, V.P. Nguyen, and J.-Y. Wu, Length scale and mesh bias sensitivity of phase-field models for brittle and cohesive fracture, Eng. Fract. Mech., vol. 217, pp. 106532, 2019. DOI: https://doi.org/10.1016/j.engfracmech.2019.106532.
- J.-Y. Wu, Robust numerical implementation of non-standard phase-field damage models for failure in solids, Comput. Methods Appl. Mech. Eng., vol. 340, pp. 767–797, 2018. DOI: https://doi.org/10.1016/j.cma.2018.06.007.
- A. Quintanas-Corominas, J. Reinoso, E. Casoni, A. Turon, and J. Mayugo, A phase field approach to simulate intralaminar and translaminar fracture in long fiber composite materials, Compos. Struct., vol. 220, pp. 899–911, 2019. DOI: https://doi.org/10.1016/j.compstruct.2019.02.007.
- E. Martínez-Pañeda, A. Golahmar, and C.F. Niordson, A phase field formulation for hydrogen assisted cracking, Comput. Methods Appl. Mech. Eng., vol. 342, pp. 742–761, 2018. DOI: https://doi.org/10.1016/j.cma.2018.07.021.
- M. Paggi and J. Reinoso, Revisiting the problem of a crack impinging on an interface: a modeling framework for the interaction between the phase field approach for brittle fracture and the interface cohesive zone model, Comput. Methods Appl. Mech. Eng., vol. 321, pp. 145–172, 2017. DOI: https://doi.org/10.1016/j.cma.2017.04.004.
- T.T. Nguyen, J. Yvonnet, Q.Z. Zhu, M. Bornert, and C. Chateau, A phase field method to simulate crack nucleation and propagation in strongly heterogeneous materials from direct imaging of their microstructure, Eng. Fract. Mech., vol. 139, pp. 18–39, 2015. DOI: https://doi.org/10.1016/j.engfracmech.2015.03.045.
- V. Carollo, J. Reinoso, and M. Paggi, A 3D finite strain model for intralayer and interlayer crack simulation coupling the phase field approach and cohesive zone model, Compos. Struct., vol. 182, pp. 636–651, 2017. DOI: https://doi.org/10.1016/j.compstruct.2017.08.095.
- M. Ambati, T. Gerasimov, and LDe Lorenzis, Phase-field modeling of ductile fracture, Comput. Mech., vol. 55, no. 5, pp. 1017–1040, 2015. DOI: https://doi.org/10.1007/s00466-015-1151-4.
- S. Natarajan, R.K. Annabattula, and E. Martínez-Pañeda, Phase field modelling of crack propagation in functionally graded materials, Compos. Part B. Eng., vol. 169, pp. 239–248, 2019. DOI: https://doi.org/10.1016/j.compositesb.2019.04.003.
- F. Amiri, D. Millán, Y. Shen, T. Rabczuk, and M. Arroyo, Phase-field modeling of fracture in linear thin shells, Theor. Appl. Fract. Mech., vol. 69, pp. 102–109, 2014. DOI: https://doi.org/10.1016/j.tafmec.2013.12.002.
- J. Kiendl, M. Ambati, L. De Lorenzis, H. Gomez, and A. Reali, Phase-field description of brittle fracture in plates and shells, Comput. Methods Appl. Mech. Eng., vol. 312, pp. 374–394, 2016. DOI: https://doi.org/10.1016/j.cma.2016.09.011.
- J. Reinoso, A. Arteiro, M. Paggi, and P. Camanho, Strength prediction of notched thin ply laminates using finite fracture mechanics and the phase field approach, Compos. Sci. Technol., vol. 150, pp. 205–216, 2017. DOI: https://doi.org/10.1016/j.compscitech.2017.07.020.
- J. Reinoso and A. Blázquez, Geometrically nonlinear analysis of functionally graded power-based and carbon nanotubes reinforced composites using a fully integrated solid shell element, Compos. Struct., vol. 152, pp. 277–294, 2016. DOI: https://doi.org/10.1016/j.compstruct.2016.05.036.
- R.L. Taylor, FEAP - finite element analysis program URL, 2014. http://www.ce.berkeley/feap
- J.-Y. Wu, Y. Huang, and V.P. Nguyen, On the BFGS monolithic algorithm for the unified phase field damage theory, Comput. Methods Appl. Mech. Eng., vol. 360, pp. 112704, 2020. DOI: https://doi.org/10.1016/j.cma.2019.112704.