Abstract
A novel phase field formulation implemented within a material point method setting is developed to address brittle fracture simulation in anisotropic media. The case of strong anisotropy in the crack surface energy is treated by considering an appropriate variational, i.e. phase field approach. Material point method is utilized to efficiently treat the resulting coupled governing equations. The brittle fracture governing equations are defined at a set of Lagrangian material points and subsequently interpolated at the nodes of a fixed Eulerian mesh where solution is performed. As a result, the quality of the solution does not depend on the quality of the underlying finite element mesh and is relieved from mesh distortion errors. The efficiency and validity of the proposed method are assessed through a set of benchmark problems.
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References
Ambati, M., Kruse, R., De Lorenzis, L.: A phase-field model for ductile fracture at finite strains and its experimental verification. Comput. Mech. 57(1), 149–167 (2016)
Ambrosio, L., Tortorelli, V.M.: Approximation of functional depending on jumps by elliptic functional via t-convergence. Commun. Pure Appl. Math. 43(8), 999–1036 (1990)
Bandara, S., Soga, K.: Coupling of soil deformation and pore fluid flow using material point method. Comput. Geotech. 63, 199–214 (2015)
Bardenhagen, S., Kober, E.: The generalized interpolation material point method. Comput. Model. Eng. Sci. 5(6), 477–495 (2004)
Bardenhagen, S.G., Nairn, J.A., Lu, H.: Simulation of dynamic fracture with the material point method using a mixed J-integral and cohesive law approach. Int. J. Fract. 170(1), 49–66 (2011)
Bathe, K.J.: Finite Element Procedures. Prentice Hall, Upper Saddle River, NJ (2007)
Batra, R., Zhang, G.: Search algorithm, and simulation of elastodynamic crack propagation by modified smoothed particle hydrodynamics (MSPH) method. Comput. Mech. 40(3), 531–546 (2007)
Bobaru, F., Hu, W.: The meaning, selection, and use of the peridynamic horizon and its relation to crack branching in brittle materials. Int. J. Fract. 176(2), 215–222 (2012)
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. 217–220, 77–95 (2012)
Borden, M.J., Hughes, T.J., Landis, C.M., Verhoosel, C.V.: 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 (2014)
Bourdin, B., Francfort, G.A., Marigo, J.J.: The variational approach to fracture. J. Elast. 91(1–3), 5–148 (2008)
Clayton, J., Knap, J.: Phase field modeling and simulation of coupled fracture and twinning in single crystals and polycrystals. Comput. Methods Appl. Mech. Eng. 312, 447–467 (2016)
Daphalapurkar, N.P., Lu, H., Coker, D., Komanduri, R.: Simulation of dynamic crack growth using the generalized interpolation material point (GIMP) method. Int. J. Fract. 143(1), 79–102 (2007)
Francfort, G.A., Marigo, J.J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46(8), 1319–1342 (1998)
Griffith, A.A.: The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. Lond. A 221, 163–198 (1921)
Gültekin, O., Dal, H., Holzapfel, G.A.: A phase-field approach to model fracture of arterial walls: theory and finite element analysis. Comput. Methods Appl. Mech. Eng. 312, 542–566 (2016)
Homel, M.A., Herbold, E.B.: Field-gradient partitioning for fracture and frictional contact in the material point method. Int. J. Numer. Methods Eng. 109(7), 1013–1044 (2017)
Hughes, T., Reali, A., Sangalli, G.: Efficient quadrature for NURBS-based isogeometric analysis. Comput. Methods Appl. Mech. Eng. 199(5–8), 301–313 (2010)
Jassim, I., Stolle, D., Vermeer, P.: Two-phase dynamic analysis by material point method. Int. J. Numer. Anal. Methods Geomech. 37(15), 2502–2522 (2013)
Kakouris, E.G., Triantafyllou, S.P.: Phase-field material point method for brittle fracture. Int. J. Numer. Methods Eng. (2017). doi:10.1002/nme.5580
Li, B., Peco, C., Milln, D., Arias, I., Arroyo, M.: Phase-field modeling and simulation of fracture in brittle materials with strongly anisotropic surface energy. Int. J. Numer. Methods Eng. 102(3–4), 711–727 (2015)
Li, T., Marigo, J.J., Guilbaud, D., Potapov, S.: Gradient damage modeling of brittle fracture in an explicit dynamics context. Int. J. Numer. Methods Eng. 108(11), 1381–1405 (2016)
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(45–48), 2765–2778 (2010)
Miehe, C., Welschinger, F., Hofacker, M.: Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations. Int. J. Numer. Methods Eng. 83(10), 1273–1311 (2010)
Miehe, C., Aldakheel, F., Raina, A.: Phase field modeling of ductile fracture at finite strains: a variational gradient-extended plasticity-damage theory. Int. J. Plast. 84, 1–32 (2016)
Nairn, J.A.: Material point method calculations with explicit cracks. Comput. Model. Eng. Sci. 4(6), 649–664 (2003)
Nairn, J.A., Hammerquist, C., Aimene, Y.E.: Numerical implementation of anisotropic damage mechanics. Int. J. Numer. Methods Eng. (2017). doi:10.1002/nme.5585
Nguyen, V.P., Rabczuk, T., Bordas, S., Duflot, M.: Meshless methods: a review and computer implementation aspects. Math. Comput. Simul. 79(3), 763–813 (2008)
Sadeghirad, A., Brannon, R.M., Burghardt, J.: A convected particle domain interpolation technique to extend applicability of the material point method for problems involving massive deformations. Int. J. Numer. Methods Eng. 86(12), 1435–1456 (2011)
Sadeghirad, A., Brannon, R., Guilkey, J.: Second-order convected particle domain interpolation (CPDI2) with enrichment for weak discontinuities at material interfaces. Int. J. Numer. Methods Eng. 95(11), 928–952 (2013)
Sanchez, J., Schreyer, H., Sulsky, D., Wallstedt, P.: Solving quasi-static equations with the material-point method. Int. J. Numer. Methods Eng. 103(1), 60–78 (2015)
Scholtès, L., Donzé, F.V.: Modelling progressive failure in fractured rock masses using a 3D discrete element method. Int. J. Rock Mech. Min. Sci. 52, 18–30 (2012)
Schreyer, H., Sulsky, D., Zhou, S.J.: Modeling delamination as a strong discontinuity with the material point method. Comput. Methods Appl. Mech. Eng. 191(23–24), 2483–2507 (2002)
Shanthraj, P., Svendsen, B., Sharma, L., Roters, F., Raabe, D.: Elasto-viscoplastic phase field modelling of anisotropic cleavage fracture. J. Mech. Phys. Solids 99, 19–34 (2017)
Steffen, M., Kirby, R.M., Berzins, M.: Analysis and reduction of quadrature errors in the material point method (MPM). Int. J. Numer. Methods Eng. 76(6), 922–948 (2008)
Steffen, M., Wallstedt, P.M., Guilkey, J., Kirby, R., Berzins, M.: Examination and analysis of implementation choices within the material point method (MPM). Comput. Model. Eng. Sci. 31(2), 107–127 (2008)
Sulsky, D., Chen, Z., Schreyer, H.L.: A particle method for history-dependent materials. Comput. Methods Appl. Mech. Eng. 118(1–2), 179–196 (1994)
Sulsky, D., Kaul, A.: Implicit dynamics in the material-point method. Comput. Methods Appl. Mech. Eng. 193(1214), 1137–1170 (2004)
Sulsky, D., Schreyer, L.: MPM simulation of dynamic material failure with a decohesion constitutive model. Eur. J. Mech. A/Solids 23(3), 423–445 (2004)
Ting, T.C.T.: Anisotropic Elasticity: Theory and Applications. Oxford University Press, New York (1996)
Yang, P., Gan, Y., Zhang, X., Chen, Z., Qi, W., Liu, P.: Improved decohesion modeling with the material point method for simulating crack evolution. Int. J. Fract. 186(1), 177–184 (2014)
Acknowledgements
The research described in this paper has been financed by the University of Nottingham through the Dean of Engineering Prize, a scheme for pump priming support for early-career academic staff. The authors are grateful to the University of Nottingham for access to its high-performance computing facility.
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Appendices
Appendix A: Variational approach of the anisotropic phase field model
In the energy balance equation (9) the rate of the kinetic energy is evaluated as
Similarly, the rate of the external work is expressed as
and the rate of the internal work is defined accordingly as
Applying the divergence theorem in Eq. (63), the rate of the internal work \(\dot{\mathcal {W}}^{int} \left( \dot{\mathbf {u}},\dot{c},\nabla \dot{c} \right) \) assumes the following form
where the components \(B_i\), \(i=1\dots 4\), assume the following expressions
and
respectively, where
and
Components \(\mathcal {T}_1\) and \(\mathcal {T}_2\) are further expanded employing the divergence theorem into
and
respectively.
Substituting Eqs. (71) and (72) in Eq. (68) the following expression is derived for \(\mathcal {B}_4\)
Substituting Eq. (73) in the energy balance equation (9) expression (14) is finally established.
Appendix B: Transformation of surface energy density to polar coordinates
The surface energy density and their corresponding reciprocal expression polar plots are evaluated according to the methodology introduced in [21]. In this, the Cartesian coordinate system \(\mathbf {x}\left( x_1 , x_2 \right) \) is transformed to \(\mathbf {x}_{\theta } \left( {x}_{1_{\theta }} , {x}_{2_{\theta }} \right) \) where the \({x}_{1_{\theta }}\) axis is defined along the crack path \({\varGamma }\) and \({x}_{2_{\theta }}\) axis is the axis normal to the crack interface as shown in Fig. 20. Angle \(\theta \) is the counterclockwise angle between \({x}_{1}\) axis and \({x}_{1_{\theta }}\).
Thus, coordinate transformation from \(\mathbf {x}\left( x_1 , x_2 \right) \) to \(\mathbf {x}_{\theta } \left( {x}_{1_{\theta }}, {x}_{2_{\theta }} \right) \) is performed through the transformation equation (74)
with the inverse transformation defined as
Assuming that \(c \left( \mathbf {x} \left( \mathbf {x}_{\theta } \right) \right) \approx c \left( \mathbf {x} \left( x_{2_{\theta }} \right) \right) \) and applying the chain rule, the phase field first spatial derivatives are expressed as
and
respectively. Similarly, the second spatial derivatives are expressed as
and
respectively. Higher-order spatial derivatives are defined accordingly as
and
Employing Eqs. (76)–(82), the functional \(\mathcal {Z}_{c,Anis}\) of Eq. (4) is expressed in polar coordinates as
where
Furthermore, the Euler–Lagrange equation is rewritten in the form
Equation (85) can then be numerically solved subject to the following boundary conditions, i.e.
where \(x_{lb}\) is the distance from the boundary, assuming that \(x_{lb}=20 l_0\). Finally, the surface energy density is numerically evaluated as
The maximum and minimum values of \(\mathcal {G}_{c} \left( \theta \right) \) for \(\theta \in \left[ 0,2\pi \right] \) define the \(\mathcal {G}_{c_{\max }}\) and \(\mathcal {G}_{c_{\min }}\), respectively. The polar plot of surface energy density \(\mathcal {G}_{c} \left( \theta \right) \) can be rotated by angle \(\phi \) through relation (88) below
The rotation matrix \(\varvec{Q}_{\phi }\) is defined as
where \(c=\cos \left( \phi \right) \) and \(s=\sin \left( \phi \right) \). The angle \(\phi \) goes clockwise. In the cases of cubic and orthotropic symmetries the fourth-order tensor \(\varvec{\gamma }\) is expressed, in global axes, as
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Kakouris, E.G., Triantafyllou, S.P. Material point method for crack propagation in anisotropic media: a phase field approach. Arch Appl Mech 88, 287–316 (2018). https://doi.org/10.1007/s00419-017-1272-7
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DOI: https://doi.org/10.1007/s00419-017-1272-7