Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2015 | 2 | 1 |
Tytuł artykułu

Isogeometric treatment of large deformation contact and debonding problems with T-splines: a review

Treść / Zawartość
Warianty tytułu
Języki publikacji
Within a setting where the isogeometric analysis (IGA) has been successful at bringing two different research fields together, i.e. Computer Aided Design (CAD) and numerical analysis, T-spline IGA is applied in this work to frictionless contact and mode-I debonding problems between deformable bodies in the context of large deformations. Based on the concept of IGA, the smooth basis functions are adopted to describe surface geometries and approximate the numerical solutions, leading to higher accuracy in the contact integral evaluation. The isogeometric discretizations are here incorporated into an existing finite element framework by using Bézier extraction, i.e. a linear operator which maps the Bernstein polynomial basis on Bézier elements to the global isogeometric basis. A recently released commercial T-spline plugin for Rhino is herein used to build the analysis models adopted in this study. In such context, the continuum is discretized with cubic T-splines, as well as with Non Uniform Rational B-Splines (NURBS) and Lagrange polynomial elements for comparison purposes, and a Gauss-point-to-surface (GPTS) formulation is combined with the penalty method to treat the contact constraints. The purely geometric enforcement of the non-penetration condition in compression is generalized to encompass both contact and mode-I debonding of interfaces which is approached by means of cohesive zone (CZ) modeling, as commonly done by the scientific community to analyse the progressive damage of materials and interfaces. Based on these models, non-linear relationships between tractions and relative displacements are assumed. These relationships dictate both the work of separation per unit fracture surface and the peak stress that has to be reached for the crack formation. In the generalized GPTS formulation an automatic switching procedure is used to choose between cohesive and contact models, depending on the contact status. Some numerical results are first presented and compared in 2D for varying resolutions of the contact and/or cohesive zone, including frictionless sliding and cohesive debonding, all featuring the competitive accuracy and performance of T-spline IGA. The superior accuracy of T-spline interpolations with respect to NURBS and Lagrange interpolations for a given number of degrees of freedom (Dofs) is always verified. The isogeometric formulation is also extended to 3D bodies, where some examples in large deformations based on T-spline discretizations show an high smoothness of the reaction history curves.
Słowa kluczowe
Opis fizyczny
  • Department of Innovation
    Engineering, University of Salento, Lecce, Italy
  • [1] Heegaard J.H., Curnier A., An augmented Lagrange method for discrete large slip contact problems, Int. J. Numer. Meth Eng., 1993, 36, 569–593. [Crossref]
  • [2] Zavarise G., De Lorenzis L., The node-to-segment algorithm for 2D frictionless contact: classical formulation and special cases, Comput. Method. Appl. M., 2009, 198(41-44), 3428–3451. [Crossref]
  • [3] Pietrzak G., Curnier A., Large deformation frictional contact mechanics: continuum formulation and augmented Lagrangean treatment, Comput. Method. Appl. M., 1999, 177, 351–381.
  • [4] Taylor R.L., Wriggers P., Smooth surface discretization for large deformation frictionless contact, Technical report, University of California, Berkeley, 1999, Report No. UCB/SEMM-99–04.
  • [5] Padmanabhan V., Laursen T.A., A framework for development of surface smoothing procedures in large deformation frictional contact analysis, Finite Elem. Anal. Des., 2001, 37, 173–198.
  • [6] Wriggers P., Krstulovic-Opara L., Korelc J., Smooth C1–interpolations for twodimensional frictional contact problems, Int. J. Numer. Methods Eng., 2001, 51, 1469–1495.
  • [7] Krstulovic-Opara L., Wriggers P., Korelc J., A C1–continuous formulation for 3D finite deformation frictional contact, Comput. Mech., 2002, 29, 27–42.
  • [8] Lengiewicz J., Korelc J., Stupkiewicz S., Automation of finite element formulations for large deformation contact problems, Int. J. Numer. Methods Eng., 2010, 85, 1252–1279.
  • [9] Stadler M., Holzapfel G.A., Korelc J., Cn–continuous modelling of smooth contact surfaces using NURBS and application to 2D problems, Int. J. Numer. Methods Eng., 2003, 57, 2177–2203. [Crossref]
  • [10] Stadler M.,Holzapfel G.A., Subdivision schemes for smooth contact surfaces of arbitrary mesh topology in 3D, Int. J. Numer. Methods Eng., 2004, 60, 1161–1195. [Crossref]
  • [11] Landon R.L., Hast M.W., Piazza S.J., Robust contact modeling using trimmed nurbs surfaces for dynamic simulations of articular contact, Comput. Meth. Appl. Mech. Eng., 2009, 198, 2339–2346. [Crossref]
  • [12] Hughes T.J.R., Cottrell J. A., Bazilevs Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement, Comput. Meth. Appl. Mech. Eng., 2005, 194, 4135–4195. [Crossref]
  • [13] Temizer İ., Wriggers P., Hughes T.J.R., Contact treatment in isogeometric analysis with NURBS, Comput. Meth. Appl. Mech. Eng., 2011, 200(9–12), 1100–1112.
  • [14] Temizer İ., Wriggers P., Hughes T.J.R., Three-dimensional mortar-based frictional contact treatment in isogeometric analysis with NURBS, Comput. Meth. Appl. Mech. Eng., 2012, 209–212, 115–128.
  • [15] Lu J., Isogeometric contact analysis: geometric basis and formulation for frictionless contact, Comput. Meth. Appl. Mech. Eng., 2011, 200, 726–74.
  • [16] De Lorenzis L., Temizer İ.,Wriggers P., Zavarise G., A large deformation frictional contact formulation using NURBS-based isogeometric analysis, Int. J. Numer. Methods Eng., 2011, 87(13), 1278–1300.
  • [17] De Lorenzis L.,Wriggers P., Zavarise G., A mortar formulation for 3D large deformation contact using NURBS-based isogeometric analysis and the augmented Lagrangian method, Comput. Mech., 2012, 49(1), 1–20. [Crossref]
  • [18] Matzen M.E., Cichosz T., Bischoff M., A point to segment contact formulation for isogeometric, NURBS based finite elements, Comput. Meth. Appl. Mech. Eng., 2013, 255, 27–39.
  • [19] Dimitri R., De Lorenzis L., Scott M.A., Wriggers P., Taylor R.L., Zavarise G., Isogeometric large deformation frictionless contact using T-splines, Comput. Meth. Appl. Mech. Eng., 2014, 269, 394–414.
  • [20] De Lorenzis L., Evans J.A., Hughes T.J.R., Reali A., Isogeometric collocation: Neumann boundary conditions and contact, Comput. Meth. Appl. Mech. Eng., 2015, 284, 21–54.
  • [21] De Lorenzis L., Wriggers P., Hughes T.J.R., Isogeometric contact: a review, GAMM-Mitt. 37, No.1, 2014, 85–123, /DOI 10.1002/gamm.201410005. [Crossref]
  • [22] Alart P., Curnier A., A mixed formulation for frictional contact problems prone to Newton like solution methods, Comput. Meth. Appl. Mech. Eng., 1991, 92, 353–375. [Crossref]
  • [23] Cottrell J. A., Hughes T.J.R., Bazilevs Y., Isogeometric Analysis: Toward Integration of CAD and FEA, Wiley, Chichester, 2009.
  • [24] Bazilevs Y., Calo V.M., Cottrell J.A., Evans J.A., Hughes T.J.R., Lipton S., Scott M.A., Sederberg T.W., Isogeometric analysis using T-splines, Comput. Meth. Appl. Mech. Eng., 2010, 199 (5-8), 229–263. [Crossref]
  • [25] Fischer K.A., Wriggers P., Frictionless 2D contact formulations for finite deformations based on the mortar method, Comput. Mech., 2005, 36, 226–244. [Crossref]
  • [26] Kanninen M.F., Popelar C.H., Advanced Fracture Mechanics, Oxford University Press, New York, 1985.
  • [27] Bazˇant Z.P., Planas J., Fracture and Size Effects in Concrete and Other Quasi-Brittle Materials, CRC Press, Boca Raton, 1998.
  • [28] Rice J.R., A path independent integral and the approximate analysis of strain concentration by notches and cracks, J. Appl. Mech., 1968, 35, 379–386. [Crossref]
  • [29] Rybicki E.F., Kanninen M.F., A finite element calculation of stress intensity factors by a modified crack closure integral, Eng. Fract. Mech., 1977, 9, 931–938. [Crossref]
  • [30] Raju I.S., Calculation of strain-energy release rates with higher order and singular finite elements, Eng. Fract. Mech., 1987, 28, 251–274. [Crossref]
  • [31] Hellen T.K., On the method of virtual crack extensions, Int. J. Numer. Methods Eng., 1975, 9, 187–207. [Crossref]
  • [32] Griflth A., The phenomena of rupture and flow in solids, Phil. Trans. R. Soc. A., 1921, 221, 163–198. [Crossref]
  • [33] Thouless M.D., Fracture of a model interface under mixed-mode loading, Acta Metall. Mater., 1990, 38, 1135–1140. [Crossref]
  • [34] Thouless M.D., Hutchinson J.W., Liniger E.G., Plane-strain, buckling driven delamination of thin films: model experiments and mode-II fracture, Acta Metall. Mater., 1992, 40, 2639–2649. [Crossref]
  • [35] Natarajan S., Ferreira A.J.M., Nguyen-Xuan H., Analysis of crossply laminated plates using isogeometric analysis and unified formulation, Curved and Layer. Struct., 2014, 1, 1–10.
  • [36] Benzley S.E., Representation of singularitieswith isoparametric finite elements, Int. J. Numer. Methods Eng., 1974, 8, 537–545. [Crossref]
  • [37] Gifford L.N., Hilton P.D., Stress intensity factors by enriched finite elements, Eng. Fract. Mech., 1978, 10, 485–496. [Crossref]
  • [38] Sukumar N., Moran B., Black T., Belytschko T., An elementfree galerkin method for three-dimensional fracture mechanics, Comput. Mech., 1997, 20, 170–175. [Crossref]
  • [39] Jirasek M., Zimmerman T., Embedded crack model: I. basic formulation, Int. J. Numer. Methods Eng., 2001, 50, 1269–1290. [Crossref]
  • [40] Wells G.N., Sluys L.J., A new method for modelling cohesive cracks using finite elements, Int. J. Numer. Methods Eng., 2001, 50, 2667–2682. [Crossref]
  • [41] Moës N., Belytschko T., Extended finite element method for cohesive crack growth, Eng. Fract. Mech., 2002, 69, 813–833. [Crossref]
  • [42] Moës N., Dolbow J., Belytschko T., A finite element method for crack growth without remeshing, Int. J. Numer. Methods Eng., 1999, 46, 131–150. [Crossref]
  • [43] Sukumar N., Huang Z.Y., Prevost J.H., Suo Z., Partition of unity enrichment for bimaterial interface cracks, Int. J. Numer. Methods Eng., 2004, 59, 1075–1102. [Crossref]
  • [44] Allix O., Ladeveze P., Interlaminar interface modelling for the prediction of delamination, Compos. Struct., 1992, 22, 235-242. [Crossref]
  • [45] Schellekens J.C.J., de Borst R., A non-linear finite element approach for the analysis of mode-I free edge delamination in composites, Int. J. Solids Struct., 1993, 30, 1239-1253. [Crossref]
  • [46] Barenblatt G.I., The formation of equilibrium cracks during brittle fracture. General ideas and hypotheses, Axially-symmetric cracks, J. Appl. Math. Mech., 1959, 23, 622–636. [Crossref]
  • [47] Dugdale D.S., Yielding of steel sheets containing slits, J. Mech. Phys. Solids, 1960, 8, 100–104. [Crossref]
  • [48] Needleman A., A continuum model for void nucleation by inclusion debonding, J. Appl. Mech., 1987, 54, 525–531. [Crossref]
  • [49] Needleman A., An analysis of tensile decohesion along an interface, J. Mech. Phys. Solids, 1990, 38, 289– 324. [Crossref]
  • [50] Tvergaard V., Hutchinson J.W., The relation between crack growth resistance and fracture process parameters in elasticplastic solids, J. Mech. Phys. Solids, 1992, 40(6), 1377–1397. [Crossref]
  • [51] Tvergaard V., Hutchinson J.W., The influence of plasticity on mixed mode interface toughness, J. Mech. Phys. Solids, 1993, 41(6), 1119–1135. [Crossref]
  • [52] Wei Y., Hutchinson J.W., Interface strength, work of adhesion and plasticity in the peel test, Int. J. Fract., 1998, 93, 315–333. [Crossref]
  • [53] Corigliano A., Formulation, identification and use of interface models in the numerical analysis of composite delamination, Int. J. Solids Struct., 1993, 30(20), 2779–2811. [Crossref]
  • [54] AllixO., Ladeveze P., Corigliano A.,Damage analysis of interlaminar fracture specimens, Compos. Struct., 1995, 31(1), 61–74. [Crossref]
  • [55] Point N., Sacco E., A delamination model for laminated composites, Int. J. Solids Struct., 1996, 33(4), 483–509. [Crossref]
  • [56] Bolzon G., Corigliano A., A discrete formulation for elastic solids with damaging interfaces, Comput. Meth. Appl. Mech. Eng., 1997, 140, 329–359. [Crossref]
  • [57] Allix O., Corigliano A., Geometrical and interfacial nonlinearities in the analysis of delamination in composites, Int. J. Solids Struct., 1999, 36(15), 2189–2216. [Crossref]
  • [58] Alfano G., Crisfield M.A., Finite element interface models for the delamination analysis of laminated composites: mechanical and computational issues, Int. J. Numer. Methods Eng., 2001, 50, 1701–1736. [Crossref]
  • [59] Criesfield M.A., Alfano G., Adaptive hierarchical enrichment for delamination fracture using a decohesive zone model, Int. J. Numer. Methods Eng., 2002, 54, 1369–1390. [Crossref]
  • [60] Guimatsia I., Ankersen J.K., Davies G.A.O., Iannucci L., Decohesion finite element with enriched basis functions for delamination, Compos. Sci. Technol., 2009, 69(15-16), 2616–2624. [Crossref]
  • [61] Scott M.A., Borden M.J., Verhoosel C.V., Sederberg T.W., Hughes T.J.R., Isogeometric finite element data structures based on Bézier extraction of T-splines, Int. J. Numer. Methods Eng., 2011, 88(2), 126–156. [Crossref]
  • [62] Scott M.A., Li X., Sederberg T.W., Hughes T.J.R., Local refinement of analysis-suitable T-splines, Comput. Meth. Appl. Mech. Eng., 2012, 213-216, 206–222.
  • [63] Li X., Zheng J., Sederberg T.W., Hughes T.J.R., Scott M.A., On linear independence of T-spline blending functions, Comput. Aided Geom. Des., 2012, 29(1), 63–76. [Crossref]
  • [64] Scott M.A., Simpson R.N., Evans J.A., Lipton S., Bordas S.P.A., Hughes T.J.R., Sederberg T.W., Isogeometric boundary element analysis using unstructured T-splines, Comput. Meth. Appl. Mech. Eng., 2013, 254, 197–221.
  • [65] Giannelli C., Jüttler B., Speleers H., THB-splines: The truncated basis for hierarchical splines, Comput. Aided Geom. Des., 2012, 29(7), 485–498. [Crossref]
  • [66] Deng J., Chen F., Li X., Hu C., Tong W., Yang Z., Feng Y., Polynomial splines over hierarchical T-meshes, Graph. Models, 2008, 70, 76–86.
  • [67] Dokken T., Lyche T., Pettersen K.F., Polynomial splines over locally refined box-partitions, Comput. Aided Geom. Des., 2013, 30(3), 331–356. [Crossref]
  • [68] Dimitri R., De Lorenzis L., Wriggers P., Zavarise G., NURBS- and T-spline-based isogeoemtric cohesive zone modeling of interface debonding, Comput. Mech., 2014, 54, 369–388. [Crossref]
  • [69] Sederberg T.W., Zheng J., Bakenov A., Nasri A., T-splines and TNURCCSs, ACM T. Graphic., 2003, 22(3), 477–484. [Crossref]
  • [70] Sederberg T.W., Zheng J., Song X., Knot intervals and multidegree splines, Comput. Aided Geom. Des., 2003, 20, 455–468. [Crossref]
  • [71] Sederberg T.W., Cardon D.L., Finnigan G.T., North N.S., Zheng J., Lyche T., T-spline simplification and local refinement, ACM T. Graphic., 2004, 23 (3), 276–283. [Crossref]
  • [72] Cottrell J.A., Reali A., Bazilevs Y., Hughes T.J.R., Isogeometric analysis of structural vibrations, Comput. Meth. Appl. Mech. Eng., 2006, 195, 5257–5296. [Crossref]
  • [73] Hughes T.J.R., Reali A., Sangalli G., Duality and unied analysis of discrete approximations in structural dynamics and wave propagation: Comparison of p-method nite elements with k-method NURBS, Comput. Meth. Appl. Mech. Eng., 2008, 197(49-50), 4104–4124. [Crossref]
  • [74] Lorentz G. G., Bernstein Polynomials, Chelsea Publishing Co., New York, 1986.
  • [75] Piegl L. A., Tiller W., The NURBS Book, Springer, 1996.
  • [76] Autodesk, 2011, Inc. .
  • [77] Schillinger D., Rank E., An unfitted hp-adaptive finite element method based on hierarchical B-splines for interface problems of complex geometry, Comput. Meth. Appl. Mech. Eng., 2011, 200(47-48), 3358– 3380. [Crossref]
  • [78] Vuong A.V., Giannelli C., Jüttler B., Simeon B., A hierarchical approach to adaptive local refinement in isogeometric analysis, Comput. Meth. Appl. Mech. Eng., 2011, 49-52, 3554– 3567.
  • [79] Samet H., Foundations of Multidimensional and Metric Data Structures, Morgan Kaufmann Publishers: San Francisco, 2006.
  • [80] Burstedde C., Wilcox L.C., Ghattas O., p4est: Scalable Algorithms for Parallel Adaptive Mesh Refinement on Forests of Octrees, SIAM J. Sci. Comput., 2011, 33(3), 1103–1133.
  • [81] Yserantant H., On the multi-level splitting of finite element spaces, Numer. Math., 1986, 49, 379–412. [Crossref]
  • [82] Krysl P., Grinspun E., Schröder P., Natural hierarchical refinement for finite element methods, Int. J. Numer. Methods Eng., 2003, 56, 1109–1124. [Crossref]
  • [83] Bungartz H.J., Griebel M., Sparse grids, Acta Numer., 2004, 13, 147–269.
  • [84] Taylor R.L., FEAP – Finite Element Analysis Program, 2013, www.ce.berkeley/feap, University of California, Berkeley.
  • [85] Laursen T.A., Computational contact and impact mechanics, 2002, Springer, Berlin.
  • [86] P. Wriggers, Computational contact mechanics, 2nd edition, 2006, Springer, Berlin.
  • [87] Puso M.A., Laursen T.A., A mortar segment-to-segment frictional contact method for large deformations, Comput. Meth. Appl. Mech. Eng., 2004, 193, 4891–4913. [Crossref]
  • [88] Papadopoulos P., Taylor R.L., A mixed formulation for the finite element solution of contact problems, Comput. Meth. Appl. Mech. Eng., 1992, 94(3), 373–389. [Crossref]
  • [89] Zienkiewicz O.C., Taylor R.L., The Finite Element Method for Solid and Structural Mechanics, 2005, Butterworth-Heinemann, 6th edition.
  • [90] Kiendl J., Bletzinger K.U., Linhard J., Wüchner R., Isogeometric shell analysis with Kirchhoff-Love elements, Comput. Meth. Appl. Mech. Eng., 2009, 198(49-52), 3902–3914. [Crossref]
  • [91] Hermes F.H., Process zone and cohesive element size in numerical simulations of delamination in bi-layers, Master thesis, September 24th 2010, MT 10.21, Eindhoven.
Typ dokumentu
Identyfikator YADDA
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.