# American Institute of Mathematical Sciences

December  2017, 10(6): 1303-1327. doi: 10.3934/dcdss.2017070

## Averaging of time-periodic dissipation potentials in rate-independent processes

 1 Weierstraẞ-Institut für Angewandte Analysis und Stochastik, Mohrenstr. 39,10117 Berlin, Germany 2 Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25,12489 Berlin, Germany

Dedicated to Professor T. Roubíčcek on the occasion of his 60th birthday.

Received  November 2016 Revised  January 2017 Published  June 2017

Fund Project: M.H. was financed by Deutsche Forschungsgemeinschaft (DFG) through Grant CRC 1114 Scaling Cascades in Complex Systems, Project C05 Effective models for interfaces with many scales.
A.M. was partially supported by ERC through AdG 267802 AnaMultiScale.

We study the existence and well-posedness of rate-independent systems (or hysteresis operators) with a dissipation potential that oscillates in time with period $\varepsilon$. In particular, for the case of quadratic energies in a Hilbert space, we study the averaging limit $\varepsilon \to 0$ and show that the effective dissipation potential is given by the minimum of all friction thresholds in one period, more precisely as the intersection of all the characteristic domains. We show that the rates of the process do not converge weakly, hence our analysis uses the notion of energetic solutions and relies on a detailed estimates to obtain a suitable equi-continuity of the solutions in the limit $\varepsilon \to 0$.

Citation: Martin Heida, Alexander Mielke. Averaging of time-periodic dissipation potentials in rate-independent processes. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1303-1327. doi: 10.3934/dcdss.2017070
##### References:
 [1] M. Al Janaideh, A time-dependent stop operator for modeling a class of singular hysteresis loops in a piezoceramic actuator, Physica B, 413 (2013), 100-104.  doi: 10.1016/j.physb.2012.12.021. [2] M. Al Janaideh and P. Krejčí, An inversion formula for a Prandtl-Ishlinskii operator with time dependent thresholds, Physica B, 406 (2011), 1528-1532.  doi: 10.1016/j.physb.2011.01.062. [3] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows: In Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005. [4] M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4048-8. [5] G. Dal Maso, G. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Rational Mech. Anal., 176 (2005), 165-225.  doi: 10.1007/s00205-004-0351-4. [6] A. DeSimone, P. Gidoni and G. Noselli, Liquid crystal elastomer strips as soft crawlers, J. Mech. Physics Solids, 84 (2015), 254-272.  doi: 10.1016/j.jmps.2015.07.017. [7] P. Gidoni and A. DeSimone, On the genesis of directional friction through bristle-like mediating elements crawler, arXiv: 1602.05611. [8] P. Gidoni and A. DeSimone, Stasis domains and slip surfaces in the locomotion of a bio-inspired two-segment crawler, Meccanica, 52 (2017), 587-601.  doi: 10.1007/s11012-016-0408-0. [9] P. Gidoni, G. Noselli and A. DeSimone, Crawling on directional surfaces, Int. J. Non-Linear Mech., 61 (2014), 65-73.  doi: 10.1016/j.ijnonlinmec.2014.01.012. [10] J. Kopfová and V. Recupero, Bv-norm continuity of sweeping processes driven by a set with constant shape, Journal of Differential Equations, 261 (2016), 5875-5899, arXiv: 1512.08711. doi: 10.1016/j.jde.2016.08.025. [11] P. Krejčí, Evolution variational inequalities and multidimensional hysteresis operators, in Nonlinear differential equations (eds. P. Drábek, P. Krejčí and P. Takáč), Chapman & Hall/CRC, Boca Raton, FL, 404 (1999), 47-110. [12] P. Krejčí and M. Liero, Rate independent Kurzweil processes, Appl. Math., 54 (2009), 117-145.  doi: 10.1007/s10492-009-0009-5. [13] P. Krejči and T. Roche, Lipschitz continuous data dependence of sweeping processes in BV spaces, Discr. Cont. Dynam. Systems, Sec. B, 15 (2011), 637-650.  doi: 10.3934/dcdsb.2011.15.637. [14] M. Kunze and M. D. P. Monteiro Marques, On parabolic quasi-variational inequalities and state-dependent sweeping processes, Topol. Methods Nonlinear Anal., 12 (1998), 179-191.  doi: 10.12775/TMNA.1998.036. [15] A. Mielke, Evolution in rate-independent systems (Ch. 6), in Handbook of Differential Equations, Evolutionary Equations, vol. 2 (eds. C. Dafermos and E. Feireisl), Elsevier B. V. , Amsterdam, 2 (2005), 461-559. [16] A. Mielke and F. Theil, On rate-independent hysteresis models, Nonl. Diff. Eqns. Appl. (NoDEA), 11 (2004), 151-189, (Accepted July 2001).  doi: 10.1007/s00030-003-1052-7. [17] A. Mielke and T. Roubíček, Rate-Independent Systems: Theory and Application, Applied Mathematical Sciences, Vol. 193, Springer-Verlag New York, 2015. doi: 10.1007/978-1-4939-2706-7. [18] A. Mielke, T. Roubíček and U. Stefanelli, Γ-limits and relaxations for rate-independent evolutionary problems, Calc. Var. Part. Diff. Eqns., 31 (2008), 387-416.  doi: 10.1007/s00526-007-0119-4. [19] J.-J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Equations, 26 (1977), 347-374.  doi: 10.1016/0022-0396(77)90085-7. [20] F. Pfeiffer, Mechanische Systeme mit unstetigen Übergängen, Ingenieur-Archiv, 54 (1984), 232-240, (In German). [21] M. Radtke and R. R. Netz, Shear-induced dynamics of polymeric globules at adsorbing homogeneous and inhomogeneous surfaces The European Physical Journal E, 37 (2014), p20. doi: 10.1140/epje/i2014-14020-7. [22] V. Recupero, A continuity method for sweeping processes, Journal of Differential Equations, 251 (2011), 2125-2142.  doi: 10.1016/j.jde.2011.06.018. [23] T. Roche, Uniqueness of a quasivariational sweeping process on functions of bounded variation, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze-Serie V, 11 (2012), 363-394. [24] A. Visintin, Differential Models of Hysteresis, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-662-11557-2.

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Dedicated to Professor T. Roubíčcek on the occasion of his 60th birthday.

##### References:
 [1] M. Al Janaideh, A time-dependent stop operator for modeling a class of singular hysteresis loops in a piezoceramic actuator, Physica B, 413 (2013), 100-104.  doi: 10.1016/j.physb.2012.12.021. [2] M. Al Janaideh and P. Krejčí, An inversion formula for a Prandtl-Ishlinskii operator with time dependent thresholds, Physica B, 406 (2011), 1528-1532.  doi: 10.1016/j.physb.2011.01.062. [3] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows: In Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005. [4] M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4048-8. [5] G. Dal Maso, G. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Rational Mech. Anal., 176 (2005), 165-225.  doi: 10.1007/s00205-004-0351-4. [6] A. DeSimone, P. Gidoni and G. Noselli, Liquid crystal elastomer strips as soft crawlers, J. Mech. Physics Solids, 84 (2015), 254-272.  doi: 10.1016/j.jmps.2015.07.017. [7] P. Gidoni and A. DeSimone, On the genesis of directional friction through bristle-like mediating elements crawler, arXiv: 1602.05611. [8] P. Gidoni and A. DeSimone, Stasis domains and slip surfaces in the locomotion of a bio-inspired two-segment crawler, Meccanica, 52 (2017), 587-601.  doi: 10.1007/s11012-016-0408-0. [9] P. Gidoni, G. Noselli and A. DeSimone, Crawling on directional surfaces, Int. J. Non-Linear Mech., 61 (2014), 65-73.  doi: 10.1016/j.ijnonlinmec.2014.01.012. [10] J. Kopfová and V. Recupero, Bv-norm continuity of sweeping processes driven by a set with constant shape, Journal of Differential Equations, 261 (2016), 5875-5899, arXiv: 1512.08711. doi: 10.1016/j.jde.2016.08.025. [11] P. Krejčí, Evolution variational inequalities and multidimensional hysteresis operators, in Nonlinear differential equations (eds. P. Drábek, P. Krejčí and P. Takáč), Chapman & Hall/CRC, Boca Raton, FL, 404 (1999), 47-110. [12] P. Krejčí and M. Liero, Rate independent Kurzweil processes, Appl. Math., 54 (2009), 117-145.  doi: 10.1007/s10492-009-0009-5. [13] P. Krejči and T. Roche, Lipschitz continuous data dependence of sweeping processes in BV spaces, Discr. Cont. Dynam. Systems, Sec. B, 15 (2011), 637-650.  doi: 10.3934/dcdsb.2011.15.637. [14] M. Kunze and M. D. P. Monteiro Marques, On parabolic quasi-variational inequalities and state-dependent sweeping processes, Topol. Methods Nonlinear Anal., 12 (1998), 179-191.  doi: 10.12775/TMNA.1998.036. [15] A. Mielke, Evolution in rate-independent systems (Ch. 6), in Handbook of Differential Equations, Evolutionary Equations, vol. 2 (eds. C. Dafermos and E. Feireisl), Elsevier B. V. , Amsterdam, 2 (2005), 461-559. [16] A. Mielke and F. Theil, On rate-independent hysteresis models, Nonl. Diff. Eqns. Appl. (NoDEA), 11 (2004), 151-189, (Accepted July 2001).  doi: 10.1007/s00030-003-1052-7. [17] A. Mielke and T. Roubíček, Rate-Independent Systems: Theory and Application, Applied Mathematical Sciences, Vol. 193, Springer-Verlag New York, 2015. doi: 10.1007/978-1-4939-2706-7. [18] A. Mielke, T. Roubíček and U. Stefanelli, Γ-limits and relaxations for rate-independent evolutionary problems, Calc. Var. Part. Diff. Eqns., 31 (2008), 387-416.  doi: 10.1007/s00526-007-0119-4. [19] J.-J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Equations, 26 (1977), 347-374.  doi: 10.1016/0022-0396(77)90085-7. [20] F. Pfeiffer, Mechanische Systeme mit unstetigen Übergängen, Ingenieur-Archiv, 54 (1984), 232-240, (In German). [21] M. Radtke and R. R. Netz, Shear-induced dynamics of polymeric globules at adsorbing homogeneous and inhomogeneous surfaces The European Physical Journal E, 37 (2014), p20. doi: 10.1140/epje/i2014-14020-7. [22] V. Recupero, A continuity method for sweeping processes, Journal of Differential Equations, 251 (2011), 2125-2142.  doi: 10.1016/j.jde.2011.06.018. [23] T. Roche, Uniqueness of a quasivariational sweeping process on functions of bounded variation, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze-Serie V, 11 (2012), 363-394. [24] A. Visintin, Differential Models of Hysteresis, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-662-11557-2.
Because of the in-built unbalance, the plate compactor vibrates vertically leading to an oscillatory normal pressure. When pushing the plate compactor horizontally it will move only when the normal pressure is very low
(A) In rest, the woodpecker sticks to the metal rod by dry friction, when oscillating the reduction in friction allows for a slow sliding downwards, cf. [20]. (B) Toy ramp walker: the frog walks down only, when alternating the weight between the rigid downhill leg and the hinged uphill leg. (C) Rocking animal: A weight beyond the table edge pulls the cow forward, while the perpendicular rocking motions allows the lifted legs to swing forward because of the reduced normal pressure
The bold, red curve is the solution of $0\in \rho(t/\varepsilon)\mathrm{Sign}(\dot y(t)) + y(t) -5t +t^2$ with $y(0)=0$ for $\varepsilon =0.04$. The shaded, wavy area indicates the stable regions
Plots for the solution of (2.2). (A) The positions $y_j(t)$ of the two legs move by alternating between plateaus (sticking phase) and fast motion. (B) The derivatives $\dot y_j(t)$ show that the motion is alternating, i.e. at most one of the legs moves at a time. (C) The path $t\mapsto y(t)=(y_1(t),y_2(t)) \in \mathbb{R}^2$ shows a microscopic zigzag pattern.
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