Skip to main content

2018 | OriginalPaper | Buchkapitel

3D Dynamic Crack Propagation by the Extended Finite Element Method and a Gradient-Enhanced Damage Model

verfasst von : M. Pezeshki, S. Loehnert, P. Wriggers, P. A. Guidault, E. Baranger

Erschienen in: Multiscale Modeling of Heterogeneous Structures

Verlag: Springer International Publishing

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

A combined continuous-discontinuous approach to fracture is presented to model crack propagation under dynamic loading. A gradient-enhanced damage model is used to evaluate degradation of the material ahead of the crack. This type of model avoids mesh dependency and pathological effects of local damage models. Discrete cracks are reflected by means of extended finite elements (XFEM) and level sets. For the transition between damage and discrete fracture a damage based criterion is utilized. A discrete crack propagates if a critical damage value at the crack front is reached. The propagation direction is also determined through the damage field. Finally a dynamic mode II crack propagation example is simulated to show the capabilities and robustness of the employed approach.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Literatur
1.
Zurück zum Zitat Martha, L.F., Wawrzynek, P.A., Ingraffea, A.R.: Arbitrary crack representation using solid modeling. Eng. Comput. 9(2), 63–82 (1993)CrossRef Martha, L.F., Wawrzynek, P.A., Ingraffea, A.R.: Arbitrary crack representation using solid modeling. Eng. Comput. 9(2), 63–82 (1993)CrossRef
2.
Zurück zum Zitat Potyondy, D.O., Wawrzynek, P.A., Ingraffea, A.R.: An algorithm to generate quadrilateral or triangular element surface meshes in arbitrary domains with applications to crack propagation. Int. J. Numer. Methods Eng. 38(16), 2677–2701 (1995)CrossRefMATH Potyondy, D.O., Wawrzynek, P.A., Ingraffea, A.R.: An algorithm to generate quadrilateral or triangular element surface meshes in arbitrary domains with applications to crack propagation. Int. J. Numer. Methods Eng. 38(16), 2677–2701 (1995)CrossRefMATH
3.
Zurück zum Zitat Belytschko, T., Chen, H., Xu, J., Zi, G.: Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment. Int. J. Numer. Methods Eng. 58(12), 1873–1905 (2003)CrossRefMATH Belytschko, T., Chen, H., Xu, J., Zi, G.: Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment. Int. J. Numer. Methods Eng. 58(12), 1873–1905 (2003)CrossRefMATH
4.
Zurück zum Zitat Camacho, G., Ortiz, M.: Computational modelling of impact damage in brittle materials. Int. J. Solids Struct. 33(20–22), 2899–2938 (1996)CrossRefMATH Camacho, G., Ortiz, M.: Computational modelling of impact damage in brittle materials. Int. J. Solids Struct. 33(20–22), 2899–2938 (1996)CrossRefMATH
5.
Zurück zum Zitat Song, J.H., Areias, P.M.A., Belytschko, T.: A method for dynamic crack and shear band propagation with phantom nodes. Int. J. Numer. Methods Eng. 67(6), 868–893 (2006)CrossRefMATH Song, J.H., Areias, P.M.A., Belytschko, T.: A method for dynamic crack and shear band propagation with phantom nodes. Int. J. Numer. Methods Eng. 67(6), 868–893 (2006)CrossRefMATH
6.
Zurück zum Zitat Xu, X.P., Needleman, A.: Numerical simulations of fast crack growth in brittle solids. J. Mech. Phys. Solids 42(9), 1397–1434 (1994)CrossRefMATH Xu, X.P., Needleman, A.: Numerical simulations of fast crack growth in brittle solids. J. Mech. Phys. Solids 42(9), 1397–1434 (1994)CrossRefMATH
7.
Zurück zum Zitat Belytschko, T., Fish, J., Engelmann, B.E.: A finite element with embedded localization zones. Comput. Methods Appl. Mech. Eng. 70(1), 59–89 (1988)CrossRefMATH Belytschko, T., Fish, J., Engelmann, B.E.: A finite element with embedded localization zones. Comput. Methods Appl. Mech. Eng. 70(1), 59–89 (1988)CrossRefMATH
8.
Zurück zum Zitat Lloberas-Valls, O., Huespe, A., Oliver, J., Dias, I.: Strain injection techniques in dynamic fracture modeling. Comput. Methods Appl. Mech. Eng. 308, 499–534 (2016)MathSciNetCrossRef Lloberas-Valls, O., Huespe, A., Oliver, J., Dias, I.: Strain injection techniques in dynamic fracture modeling. Comput. Methods Appl. Mech. Eng. 308, 499–534 (2016)MathSciNetCrossRef
9.
Zurück zum Zitat Silling, S.: Dynamic fracture modeling with a meshfree peridynamic code. Comput. Fluid Solid Mech. 2003, 641–644 (2003) Silling, S.: Dynamic fracture modeling with a meshfree peridynamic code. Comput. Fluid Solid Mech. 2003, 641–644 (2003)
10.
Zurück zum Zitat Silling, S., Askari, E.: A meshfree method based on the peridynamic model of solid mechanics. Comput. Struct. 83(17), 1526–1535 (2005)CrossRef Silling, S., Askari, E.: A meshfree method based on the peridynamic model of solid mechanics. Comput. Struct. 83(17), 1526–1535 (2005)CrossRef
11.
Zurück zum Zitat Belytschko, T., Organ, D., Gerlach, C.: Element-free galerkin methods for dynamic fracture in concrete. Comput. Methods Appl. Mech. Eng. 187(3–4), 385–399 (2000)CrossRefMATH Belytschko, T., Organ, D., Gerlach, C.: Element-free galerkin methods for dynamic fracture in concrete. Comput. Methods Appl. Mech. Eng. 187(3–4), 385–399 (2000)CrossRefMATH
12.
Zurück zum Zitat Raymond, S., Lemiale, V., Ibrahim, R., Lau, R.: A meshfree study of the Kalthoffwinkler experiment in 3D at room and low temperatures under dynamic loading using viscoplastic modelling. Eng. Anal. Boundary Elem. 42, 20–25 (2014) Raymond, S., Lemiale, V., Ibrahim, R., Lau, R.: A meshfree study of the Kalthoffwinkler experiment in 3D at room and low temperatures under dynamic loading using viscoplastic modelling. Eng. Anal. Boundary Elem. 42, 20–25 (2014)
13.
Zurück zum Zitat Fries, T.: A corrected XFEM approximation without problems in blending elements. Int. J. Numer. Methods Eng. 75(5), 503–532 (2008)MathSciNetCrossRefMATH Fries, T.: A corrected XFEM approximation without problems in blending elements. Int. J. Numer. Methods Eng. 75(5), 503–532 (2008)MathSciNetCrossRefMATH
14.
Zurück zum Zitat Moes, N., Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng. 46(1), 131–150 (1999)CrossRefMATH Moes, N., Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng. 46(1), 131–150 (1999)CrossRefMATH
15.
Zurück zum Zitat Moës, N., Gravouil, A., Belytschko, T.: Non-planar 3D crack growth by the extended finite element and level sets-part I: mechanical model. Int. J. Numer. Methods Eng. 53(11), 2549–2568 (2002)CrossRefMATH Moës, N., Gravouil, A., Belytschko, T.: Non-planar 3D crack growth by the extended finite element and level sets-part I: mechanical model. Int. J. Numer. Methods Eng. 53(11), 2549–2568 (2002)CrossRefMATH
16.
Zurück zum Zitat Stolarska, M., Chopp, D.L., Moës, N., Belytschko, T.: Modelling crack growth by level sets in the extended finite element method. Int. J. Numer. Methods Eng. 51(8), 943–960 (2001)CrossRefMATH Stolarska, M., Chopp, D.L., Moës, N., Belytschko, T.: Modelling crack growth by level sets in the extended finite element method. Int. J. Numer. Methods Eng. 51(8), 943–960 (2001)CrossRefMATH
17.
Zurück zum Zitat Strouboulis, T., Babuška, I., Copps, K.: The design and analysis of the generalized finite element method. Comput. Methods Appl. Mech. Eng. 181, 43–69 (2000)MathSciNetCrossRefMATH Strouboulis, T., Babuška, I., Copps, K.: The design and analysis of the generalized finite element method. Comput. Methods Appl. Mech. Eng. 181, 43–69 (2000)MathSciNetCrossRefMATH
18.
Zurück zum Zitat Strouboulis, T., Copps, K., Babuška, I.: The generalized finite element method. Comput. Methods Appl. Mech. Eng. 190, 4081–4193 (2001)MathSciNetCrossRefMATH Strouboulis, T., Copps, K., Babuška, I.: The generalized finite element method. Comput. Methods Appl. Mech. Eng. 190, 4081–4193 (2001)MathSciNetCrossRefMATH
19.
Zurück zum Zitat Pijaudier-Cabot, G., Baẑant, Z.P.: Nonlocal damage theory. J. Eng. Mech. 113(0), 1512–1533 (1987) Pijaudier-Cabot, G., Baẑant, Z.P.: Nonlocal damage theory. J. Eng. Mech. 113(0), 1512–1533 (1987)
20.
Zurück zum Zitat Peerlings, R.H.J., De Borst, R., Brekelmans, W.A.M., De Vree, J.H.P.: Gradient enhanced damage for quasi-brittle materials. Int. J. Numer. Methods Eng. 39(19), 3391–3403 (1996)CrossRefMATH Peerlings, R.H.J., De Borst, R., Brekelmans, W.A.M., De Vree, J.H.P.: Gradient enhanced damage for quasi-brittle materials. Int. J. Numer. Methods Eng. 39(19), 3391–3403 (1996)CrossRefMATH
21.
Zurück zum Zitat Borden, M.J., Verhoosel, C.V., Scott, M.A., Hughes, T.J., Landis, C.M.: A phase-field description of dynamic brittle fracture. Comput. Methods Appl. Mech. Eng. 217, 77–95 (2012)MathSciNetCrossRefMATH Borden, M.J., Verhoosel, C.V., Scott, M.A., Hughes, T.J., Landis, C.M.: A phase-field description of dynamic brittle fracture. Comput. Methods Appl. Mech. Eng. 217, 77–95 (2012)MathSciNetCrossRefMATH
22.
Zurück zum Zitat Miehe, C., Hofacker, M., Welschinger, F.: A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. Comput. Methods Appl. Mech. Eng. 199(45), 2765–2778 (2010)MathSciNetCrossRefMATH Miehe, C., Hofacker, M., Welschinger, F.: A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. Comput. Methods Appl. Mech. Eng. 199(45), 2765–2778 (2010)MathSciNetCrossRefMATH
23.
Zurück zum Zitat Gerasimov, T., De Lorenzis, L.: A line search assisted monolithic approach for phase-field computing of brittle fracture. Comput. Methods Appl. Mech. Eng. 312, 276–303 (2016)MathSciNetCrossRef Gerasimov, T., De Lorenzis, L.: A line search assisted monolithic approach for phase-field computing of brittle fracture. Comput. Methods Appl. Mech. Eng. 312, 276–303 (2016)MathSciNetCrossRef
24.
Zurück zum Zitat Broumand, P., Khoei, A.: X-FEM modeling of dynamic ductile fracture problems with a nonlocal damage-viscoplasticity model. Finite Elem. Anal. Des. 99, 49–67 (2015)CrossRef Broumand, P., Khoei, A.: X-FEM modeling of dynamic ductile fracture problems with a nonlocal damage-viscoplasticity model. Finite Elem. Anal. Des. 99, 49–67 (2015)CrossRef
25.
Zurück zum Zitat Mazars, J.: PijaudierCabot, G.: Continuum damage theory application to concrete. J. Eng. Mech. 115(2), 345–365 (1989) Mazars, J.: PijaudierCabot, G.: Continuum damage theory application to concrete. J. Eng. Mech. 115(2), 345–365 (1989)
26.
Zurück zum Zitat Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988)MathSciNetCrossRefMATH Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988)MathSciNetCrossRefMATH
27.
Zurück zum Zitat Fries, T.P., Baydoun, M.: Crack propagation with the extended finite element method and a hybrid explicit-implicit crack description. Int. J. Numer. Methods Eng. 89(12), 1527–1558 (2012)MathSciNetCrossRefMATH Fries, T.P., Baydoun, M.: Crack propagation with the extended finite element method and a hybrid explicit-implicit crack description. Int. J. Numer. Methods Eng. 89(12), 1527–1558 (2012)MathSciNetCrossRefMATH
28.
Zurück zum Zitat Gravouil, A., Moës, N., Belytschko, T.: Non-planar 3D crack growth by the extended finite element and level sets-part II: level set update. Int. J. Numer. Methods Eng. 53(11), 2569–2586 (2002)CrossRefMATH Gravouil, A., Moës, N., Belytschko, T.: Non-planar 3D crack growth by the extended finite element and level sets-part II: level set update. Int. J. Numer. Methods Eng. 53(11), 2569–2586 (2002)CrossRefMATH
29.
Zurück zum Zitat Loehnert, S., Mueller-Hoeppe, D.S., Wriggers, P.: 3D corrected XFEM approach and extension to finite deformation theory. Int. J. Numer. Methods Eng. 86(4–5), 431–452 (2011)MathSciNetCrossRefMATH Loehnert, S., Mueller-Hoeppe, D.S., Wriggers, P.: 3D corrected XFEM approach and extension to finite deformation theory. Int. J. Numer. Methods Eng. 86(4–5), 431–452 (2011)MathSciNetCrossRefMATH
30.
Zurück zum Zitat Sukumar, N., Chopp, D., Moran, B.: Extended finite element method and fast marching method for three-dimensional fatigue crack propagation. Eng. Fract. Mech. 70(1), 29–48 (2003)CrossRef Sukumar, N., Chopp, D., Moran, B.: Extended finite element method and fast marching method for three-dimensional fatigue crack propagation. Eng. Fract. Mech. 70(1), 29–48 (2003)CrossRef
31.
Zurück zum Zitat Oliver, J., Huespe, A.: Continuum approach to material failure in strong discontinuity settings. Comput. Methods Appl. Mech. Eng. 193(30), 3195–3220 (2004)MathSciNetCrossRefMATH Oliver, J., Huespe, A.: Continuum approach to material failure in strong discontinuity settings. Comput. Methods Appl. Mech. Eng. 193(30), 3195–3220 (2004)MathSciNetCrossRefMATH
32.
Zurück zum Zitat Sukumar, N., Mos, N., Moran, B., Belytschko, T.: Extended finite element method for three-dimensional crack modelling. Int. J. Numer. Methods Eng. 48(11), 1549–1570 (2000) Sukumar, N., Mos, N., Moran, B., Belytschko, T.: Extended finite element method for three-dimensional crack modelling. Int. J. Numer. Methods Eng. 48(11), 1549–1570 (2000)
33.
Zurück zum Zitat Newmark, N.M., Asce, F.: A method of computation for structural dynamics. J. Eng. Mech. Div. 85, 67–94 (1959) Newmark, N.M., Asce, F.: A method of computation for structural dynamics. J. Eng. Mech. Div. 85, 67–94 (1959)
34.
Zurück zum Zitat Menouillard, T., Réthoré, J., Combescure, A., Bung, H.: Efficient explicit time stepping for the eXtended Finite Element Method (X-FEM). Inter. J. Numer. Methods Eng. 68(9), 911–939 (2006)MathSciNetCrossRefMATH Menouillard, T., Réthoré, J., Combescure, A., Bung, H.: Efficient explicit time stepping for the eXtended Finite Element Method (X-FEM). Inter. J. Numer. Methods Eng. 68(9), 911–939 (2006)MathSciNetCrossRefMATH
35.
Zurück zum Zitat Menouillard, T., Réthoré, J., Moës, N., Combescure, A., Bung, H.: Mass lumping strategies for X-FEM explicit dynamics: application to crack propagation. Int. J. Numer. Methods Eng. 74(3), 447–474 (2008)MathSciNetCrossRefMATH Menouillard, T., Réthoré, J., Moës, N., Combescure, A., Bung, H.: Mass lumping strategies for X-FEM explicit dynamics: application to crack propagation. Int. J. Numer. Methods Eng. 74(3), 447–474 (2008)MathSciNetCrossRefMATH
36.
Zurück zum Zitat Shahbeyk, S., Yaghoobi, M., Vafai, A.: Explicit dynamics X-FEM simulation of heterogeneous materials. Finite Elem. Anal. Des. 56, 52–79 (2012)CrossRef Shahbeyk, S., Yaghoobi, M., Vafai, A.: Explicit dynamics X-FEM simulation of heterogeneous materials. Finite Elem. Anal. Des. 56, 52–79 (2012)CrossRef
37.
Zurück zum Zitat Holl, M.: Multiscale crack-propagation and crack coalescence using the XFEM. Ph.D. thesis, Leibniz Universität Hannover (2014) Holl, M.: Multiscale crack-propagation and crack coalescence using the XFEM. Ph.D. thesis, Leibniz Universität Hannover (2014)
38.
Zurück zum Zitat Holl, M., Rogge, T., Loehnert, S., Wriggers, P., Rolfes, R.: 3D multiscale crack propagation using the XFEM applied to a gas turbine blade. Comput. Mech. 53(1), 173–188 (2014)MathSciNetCrossRefMATH Holl, M., Rogge, T., Loehnert, S., Wriggers, P., Rolfes, R.: 3D multiscale crack propagation using the XFEM applied to a gas turbine blade. Comput. Mech. 53(1), 173–188 (2014)MathSciNetCrossRefMATH
39.
Zurück zum Zitat Lemaitre, J.: A Course on Damage Mechanics. Springer, Berlin (1996) Lemaitre, J.: A Course on Damage Mechanics. Springer, Berlin (1996)
40.
Zurück zum Zitat Comi, C., Mariani, S., Perego, U.: An extended FE strategy for transition from continuum damage to mode I cohesive crack propagation. Int. J. Numer. Anal. Methods Geomech. (2007) Comi, C., Mariani, S., Perego, U.: An extended FE strategy for transition from continuum damage to mode I cohesive crack propagation. Int. J. Numer. Anal. Methods Geomech. (2007)
41.
Zurück zum Zitat Seabra, M.R.R., Šuštarič, P., Cesar de Sa, J.M.A., Rodič, T.: Damage driven crack initiation and propagation in ductile metals using XFEM. Comput. Mech. 52(1), 161–179 (2013)MathSciNetCrossRefMATH Seabra, M.R.R., Šuštarič, P., Cesar de Sa, J.M.A., Rodič, T.: Damage driven crack initiation and propagation in ductile metals using XFEM. Comput. Mech. 52(1), 161–179 (2013)MathSciNetCrossRefMATH
42.
Zurück zum Zitat Pijaudier-Cabot, G., Benallal, A.: Strain localization and bifurcation in a nonlocal continuum. Int. J. Solids Struct. 30(13), 1761–1775 (1993)CrossRefMATH Pijaudier-Cabot, G., Benallal, A.: Strain localization and bifurcation in a nonlocal continuum. Int. J. Solids Struct. 30(13), 1761–1775 (1993)CrossRefMATH
43.
Zurück zum Zitat Areias, P.M.A., Belytschko, T.: Analysis of three-dimensional crack initiation and propagation using the extended finite element method. Int. J. Numer. Meth. Eng. 63(5), 760–788 (2005)CrossRefMATH Areias, P.M.A., Belytschko, T.: Analysis of three-dimensional crack initiation and propagation using the extended finite element method. Int. J. Numer. Meth. Eng. 63(5), 760–788 (2005)CrossRefMATH
44.
Zurück zum Zitat Combescure, A., Gravouil, A., Grégoire, D., Réthoré, J.: X-FEM a good candidate for energy conservation in simulation of brittle dynamic crack propagation. Comput. Methods Appl. Mech. Eng. 197(5), 309–318 (2008)CrossRefMATH Combescure, A., Gravouil, A., Grégoire, D., Réthoré, J.: X-FEM a good candidate for energy conservation in simulation of brittle dynamic crack propagation. Comput. Methods Appl. Mech. Eng. 197(5), 309–318 (2008)CrossRefMATH
45.
Zurück zum Zitat Grégoire, D., Maigre, H., Réthoré, J., Combescure, A.: Dynamic crack propagation under mixed-mode loading comparison between experiments and X-FEM simulations. Int. J. Solids Struct. 44(20), 6517–6534 (2007) Grégoire, D., Maigre, H., Réthoré, J., Combescure, A.: Dynamic crack propagation under mixed-mode loading comparison between experiments and X-FEM simulations. Int. J. Solids Struct. 44(20), 6517–6534 (2007)
46.
Zurück zum Zitat Wells, G.N., Sluys, L.J., de Borst, R.: Simulating the propagation of displacement discontinuities in a regularized strain-softening medium. Int. J. Numer. Methods Eng. 53(5), 1235–1256 (2002)CrossRef Wells, G.N., Sluys, L.J., de Borst, R.: Simulating the propagation of displacement discontinuities in a regularized strain-softening medium. Int. J. Numer. Methods Eng. 53(5), 1235–1256 (2002)CrossRef
47.
Zurück zum Zitat Kalthoff, J.F.: Shadow optical analysis of dynamic shear fracture. Opt. Eng. 27(10), 271035–271035 (1988) Kalthoff, J.F.: Shadow optical analysis of dynamic shear fracture. Opt. Eng. 27(10), 271035–271035 (1988)
48.
Zurück zum Zitat Kalthoff, J.F.: Modes of dynamic shear failure in solids. Int. J. Fract. 101(1/2), 1–31 (2000)CrossRef Kalthoff, J.F.: Modes of dynamic shear failure in solids. Int. J. Fract. 101(1/2), 1–31 (2000)CrossRef
49.
Zurück zum Zitat Elguedj, T., Gravouil, A., Maigre, H.: An explicit dynamics extended finite element method. Part 1: mass lumping for arbitrary enrichment functions. Comput. Methods Appl. Mech. Eng. 198(30), 2297–2317 (2009)CrossRefMATH Elguedj, T., Gravouil, A., Maigre, H.: An explicit dynamics extended finite element method. Part 1: mass lumping for arbitrary enrichment functions. Comput. Methods Appl. Mech. Eng. 198(30), 2297–2317 (2009)CrossRefMATH
50.
Zurück zum Zitat Moreau, K., Moës, N., Picart, D., Stainier, L.: Explicit dynamics with a non-local damage model using the thick level set approach. Int. J. Numer. Methods Eng. 102(3–4), 808–838 (2015)MathSciNetCrossRefMATH Moreau, K., Moës, N., Picart, D., Stainier, L.: Explicit dynamics with a non-local damage model using the thick level set approach. Int. J. Numer. Methods Eng. 102(3–4), 808–838 (2015)MathSciNetCrossRefMATH
51.
Zurück zum Zitat Borden, M.J.: Isogeometric analysis of phase-field models for dynamic brittle and ductile fracture. Ph.D. thesis (2012) Borden, M.J.: Isogeometric analysis of phase-field models for dynamic brittle and ductile fracture. Ph.D. thesis (2012)
52.
Zurück zum Zitat Zhou, M., Rosakis, A., Ravichandran, G.: On the growth of shear bands and failure-mode transition in prenotched plates: a comparison of singly and doubly notched specimens. Int. J. Plast. 14(4), 435–451 (1998)CrossRefMATH Zhou, M., Rosakis, A., Ravichandran, G.: On the growth of shear bands and failure-mode transition in prenotched plates: a comparison of singly and doubly notched specimens. Int. J. Plast. 14(4), 435–451 (1998)CrossRefMATH
53.
Zurück zum Zitat Ravi-Chandar, K., Lu, J., Yang, B., Zhu, Z.: Failure mode transitions in polymers under high strain rate loading. Int. J. Fract. 101(1/2), 33–72 (2000)CrossRef Ravi-Chandar, K., Lu, J., Yang, B., Zhu, Z.: Failure mode transitions in polymers under high strain rate loading. Int. J. Fract. 101(1/2), 33–72 (2000)CrossRef
54.
Zurück zum Zitat Batra, R., Ravinsankar, M.: Three-dimensional numerical simulation of the Kalthoff experiment. Int. J. Fract. 105(2), 161–186 (2000)CrossRef Batra, R., Ravinsankar, M.: Three-dimensional numerical simulation of the Kalthoff experiment. Int. J. Fract. 105(2), 161–186 (2000)CrossRef
Metadaten
Titel
3D Dynamic Crack Propagation by the Extended Finite Element Method and a Gradient-Enhanced Damage Model
verfasst von
M. Pezeshki
S. Loehnert
P. Wriggers
P. A. Guidault
E. Baranger
Copyright-Jahr
2018
DOI
https://doi.org/10.1007/978-3-319-65463-8_14

    Marktübersichten

    Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen.