On a class of limit states of frictional joints: formulation and existence theorem

Authors:
Lars-Erik Andersson and Anders Klarbring

Journal:
Quart. Appl. Math. **55** (1997), 69-87

MSC:
Primary 73C99; Secondary 35Q72, 49J40, 73T05, 73V25

DOI:
https://doi.org/10.1090/qam/1433753

MathSciNet review:
MR1433753

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The model dealt with is a linear elastic body in frictional contact with a rigid support. Limit states of such an assemblage are characterized by deformations and forces such that a small perturbation may introduce a large change in configuration. The class of limit states considered here is specified by the possibility of superposing a time constant rigid body velocity field to a static deformation. The problem of finding such states (i.e., forces and static deformations) for a prescribed rigid body velocity is formulated, and for the case when the geometrically admissible rigid body displacements form a linear space an existence result is given. It is proved that under restrictions on the magnitude of the friction coefficient and in the case that an intuitively clear condition on the direction of the forces is satisfied, there exist a load multiplier and a corresponding static displacement.

**[1]**A. Klarbring, A. Mikelić, and M. Shillor,*The rigid punch problem with friction*, Internat. J. Engrg. Sci.**29**(1991), no. 6, 751–768. MR**1107199**, https://doi.org/10.1016/0020-7225(91)90104-B**[2]**Claudio Baiocchi, Giuseppe Buttazzo, Fabio Gastaldi, and Franco Tomarelli,*General existence theorems for unilateral problems in continuum mechanics*, Arch. Rational Mech. Anal.**100**(1988), no. 2, 149–189. MR**913962**, https://doi.org/10.1007/BF00282202**[3]**Philippe G. Ciarlet and Jindřich Nečas,*Unilateral problems in nonlinear, three-dimensional elasticity*, Arch. Rational Mech. Anal.**87**(1985), no. 4, 319–338 (English, with French summary). MR**767504**, https://doi.org/10.1007/BF00250917**[4]**I. Hlaváček and J. Nečas,*On inequalities of Korn's type*II.*Applications to linear elasticity*, Arch. Rational Mech. Anal.**36**, 312-334 (1970)**[5]**J. Nečas and I. Hlaváček,*Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction*, Elsevier, Amsterdam, 1981**[6]**Marius Cocu,*Existence of solutions of Signorini problems with friction*, Internat. J. Engrg. Sci.**22**(1984), no. 5, 567–575. MR**750004**, https://doi.org/10.1016/0020-7225(84)90058-2**[7]**Ky Fan,*A minimax inequality and applications*, Inequalities, III (Proc. Third Sympos., Univ. California, Los Angeles, Calif., 1969; dedicated to the memory of Theodore S. Motzkin), Academic Press, New York, 1972, pp. 103–113. MR**0341029****[8]**H. Brézis, L. Nirenberg, and G. Stampacchia,*A remark on Ky Fan's minimax principle*, Bolletino Un. Mat. Ital.**6**, no. 293(4), 293-300 (1972)**[9]**A. Klarbring,*Examples of non-uniqueness and non-existence of solutions to quasistatic contact problems with friction*, Ing. Arch.**60**, 529-542 (1990)**[10]**M. Frémond,*Yield theory in physics*, Topics in nonsmooth mechanics, Birkhäuser, Basel, 1988, pp. 187–240. MR**957091****[11]**D. C. Drucker,*Coulomb friction, plasticity and limit loads*, J. Applied Mech., March, 71-74 (1954)**[12]**I. F. Collins,*The upper bound theorem for rigid/plastic solids generalized to include Coulomb friction*, J. Mech. Phys. Solids**17**, 323-338 (1969)**[13]**J. J. Telega,*Limit analysis theorems in the case of Signorini’s boundary conditions and friction*, Arch. Mech. (Arch. Mech. Stos.)**37**(1985), no. 4-5, 549–562 (English, with Russian and Polish summaries). MR**844137****[14]**F. Gastaldi and J. A. C. Martins,*A Noncoercive Steady-Sliding Problem with Friction*, Istituto di Analisi Numerica, Pubblicazioni N. 650, Pavia, 1988

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
73C99,
35Q72,
49J40,
73T05,
73V25

Retrieve articles in all journals with MSC: 73C99, 35Q72, 49J40, 73T05, 73V25

Additional Information

DOI:
https://doi.org/10.1090/qam/1433753

Article copyright:
© Copyright 1997
American Mathematical Society