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).
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2017, 37(11): 5883-5912.
doi: 10.3934/dcds.2017256
Visco-Energetic solutions to one-dimensional rate-independent problems
Dipartimento di Matematica "F. Casorati", Università di Pavia, Via Ferrata 1, I-27100 Pavia, Italy |
Visco-Energetic solutions of rate-independent systems (recently introduced in [
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 $δ$.
Keywords:
Abstract evolution equations,
rate-independent systems,
energetic and BV solutions,
Visco-Energetic solutions.
Mathematics Subject Classification:
Primary: 34C55, 47J20, 49J40; Secondary: 74N30.
Citation:
Luca Minotti. Visco-Energetic solutions to one-dimensional rate-independent problems. Discrete and Continuous Dynamical Systems,
2017, 37
(11)
: 5883-5912.
doi: 10.3934/dcds.2017256
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References:
| [1] |
L. Ambrosio, N. Fusco and D. Pallara,
Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford, 2000. |
| [2] |
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. |
| [3] |
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.
|
| [4] |
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. |
| [5] |
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. |
| [6] |
K. Kuratowski,
Sur l'espace des fonctions partielles, Ann. Mat. Pura Appl.(4), 40 (1955), 61-67.
doi: 10.1007/BF02416522. |
| [7] |
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. |
| [8] |
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. |
| [9] |
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. |
| [10] |
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. |
| [11] |
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. |
| [12] |
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. |
| [13] |
A. Mielke and T. Roubíček,
Rate-Independent Systems: Theory and Application, Springer, New York, 2015.
doi: 10.1007/978-1-4939-2706-7. |
| [14] |
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. |
| [15] |
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. |
| [16] |
L. Minotti,
Visco-Energetic Solutions to Rate-Independent Evolution Problems, PhD thesis, Pavia, 2016. |
| [17] |
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. |
| [18] |
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. |
| [19] |
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.
|
| [20] |
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.
|
| [21] |
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. |
show all references
References:
| [1] |
L. Ambrosio, N. Fusco and D. Pallara,
Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford, 2000. |
| [2] |
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. |
| [3] |
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.
|
| [4] |
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. |
| [5] |
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. |
| [6] |
K. Kuratowski,
Sur l'espace des fonctions partielles, Ann. Mat. Pura Appl.(4), 40 (1955), 61-67.
doi: 10.1007/BF02416522. |
| [7] |
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. |
| [8] |
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. |
| [9] |
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. |
| [10] |
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. |
| [11] |
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. |
| [12] |
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. |
| [13] |
A. Mielke and T. Roubíček,
Rate-Independent Systems: Theory and Application, Springer, New York, 2015.
doi: 10.1007/978-1-4939-2706-7. |
| [14] |
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. |
| [15] |
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. |
| [16] |
L. Minotti,
Visco-Energetic Solutions to Rate-Independent Evolution Problems, PhD thesis, Pavia, 2016. |
| [17] |
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. |
| [18] |
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. |
| [19] |
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.
|
| [20] |
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.
|
| [21] |
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. |
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.
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