%0 Journal Article %J Discrete & Continuous Dynamical Systems - A %D 2012 %T Second order approximations of quasistatic evolution problems in finite dimension %A Virginia Agostiniani %K discrete approximations %K perturbation methods %K saddle-node bifurcation %K Singular perturbations %K vanishing viscosity %X

In this paper, we study the limit, as ε goes to zero, of a particular solution of the equation $\epsilon^2A\ddot u^ε(t)+εB\dot u^ε(t)+\nabla_xf(t,u^ε(t))=0$, where $f(t,x)$ is a potential satisfying suitable coerciveness conditions. The limit $u(t)$ of $u^ε(t)$ is piece-wise continuous and verifies $\nabla_xf(t,u(t))=0$. Moreover, certain jump conditions characterize the behaviour of $u(t)$ at the discontinuity times. The same limit behaviour is obtained by considering a different approximation scheme based on time discretization and on the solutions of suitable autonomous systems.

%B Discrete & Continuous Dynamical Systems - A %V 32 %P 1125 %G eng %U http://aimsciences.org//article/id/560b82d9-f289-498a-a619-a4b132aaf9f8 %R 10.3934/dcds.2012.32.1125 %0 Journal Article %J {MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES} %D 2011 %T A MODEL FOR CRACK PROPAGATION BASED ON VISCOUS APPROXIMATION %A Giuliano Lazzaroni %A Rodica Toader %K Brittle fracture %K Crack propagation %K energy derivative %K energy release rate %K free-discontinuity problems %K Griffith's criterion %K local minimizers %K stress intensity factor} %K vanishing viscosity %K {Variational models %X

{In the setting of antiplane linearized elasticity, we show the existence of quasistatic evolutions of cracks in brittle materials by using a vanishing viscosity approach, thus taking into account local minimization. The main feature of our model is that the path followed by the crack need not be prescribed a priori: indeed, it is found as the limit (in the sense of Hausdorff convergence) of curves obtained by an incremental procedure. The result is based on a continuity property for the energy release rate in a suitable class of admissible cracks.}

%B {MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES} %I {WORLD SCIENTIFIC PUBL CO PTE LTD} %C {5 TOH TUCK LINK, SINGAPORE 596224, SINGAPORE} %V {21} %P {2019-2047} %8 {OCT} %G eng %9 {Article} %R {10.1142/S0218202511005647}