A 3 D contact investigation of rough surfaces considering elastoplasticity

In this work, the non-penetration condition and the interface models for contact, taking into account the surface microstructure, are investigated in detail. It is done using a homogenization procedures presented by Bandeira, Wriggers and Pimenta (2001a), to obtain by numerical simulation the interface behavior for the normal and tangential contact pressures, based on statistical surface models. The contact surfaces of both bodies are rough. This paper can be regarded as a complementary study to that presented by Bandeira, Wriggers and Pimenta (2006). Here, the plasticity of the asperities is taken into account by assuming a constitutive equation based on an associated von Mises yield function formulated in principal axes. The plastic zones in the microstructure are shown to study in detail the contact interface. Numerical examples are selected to show the ability of the algorithm to represent interface law for rough surfaces, considering elastoplastic behaviour of the asperities.


Introduction
Several formulations concerning the treatment of the contact interface have been presented in the literature, especially when the contact interface of two contacting bodies is rough.Constitutive equations for the normal contact have been developed by investigating micromechanical behavior within the contact surface.Associated models have been developed based on experiments (GREENWOOD; WILLIAMSON, 1966;EVSEEV;MEDVEDEV;GRIGORIYAN, 1991;ZAVARISE;SCHREFLER;Wriggers, 1992;YOVANOVICH, 1981;KRAGELSKY;DOBYCHIN;KOMBALOV, 1982).In general, the micromechanical behavior depends on material parameters, like hardness, and on geometrical parameters, like surface roughness.
It should be noted that the real micromechanical phenomena are extremely complex due to extremely high local pressure at the asperities.The model used in this paper attempts only to capture the most important phenomena and assume either elastic or plastic deformation of the asperities having real contact in the interface.This article concentrates on the behavior of the contact interface.The idea is to study the interface behavior by modeling the contact surfaces, using a finite element discretization, to take into account the geometrical properties of the microstructure.The probabilistic theory is applied based on a statistical model of the micro geometry, like in the microscopic contact mechanics developed by Greenwood and Williamson (1966), and by Wriggers and Vu Van and Stein (1990).Finally, a simple homogenization leads to a contact interface law.
The finite element program is based on a C++ code, developed by Bandeira, Wriggers and Pimenta (2006) , 1982).
In this paper, the current normal approach g N is investigated in detail.It can be defined by where ξ denotes the maximum initial asperities height and d the current mean plane distance.
Thefigureal interpretation of equation (2.1) is illustrated in Figure 1.
The normal contact force F N is obtained as a product of the apparent pressure by the apparent contact area A, as follows: (2.2) where c N is defined by the penalty parameters of the augmented Lagrangian algorithmic.Following relationship correlates the current normal approach g N with the apparent mechanical pressure P N : (2.3) The mechanical constants mentioned in (2.2) depend on the micromechanics of the surface.This constitutive equation was presented in Zavarise, Schrefler and Wriggers (1992).

A simple homogenization method for contact interface
The basic aim of this paper is to derive constitutive contact laws, as stated in Section 2, for a rough surface by using the finite element method.
For this purpose, one has to model and discretize the rough surface and then, by homogenization procedures, to develop an interface law for contact.This section summarizes the homogenization method leading to the contact interface laws.The interface law is obtained from numerical simulation, using a model that consists of two deformable bodies in contact (Figure 2).
The contact surfaces of both bodies are rough.A representative surface at the contact interface is represented by the Figure 3.
The procedure to find the associated interface law is performed in several steps.The bodies are discretized using standard hexahedral finite elements.One body is placed above another with initial distance to separate them.The inferior body is kept fixed in position and superior body is moved   towards the inferior body by a displacement imposed on the top surface A. The fixed block is defined as master body and one that is in motion is defined as slave body.The prescribed displacement is applied in several increments.In each increment, the resultant force R N at the top of superior body is calculated by summing up the reaction forces R Nk on the surface related to each node k.Then, The existence of a reaction force R N indicates the first contact between the bodies.The maximum initial asperities height ξ between the two middle planes, which are in contact, is determined at this step.See Figure 1 for the geometrical relations.
In general it is difficult to determine the actual discrete contact areas and to compute F N from (3.2).Here a different procedure is followed.
Considering the equilibrium of the bodies, it is clear that the normal contact force developed at the interface is equal to the normal reactions developed at the top of superior body.Therefore, within the finite element treatment it is sufficient, instead of computing (3.2), to calculate the normal reaction force R N .Hence: The total contact force F N at the interface can be distributed on the apparent contact area A to yield a uniform apparent contact pressure, p N .
Therefore, with equation (3.3), Since R N depends on the current mean plane distance d, the penetration law is displayed as  The generation of the smoothen contact surfaces with Bézier interpolations (FARIN, 1988), the procedures to obtain a statistical law and the techniques to obtain the maximum initial asperities height ξ for a generated surface were presented in detail in Bandeira, Wriggers and Pimenta (2001b).

Constitutive equation
The von Mises elastic-plastic constitutive law is based on the following multiplicative decomposition of the deformation gradient: where the superscript e and p describe the elastic and the plastic part, respectively.The elastic logarithmic strain tensor is given by The logarithmic isotropic linear elastic material simplifies the volumetric-isochoric splitting.
Note also that (4.5) is similar to the expression of the small strain Hooke's Law.The fourth-order tangent tensor is obtained from (4.5), as shown in detail in Pimenta (1992).
For computational purposes, the classical radial return algorithm along with the von Mises plasticity; with linear isotropic hardening is summarized below.
1) Trial step: 1.The assumed linear isotropic hardening used in this paper is presented in the Figure 5.
Its function behavior is defined as following: where s yo is the initial yield stress, a = e p is the equivalent plastic strain and h is the linear hardening parameter.

Example 1
In this first example, the number of mas-  After all generated surfaces were analyzed, the mean value curve of the normal pressures is depicted (Graphic 1), which represents the constitutive interface law for different hardness.
The plastic zone developed at the master surface can be analyzed in each increment of loads by the equivalent plastic strain, presented in the algorithmic for elastoplasticity stated in Section 4 (see Figure 7).

Example 2
The second example is the same of the first one.The differences are that the maximum initial asperities height ξ is 0,180394 mm, and the load consists of a uniform displacement of 8 mm prescribed at the top of the slave block in several increments.Each analysis ends when the current mean plane distance d approaches zero.The mean plane distance goes to zero in the 35 th increment of loads.After all generated surfaces were analyzed, the mean value curve of the normal pressures is depicted in Graphic 2, which represents the constitutive interface law for different hardness.
The plastic zone developed at the master surface can be analyzed in each increment of loads by the equivalent plastic strain (Figure 9).

Example 3
The last example has the same discretization of the first one.The difference is that the maximum initial asperity height ξ is 0,226265 mm and the applied load is done in two steps.
A uniform displacement of 0,24 mm is prescribed at the top of the slave block in several increments in the vertical direction.Each analysis ends when the current mean plane distance d approaches zero.The mean plane distance goes to zero in the 35 th increment of load.After that, another increment of load, of 0,10 mm, is applied in the tangential direction, paralleled  References ALART, P.; CURNIER, A. A mixed formulation for frictional contact problems prone to Newton like solution methods.Computational Methods in Applied Mechanics and Engineering, Lausanne, Switzerland, v. 92, Issue 3, p. 353-375, 1991.BANDEIRA, A. A.;WRIGGERS, P.; PIMENTA, P. M. Computational analysis of contact mechanics undergoing large 3D deformation.In: EUROPEAN CONFERENCE ON COMPUTATIONAL MECHANICS0, 2001, Krakow, Poland: ECCM, 2001a.
The normal contact force F N at the contact interface is obtained by taking into account the real contact area A r .The actual contact occurs at n-discrete areas A n on the discrete boundary G n c , as shown in Figure 4.This yields A r = n c ∑ n = 1 A n when n c asperities are in contact.The real contact pressure t Nn occurs on the discrete contact surface G n c .The actual normal contact force F N is obtained by summing up the integrals of the real normal contact pressures t Nn related to the discrete area A n at each boundary G n c .Then: At the end of each step, the current mean plane distance d, the total reaction force R N and the apparent contact pressure p N are calculated.The analysis ends when the current mean plane distance d goes to zero.The numerical results after homogenization yield a microscopic contact law analogous to the theoretical law, presented in Section 2. The homogenization is computed in terms of the average normal pressure p N .The plotted curve describes the penetration law relating the
lnV e , where V e = F e F eT ) stretching tensor.The elasticity is described by the following strain energy function y(E e ) = 1 2 kj 2 + m(E e : E e ) e and E e = DevE e (4.4) that leads to the following Kirchhoff-Treftz stress tensor S = D e E e , where D e = kI I + 2m( In this section, three numerical examples are presented to obtain an interface law for rough surfaces numerically.In these examples two blocks are considered in contact as shown in Figure 6.This is done for three-dimensional bodies in contact.The homogenization method used was presented in Section 3. The Preconditioned Bi-Conjugate Gradient Method (PBCG) is used to solve the linear equations system (PRESS, 1995).The PBCG method is used because it is more efficient, spend less computer memory to allocate the matrixes and solve linear system with non-symmetrical matrixes.The elastic-plastic material law presented in Section 4 is used in all examples.It is important to mention that each numerical laws are statistically computed curves resulted by 20 different random generated contact surfaces.
of 90 mm, in thirty increments.The penalty parameters used for the normal direction is around 10 5 and for the tangential direction is around 5×10 4 .The friction coefficient is 0,20.After this analysis, the mean value curve of the normal pressures is depicted in Graphic 11, which represents the constitutive interface law for different hardness.In this analysis the normal and tangential pressures are plotted over the displacement in each increment of load.The contact surfaces are smooth that makes the pressure very sensitive regarding the hardness.The normal and the tangential pressures behaviour are presented in Graphic 12a and 12b, respectively.6 Final considerations This work can be regarded as a complementary study of Bandeira et al. (2006).In that paper, the plasticity of the asperities is taken into account by assuming a constitutive equation based on associated von Mises yield function formulated in principal axes.The authors can concluded that is possible to modeling the micromechanical phenomena developed at the contact surfaces and obtained results for constitutive equations derived by numerical simulations with good agreements with the theoretical laws.In the numerical examples, the normal and tangential contact pressures are plotted to show the contact surfaces behaviour.The plasticity evolution developed at the contact surfaces are also presented in the numerical results.It can be seen that the contact surfaces developed high plasticity effects during the contact mechanics.