×

zbMATH — the first resource for mathematics

A full-discontinuous Galerkin formulation of nonlinear Kirchhoff-Love shells: elasto-plastic finite deformations, parallel computation, and fracture applications. (English) Zbl 1352.74170
Summary: Because of its ability to take into account discontinuities, the discontinuous Galerkin (DG) method presents some advantages for modeling cracks initiation and propagation. This concept has been recently applied to three-dimensional simulations and to elastic thin bodies. In this last case, the assumption of small elastic deformations before cracks initiation or propagation reduces drastically the applicability of the framework to a reduced number of materials.
To remove this limitation, a full-DG formulation of nonlinear Kirchhoff–Love shells is presented and is used in combination with an elasto-plastic finite deformations model. The results obtained by this new formulation are in agreement with other continuum elasto-plastic shell formulations.
Then, this full-DG formulation of Kirchhoff–Love shells is coupled with the cohesive zone model to perform thin body fracture simulations. As this method considers elasto-plastic constitutive laws in combination with the cohesive model, accurate results compared with the experiments are found. In particular, the crack path and propagation rate of a blasted cylinder are shown to match experimental results. One of the main advantages of this framework is its ability to run in parallel with a high speed-up factor, allowing the simulation of ultra fine meshes.

MSC:
74K25 Shells
74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74C15 Large-strain, rate-independent theories of plasticity (including nonlinear plasticity)
74R10 Brittle fracture
Software:
Gmsh; PETSc; VTF
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Hillerborg, Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements, Cement and Concrete Research 6 (6) pp 773– (1976) · doi:10.1016/0008-8846(76)90007-7
[2] Dugdale, Yielding of steel sheets containing slits, Journal of the Mechanics and Physics of Solids 8 (2) pp 100– (1960) · doi:10.1016/0022-5096(60)90013-2
[3] Barenblatt, The Mathematical Theory of Equilibrium Cracks in Brittle Fracture pp 55– (1962)
[4] Needleman, A continuum model for void nucleation by inclusion debonding, Journal of Applied Mechanics 54 pp 525– (1987) · Zbl 0626.73010 · doi:10.1115/1.3173064
[5] Tvergaard, Effect of fibre debonding in a whisker-reinforced metal, Materials Science and Engineering: A 125 (2) pp 203– (1990) · doi:10.1016/0921-5093(90)90170-8
[6] Needleman, An analysis of decohesion along an imperfect interface, International Journal of Fracture 42 pp 21– (1990) · doi:10.1007/BF00018611
[7] Needleman, An analysis of tensile decohesion along an interface, Journal of the Mechanics and Physics of Solids 38 (3) pp 289– (1990) · doi:10.1016/0022-5096(90)90001-K
[8] Tvergaard, The relation between crack growth resistance and fracture process parameters in elastic-plastic solids, Journal of the Mechanics and Physics of Solids 40 (6) pp 1377– (1992) · Zbl 0775.73218 · doi:10.1016/0022-5096(92)90020-3
[9] Tvergaard, The influence of plasticity on mixed mode interface toughness, Journal of the Mechanics and Physics of Solids 41 (6) pp 1119– (1993) · Zbl 0775.73219 · doi:10.1016/0022-5096(93)90057-M
[10] Scheider, Simulation of cup-cone fracture using the cohesive model, Engineering Fracture Mechanics 70 (14) pp 1943– (2003) · doi:10.1016/S0013-7944(03)00133-4
[11] Tvergaard, Effect of strain-dependent cohesive zone model on predictions of crack growth resistance, International Journal of Solids and Structures 33 (20-22) pp 3297– (1996) · Zbl 0905.73056 · doi:10.1016/0020-7683(95)00261-8
[12] Klein, Physics-based modeling of brittle fracture: cohesive formulations and the application of meshfree methods, Theoretical and Applied Fracture Mechanics 37 (1-3) pp 99– (2001) · doi:10.1016/S0167-8442(01)00091-X
[13] Seagraves, Dynamic Failure of Materials and Structures pp 349– (2010)
[14] Camacho, Computational modelling of impact damage in brittle materials, International Journal of Solids and Structures 33 (20-22) pp 2899– (1996) · Zbl 0929.74101 · doi:10.1016/0020-7683(95)00255-3
[15] Pandolfi, Finite element simulation of ring expansion and fragmentation: the capturing of length and time scales through cohesive models of fracture, International Journal of Fracture 95 (1) pp 279– (1999) · doi:10.1023/A:1018672922734
[16] Ortiz, Finite-deformation irreversible cohesive elements for three-dimensional crack propagation analysis, International Journal for Numerical Methods in Engineering 44 pp 44– (2000)
[17] Cirak, Large-scale fluid-structure interaction simulation of viscoplastic and fracturing thin-shells subjected to shocks and detonations, Computers & Structures 85 (11-14) pp 1049– (2007) · doi:10.1016/j.compstruc.2006.11.014
[18] Cirak, A cohesive approach to thin-shell fracture and fragmentation, Computer Methods in Applied Mechanics and Engineering 194 (21-24) pp 2604– (2005) · Zbl 1082.74052 · doi:10.1016/j.cma.2004.07.048
[19] Becker, A one field full discontinuous Galerkin method for Kirchhoff-Love shells applied to fracture mechanics, Computer Methods in Applied Mechanics and Engineering 200 pp 3223– (2011) · Zbl 1230.74168 · doi:10.1016/j.cma.2011.07.008
[20] Cocburn, Lecture Notes in Computational Sciences and Engineering 11, in: High-Order Methods for Computational Physics (1999)
[21] Cockburn, Discontinuous Galerkin methods, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift far Angewandte Mathematik und Mechanik 83 (11) pp 731– (2003) · Zbl 1036.65079 · doi:10.1002/zamm.200310088
[22] Arnold, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM Journal on Numerical Analysis 39 (5) pp 1749– (2002) · Zbl 1008.65080 · doi:10.1137/S0036142901384162
[23] Hansbo, A discontinuous Galerkin method for the plate equation, Calcolo 39 (1) pp 41– (2002) · Zbl 1012.74066 · doi:10.1007/s100920200001
[24] Lew, Optimal bv estimates for a discontinuous Galerkin method for linear elasticity, Applied Mathematics Research Express 2004 (3) pp 73– (2004) · Zbl 1115.74021 · doi:10.1155/S1687120004020052
[25] Celiker, Locking-free optimal discontinuous Galerkin methods for Timoshenko beams, SIAM Journal on Numerical Analysis 44 (6) pp 2297– (2006) · Zbl 1127.74041 · doi:10.1137/050635821
[26] Guzey, Design and development of a discontinuous Galerkin method for shells, Computer Methods in Applied Mechanics and Engineering 195 (25-28) pp 3528– (2006) · Zbl 1157.74041 · doi:10.1016/j.cma.2005.08.001
[27] Eyck, Discontinuous Galerkin methods for non-linear elasticity, International Journal for Numerical Methods in Engineering 67 (9) pp 1204– (2006) · Zbl 1113.74068 · doi:10.1002/nme.1667
[28] Güzey, The embedded discontinuous Galerkin method: application to linear shell problems, International Journal for Numerical Methods in Engineering 70 (7) pp 757– (2007) · Zbl 1194.74403 · doi:10.1002/nme.1893
[29] Wells, A c0 discontinuous Galerkin formulation for kirchhoff plates, Computer Methods in Applied Mechanics and Engineering 196 (35-36) pp 3370– (2007) · Zbl 1173.74447 · doi:10.1016/j.cma.2007.03.008
[30] Lew, Some applications of discontinuous Galerkin methods in solid mechanics, IUTAM Symposium on Theoretical, Computational and Modelling Aspects of Inelastic Media 11 pp 227– (2008) · Zbl 1209.74051 · doi:10.1007/978-1-4020-9090-5_21
[31] Eyck, Adaptive stabilization of discontinuous Galerkin methods for nonlinear elasticity: motivation, formulation, and numerical examples, Computer Methods in Applied Mechanics and Engineering 197 (45-48) pp 3605– (2008) · Zbl 1194.74389 · doi:10.1016/j.cma.2008.02.020
[32] Noels, An explicit discontinuous Galerkin method for non-linear solid dynamics: formulation, parallel implementation and scalability properties, International Journal for Numerical Methods in Engineering 74 (9) pp 1393– (2008) · Zbl 1158.74496 · doi:10.1002/nme.2213
[33] Noels, A new discontinuous Galerkin method for Kirchhoff-Love shells, Computer Methods in Applied Mechanics and Engineering 197 (33-40) pp 2901– (2008) · Zbl 1194.74456 · doi:10.1016/j.cma.2008.01.018
[34] Noels, A discontinuous Galerkin formulation of non-linear Kirchhoff-Love shells, International Journal for Numerical Methods in Engineering 78 (3) pp 296– (2009) · Zbl 1183.74302 · doi:10.1002/nme.2489
[35] Shen, An optimally convergent discontinuous Galerkin-based extended finite element method for fracture mechanics, International Journal for Numerical Methods in Engineering 82 (6) pp 716– (2010) · Zbl 1188.74070
[36] Engel, Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity, Computer Methods in Applied Mechanics and Engineering 191 (34) pp 3669– (2002) · Zbl 1086.74038 · doi:10.1016/S0045-7825(02)00286-4
[37] Wells, A discontinuous Galerkin formulation for a strain gradient-dependent damage model, Computer Methods in Applied Mechanics and Engineering 193 (33-35) pp 3633– (2004) · Zbl 1068.74084 · doi:10.1016/j.cma.2004.01.020
[38] Molari, A discontinuous Galerkin method for strain gradient-dependent damage: study of interpolations and convergence, Computer Methods in Applied Mechanics and Engineering 195 (13-16) pp 1480– (2006) · Zbl 1116.74065 · doi:10.1016/j.cma.2005.05.026
[39] Becker, A fracture framework for euler-bernoulli beams based on a full discontinuous Galerkin formulation/extrinsic cohesive law combination, International Journal for Numerical Methods in Engineering 85 (10) pp 1227– (2011) · Zbl 1217.74115
[40] Mergheim, A hybrid discontinuous Galerkin/interface method for the computational modelling of failure, Communications in Numerical Methods in Engineering 20 (7) pp 511– (2004) · Zbl 1302.74166 · doi:10.1002/cnm.689
[41] Prechtel, Simulation of fracture in heterogeneous elastic materials with cohesive zone models, International Journal of Fracture 168 pp 15– (2011) · Zbl 1283.74073 · doi:10.1007/s10704-010-9552-z
[42] Radovitzky, A scalable 3D fracture and fragmentation algorithm based on a hybrid, discontinuous Galerkin, cohesive element method, Computer Methods in Applied Mechanics and Engineering 200 pp 326– (2011) · Zbl 1225.74105 · doi:10.1016/j.cma.2010.08.014
[43] Larsson, Dynamic fracture modeling in shell structures based on XFEM, International Journal for Numerical Methods in Engineering 86 (4-5) pp 499– (2011) · Zbl 1216.74025 · doi:10.1002/nme.3086
[44] Simo, On stress resultant geometrically exact shell model. Part i: formulation and optimal parametrization, Computer Methods in Applied Mechanics and Engineering 72 (3) pp 267– (1989) · Zbl 0692.73062 · doi:10.1016/0045-7825(89)90002-9
[45] Simo, On a stress resultant geometrically exact shell model. Part ii: the linear theory; computational aspects, Computer Methods in Applied Mechanics and Engineering 73 (1) pp 53– (1989) · Zbl 0724.73138 · doi:10.1016/0045-7825(89)90098-4
[46] Simo, On a stress resultant geometrically exact shell model. Part iii: computational aspects of the nonlinear theory, Computer Methods in Applied Mechanics and Engineering 79 (1) pp 21– (1990) · Zbl 0746.73015 · doi:10.1016/0045-7825(90)90094-3
[47] Simo, On a stress resultant geometrically exact shell model. Part iv: variable thickness shells with through-the-thickness stretching, Computer Methods in Applied Mechanics and Engineering 81 (1) pp 91– (1990) · Zbl 0746.73016 · doi:10.1016/0045-7825(90)90143-A
[48] Simo, On a stress resultant geometrically exact shell model. Part v. nonlinear plasticity: formulation and integration algorithms, Computer Methods in Applied Mechanics and Engineering 96 (2) pp 133– (1992) · Zbl 0754.73042 · doi:10.1016/0045-7825(92)90129-8
[49] Cirak, Subdivision surfaces: a new paradigm for thin-shell finite-element analysis, International Journal for Numerical Methods in Engineering 47 (12) pp 2039– (2000) · Zbl 0983.74063 · doi:10.1002/(SICI)1097-0207(20000430)47:12<2039::AID-NME872>3.0.CO;2-1
[50] Cirak, Fully c1-conforming subdivision elements for finite deformation thin-shell analysis, International Journal for Numerical Methods in Engineering 51 (7) pp 813– (2001) · Zbl 1039.74045 · doi:10.1002/nme.182.abs
[51] Deiterding, A virtual test facility for the efficient simulation of solid material response under strong shock and detonation wave loading, Engineering with Computers 22 (3) pp 325– (2006) · Zbl 05192772 · doi:10.1007/s00366-006-0043-9
[52] Cuitino, A material-independent method for extending stress update algorithms from small-strain plasticity to finite plasticity with multiplicative kinematics, Engineering Computations 9 pp 437– (1992) · doi:10.1108/eb023876
[53] Areias, Non-linear analysis of shells with arbitrary evolving cracks using XFEM, International Journal for Numerical Methods in Engineering 62 (3) pp 384– (2005) · Zbl 1080.74043 · doi:10.1002/nme.1192
[54] Areias, Analysis of fracture in thin shells by overlapping paired elements, Computer Methods in Applied Mechanics and Engineering 195 (41-43) pp 5343– (2006) · Zbl 1120.74048 · doi:10.1016/j.cma.2005.10.024
[55] Song, Dynamic fracture of shells subjected to impulsive loads, Journal of Applied Mechanics 76 (5) pp 051301– (2009) · doi:10.1115/1.3129711
[56] Zavattieri, Modeling of crack propagation in thin-walled structures, Mecanica Computacional XXIII: pp 209– (2004) · Zbl 1111.74738
[57] Zavattieri, Modeling of crack propagation in thin-walled structures using a cohesive model for shell elements, Journal of Applied Mechanics 73 (6) pp 948– (2006) · Zbl 1111.74738 · doi:10.1115/1.2173286
[58] Corigliano, Numerical modelling of impact rupture in polysilicon microsystems, Computational Mechanics 42 pp 251– (2008) · Zbl 05344528 · doi:10.1007/s00466-007-0231-5
[59] Molinari, The cohesive element approach to dynamic fragmentation: the question of energy convergence, International Journal for Numerical Methods in Engineering 69 (3) pp 484– (2007) · Zbl 1194.74450 · doi:10.1002/nme.1777
[60] Li, Analysis of crack growth and crack-tip plasticity in ductile materials using cohesive zone models, International Journal of Plasticity 19 (6) pp 849– (2003) · Zbl 1090.74670 · doi:10.1016/S0749-6419(02)00008-6
[61] Pandolfi, An efficient adaptive procedure for three-dimensional fragmentation simulations, Engineering with Computers 18 (2) pp 148– (2002) · Zbl 01993863 · doi:10.1007/s003660200013
[62] Papoulia, Time continuity in cohesive finite element modeling, International Journal for Numerical Methods in Engineering 58 (5) pp 679– (2003) · Zbl 1032.74676 · doi:10.1002/nme.778
[63] Zhang, Extrinsic cohesive modelling of dynamic fracture and microbranching instability in brittle materials, International Journal for Numerical Methods in Engineering 72 (8) pp 893– (2007) · Zbl 1194.74319 · doi:10.1002/nme.2030
[64] Geuzaine, Gmsh: A 3-d finite element mesh generator with built-in pre- and post-processing facilities, International Journal for Numerical Methods in Engineering 79 (11) pp 1309– (2009) · Zbl 1176.74181 · doi:10.1002/nme.2579
[65] Hulbert, Explicit time integration algorithms for structural dynamics with optimal numerical dissipation, Computer Methods in Applied Mechanics and Engineering 137 (2) pp 175– (1996) · Zbl 0881.73134 · doi:10.1016/S0045-7825(96)01036-5
[66] Karypis, A fast and high quality multilevel scheme for partitioning irregular graphs, SIAM Journal on Scientific Computing 20 (1) pp 359– (1998) · Zbl 0915.68129 · doi:10.1137/S1064827595287997
[67] Balay S Brown J Buschelman K Eijkhout V Gropp WD Kaushik D Knepley MG McInnes LC Smith BF et al. PETSc users manual Technical Report ANL-95/11 - Revision 3.1 2010
[68] Balay S Brown J Buschelman K Gropp WD Kaushik D Knepley MG McInnes LC Smith BF Zhang H PETSc Web page 2011
[69] Basar, Finite-rotation shell elements for the analysis of finite-rotation shell problems, International Journal for Numerical Methods in Engineering 34 (1) pp 165– (1992) · doi:10.1002/nme.1620340109
[70] Buechter, Shell theory versus degeneration-a comparison in large rotation finite element analysis, International Journal for Numerical Methods in Engineering 34 (1) pp 39– (1992) · Zbl 0760.73041 · doi:10.1002/nme.1620340105
[71] Sansour, On hybrid stress, hybrid strain and enhanced strain finite element formulations for a geometrically exact shell theory with drilling degrees of freedom, International Journal for Numerical Methods in Engineering 43 (1) pp 175– (1998) · Zbl 0936.74072 · doi:10.1002/(SICI)1097-0207(19980915)43:1<175::AID-NME448>3.0.CO;2-9
[72] Sansour, Families of 4-node and 9-node finite elements for a finite deformation shell theory. An assesment of hybrid stress, hybrid strain and enhanced strain elements, Computational Mechanics 24 pp 435– (2000) · Zbl 0959.74072 · doi:10.1007/s004660050003
[73] Areias, A finite-strain quadrilateral shell element based on discrete Kirchhoff-Love constraints, International Journal for Numerical Methods in Engineering 64 (9) pp 1166– (2005) · Zbl 1113.74063 · doi:10.1002/nme.1389
[74] Belytschko, Explicit algorithms for the nonlinear dynamics of shells, Computer Methods in Applied Mechanics and Engineering 42 (2) pp 225– (1984) · Zbl 0512.73073 · doi:10.1016/0045-7825(84)90026-4
[75] Swaddiwudhipong, Dynamic response of large strain elasto-plastic plate and shell structures, Thin-Walled Structures 26 (4) pp 223– (1996) · doi:10.1016/0263-8231(96)00031-6
[76] Betsch, Numerical implementation of multiplicative elasto-plasticity into assumed strain elements with application to shells at large strains, Computer Methods in Applied Mechanics and Engineering 179 (3-4) pp 215– (1999) · Zbl 0965.74060 · doi:10.1016/S0045-7825(99)00063-8
[77] Belytschko, Fission-fusion adaptivity in finite elements for nonlinear dynamics of shells, Computers & Structures 33 (5) pp 1307– (1989) · Zbl 0724.73224 · doi:10.1016/0045-7949(89)90468-9
[78] Zhou, Stochastic fracture of ceramics under dynamic tensile loading, International Journal of Solids and Structures 41 (22-23) pp 6573– (2004) · Zbl 1181.74114 · doi:10.1016/j.ijsolstr.2004.05.029
[79] Papadrakakis, A method for the automatic evaluation of the dynamic relaxation parameters, Computer Methods in Applied Mechanics and Engineering 25 (1) pp 35– (1981) · Zbl 0444.73067 · doi:10.1016/0045-7825(81)90066-9
[80] Rice, Mathematical Analysis in the Mechanics of Fracture 2 (1968) · Zbl 0214.51802
[81] Chao, Fracture response of externally flawed aluminum cylindrical shells under internal gaseous detonation loading, International Journal of Fracture 134 pp 59– (2005) · doi:10.1007/s10704-005-5462-x
[82] Oakley, Adaptive dynamic relaxation algorithm for non-linear hyperelastic structures part i. formulation, Computer Methods in Applied Mechanics and Engineering 126 (1-2) pp 67– (1995) · Zbl 1067.74602 · doi:10.1016/0045-7825(95)00805-B
[83] Zhang, Development of the MADR method, Computers & Structures 52 (1) pp 1– (1994) · Zbl 0900.73933 · doi:10.1016/0045-7949(94)90249-6
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.