Thermodynamically consistent phase‐field models of fracture: Variational principles and multi‐field FE implementations

International Journal for Numerical Methods in Engineering - Tập 83 Số 10 - Trang 1273-1311 - 2010
Christian Miehé1, Fabian Welschinger1, Martina Hofacker1
1Institut für Mechanik (Bauwesen) Lehrstuhl I, Universität Stuttgart, 70550 Stuttgart, Pfaffenwaldring 7, Germany

Tóm tắt

AbstractThe computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase‐field. In this paper, we outline a thermodynamically consistent framework for phase‐field models of crack propagation in elastic solids, develop incremental variational principles and consider their numerical implementations by multi‐field finite element methods. We start our investigation with an intuitive and descriptive derivation of a regularized crack surface functional that Γ‐converges for vanishing length‐scale parameter to a sharp crack topology functional. This functional provides the basis for the definition of suitable convex dissipation functions that govern the evolution of the crack phase‐field. Here, we propose alternative rate‐independent and viscous over‐force models that ensure the local growth of the phase‐field. Next, we define an energy storage function whose positive tensile part degrades with increasing phase‐field. With these constitutive functionals at hand, we derive the coupled balances of quasi‐static stress equilibrium and gradient‐type phase‐field evolution in the solid from the argument of virtual power. Here, we consider a canonical two‐field setting for rate‐independent response and a time‐regularized three‐field formulation with viscous over‐force response. It is then shown that these balances follow as the Euler equations of incremental variational principles that govern the multi‐field problems. These principles make the proposed formulation extremely compact and provide a perfect base for the finite element implementation, including features such as the symmetry of the monolithic tangent matrices. We demonstrate the performance of the proposed phase‐field formulations of fracture by means of representative numerical examples. Copyright © 2010 John Wiley & Sons, Ltd.

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Tài liệu tham khảo

10.1098/rsta.1921.0006

Irwin GR, 1958, Encyclopedia of Physics, 551

10.1016/S0065-2156(08)70121-2

10.1016/S0022-5096(98)00034-9

10.1007/978-1-4020-6395-4

10.1007/s002050100187

10.1023/A:1007545213010

10.1016/S0022-5096(99)00028-9

10.1002/cpa.3160420503

10.1002/cpa.3160430805

10.1007/978-1-4612-0327-8

10.1007/BFb0097344

10.1093/acprof:oso/9780198507840.001.0001

10.1016/j.jmps.2008.10.012

Karma A, 2001, Phase‐field model of mode III dynamic fracture, Physical Review Letters, 92, 8704.045501

10.1103/PhysRevE.65.036117

10.1002/nme.941

10.1002/nme.2415

10.1016/j.cma.2008.12.028

10.1002/nme.1999

10.1007/s10704-007-9078-1

10.1007/978-1-4612-3584-2

10.1016/S0065-2156(02)80102-8

10.1007/978-3-662-04800-9

10.1007/978-94-017-1957-5

10.1016/0020-7683(95)00074-7

10.1016/0167-2789(95)00173-5

10.1016/0013-7944(95)00247-2

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Thông tin xuất bản

International Journal for Numerical Methods in Engineering

Tập 83 Số 10

1273-1311

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