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Irreversibility and alternate minimization in phase field fracture: a viscosity approach
No full textThis work is devoted to the analysis of convergence of an alternate (staggered) minimization algorithm in the framework of phase field models of fracture. The energy of the system is characterized by a nonlinear splitting of tensile and compressive strains, featuring non-interpenetration of the fracture lips. The alternating scheme is coupled with an L2-penalization in the phase field variable, driven by a viscous parameter δ> 0 , and with an irreversibility constraint, forcing the monotonicity of the phase field only w.r.t. time, but not along the whole iterative minimization. We show first the convergence of such a scheme to a viscous evolution for δ> 0 and then consider the vanishing viscosity limit δ→ 0
Analysis of Staggered Evolutions for Nonlinear Energies in Phase Field Fracture
No full textWe consider a class of separately convex phase field energies employed in fracture mechanics, featuring non-interpenetration and a general softening behavior. We analyze the time-discrete evolutions generated by a staggered minimization scheme, where fracture irreversibility is modeled by a monotonicity constraint on the phase field variable. After recasting the staggered scheme by means of gradient flows, we characterize the time-continuous limits of the discrete solutions in terms of balanced viscosity evolutions, parametrized by their arc-length with respect to the L2-norm (for the phase field) and the H1-norm (for the displacement field). By a careful study of the energy balance we deduce that time-continuous evolutions may still exhibit an alternate behavior in discontinuity times
Energy release rate and stress intensity factors in planar elasticity in presence of smooth cracks
No full textIn this work we first analyze the singular behavior of the displacement u of a linearly elastic body in dimension 2 close to the tip of a smooth crack, extending the well-known results for straight fractures to general smooth ones. As conjectured by Griffith (Phys Eng Sci 221:163–198, 1921), u behaves as the sum of an H2-function and a linear combination of two singular functions whose profile is similar to the square root of the distance from the tip. The coefficients of the linear combination are the so called stress intensity factors. Afterwards, we prove the differentiability of the elastic energy with respect to an infinitesimal fracture elongation and we compute the energy release rate, enlightening its dependence on the stress intensity factors
Energy release rate and quasi-static evolution via vanishing viscosity in a fracture model depending on the crack opening
Full text linkAlmi S., Urbanisme et colonisation, présence française en Algérie
No full textMerlin Pierre. Almi S., Urbanisme et colonisation, présence française en Algérie. In: Annales de Géographie, t. 112, n°630, 2003. p. 214
Crack growth by vanishing viscosity in planar elasticity
Full text linkWe show the existence of quasistatic evolutions in a fracture model for brittle materials by a vanishing viscosity approach, in the setting of planar linearized elasticity. Differently from previous works, the crack is not prescribed a priori and is selected in a class of (unions of) regular curves. To prove the result, it is crucial to analyze the properties of the energy release rate showing that it is independent of the crack extension
A multi-step Lagrangian scheme for spatially inhomogeneous evolutionary games
Full text linkA multi-step Lagrangian scheme at discrete times is proposed for the approximation of a nonlinear continuity equation arising as a mean-field limit of spatially inhomogeneous evolutionary games, describing the evolution of a system of spatially distributed agents with strategies, or labels, whose payoff depends also on the current position of the agents. The scheme is Lagrangian, as it traces the evolution of position and labels along characteristics, and is a multi-step scheme, as it develops on the following two stages: First, the distribution of strategies or labels is updated according to a best performance criterion, and then, this is used by the agents to evolve their position. A general convergence result is provided in the space of probability measures. In the special cases of replicator-type systems and reversible Markov chains, variants of the scheme, where the explicit step in the evolution of the labels is replaced by an implicit one, are also considered and convergence results are provided
Lower semicontinuity and relaxation for free discontinuity functionals with non-standard growth
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