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Optimal control of a rate-independent evolution equation via viscous regularization
Cohesive zone-type delamination in visco-elasticity
1. | Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39,10117 Berlin, Germany |
2. | Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy |
We study a model for the rate-independent evolution of cohesive zone delamination in a visco-elastic solid, also exposed to dynamics effects. The main feature of this model, inspired by [
Due to the presence of multivalued and unbounded operators featuring non-penetration and the 'memory'-constraint in the strong formulation of the problem, we prove existence of a weaker notion of solution, known as semistable energetic solution, pioneered in [
References:
[1] |
L. Adam, J. Outrata and T. Roubíček,
Identification of some nonsmooth evolution systems with illustration on adhesive contacts at small strains, A Journal of Mathematical Programming and Operations Research Latest Articles, (2015), 1-25.
doi: 10.1080/02331934.2015.1111364. |
[2] |
R. Alessi, J.-J. Marigo and S. Vidoli,
Gradient damage models coupled with plasticity and nucleation of cohesive cracks, Arch. Ration. Mech. Anal., 214 (2014), 575-615.
doi: 10.1007/s00205-014-0763-8. |
[3] |
S. Almi, Energy release rate and quasistatic evolution via vanishing viscosity in a cohesive fracture model with an activation threshold,
ESAIM: Control Optim. Calc. Var. Published online. |
[4] |
M. Artina, F. Cagnetti, M. Fornasier and F. Solombrino,
Linearly constrained evolutions of critical points and an application to cohesive fractures, Math. Models Methods Appl. Sci., 27 (2017), 231-290.
doi: 10.1142/S0218202517500014. |
[5] |
H. Attouch,
Variational Convergence for Functions and Operators Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984. |
[6] |
G. Barenblatt,
The mathematical theory of equilibrium of cracks in brittle fracture, Adv. Appl. Mech., 7 (1962), 55-129.
|
[7] |
E. Bonetti, G. Bonfanti and R. Rossi,
Global existence for a contact problem with adhesion, Math. Meth. Appl. Sci., 31 (2008), 1029-1064.
doi: 10.1002/mma.957. |
[8] |
E. Bonetti, G. Bonfanti and R. Rossi,
Thermal effects in adhesive contact: Modelling and analysis, Nonlinearity, 22 (2009), 2697-2731.
doi: 10.1088/0951-7715/22/11/007. |
[9] |
E. Bonetti, E. Rocca, R. Scala and G. Schimperna, On the strongly damped wave equation with constraint,
WIAS-Preprint 2094. |
[10] |
G. Bouchitté, A. Braides and G. Buttazzo,
Relaxation results for some free discontinuity problems, J. Reine Angew. Math., 458 (1995), 1-18.
doi: 10.1515/crll.1995.458.1. |
[11] |
H. Brézis,
Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert North-Holland Publishing Co. , Amsterdam-London; American Elsevier Publishing Co. , Inc. , New York, 1973, North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). |
[12] |
F. Cagnetti,
A vanishing viscosity approach to fracture growth in a cohesive zone model with prescribed crack path, Math. Models Methods Appl. Sci., 18 (2008), 1027-1071.
doi: 10.1142/S0218202508002942. |
[13] |
F. Cagnetti and R. Toader,
Quasistatic crack evolution for a cohesive zone model with different response to loading and unloading: A Young measures approach, ESAIM Control Optim. Calc. Var., 17 (2011), 1-27.
doi: 10.1051/cocv/2009037. |
[14] |
V. Crismale, G. Lazzaroni and G. Orlando, Cohesive fracture with irreversibility: quasistatic evolution for a model subject to fatigue,
SISSA-Preprint SISSA 40/2016/MATE 1508. 02965. |
[15] |
G. Dal Maso, G. Francfort and R. Toader,
Quasistatic crack growth in nonlinear elasticity, Arch. Rat. Mech. Anal., 176 (2005), 165-225.
doi: 10.1007/s00205-004-0351-4. |
[16] |
G. Dal Maso, G. Orlando and R. Toader,
Lower semicontinuity of a class of integral functionals on the space of functions of bounded deformation, Adv. Cal. Var., 10 (2017), 183-207.
doi: 10.1515/acv-2015-0036. |
[17] |
G. Dal Maso, G. Orlando and R. Toader, Fracture models for elasto-plastic materials as limits of gradient damage models coupled with plasticity: The anticline case Calc. Var. Partial Differential Equations 55 (2016), 39pp.
doi: 10.1007/s00526-016-0981-z. |
[18] |
G. Dal Maso and C. Zanini,
Quasi-static crack growth for a cohesive zone model with prescribed crack path, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 253-279.
doi: 10.1017/S030821050500079X. |
[19] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhikers guide to the fractional sobolev spaces, Bulletin des Sciences Mathématiques, 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[20] |
D. Dugdale,
Yielding of steel sheets containing clits, J. Mech. Phys. Solids, 8 (1960), 100-104.
|
[21] |
M. Fr´emond, Contact with adhesion, in Topics in Nonsmooth Mechanics (eds. J. Moreau,
P. Panagiotopoulos and G. Strang), Birkhäuser, 1988,157–186. |
[22] |
M. Frémond,
Non-Smooth Thermomechanics Springer-Verlag Berlin Heidelberg, 2002. |
[23] |
A. D. Ioffe and V. M. Tihomirov,
Theory of Extremal Problems vol. 6 of Studies in Mathematics and its Applications, North-Holland Publishing Co. , Amsterdam, 1979, Translated from the Russian by Karol Makowski. |
[24] |
A. Ioffe,
On lower semicontinuity of integral functionals. I, SIAM J. Control Optimization, 15 (1977), 521-538.
doi: 10.1137/0315035. |
[25] |
M. Kočvara, A. Mielke and T. Roubíček,
A rate-independent approach to the delamination problem, Math. Mech. Solids, 11 (2006), 423-447.
doi: 10.1177/1081286505046482. |
[26] |
R. Kregting,
Cohesive Zone Models Towards a Robust Implementation of Irreversible Behavior Technical Report MT05. 11, TU Eindhoven, Materials Technology, 2005. |
[27] |
M. Kružík, C. Panagiotopoulos and T. Roubíček,
Quasistatic adhesive contact delaminating in mixed mode and its numerical treatment, Math. Mech. Solids, 20 (2015), 582-599.
doi: 10.1177/1081286513507942. |
[28] |
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. |
[29] |
M. Marcus and V. Mizel,
Every superposition operator mapping one Sobolev space into another is continuous, J. Functional Analysis, 33 (1979), 217-229.
doi: 10.1016/0022-1236(79)90113-7. |
[30] |
A. Mielke and T. Roubíček,
Rate-independent Systems: Theory and Application vol. 193 of Applied Mathematical Sciences, Springer, 2015.
doi: 10.1007/978-1-4939-2706-7. |
[31] |
A. Mielke, T. Roubíček and M. Thomas,
From damage to delamination in nonlinearly elastic materials at small strains, J. Elasticity, 109 (2012), 235-273.
doi: 10.1007/s10659-012-9379-0. |
[32] |
M. Ortiz and A. Pandolfi,
Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis, International Journal for Numerical Methods in Engineering, 44 (1999), 1267-1282.
|
[33] |
C. Panagiotopoulos, V. Mantič and T. Roubíček, Two adhesive-contact models for quasistatic mixed-mode delamination problems Mathematics and Computers in Simulation 2016.
doi: 10.1016/j.matcom.2016.10.004. |
[34] |
K. Park and G. Paulino, Cohesive zone models: A critical review of traction-separation relationships across fracture surfaces Applied Mechanics Reviews 64 (2013), 060802.
doi: 10.1115/1.4023110. |
[35] |
J. Rice, Fracture, chapter Mathematical analysis in the mechanics of fracture, , (1968): 191-311.
|
[36] |
R. Rossi and T. Roubíček,
Thermodynamics and analysis of rate-independent adhesive contact at small strains, Nonlinear Anal., 74 (2011), 3159-3190.
doi: 10.1016/j.na.2011.01.031. |
[37] |
R. Rossi and T. Roubíček,
Adhesive contact delaminating at mixed mode, its thermodynamics and analysis, Interfaces Free Bound., 15 (2013), 1-37.
doi: 10.4171/IFB/293. |
[38] |
R. Rossi and M. Thomas,
Coupling rate-independent and rate-dependent processes: Existence results, SIAM J. Math. Anal., 49 (2017), 1419-1494.
doi: 10.1137/15M1051567. |
[39] |
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. |
[40] |
R. Rossi and M. Thomas, From adhesive to brittle delamination in visco-elastodynamics Math. Models Methods Appl. Sci. , 2017.
doi: 10.1142/S0218202517500257. |
[41] |
T. Roubíček,
Rate-independent processes in viscous solids at small strains, Math. Methods Appl. Sci., 32 (2009), 825-862.
doi: 10.1002/mma.1069. |
[42] |
T. Roubíček,
Thermodynamics of rate-independent processes in viscous solids at small strains, SIAM J. Math. Anal., 42 (2010), 256-297.
doi: 10.1137/080729992. |
[43] |
T. Roubíček, M. Kružík and J. Zeman, Delamination and adhesive contact models and their
mathematical analysis and numerical treatment, in Mathematical methods and models in
composites, vol. 5 of Comput. Exp. Methods Struct. , Imp. Coll. Press, London, 2014,349–
400.
doi: 10.1142/9781848167858_0009. |
[44] |
T. Roubíček, V. Mantič and C. Panagiotopoulos,
A quasistatic mixed-mode delamination model, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 591-610.
|
[45] |
T. Roubíček, C. Panagiotopoulos and V. Mantič,
Quasistatic adhesive contact of visco-elastic bodies and its numerical treatment for very small viscosity, ZAMM Z. Angew. Math. Mech., 93 (2013), 823-840.
doi: 10.1002/zamm.201200239. |
[46] |
T. Roubíček, L. Scardia and C. Zanini,
Quasistatic delamination problem, Continuum Mech. Thermodynam., 21 (2009), 223-235.
doi: 10.1007/s00161-009-0106-4. |
[47] |
T. Roubíček, M. Thomas and C. 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. |
[48] |
R. Scala, A weak formulation for a rate-independent delamination evolution with inertial and viscosity effects subjected to unilateral constraint,
WIAS-Preprint 2172. |
[49] |
R. Scala and G. Schimperna,
A contact problem for viscoelastic bodies with inertial effects and unilateral boundary constraints, European J. Appl. Math., 28 (2017), 91-122.
doi: 10.1017/S0956792516000097. |
[50] |
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. |
[51] |
M. Thomas and A. Mielke,
Damage of nonlinearly elastic materials at small strain: Existence and regularity results, Zeit. angew. Math. Mech., 90 (2010), 88-112.
doi: 10.1002/zamm.200900243. |
[52] |
R. Vodička, V. Mantič and T. Roubíček,
Energetic versus maximally-dissipative local solutions of a quasi-static rate-independent mixed-mode delamination model, Meccanica, 49 (2014), 2933-2963.
doi: 10.1007/s11012-014-0045-4. |
show all references
To the occasion of the 60th anniversary of Tomáš Roubíček
References:
[1] |
L. Adam, J. Outrata and T. Roubíček,
Identification of some nonsmooth evolution systems with illustration on adhesive contacts at small strains, A Journal of Mathematical Programming and Operations Research Latest Articles, (2015), 1-25.
doi: 10.1080/02331934.2015.1111364. |
[2] |
R. Alessi, J.-J. Marigo and S. Vidoli,
Gradient damage models coupled with plasticity and nucleation of cohesive cracks, Arch. Ration. Mech. Anal., 214 (2014), 575-615.
doi: 10.1007/s00205-014-0763-8. |
[3] |
S. Almi, Energy release rate and quasistatic evolution via vanishing viscosity in a cohesive fracture model with an activation threshold,
ESAIM: Control Optim. Calc. Var. Published online. |
[4] |
M. Artina, F. Cagnetti, M. Fornasier and F. Solombrino,
Linearly constrained evolutions of critical points and an application to cohesive fractures, Math. Models Methods Appl. Sci., 27 (2017), 231-290.
doi: 10.1142/S0218202517500014. |
[5] |
H. Attouch,
Variational Convergence for Functions and Operators Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984. |
[6] |
G. Barenblatt,
The mathematical theory of equilibrium of cracks in brittle fracture, Adv. Appl. Mech., 7 (1962), 55-129.
|
[7] |
E. Bonetti, G. Bonfanti and R. Rossi,
Global existence for a contact problem with adhesion, Math. Meth. Appl. Sci., 31 (2008), 1029-1064.
doi: 10.1002/mma.957. |
[8] |
E. Bonetti, G. Bonfanti and R. Rossi,
Thermal effects in adhesive contact: Modelling and analysis, Nonlinearity, 22 (2009), 2697-2731.
doi: 10.1088/0951-7715/22/11/007. |
[9] |
E. Bonetti, E. Rocca, R. Scala and G. Schimperna, On the strongly damped wave equation with constraint,
WIAS-Preprint 2094. |
[10] |
G. Bouchitté, A. Braides and G. Buttazzo,
Relaxation results for some free discontinuity problems, J. Reine Angew. Math., 458 (1995), 1-18.
doi: 10.1515/crll.1995.458.1. |
[11] |
H. Brézis,
Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert North-Holland Publishing Co. , Amsterdam-London; American Elsevier Publishing Co. , Inc. , New York, 1973, North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). |
[12] |
F. Cagnetti,
A vanishing viscosity approach to fracture growth in a cohesive zone model with prescribed crack path, Math. Models Methods Appl. Sci., 18 (2008), 1027-1071.
doi: 10.1142/S0218202508002942. |
[13] |
F. Cagnetti and R. Toader,
Quasistatic crack evolution for a cohesive zone model with different response to loading and unloading: A Young measures approach, ESAIM Control Optim. Calc. Var., 17 (2011), 1-27.
doi: 10.1051/cocv/2009037. |
[14] |
V. Crismale, G. Lazzaroni and G. Orlando, Cohesive fracture with irreversibility: quasistatic evolution for a model subject to fatigue,
SISSA-Preprint SISSA 40/2016/MATE 1508. 02965. |
[15] |
G. Dal Maso, G. Francfort and R. Toader,
Quasistatic crack growth in nonlinear elasticity, Arch. Rat. Mech. Anal., 176 (2005), 165-225.
doi: 10.1007/s00205-004-0351-4. |
[16] |
G. Dal Maso, G. Orlando and R. Toader,
Lower semicontinuity of a class of integral functionals on the space of functions of bounded deformation, Adv. Cal. Var., 10 (2017), 183-207.
doi: 10.1515/acv-2015-0036. |
[17] |
G. Dal Maso, G. Orlando and R. Toader, Fracture models for elasto-plastic materials as limits of gradient damage models coupled with plasticity: The anticline case Calc. Var. Partial Differential Equations 55 (2016), 39pp.
doi: 10.1007/s00526-016-0981-z. |
[18] |
G. Dal Maso and C. Zanini,
Quasi-static crack growth for a cohesive zone model with prescribed crack path, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 253-279.
doi: 10.1017/S030821050500079X. |
[19] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhikers guide to the fractional sobolev spaces, Bulletin des Sciences Mathématiques, 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[20] |
D. Dugdale,
Yielding of steel sheets containing clits, J. Mech. Phys. Solids, 8 (1960), 100-104.
|
[21] |
M. Fr´emond, Contact with adhesion, in Topics in Nonsmooth Mechanics (eds. J. Moreau,
P. Panagiotopoulos and G. Strang), Birkhäuser, 1988,157–186. |
[22] |
M. Frémond,
Non-Smooth Thermomechanics Springer-Verlag Berlin Heidelberg, 2002. |
[23] |
A. D. Ioffe and V. M. Tihomirov,
Theory of Extremal Problems vol. 6 of Studies in Mathematics and its Applications, North-Holland Publishing Co. , Amsterdam, 1979, Translated from the Russian by Karol Makowski. |
[24] |
A. Ioffe,
On lower semicontinuity of integral functionals. I, SIAM J. Control Optimization, 15 (1977), 521-538.
doi: 10.1137/0315035. |
[25] |
M. Kočvara, A. Mielke and T. Roubíček,
A rate-independent approach to the delamination problem, Math. Mech. Solids, 11 (2006), 423-447.
doi: 10.1177/1081286505046482. |
[26] |
R. Kregting,
Cohesive Zone Models Towards a Robust Implementation of Irreversible Behavior Technical Report MT05. 11, TU Eindhoven, Materials Technology, 2005. |
[27] |
M. Kružík, C. Panagiotopoulos and T. Roubíček,
Quasistatic adhesive contact delaminating in mixed mode and its numerical treatment, Math. Mech. Solids, 20 (2015), 582-599.
doi: 10.1177/1081286513507942. |
[28] |
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. |
[29] |
M. Marcus and V. Mizel,
Every superposition operator mapping one Sobolev space into another is continuous, J. Functional Analysis, 33 (1979), 217-229.
doi: 10.1016/0022-1236(79)90113-7. |
[30] |
A. Mielke and T. Roubíček,
Rate-independent Systems: Theory and Application vol. 193 of Applied Mathematical Sciences, Springer, 2015.
doi: 10.1007/978-1-4939-2706-7. |
[31] |
A. Mielke, T. Roubíček and M. Thomas,
From damage to delamination in nonlinearly elastic materials at small strains, J. Elasticity, 109 (2012), 235-273.
doi: 10.1007/s10659-012-9379-0. |
[32] |
M. Ortiz and A. Pandolfi,
Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis, International Journal for Numerical Methods in Engineering, 44 (1999), 1267-1282.
|
[33] |
C. Panagiotopoulos, V. Mantič and T. Roubíček, Two adhesive-contact models for quasistatic mixed-mode delamination problems Mathematics and Computers in Simulation 2016.
doi: 10.1016/j.matcom.2016.10.004. |
[34] |
K. Park and G. Paulino, Cohesive zone models: A critical review of traction-separation relationships across fracture surfaces Applied Mechanics Reviews 64 (2013), 060802.
doi: 10.1115/1.4023110. |
[35] |
J. Rice, Fracture, chapter Mathematical analysis in the mechanics of fracture, , (1968): 191-311.
|
[36] |
R. Rossi and T. Roubíček,
Thermodynamics and analysis of rate-independent adhesive contact at small strains, Nonlinear Anal., 74 (2011), 3159-3190.
doi: 10.1016/j.na.2011.01.031. |
[37] |
R. Rossi and T. Roubíček,
Adhesive contact delaminating at mixed mode, its thermodynamics and analysis, Interfaces Free Bound., 15 (2013), 1-37.
doi: 10.4171/IFB/293. |
[38] |
R. Rossi and M. Thomas,
Coupling rate-independent and rate-dependent processes: Existence results, SIAM J. Math. Anal., 49 (2017), 1419-1494.
doi: 10.1137/15M1051567. |
[39] |
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. |
[40] |
R. Rossi and M. Thomas, From adhesive to brittle delamination in visco-elastodynamics Math. Models Methods Appl. Sci. , 2017.
doi: 10.1142/S0218202517500257. |
[41] |
T. Roubíček,
Rate-independent processes in viscous solids at small strains, Math. Methods Appl. Sci., 32 (2009), 825-862.
doi: 10.1002/mma.1069. |
[42] |
T. Roubíček,
Thermodynamics of rate-independent processes in viscous solids at small strains, SIAM J. Math. Anal., 42 (2010), 256-297.
doi: 10.1137/080729992. |
[43] |
T. Roubíček, M. Kružík and J. Zeman, Delamination and adhesive contact models and their
mathematical analysis and numerical treatment, in Mathematical methods and models in
composites, vol. 5 of Comput. Exp. Methods Struct. , Imp. Coll. Press, London, 2014,349–
400.
doi: 10.1142/9781848167858_0009. |
[44] |
T. Roubíček, V. Mantič and C. Panagiotopoulos,
A quasistatic mixed-mode delamination model, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 591-610.
|
[45] |
T. Roubíček, C. Panagiotopoulos and V. Mantič,
Quasistatic adhesive contact of visco-elastic bodies and its numerical treatment for very small viscosity, ZAMM Z. Angew. Math. Mech., 93 (2013), 823-840.
doi: 10.1002/zamm.201200239. |
[46] |
T. Roubíček, L. Scardia and C. Zanini,
Quasistatic delamination problem, Continuum Mech. Thermodynam., 21 (2009), 223-235.
doi: 10.1007/s00161-009-0106-4. |
[47] |
T. Roubíček, M. Thomas and C. 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. |
[48] |
R. Scala, A weak formulation for a rate-independent delamination evolution with inertial and viscosity effects subjected to unilateral constraint,
WIAS-Preprint 2172. |
[49] |
R. Scala and G. Schimperna,
A contact problem for viscoelastic bodies with inertial effects and unilateral boundary constraints, European J. Appl. Math., 28 (2017), 91-122.
doi: 10.1017/S0956792516000097. |
[50] |
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. |
[51] |
M. Thomas and A. Mielke,
Damage of nonlinearly elastic materials at small strain: Existence and regularity results, Zeit. angew. Math. Mech., 90 (2010), 88-112.
doi: 10.1002/zamm.200900243. |
[52] |
R. Vodička, V. Mantič and T. Roubíček,
Energetic versus maximally-dissipative local solutions of a quasi-static rate-independent mixed-mode delamination model, Meccanica, 49 (2014), 2933-2963.
doi: 10.1007/s11012-014-0045-4. |

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