Finite Element Analysis of Frictional Contact Problem During the Process of Metal Working

The quasistatic contact problem with dry friction, contributes to the achievement of a numerical model for the simultaneity of different kinds of viscoplastic, incompressible materials forming. Starting from a variational formulation of the problem, which extends the static case, a perturbed Lagrangean, discrete formulation within the framework of finite element method is obtained. On the contact boundary we have used a method based on contact finite elements, with three nodes, for the unilateral contact conditions with the friction law. One numerical example is presented.


INTRODUCTION
The paper presents a formulation of the basic approach for finite element modeling of material forming processes with a viscoplastic, incompressible and frictional contact approximation.
Assume that the body is subject to volume forces while the input and output velocities are given and on a part of the boundary it is in a unilateral frictional contact with a rigid fixed support. Our interest is focused on the discretization by finite element of the contact area, such as the geometry of the contact area, the contact conditions and the frictional law is well approximated. The main numerical problems presented are: variational problem, evolving contact friction, modeling with finite contact element, incremental approach, meshing and remeshing of the domain, solving the non-linear set of equations using Newton-Raphson method.
The problem is also reduced to a set of optimization constrained problems. We solve this problem by using the Lagrange multipliers method. In order to illustrate the physical model the quasistatic problem is considered. The purpose is to find such velocities for the given boundary and constrained conditions that the forming processes are steady-state Γ it is in unilateral contact with a rigid fixed support.

MATERIALS and METHODS
We use the following notations for the normal and tangential components of the velocities and of the stress vector: ( , ) n n n = are the outward normal unit vector on Γ and the summation convention are used for i and j.
The contact problem with friction law of elastoviscoplastic incompressible material model as the following classical formulation [1] : Find the field of velocities , defined Ω , which satisfy the following equations and conditions}: the equilibrium equation -the constitutive equation Open Access

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-the boundary conditions for field velocities: The classical formulation of this problem is as follows: Problem P1: Find the velocities v v µ f and η are given.
It is known [2] that a variational formulation of the Problem P1 is the following inequality: a .i. on , a .i. on , div 0 on , and 0 a .e. on }, , the penalty parameter of the unsatisfactory of the incompressibility condition.
Finite element discretization of the problem: We consider a discrete variational problem P2 using four nodes isoparametric finite elements on Ω , and three nodes contact element [3,4] on 1 Γ .
The contact finite elements with 3 nodes (see Figure 1) is a bilinear isoparametric element which associates one gap with a typical slave node s is given   In all numerical applications we have derived a perturbed Langrangean formulation for the case of frictional stick and for the case of frictional slide. For the case of frictional stick the perturbed Lagrangean functional for bodies in contact has the following form: ( , , , ) ( ) where v is the vector of nodal velocities n , t , are the vectors of normal and tangential nodal contact forces, respectively, G n and G t are the vectors of normal and tangential nodal gaps, and n ω , t ω are the normal and tangential penalty parameters respectively, ( ) v ∏ are the total potential energy, ε is the penalty parameter of the unsatisfaction of the incompressibility condition and D v is the vector obtained from the incompressibility condition.
For the case of frictional slide the relation t n µ = must be considered, as a direct consequence of the Coulomb's friction law.
The Newton-Raphson method was applied to the discrete variational formulations that can be derived from these perturbed Lagrangean functional.
In the three dimensional case [7] , a four node contact element consisting of three "master" nodes 1, 2, 3 and one "slave" node's, is employed (see Figure 2). Normal vectors on a defined plane by the nodes 1, 2 and 3 and respectively the vectors, defined by directions of the nodes 1-2 and 1-3 will be: 3 3 x X u = + signify the current positions of master nodes; X 1 , X 2 , X 3 are reference coordinates and u 1 , u 2 , u 3 are current nodal displacements of points 1, 2 and 3.
In addition, we define the current "surface coordinate" as following: Note that the gap g depends on the slave node s as well as on the master nodes 1, 2 and 3. Thus, the variation of the gap is obtained according to 1 With respect to finite element implementations, explicit matrix expressions for the Lagrangean multiplier formulation and the penalty formulation are derived as follows. The discrete variational equation associated with (12) take the form: where 1 2 ( ) ( ) ( ) Π = , − u a u u L u is the total potential energy of the bodies in contact, 1 2 , T s u n u n u n u n G g g … g δ δ δ δ = , , ,  Similarly, the variation of a typical nodal tangential gap t t g G ∈ , g G τ τ ∈ can be obtained according to Moreover, the residual vector B R and the tangent stiffness B K associated, with the total potential energy of the contacting bodies simply read, result ( ) and With, the convention: Finally after the discrete formulation within the framework FEM, a standard assembly procedure can be used to add the contact contributions of each contact element to the global tangent stiffness and residual matrix and thus we obtain:   [7] .

RESULTS and DISCUSSION
Based on these matrices, a standard assembly procedure can be used to add the contact contributions of each contact node to the global tangent stiffness and residual.
We consider [5,6] a simplified model with the geometry and loading given in Figure 3, a discretization with 32 nodes and 20 finite elements with four nodes isoparametric and for contact boundary, three nodes contact element [4] . Data of the problem:   (7) in the discretization variant had been observed. Only three iterations are enough to obtain a relative error less than one percent. The problem can be extended in the three-dimensional case and for crack analysis [8] .