# zbMATH — the first resource for mathematics

On friction problems with normal compliance. (English) Zbl 0707.73068
A normal compliance condition and a modified form of Coulomb’s law are utilized to analyze the friction problem of linear elastic bodies. The problem can be expressed as a variational inequality for which a solution does exist. Utilizing the convergence of a linearized iterative scheme, the local uniqueness of a solution for sufficiently small coefficients, is proved. A two-step iterative scheme is considered and its convergence is proved. A dual formulation for the problem is presented. The usefulness of this formulation in numerical calculations is indicated. Some examples are given.
Reviewer: V.K.Arya

##### MSC:
 74A55 Theories of friction (tribology) 74M15 Contact in solid mechanics 74S30 Other numerical methods in solid mechanics (MSC2010) 74P10 Optimization of other properties in solid mechanics
##### Keywords:
two-step iterative scheme; convergence; dual formulation
Full Text:
##### References:
 [1] Bielski, W.R.; Telega, J.J., A contribution to contact problems for a class of solids and structures, Arch. mech., 37, 4-5, 303-320, (1985) · Zbl 0597.73114 [2] Capuzzo, Dolcetta I.; Matzeu, M., Duality for implicit variational problems and numerical applications, Num. funct. analysis optim., 2, 4, 231-265, (1980) · Zbl 0456.49011 [3] Cocu, M., Existence of solutions of Signorini problems with friction, Int. J. engng sci., 22, 5, 567-575, (1984) · Zbl 0554.73096 [4] Demkowicz, L.; Oden, J.T., On some existence and uniqueness results in contact problems with nonlocal friction, Nonlinear analysis, 6, 10, 1075-1093, (1982) · Zbl 0511.73122 [5] Duvaut, G.; Lions, J.L., Inequalities in mechanics and physics, (1976), Springer Berlin · Zbl 0331.35002 [6] Ekeland, I.; Temam, R., Convex analysis and variational problems, (1976), North-Holland Amsterdam [7] Hardy, G.H.; Littlewood, J.E.; Polya, G., Inequalities, (1952), Cambridge University Press Cambridge [8] Johnson, K.L., Contact mechanics, (1985), Cambridge University Press Cambridge · Zbl 0599.73108 [9] Kalker, J.J., On the contact problem in elastostatics, (), 81-118 · Zbl 0621.73127 [10] Kalker, J.J., The principle of virtual work and its dual for contact problems, Ingenieurarchiv, 56, 453-467, (1986) · Zbl 0595.73124 [11] Kalker, J.J., Mathematical models of friction for contact problems in elasticity, Wear, 113, 61-77, (1986) [12] Kikuchi, N.; Park, K.C.; Gartling, D.K., A class of Signorini problems by reciprocal variational inequalities, Computational techniques for interface problems, Vol. 30, 135-153, (1978), ADM [13] Kikuchi, N., Friction contact problems by using penalty regularization methods, Contact mechanics and wear of rail/wheel systems, (1982), University of Waterloo [14] Klarbring, A.; Mikelić, A.; Shillor, M., Frictional contact problems with normal compliance, Int. J. engng sci., 26, 811-832, (1988) · Zbl 0662.73079 [15] Lions, J.L.; Magenes, E., Problémes aux limites non homogénes, (1968), Dunod Paris · Zbl 0165.10801 [16] Martins, J.A.C., Dynamic frictional contact problems involving metallic bodies, Ph.D. thesis, (1968), The University of Texas at Austin [17] Mosco, U., Dual variational inequalities, J. math. analysis applic., 40, 202-206, (1972) · Zbl 0262.49003 [18] Mosco, U., Implicit variational problems and quasivariational inequalities, (), 83-156 · Zbl 0338.49016 [19] Nečas, J.; Jarušek, J.; Haslinger, J., On the solution of the variational inequality to the Signorini problem with small friction, Bollettino U.M.I., 5, 796-811, (1980) · Zbl 0445.49011 [20] Oden, J.T.; Martins, J.A.C., Models and computational methods for dynamic friction phenomena, Comp. meth. appl. mech. engng, 52, 527-634, (1985) · Zbl 0567.73122 [21] Panagiotopoulos, P.D., A nonlinear programming approach to the unilateral contact and friction boundary value problem in the theory of elasticity, Ing. arch., 44, 421-432, (1975) · Zbl 0332.73018 [22] Rabier, P.; Martins, J.A.C.; Oden, J.T.; Campos, L., Existence and local uniqueness of solutions to contact problems in elasticity with nonlinear friction laws, Int. J. engng. sci., 24, 1755-1768, (1986) · Zbl 0601.73113 [23] Rockafellar, R.T., Convex analysis, (1970), Princeton University Press Princeton · Zbl 0229.90020 [24] Spence, D.A.; De Pater, A.D.; Kalker, J.J., Similarity considerations for contact between dissimilar elastic bodies, Mechanics of contact between deformable bodies, proc. IUTAM conf., 67-76, (1974) [25] Turner, J.R., The frictional unloading problems on a linear elastic half-space, J. instn math. applic., 24, 439-469, (1979) · Zbl 0417.73096 [26] Rabier, P.J.; Oden, J.T., Solution to Signorini-like contact problems through interface models—I. preliminaries and formulation of a variational equality, Nonlinear analysis, 11, 1325-1350, (1987) [27] Rabier, P.J.; Oden, J.T., Solution to Signorini-like contact problems through interface models—II. existence and uniqueness theorems, Nonlinear analysis, 12, 1-17, (1988)
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.