Abstract
We present an adaptive space-time phase field formulation for dynamic fracture of brittle shells. Their deformation is characterized by the Kirchhoff–Love thin shell theory using a curvilinear surface description. All kinematical objects are defined on the shell’s mid-plane. The evolution equation for the phase field is determined by the minimization of an energy functional based on Griffith’s theory of brittle fracture. Membrane and bending contributions to the fracture process are modeled separately and a thickness integration is established for the latter. The coupled system consists of two nonlinear fourth-order PDEs and all quantities are defined on an evolving two-dimensional manifold. Since the weak form requires \(C^1\)-continuity, isogeometric shape functions are used. The mesh is adaptively refined based on the phase field using Locally Refinable (LR) NURBS. Time is discretized based on a generalized-\(\alpha \) method using adaptive time-stepping, and the discretized coupled system is solved with a monolithic Newton–Raphson scheme. The interaction between surface deformation and crack evolution is demonstrated by several numerical examples showing dynamic crack propagation and branching.
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Notes
This is a part of the Saint Venant–Kirchhoff model, see Duong et al. [19].
To avoid confusion, we write discrete arrays, such as the shape function array \({\mathbf {N}}\), in roman font, whereas continuous tensors, such as the normal vector \({\varvec{N}}\), are written in italic font.
The free nodes refer to the degrees of freedom, which are not given by boundary conditions.
\(T_0\) refers to a reference time used to obtain a dimensionless formulation, see Sect. 5.7
Note that \(\rho _0\) is the surface density and has units \([\mathrm {kg}/\mathrm {m}^2]\).
Also see the remark on stress waves at the beginning of this section.
We can compute the shear wave speed based on \(c_\mathrm {s}=\sqrt{G/\rho }\approx 6.2\,L_0/T_0\). An approximate value for the Rayleigh wave speed is then obtained as \(c_\mathrm {R}\approx 0.9162\cdot c_\mathrm {s}\approx 5.7\,L_0/T_0\). Based on the experiments by Ravi-Chandar and Knauss [58], the crack tip velocity stays below \(60\%\) of the Rayleigh wave speed. We can thus formulate a condition for the minimum time step, i.e. \(\Delta t\le \Delta t_\mathrm {max}<\Delta x_\mathrm {min}/(0.6\cdot c_\mathrm {R})\approx 1.1\cdot 10^{-3}\,T_0\), where the minimum element size is \(\Delta x_\mathrm {min}=1/256\,L_0\).
References
Ambati M, De Lorenzis L (2016) Phase-field modeling of brittle and ductile fracture in shells with isogeometric NURBS-based solid-shell elements. Comput Methods Appl Mech Eng 312:351–373
Ambati M, Gerasimov T, De Lorenzis L (2015) A review on phase-field models of brittle fracture and a new fast hybrid formulation. Comput Mech 55(2):383–405
Ambati M, Kruse R, De Lorenzis L (2016) A phase-field model for ductile fracture at finite strains and its experimental verification. Comput Mech 57(1):149–167
Amiri F, Millán D, Shen Y, Rabczuk T, Arroyo M (2014) Phase-field modeling of fracture in linear thin shells. Introducing the new features of Theoretical and Applied Fracture Mechanics through the scientific expertise of the Editorial Board. Theor Appl Fract Mech 69:102–109
Amor H, Marigo J-J, Maurini C (2009) Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments. J Mech Phys Solids 57(8):1209–1229
Areias P, Rabczuk T, Msekh M (2016) Phase-field analysis of finite-strain plates and shells including element subdivision. Comput Methods Appl Mech Eng 312:322–350
Badnava H, Msekh MA, Etemadi E, Rabczuk T (2018) An h-adaptive thermo–mechanical phase field model for fracture. Finite Elem Anal Des 138:31–47
Benson DJ, Hartmann S, Bazilevs Y, Hsu M-C, Hughes TJR (2013) Blended isogeometric shells. Comput Methods Appl Mech Eng 255:133–146
Borden MJ, Hughes TJR, Landis CM, Anvari A, Lee IJ (2016) A phase-field formulation for fracture in ductile materials: finite deformation balance law derivation, plastic degradation, and stress triaxiality effects. Comput Methods Appl Mech Eng 312:130–166
Borden MJ, Hughes TJR, Landis CM, Verhoosel CV (2014) A higher-order phase-field model for brittle fracture: formulation and analysis within the isogeometric analysis framework. Comput Methods Appl Mech Eng 273:100–118
Borden MJ, Verhoosel CV, Scott MA, Hughes TJR, Landis CM (2012) A phase-field description of dynamic brittle fracture. Comput Methods Appl Mech Eng 217–220:77–95
Bourdin B, Francfort G, Marigo J-J (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 48(4):797–826
Bourdin B, Larsen CJ, Richardson CL (2011) A time-discrete model for dynamic fracture based on crack regularization. Int J Fract 168(2):133–143
Chen L, de Borst R (2018) Locally refined T-splines. Int J Numer Methods Eng 114(6):637–659
Chen L, Verhoosel CV, de Borst R (2018) Discrete fracture analysis using locally refined T-splines. Int J Numer Methods Eng 116(2):117–140
Chung J, Hulbert GM (1993) A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-alpha method. J Appl Mech 60(2):371–375
Ciarlet PG (1993) Mathematical elasticity: three dimensional elasticity. Elsevier, North-Holland
Dokken T, Lyche T, Pettersen KF (2013) Polynomial splines over locally refined box-partitions. Comput Aided Geom Des 30(3):331–356
Duong TX, Roohbakhshan F, Sauer RA (2017) A new rotation-free isogeometric thin shell formulation and a corresponding continuity constraint for patch boundaries. Comput Methods Appl Mech Eng 316:43–83
Echter R, Oesterle B, Bischoff M (2013) A hierarchic family of isogeometric shell finite elements. Comput Methods Appl Mech Eng 254:170–180
Forsey DR, Bartels RH (1988) Hierarchical B-spline refinement. SIGGRAPH Comput Graph 22(4):205–212
Francfort G, Marigo J-J (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 46(8):1319–1342
Geelen RJ, Liu Y, Hu T, Tupek MR, Dolbow JE (2019) A phase-field formulation for dynamic cohesive fracture. Comput Methods Appl Mech Eng 348:680–711
Geelen RJM, Liu Y, Dolbow JE, Rodríguez-Ferran A (2018) An optimization-based phase-field method for continuous-discontinuous crack propagation. Int J Numer Methods Eng 116(1):1–20
Gerasimov T, Lorenzis LD (2016) A line search assisted monolithic approach for phase-field computing of brittle fracture. Comput Methods Appl Mech Eng 312:276–303
Gerasimov T, Lorenzis LD (2019) On penalization in variational phase-field models of brittle fracture. Comput Methods Appl Mech Eng 354:990–1026
Gerasimov T, Noii N, Allix O, De Lorenzis L (2018) A non-intrusive global/local approach applied to phase-field modeling of brittle fracture. Adv Model Simul Eng Sci 5:14
Gomez H, Reali A, Sangalli G (2014) Accurate, efficient, and (iso)geometrically flexible collocation methods for phase-field models. J Comput Phys 262:153–171
Griffith AA (1921) VI. The phenomena of rupture and flow in solids. Philos Trans R Soc Lond Ser A 221:163–198
Heister T, Wheeler MF, Wick T (2015) A primal-dual active set method and predictor-corrector mesh adaptivity for computing fracture propagation using a phase-field approach. Comput Methods Appl Mech Eng 290:466–495
Hesch C, Franke M, Dittmann M, Temizer İ (2016a) Hierarchical NURBS and a higher-order phase-field approach to fracture for finite-deformation contact problems. Comput Methods Appl Mech Eng 301:242–58
Hesch C, Schuß S, Dittmann M, Franke M, Weinberg K (2016b) Isogeometric analysis and hierarchical refinement for higher-order phase-field models. Comput Methods Appl Mech Eng 303:185–207
Hirmand MR, Papoulia KD (2018) A continuation method for rigid-cohesive fracture in a discontinuous Galerkin finite element setting. Int J Numer Methods Eng 115(5):627–650
Hirmand MR, Papoulia KD (2019) Block coordinate descent energy minimization for dynamic cohesive fracture. Comput Methods Appl Mech Eng 354:663–688
Hofacker M, Miehe C (2013) A phase field model of dynamic fracture: robust field updates for the analysis of complex crack patterns. Int J Numer Methods Eng 93(3):276–301
Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194(39–41):4135–4195
Johannessen KA, Kvamsdal T, Dokken T (2014) Isogeometric analysis using LR B-splines. Comput Methods Appl Mech Eng 269:471–514
Karma A, Kessler D, Levine H (2001) Phase-field model of mode III dynamic fracture. Phys Rev Lett 75:045501
Kästner M, Hennig P, Linse T, Ulbricht V (2016) Phase-field modelling of damage and fracture–convergence and local mesh refinement. In: Naumenko K, Aßmus M (eds) Advanced methods of continuum mechanics for materials and structures. Springer, Singapore, pp 307–324
Kiendl J, Ambati M, De Lorenzis L, Gomez H, Reali A (2016) Phase-field description of brittle fracture in plates and shells. Comput Methods Appl Mech Eng 312:374–394
Kiendl J, Hsu M-C, Wu MC, Reali A (2015) Isogeometric Kirchhoff–Love shell formulations for general hyperelastic materials. Comput Methods Appl Mech Eng 291:280–303
Krueger R (2004) Virtual crack closure technique: history, approach, and applications. Appl Mech Rev 57(2):109–143
Kuhn C, Müller R (2010) A continuum phase field model for fracture. Computational mechanics in fracture and damage: a special issue in Honor of Prof. Gross. Eng Fract Mech 77(18):3625–3634
Kuhn C, Schlüter A, Müller R (2015) On degradation functions in phase field fracture models. Selected articles from phase-field method 2014 international seminar. Comput Mater Sci 108:374–384
Larsen C, Ortner C, Süli E (2010) Existence of solutions to a regularized model of dynamic fracture. Math Model Methods Appl Sci 20:1021–1048
Larsen CJ (2010) Models for dynamic fracture based on Griffith’s criterion. In: Hackl K (ed) IUTAM symposium on variational concepts with applications to the mechanics of materials. Springer, Dordrecht, pp 131–140
Linse T, Hennig P, Kästner M, de Borst R (2017) A convergence study of phase-field models for brittle fracture. Eng Fract Mech 184:307–318
Miehe C, Hofacker M, Welschinger F (2010a) 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
Miehe C, Welschinger F, Hofacker M (2010b) Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations. Int J Numer Methods Eng 83(10):1273–1311
Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46:131–150
Molinari JF, Gazonas G, Raghupathy R, Rusinek A, Zhou F (2007) The cohesive element approach to dynamic fragmentation: the question of energy convergence. Int J Numer Methods Eng 69(3):484–503
Nagaraja S, Elhaddad M, Ambati M, Kollmannsberger S, De Lorenzis L, Rank E (2018) Phase-field modeling of brittle fracture with multi-level hp-FEM and the finite cell method. Comput Mech 63:1283–1300
Naghdi PM (1973) The theory of shells and plates. In: Truesdell C (ed) Linear theories of elasticity and thermoelasticity: linear and nonlinear theories of rods, plates, and shells. Springer, Berlin, pp 425–640
Ortiz M, Pandolfi A (1999) Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis. Int J Numer Methods Eng 44(9):1267–1282
Papoulia KD (2017) Non-differentiable energy minimization for cohesive fracture. Int J Fract 204(2):143–158
Parvizian J, Düster A, Rank E (2007) Finite cell method. Comput Mech 41(1):121–133
Radovitzky R, Seagraves A, Tupek M, Noels L (2011) A scalable 3d fracture and fragmentation algorithm based on a hybrid, discontinuous Galerkin, cohesive element method. Comput Methods Appl Mech Eng 200(1):326–344
Ravi-Chandar K, Knauss W G (1984) An experimental investigation into dynamic fracture: III. On steady-state crack propagation and crack branching. Int J Fract 26(2):141–154
Reali A, Hughes TJ R (2015) An introduction to isogeometric collocation methods. Springer, Vienna, pp 173–204
Reinoso J, Paggi M, Linder C (2017) Phase field modeling of brittle fracture for enhanced assumed strain shells at large deformations: formulation and finite element implementation. Comput Mech 59(6):981–1001
Remmers JJC, de Borst R, Needleman A (2003) A cohesive segments method for the simulation of crack growth. Comput Mech 31(1):69–77
Sahu A, Sauer RA, Mandadapu KK (2017) Irreversible thermodynamics of curved lipid membranes. Phys Rev E 96:042409
Sargado JM, Keilegavlen E, Berre I, Nordbotten JM (2018) High-accuracy phase-field models for brittle fracture based on a new family of degradation functions. J Mech Phys Solids 111:458–489
Sauer RA (2018) On the computational modeling of lipid bilayers using thin-shell theory. In: Steigmann DJ (ed) The role of mechanics in the study of lipid bilayers. Springer, Cham, pp 221–286
Sauer RA, Duong TX (2017) On the theoretical foundations of thin solid and liquid shells. Math Mech Solids 22(3):343–371
Sauer RA, Duong TX, Corbett CJ (2014) A computational formulation for constrained solid and liquid membranes considering isogeometric finite elements. Comput Methods Appl Mech Eng 271:48–68
Sauer RA, Duong TX, Mandadapu KK, Steigmann DJ (2017) A stabilized finite element formulation for liquid shells and its application to lipid bilayers. J Comput Phys 330:436–466
Schillinger D, Borden MJ, Stolarski HK (2015) Isogeometric collocation for phase-field fracture models. Isogeometric analysis special issue. Comput Methods Appl Mech Eng 284:583–610
Schlüter A, Willenbücher A, Kuhn C, Müller R (2014) Phase field approximation of dynamic brittle fracture. Comput Mech 54(5):1141–1161
Sederberg TW, Zheng J, Bakenov A, Nasri A (2003) T-splines and T-NURCCs. ACM Trans Graph 22(3):477–484
Steigmann DJ (1999) Fluid films with curvature elasticity. Arch Ration Mech Anal 150:127–152
Ulmer H, Hofacker M, Miehe C (2012) Phase field modeling of fracture in plates and shells. PAMM 12(1):171–172
Vavasis SA, Papoulia KD, Hirmand MR (2020) Second-order cone interior-point method for quasistatic and moderate dynamic cohesive fracture. Comput Methods Appl Mech Eng 358:112633
Zhou S, Zhuang X (2018) Adaptive phase field simulation of quasi-static crack propagation in rocks. Computational modeling of fracture in geotechnical engineering part I. Undergr Sp 3(3):190–205
Zimmermann C, Sauer RA (2017) Adaptive local surface refinement based on LR NURBS and its application to contact. Comput Mech 60:1011–1031
Zimmermann C, Toshniwal D, Landis CM, Hughes TJR, Mandadapu KK, Sauer RA (2019) An isogeometric finite element formulation for phase transitions on deforming surfaces. Comput Methods Appl Mech Eng 351:441–477
Acknowledgements
Thomas J.R. Hughes and Chad M. Landis were partially supported by the Office of Naval Research (Grant Nos. N00014-17-1-2119, N00014-13-1-0500, and N00014-17-1-2039). Kranthi K. Mandadapu acknowledges support from University of California Berkeley and from the National Institutes of Health Grant R01-GM110066. Roger A. Sauer acknowledges the support from a J. Tinsley Oden fellowship in 2016 and funding from the German Research Foundation (DFG) through project GSC 111. Christopher Zimmermann and Karsten Paul were funded by the German Research Foundation (DFG) through projects GSC 111 and 33849990/GRK2379 (IRTG Modern Inverse Problems). Simulations were performed with computing resources granted by RWTH Aachen University under Projects rwth0401 and rwth0433.
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Appendices
Appendix
Time integration scheme
The system in Eq. (87) with intermediate quantities and the quantities at time step \(n+1\)
has to be solved. Here, \(\Delta t = t_{n+1}-t_n\) refers to the time step size. Numerical dissipation is controlled by the parameters \(\gamma \), \(\beta \), \(\alpha _\mathrm {f}\) and \(\alpha _\mathrm {m}\). They are expressed in terms of \(\rho _{\infty }\in [0,1]\), which resembles an algorithmic parameter that corresponds to the spectral radius of the amplification matrix as \(\Delta t \rightarrow \infty \) (see Chung and Hulbert [16] for further details), i.e.
We have found \(\rho _\infty = 0.5\) to be a good choice and have used this in all computations. To solve the nonlinear system of equations in Eq. (87) using the Newton–Raphson procedure, it has to be linearized, i.e.
where the tangent matrix blocks are computed from
The required linearizations of the force vectors are shown in Appendix B:. The initial guess for the Newton–Raphson iteration is set to
and then updated from iteration step \(i\rightarrow i+1\) by
until convergence is achieved. At iteration i, we check for the two convergence criteria
with \(\Vert ...\Vert \) denoting the Euclidean norm and \(\text {tol}^\mathrm {dyn}=10^{-4}\) and
with \(\text {tol}^\mathrm {nrg}=10^{-25}\).
Linearization
This section presents the respective elemental contributions for the tangent blocks in Eq. (101). The linearization of the mechanical force vector \({\mathbf {f}}^e:={\mathbf {f}}^e_\mathrm {kin}+{\mathbf {f}}^e_\mathrm {int}-{\mathbf {f}}^e_\mathrm {ext}\) of finite element \(\Omega ^e\) with respect to the respective nodal positions \({\mathbf {x}}_e\) can be found in the work of Duong et al. [19]. Since we model the pressure as a function of the phase field variable, we need to linearize the external force vector with respect to \(\phi \). This linearization of the pressure part \({\mathbf {f}}_{\mathrm {ext}p}^e\) of the external elemental force vector reads
For the linearization of the internal force vector, the four material tangents
have to be defined. Since we assume the constitutive in-plane response to be fully decoupled from the out-of-plane response, it follows that \(d^{\alpha \beta \gamma \delta }=e^{\alpha \beta \gamma \delta }=0\). According to Eqs. (49) and (50), the first tangent matrix can be computed based on the contributions
Based on Eqs. (52) and (53), the tangent matrix \(f^{{\alpha \beta }{\gamma \delta }}\) can be computed from the contribution
Since we consider the fully linearized system in Eq. (100), we also need to linearize the mechanical force vector with respect to the phase field, i.e.
with
where \(\tau ^{\alpha \beta }:=J\sigma ^{\alpha \beta }\) and \(M^{\alpha \beta }_0:=JM^{\alpha \beta }\) has been used to map the integrals to the element domain in the reference configuration. According to Eq. (86), the linearization of \({{\bar{{\mathbf {f}}}}}^e\) with respect to the respective nodal positions \({\mathbf {x}}_e\) yields
with
and
The linearization of \({\bar{{\mathbf {f}}}}_\mathrm {int}^e\) with respect to the phase field variables of \(\Omega ^e\) reads
with
The matrices \({{\bar{{\mathbf {k}}}}}^e_0\) and \({{\bar{{\mathbf {k}}}}}^e_\mathrm {el}\) both contribute to the tangent block \({{\bar{{\mathbf {K}}}}}_\phi \) in Eq. (100).
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Paul, K., Zimmermann, C., Mandadapu, K.K. et al. An adaptive space-time phase field formulation for dynamic fracture of brittle shells based on LR NURBS. Comput Mech 65, 1039–1062 (2020). https://doi.org/10.1007/s00466-019-01807-y
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DOI: https://doi.org/10.1007/s00466-019-01807-y