×

Quasistatic adhesive contact delaminating in mixed mode and its numerical treatment. (English) Zbl 1327.74115

Summary: An adhesive unilateral contact between visco-elastic bodies at small strains and in a Kelvin-Voigt rheology is scrutinized, neglecting inertia. The flow-rule for debonding the adhesive is considered rate independent, unidirectional, and non-associative due to dependence on the mixity of modes of delamination, namely Mode I (opening) needs (= dissipates) less energy than Mode II (shearing). Such mode-mixity dependence of delamination is a very pronounced (and experimentally confirmed) phenomenon typically considered in engineering models. An efficient semi-implicit-in-time FEM discretization leading to recursive quadratic mathematical programs is devised. Its convergence and thus the existence of weak solutions is proved. Computational experiments implemented by BEM illustrate the modeling aspects and the numerical efficiency of the discretization.

MSC:

74M15 Contact in solid mechanics
74D05 Linear constitutive equations for materials with memory

Software:

SERBA
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Banks-Sills L, Intl J Fracture 103 pp 177– (2000) · doi:10.1023/A:1007612613338
[2] Liechti KM, J Appl Mech 59 pp 295– (1992) · doi:10.1115/1.2899520
[3] Mantič V, J Engr Mater Technology 130 pp 045501-1-2– (2008)
[4] Swadener JG, J Mech Phys Solids 47 pp 223– (1999) · Zbl 0947.74501 · doi:10.1016/S0022-5096(98)00084-2
[5] Evans AG, Metall Trans A 21 pp 2419– (1990) · doi:10.1007/BF02646986
[6] Tveergard W, Intl J Physics of Solids 41 pp 1119– (1993) · Zbl 0775.73219 · doi:10.1016/0022-5096(93)90057-M
[7] Rossi R, Interface Free Bound 14 pp 1– (2004)
[8] DOI: 10.1007/BF00253945 · Zbl 0112.16805 · doi:10.1007/BF00253945
[9] Camanho PP, J Composite Mater 37 pp 1415– (2003) · doi:10.1177/0021998303034505
[10] Hutchinson JW, Advances in Applied Mechanics pp 63– (1992)
[11] Távara L, CMES 58 pp 247– (2010)
[12] Távara L, Eng Anal Bound Elem 35 pp 207– (2011) · Zbl 1259.74069 · doi:10.1016/j.enganabound.2010.08.006
[13] Frémond M, Comptes Rendus Acad Sci Paris 295 pp 913– (1982)
[14] Frémond M, J Méch Théorique Appliq 6 pp 383– (1987)
[15] Kočvara M, Math Mech Solids 11 pp 423– (2006) · Zbl 1133.74038 · doi:10.1177/1081286505046482
[16] Roubíček T, Mathematical Methods and Models in Composites pp 349– (2013) · doi:10.1142/9781848167858_0009
[17] Roubíček T, Discrete Cont Dyn Syst-S 6 pp 591– (2013)
[18] Rossi R, Nonlin Anal 74 pp 3159– (2011) · Zbl 1217.35108 · doi:10.1016/j.na.2011.01.031
[19] Mielke A, Nonlinear PDE’s and Applications pp 87– (2011) · Zbl 1251.35003 · doi:10.1007/978-3-642-21861-3_3
[20] Toader R, Boll Unione Matem Ital 2 pp 1– (2009)
[21] Roubíček T, SIAM J Appl Math 73 pp 1460– (2013) · Zbl 1295.35283 · doi:10.1137/120870396
[22] Bonetti E, Math Methods Appl Sci 31 pp 1029– (2008) · Zbl 1145.35301 · doi:10.1002/mma.957
[23] Thomas M. Rate-independent damage processes in nonlinearly elastic materials. PhD Thesis, Institut für Mathematik, Humboldt-Universität zu Berlin, 2010.
[24] Thomas M, ZAMM-Z Angew Math Mech 90 pp 88– (2010) · Zbl 1191.35159 · doi:10.1002/zamm.200900243
[25] Mielke A, Handbook of Differential Equations, Evolutionary Equations 2 pp 461– (2005) · doi:10.1016/S1874-5717(06)80009-5
[26] Mielke A, Nonlin Diff Equations Appl 11 pp 151– (2004)
[27] Roubíček T, Math Methods Appl Sci 32 pp 825– (2009) · Zbl 1239.35158 · doi:10.1002/mma.1069
[28] Roubíček T, Nonlinear Partial Differential Equations with Applications, 2. ed. (2013) · Zbl 1270.35005 · doi:10.1007/978-3-0348-0513-1
[29] Cornetti P, Engineering Structures 33 pp 1988– (2011) · doi:10.1016/j.engstruct.2011.02.036
[30] Mukherjee S, Math Mech Solids 6 pp 47– (2001) · Zbl 1042.74055 · doi:10.1177/108128650100600103
[31] París F, Boundary Element Method, Fundamentals and Applications (1997)
[32] Roubíček T, ZAMM-Z Angew Math Mech 93 pp 823– (2013) · Zbl 1432.74197 · doi:10.1002/zamm.201200239
[33] Roubíček T, SIAM J Math Anal 45 pp 101– (2013) · Zbl 1264.35131 · doi:10.1137/12088286X
[34] Mielke A, Math Model Numer Anal 43 pp 399– (2009) · Zbl 1166.74010 · doi:10.1051/m2an/2009009
[35] Dal Maso G, Arch Rational Mech Anal 176 pp 165– (2005) · Zbl 1064.74150 · doi:10.1007/s00205-004-0351-4
[36] Visintin A, Models of Phase Transitions (1996) · doi:10.1007/978-1-4612-4078-5
[37] Dal Maso G, Arch Rational Mech Anal 189 pp 469– (2008) · Zbl 1219.35305 · doi:10.1007/s00205-008-0117-5
[38] Knees D, Math Models Meth Appl Sci 18 pp 1529– (2008) · Zbl 1151.49014 · doi:10.1142/S0218202508003121
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.