zbMATH — the first resource for mathematics

A frictionless contact problem for elastic-viscoplastic materials with normal compliance and damage. (English) Zbl 1042.74039
Summary: We study a quasistatic frictionless contact problem with normal compliance and damage for elastic-viscoplastic bodies. The mechanical damage of the material, caused by excessive stress or strain, is described by a damage function whose evolution is modelled by an inclusion of parabolic type. We provide a variational formulation for the mechanical problem and sketch a proof of the existence of a unique weak solution to the model. We then introduce and study a fully discrete scheme for the numerical solutions of the problem. An optimal order error estimate is derived for the approximate solutions under suitable solution regularity. Numerical examples are presented to show the performance of the method.

74M15 Contact in solid mechanics
74C10 Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity)
74G25 Global existence of solutions for equilibrium problems in solid mechanics (MSC2010)
74G30 Uniqueness of solutions of equilibrium problems in solid mechanics
Full Text: DOI
[1] Barbu, V., Optimal control of variational inequalities, (1984), Pitman Boston · Zbl 0574.49005
[2] Brezis, H., Equations et inéquations non linéaires dans LES espaces vectoriels en dualité, Ann. inst. Fourier, 18, 115-175, (1968) · Zbl 0169.18602
[3] Burguera, M.; Viaño, J.M., Numerical solving of frictionless contact problems in perfect plastic bodies, Comput. meth. appl. mech. engrg., 120, 303-322, (1995) · Zbl 0851.73055
[4] J. Chen, W. Han, M. Sofonea, Numerical analysis of a class of evolution systems arising in viscoplasticity, Computat. Appl. Math. 19 (2000) 279-306 · Zbl 1205.65266
[5] Ciarlet, P.G., The finite element method for elliptic problems, (1978), North Holland Amsterdam · Zbl 0445.73043
[6] Clément, P., Approximation by finite element functions using local regularization, RAIRO anal. numer., 9R2, 77-84, (1975) · Zbl 0368.65008
[7] Cristescu, N.; Suliciu, I., Viscoplasticity, () · Zbl 0514.73022
[8] Duvaut, G.; Lions, J.L., LES inéquations en Mécanique et en physique, (1972), Dunod Paris · Zbl 0298.73001
[9] J.R. Fernández-Garcı́a, W. Han, M. Sofonea, J.M. Viaño, Variational and numerical analysis of a frictionless contact problem for elastic – viscoplastic materials with internal state variable, Quart. J. Mech. Appl. Math. 54 (2001) 501-522
[10] Fernández-Garcı́a, J.R.; Sofonea, M.; Viaño, J.M., Analyse numérique d’un problème élasto-viscoplastique de contact sans frottement avec compliance normale, C.R. acad. sci. Paris, t.331, Série I, 323-328, (2000) · Zbl 0993.74043
[11] J.R. Fernández-Garcı́a, M. Sofonea, J.M. Viaño, A frictionless contact problem for elastic – viscoplastic materials with normal compliance, Numerische Mathematik, in press
[12] Frémond, M.; Kuttler, K.L.; Nedjar, B.; Shillor, M., One-dimensional models of damage, Adv. math. appl., 8, 541-570, (1998) · Zbl 0915.73041
[13] Frémond, M.; Kuttler, K.L.; Shillor, M., Existence and uniqueness of solutions for a one-dimensional damage model, Jmma, 229, 271-294, (1999) · Zbl 0920.73328
[14] Frémond, M.; Nedjar, B., Damage in concrete: the unilateral phenomenon, Nucl. engrg. des., 156, 323-335, (1995)
[15] Frémond, M.; Nedjar, B., Damage, gradient of damage and principle of virtual work, Int. J. solids struct., 33, 1083-1103, (1996) · Zbl 0910.73051
[16] Glowinski, R., Numerical methods for nonlinear variational problems, (1984), Springer-Verlag New York · Zbl 0575.65123
[17] Glowinski, R.; Lions, J.-L.; Trémolières, R., Numerical analysis of variational inequalities, (1981), North-Holland Amsterdam · Zbl 0508.65029
[18] Han, W.; Reddy, B.D., Plasticity: mathematical theory and numerical analysis, (1999), Springer-Verlag New York · Zbl 0926.74001
[19] W. Han, M. Shillor, M. Sofonea, Variational and numerical analysis of a quasistatic viscoelastic problem with normal compliance, friction and damage, J. of Comp. Appl. Math. 137 (2001) 377-398 · Zbl 0999.74087
[20] Han, W.; Sofonea, M., Numerical analysis of a frictionless contact problem for elastic – viscoplastic materials, Comput. meth. appl. mech. engrg., 190, 179-191, (2000) · Zbl 1004.74071
[21] Han, W.; Sofonea, M., Evolutionary variational inequalities arising in viscoelastic contact problems, SIAM J. numer. anal., 38, 556-579, (2000) · Zbl 0988.74048
[22] Ionescu, I.R.; Sofonea, M., Functional and numerical methods in viscoplasticity, (1993), Oxford University Press Oxford · Zbl 0787.73005
[23] Kikuchi, N.; Oden, J.T., Contact problems in elasticity: A study of variational inequalities and finite element methods, (1988), SIAM Philadelphia · Zbl 0685.73002
[24] Klarbring, A.; Mikelič, A.; Shillor, M., Frictional contact problems with normal compliance, Int. J. engrg. sci., 26, 811-832, (1988) · Zbl 0662.73079
[25] Martins, J.T.; Oden, J.T., Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws, Nonlin. anal. TMA, 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] Panagiotopoulos, P.D., Inequality problems in mechanics and applications, (1985), Birkhäuser Basel · Zbl 0579.73014
[28] Rochdi, M.; Shillor, M.; Sofonea, M., Quasistatic viscoelastic contact with normal compliance and friction, J. elasticity, 51, 105-126, (1998) · Zbl 0921.73231
[29] Rochdi, M.; Shillor, M.; Sofonea, M., Analysis of a quasistatic viscoelastic problem with friction and damage, Adv. math. sci. appl., 10, 173-189, (2000) · Zbl 0962.74044
[30] Sofonea, M., On a contact problem for elastic – viscoplastic bodies, Nonlin. anal. TMA, 29, 1037-1050, (1997) · Zbl 0918.73098
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.