Abstract. 1 Introduction

Transcription

1 Influence of the location of the contact point in node-to-point contact approaches using non conforming boundary element discretizations A. Blazquez & F. Paris Department of continuum Mechanics, E.S. de Ingenieros, Sevilla, Spain Abstract The study of contact between deformable bodies with non conforming discretizations inherently involves the possibility of a node having contact with an element at any position along it. This paper presents a study of the influence that the position along the element of this contacting point has on the results The numerical technique used is the Boundary Element Method and the classic problem of the compression of a deformable cylinder on a deformable foundation, which belongs to the class of advancing contact problems, will be used as a reference to evaluate the errors induced by the position of the contacting point. 1 Introduction The first studies on the application of the Boundary Element Method to contact problems (the papers of Andersson et al [1,2], Paris et al [3,4], Takahashi et al [5], Garndo et al [6], Foces et al [7] can be mentioned among others), required identical discretizations to be performed along the potential contact zones of the two bodies involved in the contact. In this way the contact appears between pairs of nodes of the two bodies, which enables the equations that define the contact conditions (equilibrium, compatibility and friction) to be written in terms of variables of the system of equations: displacements and components of the stress vectors of the nodes of the discretizations. However, there are problems particularly those involving large or moderate displacements, where an approach based on compatible discretizations cannot be used, as soon as a certain amount or load has been applied. From the application of the first approach based on non conforming discretizations, Blazquez et al [8], other similar procedures have been published Olukoko [9], Huesmann [10], Paris et al [11], all of them differing slightly in the manner in which equilibrium conditions are imposed along the contact zone although these conditions always have to be forced between a node of one of the

2 92 Contact Mechanics HI bodies and a point on an element of the other body. In all cases one of the bodies (called body A hereinafter) controls the stresses along the contact zone, whereas the other body (hereinafter body B) controls the displacements. A complete comparative analysis of all the variants based on this approach is presented in BMzquez et al [12], where it is concluded that all the variants of this general approach present the same type of problems, particularly in the case of advancing contact problems with friction. The purpose of this paper is to develop a further study of the errors that inherently appear as a function of the relative position of the contacting node in the contacted element, in order to obtain, if possible, a solution to the aforementioned problems. The study will be restricted to a particular problem, a cylinder compressed against a foundation, following the scheme presented by Blazquezetal[8,ll]. 2 The Boundary Element Method The basic equation of BEM is Somigliana's identity, which expressed in incremental form at boundary points of a body occupying the domain D with boundary dd takes the expression: (l) Aui y Ati being respectively the incremental values of displacements and stresses along the boundary, ^(;t,y) and Tj(x,y) the displacements and stresses of the fundamental solution and C/,(%) the tensor of coefficients of the free term whose values depend on the local geometry at point x. If the boundary dd is replaced by an approximated boundary dd^ along whose elements a certain evolution of the displacements and stresses is assumed, expression (1) can be applied at a series of collocation points x (usually coincident with die nodes that define the evolution of the functions along the elements of the boundary), obtaining a system of equations in the form: HAu = GAt (2) In this equation His a matrix whose coefficients are integrations of the terms Tj(x,y) plus the free term, and G a matrix whose coefficients are integrations of the terms *Fjj(x,y), while Au and At are vectors grouping incremental values of displacements and stresses of the nodes of the discretization. All the results presented in this paper have been obtained with linear continuous elements (elements with the nodes placed at the extreme of the elements but capable of taking into account discontinuities in the stress vector). A more detailed explanation of these elements as well as of the method itself can be found in Paris and Canas [13]. 3 Contact problems with non-conforming discretizations Figure 1 shows the configuration of a general contact problem between two bodies A and B, with a common dd^ contact zone.

3 Contact Mechanics III 93 Figure 1: The contact problem. The system of reference. Assuming the dry friction Coulomb contact law, the contact zone can be subdivided into two zones: the adhesion zone dd^ where relative displacements are not allowed and which is characterised by the fulfilment of % < /i %, and the sliding zone dd^ where relative displacements are allowed and which is characterised by the fulfilment of t^\ = fi\ti\. The equations in incremental form that establish the contact conditions expressed in the local system of reference defined in Figure 1 are: Equilibrium: At*(y) = Atf(y] i=l,2 Compatibility of normal displacements: Auf(y) + Auf*(y) = 0 Friction law: > <?/>,, \yedd.'cd K = A,B In the approach followed in this study, Blazquez et al [8,11], equilibrium is forced when assembling the system of equations corresponding to body B, due to the fact that the integration constants that multiply to the contact stresses are calculated integrating along the projection of the elements A over the boundary of B, and not along the elements of B. Thus, system (2) for the body B adopts the form: (3) (4) (5) Hf < At (6) where the subindex / makes reference to the contact-free zone, and the matrices GOZ y Gcd represent the integrations of )/(*, y) along the projection of the elements of body A over the boundary of body B. Compatibility is established by expressing the displacements of each node of body A as a function of the displacements of the nodes of the element of B with which it has contacted. This is done using the shape functions, i.e by means of expressions such as: (7)

4 where the terms that appear in the equation are defined in figure 2. Contact Mechanics HI BodyB Figure 2: Compatibility conditions. Finally, the friction law is applied to the nodes of body A (which controls the stresses), an equation of compatibility of tangential displacements being established in the case of adhesion, and a relation between the normal and tangential components of the stress vector being established in the case of sliding. The other approaches to deal with non conforming discretizations differ from that presented here in the manner of forcing equilibrium equations, which is nevertheless always done in a strong form. In the comparative study developed in Blazquez et al [12] it is shown that the results obtained are very similar with all of them, the difficulties considered in the present paper, although never having been mentioned previously, appearing when any of these approaches is used. 4 The testing problem and the method of study The testing problem, shown in Figure 3 and solved with implicit symmetry, is the compression of an elastic cylinder on a rectangular elastic foundation, the properties of both bodies being indicated in the Figure. A friction coefficient ju=0.005 will be employed, giving rise to subzones of adhesion and sliding of similar sizes. P = 545 N/mm E* = 4000 N/mm* D* =4000 N/mm o no op 09 oo p '///////////////////////// 200mm.Q.QQ < Figure 3: Definition of the problem

5 Contact Mechanics HI 95 The correct solution of the problem has been estimated employing 48 elements along the contact zone, which has been extended to afinalsize of 4mm. A basic discretization with 12 elements uniformly distributed along this contact zone has been employed as a reference to estimate the influence of the position of the contact node. The study procedure, which can be envisaged in Figure 4, has been to select a node, %\ of the cylinder (the body selected to control the stresses), which in the case of conforming discretizations will contact with a node rf, of the foundation, the distance from the node n* of the cylinder to the closest node of the foundation, the node n, being null. The scheme to be followed is to place, instead of the two elements adjacent to this node, rf, three elements whose total length is equal to that of the two original elements and in such a way that the length of the intermediate element is equal to the length of either of the two original elements. Moving the intermediate element, different distances 8 from the node n* of the cylinder to the closest node of the foundation are obtained. n QT D B / Figure 4: Study procedure. The study will be performed on three types of nodes. The first will be one placed in the adhesion zone (node 4 placed at a distance from the vertical axis of 1mm), the second will be a node placed at the limit between the two subzones (node 6 placed at a distance from the vertical axis of 5/3mm) and the third will be a node placed at the sliding zone (node 9 placed at a distance from the vertical axis of 8/3mm). The results will be compared with the solution obtained using a conforming discretization with 48 elements. 4.1 A node belonging to the adhesion zone Figure 5 represents the distributions of stresses along the contact zone for different values of the relative distance 8/1 with reference to node 4 of the discretization. It can clearly be observed that the modification of the distance 8 seriously affects the results, which become worse as the distance from the node n* to the nodes of the foundation increases. On the other hand, the local effect of this modification is very clear, the solution at nodes placed far enough from that under consideration being unaltered. Also noteworthy is the virtual absence of effect of the modifications performed on the normal stresses, a fact which remains unaltered in all the numerical experiments carried out. In Figure 6 the errors that appear at the nodes of the contact zone as a function of the different relative distances 8/1 are shown. The reference values have been taken from a case modelled with 48 conforming elements along the contact zone.

6 96 Contact Mechanics HI Figure 5: Contact stresses modifying the position of a node of the adhesion zone node 1 A B node 2 - o e node 3 - D node 4 node 5 node o- -- node 7 node 8 node Q- - node node a) Errors in the normal component b) Errors in the tangential component Figure 6: Modifications in a node in adhesion. Errors in the contact stresses. It can be observed in these figures that the normal components are only slightly influenced by the modifications introduced, with 1.5% error versus 150% in the tangential components. The reason for these errors is that the farther from a node the contacting point of node n* is, the worse the definition of the displacements at such a point will be, in accordance with the collocation-at-node nature of the method and the piece-wise approximation performed. This, in turn, will introduce an alteration in the stresses in order to achieve a compatible solution. In the case of nodes belonging to the adhesion zone, both the normal

7 Contact Mechanics III 97 and the tangential components along the contact zone are related with the displacements, both thus being influenced by the modification carried out in the study performed. 4.2 A node placed at the limit between adhesion and sliding zone Figure 7 represents the distributions of stresses along the contact zone for different values of the relative distance 5/1 with reference to node 6 of the discretization, which is placed between the sliding and the adhesion zone s/s max Figure 7: Modification in the limit node. Contact stresses _ ^% 0.01 i ; node 6 ; : ' V ' ' I \ I \p't i : : ' : : I ; ' \. - I/ ^ i i ; : Y o f " **? -q> «? * i Y, 2 : o o J2 T"~4-»J 1 [ "n % f 1 rf? -K-"^ L..!...!"^'!...!^* a) Errors in normal component b) Errors in tangential components Figure 8: Modification in the limit node. Errors in contact stresses. The local character of the error is again observed, although in relative terms the influence zone of the errors goes farther in the adhesion zone. This has a clear explanation in the fact that whereas in the nodes of the adhesion zone the jt $ *7* U.O t t T : / : : node 5 02 / ' ' / i i I I oi ^4 i i ^^^ o, : \[ T h\^37~^ : \ 1 i node 6 01,,,,!,,,,! r,?>,tt-:f, 7-,-ri /1

8 98 Contact Mechanics HI U.UZD 0.02 ^ f T~ o 0.005, 4> ' o' 5^ n m U.UJ i inode 9 i i ; 0.02 I : ^xi-- J^-"* t i-f"t" ji 0.01 :..& """ i i :..< >'"" inode 9 j...- '? -~ = i i 4>y r 1 ^f - ^k- ^_. ^ ^ - ^ 1^.- ^^- ZZJ LU 5' r^ i * " ^r ~" ^'~'~"s-i---_-jr i ct ' L [, i i _, fc 0 f i-- i j ^ L*.-...".O --d 1...JT o i ' inode 5: : '^ -..^ n ode 10 I "0.03 > 1 i j% i : node 8! ~^~~ h- p "" j : I :.n n^ / /1 a) En'ors in normal component b) Errors in tangential component Figure 10: Modification in a sliding node. Errors in contact stresses. 5 Conclusions The most important conclusion of die study carried out is that the degree of conformity (or unconformity) of a discretization is directly related to the relative distance from the nodes of the discretization of the body that controls the stresses to the nodes of the discretization of the body that controls the displacements. The lesser this distance the smaller the errors introduced. It has been observed that the normal stresses are less sensitive to the lack of conformity of the discretizations performed, with acceptable results always appearing for them, those found for the cases analysed here being less than 2%. It has also been observed that the errors in the tangential stresses are greater in the nodes belonging to the adhesion zone, due to the fact that in these nodes the tangential stresses are directly related to the tangential displacements, the errors then increasing with the lack of accuracy in the definition of the displacements of the contacting point of the node. This is the reason why these algorithms present excellent results in frictionless cases. The local character of the effect of the lack of conformity in the discretization has also been observed. The errors have only affected the nodes closest to the one where the modification is performed. A refinement in the meshes generally leads to a better definition of the variables of the problem, which in turns leads to a decrease in errors. However, in view of die results obtained, the refinement of the mesh that controls the displacement seems to be the most direct way of improvement, due to the fact that this is the quickest way to diminish the relation 5/7, as is shown in reference [12]. The problem analysed here, not previously mentioned by other authors using an algorithm based on the general node-to-point contact scheme, is inherent to the non conforming character of the discretization as well as to the local form of application of the equilibrium. Although this paper shows how to smooth this effect, it is clear that its cancellation can only be achieved by the use of a procedure forcing the equilibrium in a weak form along the whole contact zone, Blazquez [14].

9 Contact Mechanics III 99 tangential stresses are directly related to the displacements, in the sliding zone the tangential stress is proportional to the normal stress. The distribution of errors versus 8/1 for this case is presented in Figure 8. The local character of the error originated by the modifications performed is again observed, as well as the lower sensibility of the normal components to the errors in comparison with the tangential components of stress. It is clearly observed that the nodes belonging to the adhesion zone are, with regard to the tangential stress, much more affected than those of the sliding zone, where, as has been said, the value of the tangential component is directly related to the normal component, there thus being no direct relation with the displacements. 4.3 A node belonging to the sliding zone To conclude the study, Figure 9 represents the distributions of stresses along the contact zone for different values of the relative distance 8/1 with reference to node 9 of the discretization, which belongs to the sliding zone. Figure 9: Modification in a sliding node. Contact stresses. Quite acceptable results are now obtained in all cases, with small errors both in the normal and in the tangential components. The representation of errors versus 8/1, which is shown in Figure 10, again presents all the features mentioned in the two cases previously considered. As in the former cases, the errors grow with the distance from the node n of the cylinder to the closest node of the foundation, but now, in contrast to the two previous cases considered, the errors in the normal and tangential components are of a similar order. Note that the maximum error in the tangential stresses (about 4%) appears at node 5, which is in adhesion and is not affected by the modifications performed. In view of what has been adduced above, the reason for the drastic reduction observed in the errors is now clear. When the nodes are in the sliding zone, their tangential component is proportional to their normal component, and there is therefore no direct relation with the displacements. 1.2

10 700 Contact Mechanics III Key words: boundary element, contact mechanics, non-conforming discretizations, node-to-point approach. 6 References 1. Andersson, T., Fredriksson, B. & Allan Persson, E.G. The Boundary Element Method applied to two-dimensional contact problems. Proc. 2nd Sem. Rec, Adv. BEM, ed. C.A. Brebbia CML PubL, Southampton, Andersson, T. The Boundary Element Method applied to two-dimensional contact problem with friction. Boundary Elem. Meth., Springer-Verlag, , Pans, F. & Garrido, J.A. On the use of discontinuous elements in two dimensional contact problems. Boundary Elements VII, ed. C.A. Brebbia, Springer-Verlag, vol. 2, 13-27, Paris, F. & Garrido, J.A. An incremental procedure for friction contact problems with the Boundary Element Method. Int. J. for Engng. Anal with Boundary Elements, vol. 6, ir 4, , Takahashi, S. & Brebbia, C.A. A boundary element flexibility approach for solving contact problems with and without friction. Engineering Analysis with Boundary Elements, vol. 9,3-11, Garrido, J.A., Foces, A. & Paris, F. B.E.M. applied to receding contact problems with friction. Int. J. Math, and Comput. Mod., vol. 15, n^ 3, , Foces, A., Garrido, J.A. & Paris, F. Three dimensional factional conforming contact using BEM. Boundary Elements XIII, C.M.P.-Elsevier Applied Science, , BMzquez, A., Paris, F., Canas, J. & Garrido, J.A. An algorithm for friction less contact problems with non-conforming discretizations using BEM. Boundary Element XIV, Eds. C.A. Brebbia, J. Dommguez & F. Pans, , "Sevilla, Olukoko, O.A., Becker, A.A. & Fenner, R.T. A new boundary element approach for contact problems with friction. Int. J. Numer. Meth. Engng., vol. 36, , Huesmann, A. & Kuhn, G. Non-conform discretisation of the contact region applied to two-dimensional Boundary Element Method. Boundary E/g/7%. Mgfk. XW, Ed. C.A. Brebbia, , Paris, F., Blazquez, A. & Canas, J. Contact problems with nonconforming discretizations using Boundary Element Method. Comput. & Struct., vol. 57, n- 5, , Blazquez, A., Paris, F. & Canas, J. Methods of applying contact conditions in a node to point form by means of BEM with non conforming discretizations, (sent for publication to Eng. Anal with Boundary Elem.) 13. Paris, F. & Canas, J. Boundary Element Method. Fundamentals and applications, Oxford University Press (in press). 14. Bl&zquez, A., Paris, F. & Mantic, V. Weak application of contact conditions with non conforming discretizations using BEM. (to appear)