Nonlinear modelling for elasto/visco-plastic contact problem

in technological processes

Leon Kukiełka

The Technical University of Koszalin

ul. Racławicka 15-17, 75-620 Koszalin, Poland

This paper presents the modelling of a contact problem in the operation of technological production of objects. An incremental model of the contact problem for movable elasto/visco-plastic body for spatial states (3D) is being considered. Geometrical contact conditions (GCC) for the case of a deformed object and a rigid or elastic tool, with a rotation and translation of the bodies are introduced. A GCC form used in numeric calculations is determined. Dependences between increments of unit forces in the contact area of bodies is introduced. Basic incremental equations of the edge displacement in the reversible and nonreversible zone are defined. The description of a geometrical contact conditions and friction conditions in the ranges of stick-slip are considered. The models obtained are used to a variational formulation of a contact problem with the application of a method of finite elements and to a numerical analysis of the contact problem.

Introduction

One of the most important problems of the present production technologies is to form the quality of an object with proprieties set in advance. The finishing operation constitutes the most important operation of the object's technological production, during which the basic exploitation properties of the surface layer are established [1, 2÷6, 7÷9]. These properties, in definite conditions of exploitation, are decisive as regards the reliability of machines. Examples of occurrence of contact bodies (tool-object) in different technological operations are shown in Figure 1.

The contact of those bodies which either one of them or both are deformable testifies to the occurrence of the process of many mechanical phenomena. Their variety causes that contact problems occupy a special position in the mechanics of the solid body. It results from a fold character of the phenomena setting at the contact and from the difficulties of their investigation and description. The contact problem occurs in the two non-linearities: geometrical (a change of the initial geometry of the body providing a non-linear dependence strain-displacement ) and physical (a non-linear dependence stress-strain ). Moreover, boundary conditions are often changeable and definite in part only. In such cases, it is necessary to make use of an incremental description.

Figure 1. Contact problem in process of milling (a), burnishing (b) and thread rolling (c).

The analysis of the state of displacements, deformations and stress constitutes the basic problem, and the knowledge of these (both of flat and spatial problems) creates wide possibilities of the description of physical phenomena set during the mutual influence of bodies [10, 11, 12]. A particularly essential and practical group of problems is the contact between a deformable visco-plastic body (object) and a rigid body (tool). In the process of the tool's contact with the object it can step out a different degree of deformation. A different stiffness of the elastic body (the tool) corresponds to a different deformation of the object and a different resistance of the local non-reversible deformation, put mainly in the contact area. The states of small elastic deformations of the tool accompany the elastic and visco-plastic deformation of the object.

The phenomena described accompany the burnishing rolling operation. In literature, various problems are connected with burnishing rolling. Particular attention should be paid to the following issues: the geometrical structure of an object's surface after its previous processing [13], the deformation mechanism of an inequality in the burnishing process [13, 14], the durability of a burnishing element (tool) in the time of the process [1, 13, 15], the state of the displacement rate and the deformation rate of the surface layer material during the burnishing process [13] and the size of the areas of contact zones [3, 13, 16 - 20]. The publications [4, 5, 6, 21, 22] contain a modern method of burnishing rolling modelling with the use of the mechanic contact. These concern the contact problem of an ideally rigid body with a deformable body (elasto-plastic or rigid-plastic), for the cases of the occurrence of very small plastic deformations - comparable to an elastic deformation.

In the present paper, the incremental model of movable elasto/visco-plastic body contact, with mixed hardening, for spatial states (3D) is being considered. This model is used to a variational formulation of the contact problem with application of the finite element method (FEM) [23]. This then permits a numerical analysis and a simulation of contact problems [8, 9, 24, 25, 26].

Conclusions

A greater accuracy to achieve quality of objects in the technological process calls for a greater accuracy of modelling and analyses of physical concurrent phenomena process of processing. This contact constitutes the basic problem. A fold character of setting phenomena during the contact and difficulties of their examination to search solutions is on a theoretical way. Geometrical and physical nonlinearity, and only partial knowledge of boundary conditions, which move in track of the process cause the necessity to apply an incremental description.

In this work the incremental model of contact problem, for movable elasto/visco-plastic bodies and for spatial states (3D) is elaborated. The geometrical condition of contact (GCC) has been laid out for the case of a deformable object and a rigid or elastic tool, serf rotation and translation. The GCC condition applied was passed in numerical calculations. Dependences between increments of unit forces in the area of contact of bodies were introduced. Basic equations were defined onto the increment of the edge displacement in range of reversible and nonreversible displacements for an elasto/visco-plastic body. A differential form of the friction function was laid out and the conditions of slip and stick of material in area of contact were qualified. The models obtained are used in a variational formulation of the contact problem with application of the finite element method and to a numeric analysis of fabrication processes e.g. processes of burnishing rolling and grinding processes.

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