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\(\Gamma\)-convergence for high order phase field fracture: continuum and isogeometric formulations. (English) Zbl 1439.74027
Summary: We consider high order phase field functionals introduced in [M. J. Borden et al., Comput. Methods Appl. Mech. Eng. 273, 100–118 (2014; Zbl 1296.74098)] and provide a rigorous proof that these functionals converge to a sharp crack brittle fracture energy. We take into account three dimensional problems in linear elastic fracture mechanics and functionals defined both in Sobolev spaces and in spaces of tensor product \(B\)-splines. In the latter convergence holds when the mesh size vanishes faster than the internal length of the phase-field model. On the theoretical level, this condition is natural since the size of the phase field layer, around the crack, itself scales like the internal length; on the numerical level, it should be satisfied by local \(h\)-refinement.
Technically, convergence holds in the sense of \(\Gamma\)-convergence, with respect to the strong topology of \(L^1\), while the sharp crack energy is defined in \(GSBD^2\). The constraint on the phase field to take values in \([0,1]\) is taken into account both in the Sobolev setting and in the iso-geometric setting; in the latter, it requires a special treatment since the projection operator on the space of tensor product \(B\)-splines is not Lagrangian (i.e., interpolatory).
74A45 Theories of fracture and damage
74R10 Brittle fracture
74S22 Isogeometric methods applied to problems in solid mechanics
49J45 Methods involving semicontinuity and convergence; relaxation
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