A segment-to-segment contact strategy

Mathl. Comput. Modelling Vol. 28, No. 4-8, pp. 497-515, 1998 @ 1998 Else~ier Science Ltd. All rights resewed Printed in Great Britain 08957177/98 $19.00 + 0.00 PII: SO895-7177(98)00138-l

A Segment-to-Segment

Contact Strategy

G. ZAVARISE Dipartimento di Costruzioni e Trasporti Via Marzolo 9, I-35131 Padova, Italy P. WRIGGERS Institut fiir Mechanik, Technische Universitgt Hochschulstr. 1, D-64289 Darmstadt, Germany Abstract-In this paper, a method is proposed to define the geometrical contact constraints. Within this treatment one has the possibility to define locally the contact parameters for an accurate treatment of contact constraints. Local values of the geometrical variables can be determined at the integration points, hence the method permits to integrate contact constitutive laws along contact segments. The weak form for this new formulation is developed. Furthermore, also the consistent linearization is carried out. Finally a technique is proposed to reduce the large number of terms involved. In this case, an almost consistent tangent stiffness is determined. @ 1998 Elsevier Science Ltd. All rights reserved. Keywords-contact

mechanics,FEM, Consistent linearization. Contact geometry, Penalty. 1. INTRODUCTION

Numerical analysis of contact problems is moving towards more sophisticated formulations. Nonlinear contact constitutive relationships are used for mechanical and thermal problems [1,2]. Frictional phenomena are also treated with microscopical based formulations [2-51. The correct employment of such contact constitutive laws requires a detailed geometrical formulation, to obtain the same degree of accuracy for both the constitutive laws and the contact geometry discretization. If a poor contact geometry is combined with nonlinear contact constitutive laws, the results obtained will be reliable only if the contact geometry is very simple, and the geometrical simplifications involved will not affect the results strongly. The extension [2,4] of the basic node-to-segment formulation [6] has been widely used [2,4,7], and it has given good results for simple contact geometries. However, it is evident that the involved hypothesis of constant pressure within the contact element area is a rough approximation. Moreover, it is well known that due to the master and slave identification of the contacting surfaces the contact detection may fail for special geometries. The technique of double-pass which exchanges master and slave surfaces can avoid this problem. However, it cannot be easily applied when the contact stiffness matrices have a physical meaning. Alternative methods for the definition of contact geometrical variables are reported also in [8,9]. A rigorous treatment at the continuum level of contact between two surfaces has been presented in [lO,ll]. This approach still maintains the distinction between master and slave surface, and the contact detection still may fail when the discretized version is considered. Supportfor this work was partially provided by CRE, under VIGONI Program, this support is gratefully acknowledged. The authors would like to thank also R. L. Taylor of the University of California at Berkeley for providing a version of finite element code FEAP

for the development of the model.

497

G. ZAVAFUSE AND P. WRIGGERS

498

The above considerations have lead us to approach the formulation of the contact geometry directly baaed on the discretized surface.

2. PRELIMINARY

CONSIDERATIONS

The usual way to define the distance between two surfaces is obtained by determining the normal projection of each point of one surface onto the other one. This technique involves a lack of symmetry, as is clearly represented in Figure 1, where the distance of node A from the surface r2, AB, and the distance of node 3 from I’,

BC, is represented. Only when the two

surfaces are parallel the symmetry requirement is matched.

To obtain perfect symmetry we

have to consider a different definition of distance between the surfaces, e.g., we could define as local distance between two points of I” and I2 the length of the secant to the circle tangent to the surfaces in the considered points, as represented in Figure 2. In this way, the geometrical properties of both surfaces, i.e., the curvature, is involved in determining the centre and the radius of the circle. Due to the uniqueness of the identified circle, this method yields the same distance for points A and B.

Figure 1. Lack of symmetry tance between the surfaces nodal projection.

of the disdefined as

Figure 2. Example of symmetric distance between the surfaces.

definition of

Work is actually in progress to set up a discretized formulation using this approach. As a preliminary step, we will discuss here a simplified version that still leads to an asymmetric approach, but this asymmetry is compatible with the hypotheses adopted within the physical models of the microscopical contact stiffness, see later. Considering the physics of the problem, the basic step to define the contact penetration takes into account the deformation of contacting asperities. To define the contact normal stiffness we have to analyse the force-deformation behaviour with reference to the mean plane location. In this model contacting surfaces correspond to the mean asperity planes. Contact between the surfaces takes place when the highest asperities start to touch each other, as represented in Figure 3. Hence, incipient contact occurs when the two lines of the highest peaks come into contact, i.e., when the current distance, dN, of the mean planes is equal to the sum of the distances of the lines of the peaks from their mean lines

When the gap is closed, the two peak lines become the contact reference line. The relative displacement of this line with respect to the mean plane lines give us the deformation of the asperities of both surfaces. More and more deformations of contacting asperities take place

Segment-to-Segment

Figure 3. Contacting

with growing mechanical

contact

forces.

and geometrical

This method

surface and contact

We remark

that

characteristics

can be simplified

mechanisms at microscopic

only when the contacting

the contacting

considering

499

surfaces

line corresponds

that the micro-mechanical

level.

have the same

to the medium

contact

one.

model is based on

representative statistical values of both contacting surfaces. Contact between two rough surfaces can be transformed into the contact between a rough surface and a rigid plane [12], as depicted in Figure 4. In this case, the idealized rough surface is characterized by mechanical and geometrical values which take into account the characteristics of the original surfaces. If we adopt this concept at the element level, it gives a mechanical-equivalent base for a nonsymmetric treatment of the contacting surfaces.

Figure 4. Statistically equivalent transformation of the contacting surfaces.

The contact penetration plane and the rigid plane,

QN, is defined by the distance as represented in Figure 5, gN

The penetration value measures

Figure 5. Definition of penetration surfaces.

between

the surface

between the

mean

=co-dN.

asperities

(2)

function gN is negative when the gap is open. Once the gap closes its positive the amount of deformation of the contacting asperities.

3. SEGMENT-TO-SEGMENT We consider contact zones which are discretized above discussion, contact surfaces will be identified

CONTACT

GEOMETRY

by two-node segments. With reference to the by a Microscopically Rough surface (MR) and

a Microscopically Flat surface (MF). The contact area will be measured on the rough surface. To set up the geometry we define for each MR segment two lines orthogonal to the segment at nodes 1 and 2. This permits to identify the related contact area on the opposite surface, as shown in Figure 6. The choice implies that the contact surfaces are assumed to be smooth, hence the angle between adjacent segments is close to 180°, otherwise the lines bisecting the angles between adjacent segments have to be considered. However, the hypothesis is not restrictive because the difference between the two strategies diminish more and more with the contact penetration. Moreover, the relationship between penetration and segment length is usually around 10w4, hence the error is always very small. On the base of these considerations, we determine the segment zone using the two lines orthogonal to the segment passing through nodes 1 and 2. The related zone on the MF surface can be composed of one or more segments. Regarding the position of the nodes of the related segment, four possible cases can be distinguished for the

G. ZAVARISE AND P. WFUGGEFLS

500

j+3 Figure 6. Contact area related to the MR segment.

A

B I.

1

2

Figure 7. Segment-to-segment possible geometries

contact approach determination, as represented in Figure 7. In the following, they are identified by the symbols 01, 00, IO, II, where the letter 0 denotes a node outside the master segment ir& and the letter I denotes a node inside. For each segment i!?, we start identifying the geometrical characteristics of the segment, i.e., by determining the segment length 1,

l=)izI,

(3)

where i? represents the vector connecting node 1 and 2. The unit vector, tangent to the segment t, is then given by

t=E

1.

The normal vector, orthogonal to the segment, is oriented outside the segment, and thus, defined by n = es x t,

(5)

where e, ia the unit vector orthogonal to the plane. To determine the segments placed opposite to the MF surface, which are related to segment Tz, we start from node 1 and look for the closest node C,

min(Jh - d2 + (~1- d2) *

C(m,

UC),

(6)

where i is the generic node of the MF surface. The position of node C with respect to node 1 is identified by checking the projection onto segment E m*t

2 0 --) projection outside + C type is 0, < 0 -+ projection inside + C type is I,

(7)

where a is the vector connecting node C and 1. In the following, we adopt a numbering convention which increases the node numbers both on MR and MF surfaces moving along the

501

l 1

l

2

4

*

1

2

Figure 8. First segment search path examples.

contact with the material side on the right, see also Figure 6. Starting from node C, a search tree is explored to determine the first segment AB which type is OI or 00 I + search C + i with type 0 --t C type

0 + search C - i with type I +

A=C+i-1,

B=C+i, A=C-i,

09

B=C-i+l.

Some possible cases are depicted in Figure 8. The position of node A with respect to the segment E is then checked to determine the total number of MF segments related to i? A2.t

< 0 + projection outside 4 A type is 0,

(9)

> 0 --+ projection inside ---) A type is I.

If the projection is inside, then other segments should be identified. The other segments involved are determined with a search process similar to the previous one 0 -+ no more segments 0 t

A type

I+search 1

last segment

A-i (

A’“t=A-i, { Blast = A-i+l,

I 4 one more segment

A”=A-i, { Bn=A-i+l,

(10) -+ repeat.

All MF segments involved have different values of gN, (2), with respect to the MR segment. The evaluation of the contact weak form involving the constitutive laws within the MR segment hence should be performed by subintegrations. The subintegration length is computed using the projection of the MF nodes on the MR segment, and excluding the projections outside the MR segment, see also Figure 9. For each segment BA we have to define the corresponding integration limits along the segment i??, moreover, the penetration gN in a generic point should also be determined. The lower integration limit c=, is expressed by (,=(-Bid),

(11)

502

G. ZAVARISE ANDP. WRIGGERS

t

1

2

Figure 9. Example of segment-to-segments contact element geometry.

where MacAuley brackets are used to avoid negative values when the projection of node B is outside -iz. In a similar way, we define the upper integration limit 6,