Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Science, Applied Mathematics Laboratory, Ferhat Abbas- Setif 1 University, Setif, Algeria

2 Department of Mathematics, Faculty of Science, Fundamental and Numerical Mathematics Laboratory, Ferhat Abbas- Setif 1 University, Setif, Algeria

Abstract

In this work, we study two uncoupled quasistatic problems for thermo viscoplastic materials. In the model of the equation of generalised thermo viscoplasticity, both the elastic and the plastic rate of deformation depend on a parameter $\theta $ which may be interpreted as the absolute temperature. The boundary conditions considered here as displacement-traction conditions as well as unilateral contact conditions. We establish a variational formulation for the model and we prove the existence of a unique weak solution to the problem, reducing the isotherm problem to an ordinary differential equation in a Hilbert space.

Keywords

[1] K.T. Andrews, A. Klarbring, M. Shillor and S. Wright, A dynamic contact problem with friction and wear, Int. J. Engng. Sci., 35 (1997), pp. 1291-1309.
[2] K.T. Andrews, K.L. Kuttler and M. Shillor, On the dynamic behaviour of thermoviscoelastic body in frictional contact with a rigid obstacle, Euro. Jnl. appl. Math., 8 (1997), pp. 417-436.
[3] V. Barbu, Optimal Control of Variational Inequalities, Research Notes in Mathematics,
100. Pitman (Advanced Publishing Program), Boston, 1984.
[4] I. Boukaroura and S. Djabi, A dynamic Tresca's frictional contact problem with damage for thermo elastic-viscoplastic bodies, Studia Univ. Babes Bolayai Mathematica, 64 (2019), pp. 433-449.
[5] I. Boukaroura and S. Djabi, Analysis of a quasistatic contact problem with wear and damage for thermo-viscoelastic materials, Malaya Journal of Matematik, 6 (2018), pp. 299-309.
[6] S. Djabi, A monotony method in quasistatic processes for viscoplastic materials with internal state variables, Revue Roumaine de Maths Pures et Appliquées, 42 (1997), pp. 401-408.
[7] S. Djabi, A monotony method in quasistatic processes for viscoplastic materials with $ varepsilon = varepsilon(epsilon(dot{u}),k) $, Mathematical Reports, 2 (2000), pp. 9-20.
[8] S. Djabi and M. Sofonea, A fixed point method in quasistatic rate-type viscoplasticity, Appl. Math. and Comp Sci., 3 (1993), pp. 269-279 .
[9] G. Duvaut and J.L. Lions, Les inéquations en mecanique et en physique (in French). The inequalities in mechanics and physics], Springer, Berlin,1976.
[10] K.L. Kuttler, Dynamic friction contact problems for general normal and friction laws, Nonlin. Anal, 28 (1997), pp. 559-575.
[11] A. Merouani and S. Djabi, A monotony method in quasistatic processes for viscoplastic materials, Studia Univ. Babes Bolyai Mathematica, 3 (2008), pp. 25-33.
[12] M. Sofonea, On existence and behaviour of the solution of two uncoupled thermo-elastic-visco-plastic problems, An. Univ. Bucharest, 38 (1989), pp.56-65.