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Visco-Energetic solutions to one-dimensional rate-independent problems

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  • Visco-Energetic solutions of rate-independent systems (recently introduced in [17]) are obtained by solving a modified time Incremental Minimization Scheme, where at each step the dissipation is reinforced by a viscous correction $δ$, typically a quadratic perturbation of the dissipation distance. Like Energetic and Balanced Viscosity solutions, they provide a variational characterization of rate-independent evolutions, with an accurate description of their jump behaviour.

    In the present paper we study Visco-Energetic solutions in the scalar-valued case and we obtain a full characterization for a broad class of energy functionals. In particular, we prove that they exhibit a sort of intermediate behaviour between Energetic and Balanced Viscosity solutions, which can be finely tuned according to the choice of the viscous correction $δ$.

    Mathematics Subject Classification: Primary: 34C55, 47J20, 49J40; Secondary: 74N30.

    Citation:

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  • Figure 1.  The double-well potential $W$ with its convex envelope in bold (left picture) and an energetic solution $u$ in the case of a strictly increasing load $\ell$ (right picture).

    Figure 2.  BV solution for a double-well energy $W$ with an increasing load $\ell$. The blue line denotes the path described by the optimal transition $\vartheta$ solving (4).

    Figure 3.  Visco-Energetic solutions for a double-well energy $W$ with an increasing load $\ell$. When $\mu>-\min W''$ (left picture) the solution jumps when it reach the maximum of $W'$ and the transition is the ''double chain'' obtained by solving the Incremental Minimization Scheme with frozen time $t$. When $\mu$ is small (right picture) the optimal transition $\vartheta$ makes a first jump connecting $\mathit{u}_{\tiny \mathsf L}(t)$ with $u_+$ according to the modified Maxwell rule (9): $\mathit{u}_{\tiny \mathsf L}(t)$ and $u_+$ corresponds to the intersection of $W'$ with the red line, whose slope is $-\mu$.

    Figure 4.  The one-sided slopes and the stability region when $W$ is a double-well potential. For some suitable choices of $\delta$, $\mathit W_{\mathsf{i}\mathsf{r},{\delta}}'$ is intermediate between $W'_{\mathsf i\mathsf r}$ and $W'$.

    Figure 5.  $\mathsf D$-Maxwell's rule for a double well potential: when $\delta=\frac{\mu}{2}|\cdot|^2$, the ''last point'' where $W'$ and $\mathit W_{\mathsf{i}\mathsf{r},{\delta}}'$ coincide is such that the total area between the graph $W'$ and the line whose slope is $-\mu$ is zero.

    Figure 6.  Visco-Energetic solution of a double-well potential energy with an oscillating external loading and a quadratic viscous-correction $\delta(u,v)$, turned by a parameter $\mu>-\min W''$.

    Figure 7.  Visco-Energetic solutions of a nonconvex energy and an increasing loading. The optimal transition is a combination of sliding and viscous parts.

  •   L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford, 2000.
      G. Dal Maso  and  R. Toader , A model for the quasi-static growth of brittle fractures based on local minimization, Math. Models Methods Appl. Sci., 12 (2002) , 1773-1799.  doi: 10.1142/S0218202502002331.
      M. Efendiev  and  A. Mielke , On the rate-independent limit of systems with dry friction and small viscosity, J. Convex Analysis, 13 (2006) , 151-167. 
      I. S. Gál , On the fundamental theorems of the calculus, Trans. Amer. Math. Soc., 86 (1957) , 309-320.  doi: 10.1090/S0002-9947-1957-0093562-7.
      D. Knees  and  A. Schröder , Computation aspect of quasi-static crack propagation, DCDS-S, 6 (2013) , 63-99.  doi: 10.3934/dcdss.2013.6.63.
      K. Kuratowski , Sur l'espace des fonctions partielles, Ann. Mat. Pura Appl., 40 (1955) , 61-67.  doi: 10.1007/BF02416522.
      D. Leguillon , Strength or toughness? A criterion for crack onset at a notch, European J. of Mechanics A/Solids, 21 (2002) , 61-72.  doi: 10.1016/S0997-7538(01)01184-6.
      A. Mainik  and  A. Mielke , Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations, 22 (2005) , 73-99.  doi: 10.1007/s00526-004-0267-8.
      A. Mielke , Differential, energetic and metric formulations for rate-independent processes, Nonlinear PDEs and Applications, Lect. Notes Math, Springer, 2028 (2011) , 87-170.  doi: 10.1007/978-3-642-21861-3_3.
      A. Mielke , R. Rossi  and  G. Savaré , Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete and Continuous Dynamical Systems A, 25 (2009) , 585-615.  doi: 10.3934/dcds.2009.25.585.
      Alexander Mielke , Riccarda Rossi  and  Giuseppe Savaré , BV solutions and viscosity approximations of rate-independent systems, ESAIM Control Optim. Calc. Var., 18 (2012) , 36-80.  doi: 10.1051/cocv/2010054.
      Alexander Mielke , Riccarda Rossi  and  Giuseppe Savaré , Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems, J. Eur. Math. Soc., 18 (2016) , 2107-2165.  doi: 10.4171/JEMS/639.
      A. Mielke and T. Roubíček, Rate-Independent Systems: Theory and Application, Springer, New York, 2015. doi: 10.1007/978-1-4939-2706-7.
      A. Mielke  and  F. Theil , On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl., 11 (2004) , 151-189.  doi: 10.1007/s00030-003-1052-7.
      A. Mielke , F. Theil  and  V. I. Levitas , A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Ration. Mech. Anal., 162 (2002) , 137-177.  doi: 10.1007/s002050200194.
      L. Minotti, Visco-Energetic Solutions to Rate-Independent Evolution Problems, PhD thesis, Pavia, 2016.
      L. Minotti and G. Savaré, Viscous corrections of the time incremental minimization scheme and visco-energetic solutions to rate-independent evolution problems, arXiv: 1606.03359, (2016), 1-60.
      M. Negri  and  C. Ortner , Quasi-static crack propagation by Griffith's criterion, Math. Models Methods Appl. Sci., 18 (2008) , 1895-1925.  doi: 10.1142/S0218202508003236.
      R. Rossi , A. Mielke  and  G. Savaré , A metric approach to a class of doubly nonlinear evolution equations and applications, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 7 (2008) , 97-169. 
      R. Rossi  and  G. Savaré , A characterization of energetic and BV solutions to one-dimensional rate-independent systems, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013) , 167-191. 
      T. Roubíček , C. C. Panagiotopoulos  and  V. Mantic , Quasistatic adhesive contact of visco-elastic bodies and its numerical treatment for very small viscosity, Zeitschrift angew. Math. Mech., 93 (2013) , 823-840.  doi: 10.1002/zamm.201200239.
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