The bipotential method: A constructive approach to design the complete contact law with friction and improved numerical algorithms

Nowadays, the concept of convex potential of dissipation is a powerful tool customarily used to model the constitutive dissipative laws. Unfortunately, it fails when applied to Coulomb's dry friction contact, which is shown in this paper by checking the cyclic monotony condition. Next, a new approach, the bipotential method, is presented and successfully applied to the contact law. This enables us to write it in a compact form and to uncover an implicit normality rule structure. The advantages of the new approach are numerous, among which is emphasized a pretty extension of the calculus of variation. Two minimum principles of the so-called bifunctional are presented for contact problems. Next, the bipotential method can be qualified as constitutive in the sense that it suggests improved numerical algorithms. In particular, it is proved that the complete contact law can be rewritten as a projection equation onto Coulomb's cone. Numerical examples show the feasibility of the algorithm and the computer time reduction with respect to other previous numerical approaches.

[1]  J. C. Simo,et al.  Consistent tangent operators for rate-independent elastoplasticity☆ , 1985 .

[2]  Nguyen Quoc Son.  On the elastic plastic initial‐boundary value problem and its numerical integration , 1977 .

[3]  P. Germain.  Mécanique des milieux continus , 1962 .

[4]  W. Fenchel.  On Conjugate Convex Functions , 1949, Canadian Journal of Mathematics.

[5]  P. Alart,et al.  A mixed formulation for frictional contact problems prone to Newton like solution methods , 1991 .

[6]  H. Ziegler,et al.  A Possible Generalization of Onsager’s Theory , 1968 .

[7]  Zhi-Qiang Feng.  2D or 3D frictional contact algorithms and applications in a large deformation context , 1995 .

[8]  Zbigniew Mróz,et al.  Mathematical models of inelastic material behaviour , 1973 .

[9]  P. Panagiotopoulos.  Inequality Problems in Mechanics and Applications: Convex and Nonconvex Energy Functions , 1985 .

[10]  R. Rockafellar.  Characterization of the subdifferentials of convex functions , 1966 .

[11]  A. J. Durelli,et al.  Moiré analysis of strain , 1970 .

[12]  R. Luciano,et al.  Stress-penalty method for unilateral contact problems: mathematical formulation and computational aspects , 1994 .

[13]  G. Saxcé,et al.  Elastoplastic Finite Element Analysis of Soil Problems with Implicit Standard Material Constitutive Laws , 1994 .

[14]  Quoc Son Nguyen,et al.  Sur les matériaux standard généralisés , 1975 .

[15]  J. C. Simo,et al.  An augmented lagrangian treatment of contact problems involving friction , 1992 .

[16]  G. Saxcé,et al.  New Inequality and Functional for Contact with Friction: The Implicit Standard Material Approach∗ , 1991 .

[17]  J. Moreau,et al.  Topics in Nonsmooth Mechanics , 1988 .