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Smooth and non-smooth regularizations of the nonlinear diffusion equation
An energy based formulation of a quasi-static interface damage model with a multilinear cohesive law
1. | Technical University of Košice, Civil Engineering Faculty, Vysokoškolská 9,042 00 Košice, Slovakia |
2. | University of Seville, School of Engineering, Camino de los Descubrimientos s/n, 41092 Seville, Spain |
A new quasi-static and energy based formulation of an interface damage model which provides interface traction-relative displacement laws like in traditional trilinear (with bilinear softening) or generally multilinear cohesive zone models frequently used by engineers is presented. This cohesive type response of the interface may represent the behaviour of a thin adhesive layer. The level of interface adhesion or damage is defined by several scalar variables suitably defined to obtain the required traction-relative displacement laws. The weak solution of the problem is sought numerically by a semi-implicit time-stepping procedure which uses recursive double minimization in displacements and damage variables separately. The symmetric Galerkin boundary-element method is applied for the spatial discretization. Sequential quadratic programming is implemented to resolve each partial minimization in the recursive scheme applied to compute the time-space discretized solutions. Sample 2D numerical examples demonstrate applicability of the proposed model.
References:
[1] |
L. Banks-Sills and D. Askenazi,
A note on fracture criteria for interface fracture, Int. J. Fracture, 103 (2000), 177-188.
|
[2] |
O. Barani, M. Mosallanejad and S. Sadrnejad, Fracture analysis of cohesive soils using bilinear and trilinear cohesive laws,
Int. J. Geomech., 04015088,2015. |
[3] |
Z. P. Bažant and J. Planas,
Fracture and Size Effect in Concrete and Other Quasibrittle Materials, CRC Press, Boca Raton, 1998. |
[4] |
J. Besson, G. Cailletaud, J. L. Chaboche and S. Forest,
Non-Linear Mechanics of Materials, Springer, Dordrecht, 2010. |
[5] |
W. Brocks, A. Cornec and I. Scheider, Computational aspects of Nonlinear Fracture Mechanics
(Chapter 3. 03), In: Numerical and Computational Methods (Vol. 3), R. de Borst, H. A. Mang
(Vol. Eds. ), Comprehensive Structural: Fracture of Materials from Nano to Macro, I. Milne,
R. O. Ritchie, B. Karihaloo (Eds. ), pp. 127{209, Elsevier, 2003. |
[6] |
A. Carpinteri,
Post-peak and post-bifurcation analysis of catastrophic softening behaviour (snap-back instability), Engineering Fracture Mechanics, 32 (1989), 265-278.
|
[7] |
C. G. Dávila, C. A. Rose and P. P. Camanho,
A procedure for superposing linear cohesive laws to represent multiple damage mechanisms in the fracture, International Journal of Fracture, 158 (2009), 211-223.
|
[8] |
C. G. Dávila, C. A. Rose and E. V. Iarve, Modeling fracture and complex crack networks in
laminated composites, in Mathematical Methods and Models in Composites, V. Mantič (Ed. ),
Imperial College Press, 5 (2014), 297-347.
doi: 10.1142/9781848167858_0008. |
[9] |
G. Del Piero and M. Raous,
A unified model for adhesive interfaces with damage, viscosity, and friction, Europ. J. of Mechanics A/Solids, 29 (2010), 496-507.
doi: 10.1016/j.euromechsol.2010.02.004. |
[10] |
Z. Dostál,
Optimal Quadratic Programming Algorithms, volume 23 of Springer Optimization and Its Applications, Springer, Berlin, 2009. |
[11] |
M. Frémond,
Dissipation dans l'adherence des solides, C.R. Acad. Sci., Paris, Sér.Ⅱ, 300 (1985), 709-714.
|
[12] |
G. V. Guinea, J. Planas and M. Elices,
A general bilinear fit for the softening curve of concrete, Materials and Structures, 27 (1994), 99-105.
|
[13] |
R. Gutkin, M. L. Laffan, S. T. Pinho, P. Robinson and P. T. Curtis,
Modelling the R-curve effect and its specimen-dependence, Int. J. of Solids and Structures, 48 (2011), 1767-1777.
|
[14] |
J. W. Hutchinson and Z. Suo,
Mixed mode cracking in layered materials, Advances in Applied Mechanics, 29 (1992), 63-191.
|
[15] |
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. |
[16] |
J. Lemaitre and R. Desmorat,
Engineering Damage Mechanics, Springer, Berlin, 2005. |
[17] |
V. Mantič, Discussion on the reference length and mode mixity for a bimaterial interface,
J. Engr. Mater. Technology, 130 (2008), 045501. |
[18] |
V. Mantič, A. Blázquez, E. Correa and F. París, Analysis of interface cracks with contact in
composites by 2D BEM, In Fracture and Damage of Composites, M. Guagliano and M. H. Aliabadi (Eds. ), pp. 189-248. WIT Press, Southampton, 2006. |
[19] |
V. Mantič and F. París,
Relation between {SIF and ERR based measures of fracture mode mixity in interface cracks, Int. J. Fracture, 130 (2004), 557-569.
|
[20] |
V. Mantič, L. Távara, A. Blázquez, E. Graciani and F. París,
A linear elastic-brittle interface model: Application for the onset and propagation of a fibre-matrix interface crack under biaxial transverse loads, Int. J. Fracture, 195 (2015), 15-38.
|
[21] |
D. Maugis,
Contact, Adhesion and Rupture of Elastic Solids, Springer, Berlin, 2000. |
[22] |
A. Mielke, Differential, energetic and metric formulations for rate-independent processes Nonlinear PDEs and Applications, 87–170, Lecture Notes in Math. , 2028, C. I. M. E. Summer Sch. , Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-21861-3_3. |
[23] |
A. Mielke and T. Roubíček,
Rate-Independent Systems. Theory and Applications, Springer, New York, 2015.
doi: 10.1007/978-1-4939-2706-7. |
[24] |
M. Ortiz and A. Pandolfi,
Finite-deformation irreversible cohesive elements for three dimensional crack propagation analysis, Int. J. Num. Meth. Engrg., 44 (1999), 1267-1283.
|
[25] |
C. G. Panagiotopoulos, V. Mantič and T. Roubíček,
A simple and efficient BEM implementation of quasistatic linear visco-elasticity, Int. J. Solid Struct., 51 (2014), 2261-2271.
|
[26] |
K. Park and G. H. Paulino, Cohesive zone models: A critical review of traction-separation relationships across fracture surfaces,
Applied Mechanics Reviews, 64 (2011). |
[27] |
P. E. Petersson,
Crack Growth and Development of Fracture Zones in Plain Concrete and Similar Materials Crack Growth and Development of Fracture Zones in Plain Concrete, Report TVBM-1006, Lund Institute of Technology, Lund, 1981. |
[28] |
N. Pirc, F. Schmidt, M. Mongeau, F. Bugarin and F. Chinesta,
Optimization of BEM-based cooling channels injection moulding using model reduction, International Journal of Material Forming, 1 (2008), 1043-1046.
|
[29] |
M. Raous, L. Cangemi and M. Cocu,
A consistent model coupling adhesion, friction and unilateral contact, Comput. Methods Appl. Mech. Engrg., 177 (1999), 383-399.
doi: 10.1016/S0045-7825(98)00389-2. |
[30] |
T. Roubíček,
Adhesive contact of visco-elastic bodies and defect measures arising by vanishing viscosity, SIAM J. Math. Anal., 45 (2013), 101-126.
doi: 10.1137/12088286X. |
[31] |
T. Roubíček,
Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, 2nd edition, 2013. |
[32] |
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, V. Mantič (Ed. ), Imperial College Press, 5 (2014), 349{400.
doi: 10.1142/9781848167858_0009. |
[33] |
T. Roubíček, C. Panagiotopoulos and V. Mantič,
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. |
[34] |
A. Sutradhar, G. H. Paulino and L. J. Gray,
The Symmetric Galerkin Boundary Element Method, Springer-Verlag, Berlin, 2008. |
[35] |
L. Távara, V. Mantič, E. Graciani and F. París,
BEM analysis of crack onset and propagation along fiber-matrix interface under transverse tension using a linear elastic-brittle interface model, Eng. Anal. Bound. Elem., 35 (2011), 207-222.
doi: 10.1016/j.enganabound.2010.08.006. |
[36] |
L. Távara, V. Mantič, A. Salvadori, L. J. Gray and F. París,
Cohesive-zone-model formulation and implementation using the symmetric Galerkin boundary element method for homogeneous solids, Computational Mechanics, 51 (2013), 535-551.
doi: 10.1007/s00466-012-0808-5. |
[37] |
V. Tvergaard and J. W. Hutchinson,
The relation between crack growth resistance and fracture process parameters in elastic-plastic solids, J. Mech. Phys. Solids, 40 (1992), 1377-1397.
doi: 10.1016/0022-5096(92)90020-3. |
[38] |
T. Vandellos, C. Huchette and N. Carrere,
Proposition of a framework for the development of a cohesive zone model adapted to carbon-fiber reinforced plastic laminated composites, Composite Structures, 105 (2013), 199-206.
|
[39] |
A. Visintin,
Models of Phase Transition, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4078-5. |
[40] |
R. Vodička,
A quasi-static interface damage model with cohesive cracks: SQP-SGBEM implementation, Eng. Anal. Bound. Elem., 62 (2016), 123-140.
doi: 10.1016/j.enganabound.2015.09.010. |
[41] |
R. Vodička, V. Mantič and F. París,
Symmetric variational formulation of BIE for domain decomposition problems in elasticity -an SGBEM approach for nonconforming discretizations of curved interfaces, CMES -Comp. Model. Eng., 17 (2007), 173-203.
|
[42] |
R. Vodička, V. Mantič and F. París,
Two variational formulations for elastic domain decomposition problems solved by SGBEM enforcing coupling conditions in a weak form, Eng. Anal. Bound. Elem., 35 (2011), 148-155.
doi: 10.1016/j.enganabound.2010.05.002. |
[43] |
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. |
[44] |
R. Vodička, V. Mantič and T. Roubíček,
Quasistatic normal-compliance contact problem of visco-elastic bodies with Coulomb friction implemented by QP and SGBEM, J. Comp. Appl. Math., 315 (2017), 249-272.
doi: 10.1016/j.cam.2016.10.010. |
[45] |
R. Vodička, T. Roubíček and V. Mantič, General-purpose model for various adhesive frictional contacts at small strains,
Interfaces Free Bound., (submitted). |
[46] |
P. Wriggers,
Computational Contact Mechanics, Springer, Berlin, 2006. |
show all references
Dedicated to Tomáš Roubíček on the occasion of his 60th birthday
References:
[1] |
L. Banks-Sills and D. Askenazi,
A note on fracture criteria for interface fracture, Int. J. Fracture, 103 (2000), 177-188.
|
[2] |
O. Barani, M. Mosallanejad and S. Sadrnejad, Fracture analysis of cohesive soils using bilinear and trilinear cohesive laws,
Int. J. Geomech., 04015088,2015. |
[3] |
Z. P. Bažant and J. Planas,
Fracture and Size Effect in Concrete and Other Quasibrittle Materials, CRC Press, Boca Raton, 1998. |
[4] |
J. Besson, G. Cailletaud, J. L. Chaboche and S. Forest,
Non-Linear Mechanics of Materials, Springer, Dordrecht, 2010. |
[5] |
W. Brocks, A. Cornec and I. Scheider, Computational aspects of Nonlinear Fracture Mechanics
(Chapter 3. 03), In: Numerical and Computational Methods (Vol. 3), R. de Borst, H. A. Mang
(Vol. Eds. ), Comprehensive Structural: Fracture of Materials from Nano to Macro, I. Milne,
R. O. Ritchie, B. Karihaloo (Eds. ), pp. 127{209, Elsevier, 2003. |
[6] |
A. Carpinteri,
Post-peak and post-bifurcation analysis of catastrophic softening behaviour (snap-back instability), Engineering Fracture Mechanics, 32 (1989), 265-278.
|
[7] |
C. G. Dávila, C. A. Rose and P. P. Camanho,
A procedure for superposing linear cohesive laws to represent multiple damage mechanisms in the fracture, International Journal of Fracture, 158 (2009), 211-223.
|
[8] |
C. G. Dávila, C. A. Rose and E. V. Iarve, Modeling fracture and complex crack networks in
laminated composites, in Mathematical Methods and Models in Composites, V. Mantič (Ed. ),
Imperial College Press, 5 (2014), 297-347.
doi: 10.1142/9781848167858_0008. |
[9] |
G. Del Piero and M. Raous,
A unified model for adhesive interfaces with damage, viscosity, and friction, Europ. J. of Mechanics A/Solids, 29 (2010), 496-507.
doi: 10.1016/j.euromechsol.2010.02.004. |
[10] |
Z. Dostál,
Optimal Quadratic Programming Algorithms, volume 23 of Springer Optimization and Its Applications, Springer, Berlin, 2009. |
[11] |
M. Frémond,
Dissipation dans l'adherence des solides, C.R. Acad. Sci., Paris, Sér.Ⅱ, 300 (1985), 709-714.
|
[12] |
G. V. Guinea, J. Planas and M. Elices,
A general bilinear fit for the softening curve of concrete, Materials and Structures, 27 (1994), 99-105.
|
[13] |
R. Gutkin, M. L. Laffan, S. T. Pinho, P. Robinson and P. T. Curtis,
Modelling the R-curve effect and its specimen-dependence, Int. J. of Solids and Structures, 48 (2011), 1767-1777.
|
[14] |
J. W. Hutchinson and Z. Suo,
Mixed mode cracking in layered materials, Advances in Applied Mechanics, 29 (1992), 63-191.
|
[15] |
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. |
[16] |
J. Lemaitre and R. Desmorat,
Engineering Damage Mechanics, Springer, Berlin, 2005. |
[17] |
V. Mantič, Discussion on the reference length and mode mixity for a bimaterial interface,
J. Engr. Mater. Technology, 130 (2008), 045501. |
[18] |
V. Mantič, A. Blázquez, E. Correa and F. París, Analysis of interface cracks with contact in
composites by 2D BEM, In Fracture and Damage of Composites, M. Guagliano and M. H. Aliabadi (Eds. ), pp. 189-248. WIT Press, Southampton, 2006. |
[19] |
V. Mantič and F. París,
Relation between {SIF and ERR based measures of fracture mode mixity in interface cracks, Int. J. Fracture, 130 (2004), 557-569.
|
[20] |
V. Mantič, L. Távara, A. Blázquez, E. Graciani and F. París,
A linear elastic-brittle interface model: Application for the onset and propagation of a fibre-matrix interface crack under biaxial transverse loads, Int. J. Fracture, 195 (2015), 15-38.
|
[21] |
D. Maugis,
Contact, Adhesion and Rupture of Elastic Solids, Springer, Berlin, 2000. |
[22] |
A. Mielke, Differential, energetic and metric formulations for rate-independent processes Nonlinear PDEs and Applications, 87–170, Lecture Notes in Math. , 2028, C. I. M. E. Summer Sch. , Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-21861-3_3. |
[23] |
A. Mielke and T. Roubíček,
Rate-Independent Systems. Theory and Applications, Springer, New York, 2015.
doi: 10.1007/978-1-4939-2706-7. |
[24] |
M. Ortiz and A. Pandolfi,
Finite-deformation irreversible cohesive elements for three dimensional crack propagation analysis, Int. J. Num. Meth. Engrg., 44 (1999), 1267-1283.
|
[25] |
C. G. Panagiotopoulos, V. Mantič and T. Roubíček,
A simple and efficient BEM implementation of quasistatic linear visco-elasticity, Int. J. Solid Struct., 51 (2014), 2261-2271.
|
[26] |
K. Park and G. H. Paulino, Cohesive zone models: A critical review of traction-separation relationships across fracture surfaces,
Applied Mechanics Reviews, 64 (2011). |
[27] |
P. E. Petersson,
Crack Growth and Development of Fracture Zones in Plain Concrete and Similar Materials Crack Growth and Development of Fracture Zones in Plain Concrete, Report TVBM-1006, Lund Institute of Technology, Lund, 1981. |
[28] |
N. Pirc, F. Schmidt, M. Mongeau, F. Bugarin and F. Chinesta,
Optimization of BEM-based cooling channels injection moulding using model reduction, International Journal of Material Forming, 1 (2008), 1043-1046.
|
[29] |
M. Raous, L. Cangemi and M. Cocu,
A consistent model coupling adhesion, friction and unilateral contact, Comput. Methods Appl. Mech. Engrg., 177 (1999), 383-399.
doi: 10.1016/S0045-7825(98)00389-2. |
[30] |
T. Roubíček,
Adhesive contact of visco-elastic bodies and defect measures arising by vanishing viscosity, SIAM J. Math. Anal., 45 (2013), 101-126.
doi: 10.1137/12088286X. |
[31] |
T. Roubíček,
Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, 2nd edition, 2013. |
[32] |
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, V. Mantič (Ed. ), Imperial College Press, 5 (2014), 349{400.
doi: 10.1142/9781848167858_0009. |
[33] |
T. Roubíček, C. Panagiotopoulos and V. Mantič,
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. |
[34] |
A. Sutradhar, G. H. Paulino and L. J. Gray,
The Symmetric Galerkin Boundary Element Method, Springer-Verlag, Berlin, 2008. |
[35] |
L. Távara, V. Mantič, E. Graciani and F. París,
BEM analysis of crack onset and propagation along fiber-matrix interface under transverse tension using a linear elastic-brittle interface model, Eng. Anal. Bound. Elem., 35 (2011), 207-222.
doi: 10.1016/j.enganabound.2010.08.006. |
[36] |
L. Távara, V. Mantič, A. Salvadori, L. J. Gray and F. París,
Cohesive-zone-model formulation and implementation using the symmetric Galerkin boundary element method for homogeneous solids, Computational Mechanics, 51 (2013), 535-551.
doi: 10.1007/s00466-012-0808-5. |
[37] |
V. Tvergaard and J. W. Hutchinson,
The relation between crack growth resistance and fracture process parameters in elastic-plastic solids, J. Mech. Phys. Solids, 40 (1992), 1377-1397.
doi: 10.1016/0022-5096(92)90020-3. |
[38] |
T. Vandellos, C. Huchette and N. Carrere,
Proposition of a framework for the development of a cohesive zone model adapted to carbon-fiber reinforced plastic laminated composites, Composite Structures, 105 (2013), 199-206.
|
[39] |
A. Visintin,
Models of Phase Transition, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4078-5. |
[40] |
R. Vodička,
A quasi-static interface damage model with cohesive cracks: SQP-SGBEM implementation, Eng. Anal. Bound. Elem., 62 (2016), 123-140.
doi: 10.1016/j.enganabound.2015.09.010. |
[41] |
R. Vodička, V. Mantič and F. París,
Symmetric variational formulation of BIE for domain decomposition problems in elasticity -an SGBEM approach for nonconforming discretizations of curved interfaces, CMES -Comp. Model. Eng., 17 (2007), 173-203.
|
[42] |
R. Vodička, V. Mantič and F. París,
Two variational formulations for elastic domain decomposition problems solved by SGBEM enforcing coupling conditions in a weak form, Eng. Anal. Bound. Elem., 35 (2011), 148-155.
doi: 10.1016/j.enganabound.2010.05.002. |
[43] |
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. |
[44] |
R. Vodička, V. Mantič and T. Roubíček,
Quasistatic normal-compliance contact problem of visco-elastic bodies with Coulomb friction implemented by QP and SGBEM, J. Comp. Appl. Math., 315 (2017), 249-272.
doi: 10.1016/j.cam.2016.10.010. |
[45] |
R. Vodička, T. Roubíček and V. Mantič, General-purpose model for various adhesive frictional contacts at small strains,
Interfaces Free Bound., (submitted). |
[46] |
P. Wriggers,
Computational Contact Mechanics, Springer, Berlin, 2006. |
















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Hailing Xuan, Xiaoliang Cheng. Numerical analysis and simulations of a frictional contact problem with damage and memory. Mathematical Control and Related Fields, 2021 doi: 10.3934/mcrf.2021037 |
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