zbMATH — the first resource for mathematics

Numerical analysis and simulations of a dynamic frictionless contact problem with damage. (English) Zbl 1120.74651
Summary: The process of dynamic frictionless contact between a viscoelastic body and a reactive foundation, which includes material damage, is modelled, numerically analyzed, and simulated. Contact is modelled with the normal compliance condition. The damage of the material, resulting from tension or compression, is described by the damage field which measures the pointwise fractional decrease in the load carrying capacity of the material. The evolution of the damage is governed by a parabolic differential inclusion. A fully discrete scheme for the problem is proposed, and error estimates on the numerical solutions derived. The scheme has been implemented and numerical simulations of the evolution of the mechanical state of the system and its material damage are presented.

74M15 Contact in solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
74H15 Numerical approximation of solutions of dynamical problems in solid mechanics
74R20 Anelastic fracture and damage
Full Text: DOI
[1] Angelov, T.A., On a rolling problem with damage and wear, Mech. res. comm., 26, 281-286, (1999) · Zbl 0945.74660
[2] K.T. Andrews, M. Shillor, Thermomechanical behaviour of a damageable beam in contact with two stops, Appl. Anal., in press. · Zbl 1207.74057
[3] Campo, M.; Fernández, J.R.; Han, W.; Sofonea, M., A dynamic viscoelastic contact problem with normal compliance and damage, Finite elem. anal. design, 42, 1, 1-24, (2005)
[4] Dorfmann, A.; Fuller, K.N.G.; Ogden, R.W., Shear, compressive and dilatational response of rubber-like solids subject to cavitation damage, Int. J. solids struct., 39, 1825-1857, (2002) · Zbl 1006.74539
[5] Frémond, M.; Kuttler, K.L.; Nedjar, B.; Shillor, M., One-dimensional models of damage, Adv. math. sci. appl., 8, 541-570, (1998) · Zbl 0915.73041
[6] Liebe, R.; Steinmann, P.; Benallal, A., Theoretical and numerical aspects of a thermodynamically consistent framework for geometrically linear gradient damage, Comput. methods appl. mech. engrg., 190, 6555-6576, (2001) · Zbl 0991.74010
[7] Miehe, C., Discontinuous and continuous damage evolution in ogden-type large-strain elastic materials, Eur. J. mech. A solids, 14, 3, 697-720, (1995) · Zbl 0837.73054
[8] Nedjar, B., Elastoplastic-damage modelling including the gradient of damage. formulation and computational aspects, Int. J. solids struct., 38, 5421-5451, (2001) · Zbl 1013.74061
[9] Peerlings, R.H.J.; de Borst, R.; Brekelmans, W.A.M.; de Vree, J.H.P., Gradient-enhaced damage for quasi-brittle materials, Int. J. numer. methods engrg., 39, 3391-3403, (1996) · Zbl 0882.73057
[10] Steinmann, P., Formulation and computation of geometrically non-linear gradient damage, Int. J. numer. methods engrg., 4, 757-779, (1999) · Zbl 0978.74006
[11] Steinmann, P.; Miehe, C.; Stein, E., Comparison of different finite deformation inelastic damage models within multiplicative elastoplasticity for ductile materials, Comput. mech., 13, 458-474, (1994) · Zbl 0816.73021
[12] Frémond, M., Non-smooth thermomechanics, (2002), Springer Berlin · Zbl 0990.80001
[13] Shillor, M.; Sofonea, M.; Telega, J.J., Models and analysis of quasistatic contact, Lecture notes in physics, vol. 655, (2004), Springer Berlin · Zbl 1180.74046
[14] Sofonea, M.; Han, W.; Shillor, M., Analysis and approximations of contact problems with adhesion or damage, Pure applied mathematics, vol. 276, (2006), Chapman & Hall/CRC Boca Raton · Zbl 1089.74004
[15] Frémond, M.; Nedjar, B., Damage in concrete: the unilateral phenomenon, Nucl. engrg. des., 156, 323-335, (1995)
[16] Frémond, M.; Nedjar, B., Damage, gradient of damage and principle of virtual work, Int. J. solids struct., 33, 8, 1083-1103, (1996) · Zbl 0910.73051
[17] Kuttler, K.L.; Shillor, M.; Fernández, J.R., Existence and regularity for dynamic viscoelastic adhesive contact with damage, Appl. math. optim., 53, 31-66, (2006) · Zbl 1089.74040
[18] Kuttler, K.L., Quasistatic evolution of damage in an elastic – viscoplastic material, Electron. J. diff. equat., 147, 1-25, (2005) · Zbl 1082.74005
[19] Bermúdez, A.; Moreno, C., Duality methods for solving variational inequalities, Comp. math. appl., 7, 43-58, (1981) · Zbl 0456.65036
[20] Burguera, M.; Viaño, J.M., Numerical solving of frictionless contact problems in perfect plastic bodies, Comput. methods appl. mech. engrg., 120, 303-322, (1995) · Zbl 0851.73055
[21] Chau, O.; Fernández, J.R.; Han, W.; Sofonea, M., A frictionless contact problem for elastic – viscoplastic materials with normal compliance and damage, Comput. methods appl. mech. engrg., 191, 5007-5026, (2002) · Zbl 1042.74039
[22] Viaño, J.M., Análisis de un método numérico con elementos finitos para problemas de contacto unilateral sin rozamiento en elasticidad: aproximación y resolución de los problemas discretos, Rev. internac. Métod. numér. Cálc. diseñ. ingr., 2, 63-86, (1986)
[23] M. Campo, J.R. Fernández, K.L. Kuttler and M. Shillor, Quasistatic evolution of damage in an elastic body: numerical analysis and computational experiments, Preprint, 2005. · Zbl 1116.74058
[24] Klarbring, A.; Mikelić, A.; Shillor, M., Frictional contact problems with normal compliance, Int. J. engrg. sci., 26, 811-832, (1988) · Zbl 0662.73079
[25] Martins, J.A.C.; Oden, J.T., Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws, Nonlinear anal., 11, 407-428, (1987) · Zbl 0672.73079
[26] Nečas, J.; Hlavaček, I., Mathematical theory of elastic and elastoplastic bodies: an introduction, (1981), Elsevier Amsterdam · Zbl 0448.73009
[27] Glowinski, R., Numerical methods for nonlinear variational problems, (1984), Springer New York · Zbl 0575.65123
[28] Han, W.; Shillor, M.; Sofonea, M., Variational and numerical analysis of a quasistatic viscoelastic problem with normal compliance, friction and damage, J. comput. appl. math., 137, 377-398, (2001) · Zbl 0999.74087
[29] Han, W.; Sofonea, M., Quasistatic contact problems in viscoelasticity and viscoplasticity, (2002), AMS-IP Providence · Zbl 1013.74001
[30] Ciarlet, P.G., The finite element method for elliptic problems, (), 17-352 · Zbl 0198.14601
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.