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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[5]

G. Dal MasoG. 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.  Google Scholar

[6]

A. DeSimoneP. 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.  Google Scholar

[7]

P. Gidoni and A. DeSimone, On the genesis of directional friction through bristle-like mediating elements crawler, arXiv: 1602.05611. Google Scholar

[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.  Google Scholar

[9]

P. GidoniG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

P. Krejčí and M. Liero, Rate independent Kurzweil processes, Appl. Math., 54 (2009), 117-145.  doi: 10.1007/s10492-009-0009-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

A. MielkeT. 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.  Google Scholar

[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.  Google Scholar

[20]

F. Pfeiffer, Mechanische Systeme mit unstetigen Übergängen, Ingenieur-Archiv, 54 (1984), 232-240, (In German).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[24]

A. Visintin, Differential Models of Hysteresis, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-662-11557-2.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[5]

G. Dal MasoG. 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.  Google Scholar

[6]

A. DeSimoneP. 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.  Google Scholar

[7]

P. Gidoni and A. DeSimone, On the genesis of directional friction through bristle-like mediating elements crawler, arXiv: 1602.05611. Google Scholar

[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.  Google Scholar

[9]

P. GidoniG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

P. Krejčí and M. Liero, Rate independent Kurzweil processes, Appl. Math., 54 (2009), 117-145.  doi: 10.1007/s10492-009-0009-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

A. MielkeT. 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.  Google Scholar

[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.  Google Scholar

[20]

F. Pfeiffer, Mechanische Systeme mit unstetigen Übergängen, Ingenieur-Archiv, 54 (1984), 232-240, (In German).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[24]

A. Visintin, Differential Models of Hysteresis, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-662-11557-2.  Google Scholar

Figure 1.1.  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
Figure 1.2.  (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
Figure 2.1.  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
Figure 2.2.  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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