Dual formulation of a quasistatic viscoelastic contact problem with tresca's friction law

International audienceWe consider quasistatic evolution of a viscoelastic body which is in bilateral frictional contact with a rigid foundation, We derive two variational formulations for the problem: the primal formulation in terms of the displacements and the dual formulation in terms of the stress field. We prove the existence of a unique solution to each one and establish the equivalence between the two variational formulations. We also prove the continuous dependence of the solution on the friction yield limit

Thermal analysis of the grinding process

A two-dimensional mathematical model for the thermal aspects of a grinding process is presented. The model includes heat conduction in the grinding wheel, workpiece, and coolant. The heat generation through friction, heat loss to the environment as well as debris, and the interaction among the three components are described in detail. A finite-element algorithm is implemented to solve the nonlinear problem. Numerical results, such as temperatures in the grinding wheel and workpiece, are presented

A Mathematical Model for Outgassing and Contamination

A model for the mathematical description of the processes of outgassing and contamination in a vacuum system is proposed. The underlying assumptions are diffusion in the source, convection and diffusion in the cavity, mass transfer across the source-cavity interface, and a generalization of the Langmuir isotherm for the sorption kinetics on the target. Three approximations are considered where the asymptotic behavior of the model for large time is shown as well as the dependence and sensitivity of the model on some of the parameters. Some numerical examples of the full model are then presented together with a proof of the uniqueness of the solution

A one-dimensional spot welding model

A one-dimensional model is proposed for the simulations of resistance spot welding, which is a common industrial method used to join metallic plates by electrical heating. The model consists of the Stefan problem, in enthalpy form, coupled with the equation of charge conservation for the electrical potential. The temperature dependence of the density, thermal conductivity, specific heat, and electrical conductivity are taken into account, since the process generally involves a large temperature range, on the order of 1000 K. The model is general enough to allow for the welding of plates of different thicknesses or dissimilar materials and to account for variations in the Joule heating through the material thickness due to the dependence of electrical resistivity on the temperature. A novel feature in the model is the inclusion of the effects of interface resistance between the plates which is also assumed to be temperature dependent. In addition to constructing the model, a finite difference scheme for its numerical approximations is described, and representative computer simulations are depicted. These describe welding processes involving different interface resistances, different thicknesses, different materials, and different voltage forms. The differences in the process due to AC or DC currents are depicted as well

Quasistatic Problems in Contact Mechanics

form, Problem P . Find fu; ; g such that 0 + K 1 + K 2 + C 1 v + S(v; ; ) = Q in V 0 ; Bv + Au + C 2 + @ 4 j(v; ; ; v) 3 f in E 0 : Here f 2 E 0 and Q 2 V 0 are given by hf; wi = Z T 0 Z f (t)w i (t) dxdt + Z T 0 Z c ij (t)w i;j (t) dxdt + Z T 0 Z N f N (t)w i (t)d dt; hQ; i = Z T 0 hq(t); (t)i dt Z T 0 Z 0 (t)(t) dxdt Z T 0 Z C hR ((t) R (t))(t)d dt Z T 0 Z k ij ;i (t) ;j (t) dxdt; and @ 4 j(v; ; ; w) denotes the partial subdifferential with respect to w of j(v; ; ; w) = Z T 0 Z C (x; jv T (t) v (t)j; (t))jR n (t)jjw T (t) v (t)jd dt: Theorem [4]. Problem P has a unique solution when 0 is sufficiently small. 39 Future Directions and Open Problems Existence and uniqueness for quasistatic problems are in good shape. There is a need for regularity theory to remove the regularization operator R. Optimal control of frictional problems. Optimal shape design of frictional problems. Numerical a..

A dynamic thermo-mechanical actuator system with contact and Barber's heat exchange boundary conditions

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