# American Institute of Mathematical Sciences

January  2021, 14(1): 89-119. doi: 10.3934/dcdss.2020304

## Rate-independent evolution of sets

 1 DIMI, University of Brescia, Via Branze, 38, 25133 Brescia, Italy 2 Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria 3 Istituto di Matematica Applicata e Tecnologie Informatiche E. Magenes - CNR, Via Ferrata, 1, 27100 Pavia, Italy 4 Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany

* Corresponding author: Riccarda Rossi

Dedicated to Alexander Mielke on the occasion of his 60th birthday

Received  March 2019 Revised  September 2019 Published  January 2021 Early access  March 2020

The goal of this work is to analyze a model for the rate-independent evolution of sets with finite perimeter. The evolution of the admissible sets is driven by that of (the complement of) a given time-dependent set, which has to include the admissible sets and hence is to be understood as an external loading. The process is driven by the competition between perimeter minimization and minimization of volume changes.

In the mathematical modeling of this process, we distinguish the adhesive case, in which the constraint that the (complement of) the external load' contains the evolving sets is penalized by a term contributing to the driving energy functional, from the brittle case, enforcing this constraint. The existence of Energetic solutions for the adhesive system is proved by passing to the limit in the associated time-incremental minimization scheme. In the brittle case, this time-discretization procedure gives rise to evolving sets satisfying the stability condition, but it remains an open problem to additionally deduce energy-dissipation balance in the time-continuous limit. This can be obtained under some suitable quantification of data. The properties of the brittle evolution law are illustrated by numerical examples in two space dimensions.

Citation: Riccarda Rossi, Ulisse Stefanelli, Marita Thomas. Rate-independent evolution of sets. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 89-119. doi: 10.3934/dcdss.2020304
##### References:
 [1] F. Almgren, J. E. Taylor and L. Wang, Curvature-driven flows: A variational approach, SIAM J. Control Optim., 31 (1993), 387-438.  doi: 10.1137/0331020. [2] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. [3] L. Ambrosio, N. Gigli and G. Savar{é}, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. doi: 10.1007/b137080. [4] L. Ambrosio, Minimizing movements, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5), 19 (1995), 191-246. [5] D. Bucur, G. Buttazzo and A. Lux, Quasistatic evolution in debonding problems via capacitary methods, Arch. Ration. Mech. Anal., 190 (2008), 281-306.  doi: 10.1007/s00205-008-0166-9. [6] S. Campanato, Propriet{à} di h{ö}lderianit{à} di alcune classi di funzioni, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 17 (1963), 175-188. [7] S. Campanato, Propriet{à} di una famiglia di spazi funzionali, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 18 (1964), 137-160. [8] G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fractures: Existence and approximation results, Arch. Ration. Mech. Anal., 162 (2002), 101-135.  doi: 10.1007/s002050100187. [9] A. Ferriero and N. Fusco, A note on the convex hull of sets of finite perimeter in the plane, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 102-108.  doi: 10.3934/dcdsb.2009.11.103. [10] I. Fonseca and G. A. Francfort, Relaxation in BV versus quasiconvexification in ${W^{1, p}}$ ; A model for the interaction between fracture and damage, Calc. Var. Partial Differential Equations, 3 (1995), 407-446.  doi: 10.1007/BF01187895. [11] G. A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids, 46 (1998), 1319-1342.  doi: 10.1016/S0022-5096(98)00034-9. [12] M. Fr{é}mond, Contact with adhesion, in Topics in Nonsmooth Mechanics, Birkh{ä}user, Basel, 1988,157–185. [13] M. Fr{é}mond, Non-Smooth Thermomechanics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9. [14] B. Kawohl, On starshaped rearrangement and applications, Trans. Amer. Math. Soc., 296 (1986), 377-386.  doi: 10.1090/S0002-9947-1986-0837818-4. [15] M. Ko{\v c}vara, A. Mielke and T. Roub{í}{\v c}ek, A rate-independent approach to the delamination problem, Math. Mech. Solids, 11 (2006), 423-447.  doi: 10.1177/1081286505046482. [16] P. Krej{\v c}{\'\i} and M. Liero, Rate independent {K}urzweil processes, Appl. Math., 54 (2009), 117-145.  doi: 10.1007/s10492-009-0009-5. [17] S. Luckhaus and T. Sturzenhecker, Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differential Equations, 3 (1995), 253-271.  doi: 10.1007/BF01205007. [18] 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. [19] A. Mielke, T. Roub{í}{\v c}ek and U. Stefanelli, {$\Gamma$}-limits and relaxations for rate-independent evolutionary problems, Calc. Var. Partial Differential Equations, 31 (2008), 387-416.  doi: 10.1007/s00526-007-0119-4. [20] A. Mielke and T. Roub{\'\i}{\v c}ek, Rate-Independent Systems. Theory and Application, Applied Mathematical Sciences, 193, Springer, New York, 2015. doi: 10.1007/978-1-4939-2706-7. [21] A. Mielke and F. Theil, A mathematical model for rate-independent phase transformations with hysteresis, Proceedings of the Workshop on Models of Continuum Mechanics in Analysis and Engineering'', Shaker-Verlag, 1999,117–129. [22] 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. [23] R. Rossi and M. Thomas, From an adhesive to a brittle delamination model in thermo-visco-elasticity, ESAIM Control Optim. Calc. Var., 21 (2015), 1-59.  doi: 10.1051/cocv/2014015. [24] T. Roub{í}{\v c}ek, L. Scardia and C. Zanini, Quasistatic delamination problem, Contin. Mech. Thermodyn., 21 (2009), 223-235.  doi: 10.1007/s00161-009-0106-4. [25] T. Roub{\'\i}{\v c}ek, M. Thomas and C. G. Panagiotopoulos, Stress-driven local-solution approach to quasistatic brittle delamination, Nonlinear Anal. Real World Appl., 22 (2015), 645-663.  doi: 10.1016/j.nonrwa.2014.09.011. [26] M. Thomas, Quasistatic damage evolution with spatial BV-regularization, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 235-255.  doi: 10.3934/dcdss.2013.6.235. [27] M. Thomas, Uniform {P}oincaré-{S}obolev and isoperimetric inequalities for classes of domains, Discrete Contin. Dyn. Syst., 35 (2015), 2741-2761.  doi: 10.3934/dcds.2015.35.2741. [28] A. Visintin, Motion by mean curvature and nucleation, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 55-60.  doi: 10.1016/S0764-4442(97)83933-X. [29] A. Visintin, Nucleation and mean curvature flow, Comm. Partial Differential Equations, 23 (1998), 17-53.  doi: 10.1080/03605309808821337.

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##### References:
 [1] F. Almgren, J. E. Taylor and L. Wang, Curvature-driven flows: A variational approach, SIAM J. Control Optim., 31 (1993), 387-438.  doi: 10.1137/0331020. [2] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. [3] L. Ambrosio, N. Gigli and G. Savar{é}, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. doi: 10.1007/b137080. [4] L. Ambrosio, Minimizing movements, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5), 19 (1995), 191-246. [5] D. Bucur, G. Buttazzo and A. Lux, Quasistatic evolution in debonding problems via capacitary methods, Arch. Ration. Mech. Anal., 190 (2008), 281-306.  doi: 10.1007/s00205-008-0166-9. [6] S. Campanato, Propriet{à} di h{ö}lderianit{à} di alcune classi di funzioni, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 17 (1963), 175-188. [7] S. Campanato, Propriet{à} di una famiglia di spazi funzionali, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 18 (1964), 137-160. [8] G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fractures: Existence and approximation results, Arch. Ration. Mech. Anal., 162 (2002), 101-135.  doi: 10.1007/s002050100187. [9] A. Ferriero and N. Fusco, A note on the convex hull of sets of finite perimeter in the plane, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 102-108.  doi: 10.3934/dcdsb.2009.11.103. [10] I. Fonseca and G. A. Francfort, Relaxation in BV versus quasiconvexification in ${W^{1, p}}$ ; A model for the interaction between fracture and damage, Calc. Var. Partial Differential Equations, 3 (1995), 407-446.  doi: 10.1007/BF01187895. [11] G. A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids, 46 (1998), 1319-1342.  doi: 10.1016/S0022-5096(98)00034-9. [12] M. Fr{é}mond, Contact with adhesion, in Topics in Nonsmooth Mechanics, Birkh{ä}user, Basel, 1988,157–185. [13] M. Fr{é}mond, Non-Smooth Thermomechanics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9. [14] B. Kawohl, On starshaped rearrangement and applications, Trans. Amer. Math. Soc., 296 (1986), 377-386.  doi: 10.1090/S0002-9947-1986-0837818-4. [15] M. Ko{\v c}vara, A. Mielke and T. Roub{í}{\v c}ek, A rate-independent approach to the delamination problem, Math. Mech. Solids, 11 (2006), 423-447.  doi: 10.1177/1081286505046482. [16] P. Krej{\v c}{\'\i} and M. Liero, Rate independent {K}urzweil processes, Appl. Math., 54 (2009), 117-145.  doi: 10.1007/s10492-009-0009-5. [17] S. Luckhaus and T. Sturzenhecker, Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differential Equations, 3 (1995), 253-271.  doi: 10.1007/BF01205007. [18] 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. [19] A. Mielke, T. Roub{í}{\v c}ek and U. Stefanelli, {$\Gamma$}-limits and relaxations for rate-independent evolutionary problems, Calc. Var. Partial Differential Equations, 31 (2008), 387-416.  doi: 10.1007/s00526-007-0119-4. [20] A. Mielke and T. Roub{\'\i}{\v c}ek, Rate-Independent Systems. Theory and Application, Applied Mathematical Sciences, 193, Springer, New York, 2015. doi: 10.1007/978-1-4939-2706-7. [21] A. Mielke and F. Theil, A mathematical model for rate-independent phase transformations with hysteresis, Proceedings of the Workshop on `Models of Continuum Mechanics in Analysis and Engineering'', Shaker-Verlag, 1999,117–129. [22] 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. [23] R. Rossi and M. Thomas, From an adhesive to a brittle delamination model in thermo-visco-elasticity, ESAIM Control Optim. Calc. Var., 21 (2015), 1-59.  doi: 10.1051/cocv/2014015. [24] T. Roub{í}{\v c}ek, L. Scardia and C. Zanini, Quasistatic delamination problem, Contin. Mech. Thermodyn., 21 (2009), 223-235.  doi: 10.1007/s00161-009-0106-4. [25] T. Roub{\'\i}{\v c}ek, M. Thomas and C. G. Panagiotopoulos, Stress-driven local-solution approach to quasistatic brittle delamination, Nonlinear Anal. Real World Appl., 22 (2015), 645-663.  doi: 10.1016/j.nonrwa.2014.09.011. [26] M. Thomas, Quasistatic damage evolution with spatial BV-regularization, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 235-255.  doi: 10.3934/dcdss.2013.6.235. [27] M. Thomas, Uniform {P}oincaré-{S}obolev and isoperimetric inequalities for classes of domains, Discrete Contin. Dyn. Syst., 35 (2015), 2741-2761.  doi: 10.3934/dcds.2015.35.2741. [28] A. Visintin, Motion by mean curvature and nucleation, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 55-60.  doi: 10.1016/S0764-4442(97)83933-X. [29] A. Visintin, Nucleation and mean curvature flow, Comm. Partial Differential Equations, 23 (1998), 17-53.  doi: 10.1080/03605309808821337.
An example for nonconnectedness: A needle-like forcing F.
Solutions of the minimization problem (54) with forcing $F^c$ being an equilateral triangle, a square, and a regular hexagon, respectively
Partial $C^1$ regularity. The two solutions correspond to $v(x) = 3/4 - |x{-}1/2|$ (left) and $v(x) = 1/4 + |x{-}1/2|$ (right)
The C1 competitor profile
Convex forcing $F^{c}$. The two solutions correspond to $v(x) = 3/4 - \beta(x{-}1/2)^2$ for $\beta = 2$ (left) and $\beta = 1/5$ (right). The minimal set $Z$ is convex
The evolution from (60) for $M = 5$ and time $t = 2$.
An evolution of connected sets fulfilling the compatibility condition (50), time flows from left to right
The effect of changing the parameter $a$. The two solutions correspond to $v(x) = (x{-}1/2)^2+1/2$ for $a = 7$ (left) and $a = 3$ (right). The top adhesion zone is smaller for smaller $a$. Note that the parts of the boundary of $Z$ which are not in contact with $F^c$ are arcs of circles with radius $1/a$ (recall that $a$ is different in the two figures), as predicted in Subsection 4.1
Extreme configurations. The solution for $v(x) = \max\{1-5|x{-}1/2|,1/2\}$ (left) and $v(x) = \lfloor 5x\rfloor/5+1/5$ (right)
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