Advanced Search
Article Contents
Article Contents

Second order approximations of quasistatic evolution problems in finite dimension

Abstract Related Papers Cited by
  • In this paper, we study the limit, as $\epsilon$ goes to zero, of a particular solution of the equation $\epsilon^2A\ddot u^{\epsilon}(t)+\epsilon B\dot u^{\epsilon}(t)+\nabla_xf(t,u^{\epsilon}(t))=0$, where $f(t,x)$ is a potential satisfying suitable coerciveness conditions. The limit $u(t)$ of $u^{\epsilon}(t)$ 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.
    Mathematics Subject Classification: Primary: 74H10, 34K18, 34C37; Secondary: 34K26.


    \begin{equation} \\ \end{equation}
  • [1]

    F. Cagnetti, A vanishing viscosity approach to fracture growth in a cohesive zone model with prescribed crack path, Math. Models Methods Appl. Sci., 18 (2008), 1027-1071.doi: 10.1142/S0218202508002942.


    G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, A vanishing viscosity approach to quasistatic evolution in plasticity with softening, Arch. Ration. Mech. Anal., 189 (2008), 469-544.doi: 10.1007/s00205-008-0117-5.


    G. Dal Maso, A. DeSimone and F. Solombrino, Quasistatic evolution for Cam-Clay plasticity: a weak formulation via viscoplastic regularization and time rescaling, Calc. Var. Partial Differential Equations, 40 (2011), 125-181.


    G. Dal Maso, A. DeSimone and F. SolombrinoQuasistatic evolution for Cam-Clay plasticity: properties of the viscosity solution, SISSA preprint 46/2010/M.


    G. Dal Maso and F. Solombrino, Quasistatic evolution for Cam-Clay plasticity: the spatially homogeneous case, Netw. Heterog. Media, 5 (2010), 97-132.


    M. A. Efendiev and A. Mielke, On the rate-independent limit of systems with dry friction and small viscosity, J. Convex Anal., 13 (2006), 151-167.


    J. Guckenheimer and P. Holme, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations on Vector Fields," Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983.


    J. K. Hale, "Ordinary Differential Equations," Pure and Applied Mathematics, XX1, Krieger, Florida, 1980.


    M. W. Hirsch, "Differential Topology," Graduate Texts in Mathematics, 33, Springer-Verlag, New York, 1976.


    D. Knees, A. Mielke and C. Zanini, Crack growth in polyconvex materials, Phys. D, 239 (2010), 1470-1484.doi: 10.1016/j.physd.2009.02.008.


    A. Mielke, R. Rossi and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete Contin. Dyn. Syst., 25 (2009), 585-615.doi: 10.3934/dcds.2009.25.585.


    A. Mielke, R. Rossi and G. Savaré, BV solutions and viscosity approximations of rate-independent systems, ESAIM Control Optim. Calc. Var., 2011.doi: 10.1051/cocv/2010054.


    F. Solombrino, Quasistatic evolution for plasticity with softening: the spatially homogeneous case, Discrete Contin. Dyn. Syst., 27 (2010), 1189-1217.doi: 10.3934/dcds.2010.27.1189.


    R. Toader and C. Zanini, An artificial viscosity approach to quasistatic crack growth, Boll. Unione Mat. Ital. (9), 2 (2009), 1-35.


    C. Zanini, Singular perturbation of finite dimensional gradient flows, Discrete Contin. Dyn. Syst., 18 (2007), 657-675.doi: 10.3934/dcds.2007.18.657.

  • 加载中

Article Metrics

HTML views() PDF downloads(99) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint