On a geometrical approach in contact mechanics

Transcription

1 Institut für Mechanik On a geometrica approach in contact mechanics Aexander Konyukhov, Kar Schweizerhof Universität Karsruhe, Institut für Mechanik Institut für Mechanik Kaiserstr. 12, Geb Karsruhe Te.: +49 (0) 721/ Fax: +49 (0) 721/ E-Mai: ifm@uni-karsruhe.de

2 On a geometrica approach in contact mechanics Aexander Konyukhov, Kar Schweizerhof 2005 Abstract. The focus of the contribution is on a detaied discussion of the 2D formuation in contact which can be viewed either as a reduction of the genera 3D case to a specia cyindrica geometry, or as the contact of 2D bodies bounded by pane curves. In addition, typica frictiona characteristics, such as the yied surface and the update of the siding dispacements aow a geometrica interpretation in the chosen coordinate system on the contact surface. 1 Geometry and kinematics of contact In the iterature various contact descriptions adapted for an effective finite eement impementation are avaiabe, which can be characterized by the foowing: from 2D to 3D formuations, from non-frictiona to frictiona contact. One of the first contributions on finite eement soutions of 2D frictiona contact probems based on an easto-pastic anaogy has been made by Wriggers et. a. [1]. Genera overviews over contact conditions and contact agorithms which are nowadays used in practice, are covered by the books of Wriggers [2] and Laursen [3]. The covariant description, see Konyukhov and Schweizerhof [4] for the frictioness case and [5] for the frictiona contact, has been found as a universa penaty based approach for contact with various approximation of the surfaces. In the current contribution, we wi show the unity of 2D and 3D formuations, where the 2D case can be derived, from one hand, as a simpified case of the particuar 3D geometry of contact surfaces and, from the other hand, can be constructed separatey based on the differentia geometry of 2D pane curves. Considering a specia contact case contact between two cyindrica infinite bodies with pane strain deformations, see Fig. 1, eads to the definition of a 2D contact. In this case a generatrix GH of the first cyindrica body is a contact ine and corresponds to a contact ine G H which is aso a generatrix but of the second cyindrica body. Thus, 3D contact which can be seen as an interaction between two surfaces is reduced to an interaction between two boundary curves in the 2D case. One of both boundary curves resp. surfaces is chosen as the master contact curve resp. surface. A coordinate system is considered on the boundary, either for a surface in 3D or for a curve in 2D. On the pane we define a curviinear coordinate system associated with the curve by introducing two principa vectors as a basis: the tangent vector ρ ξ = 1

3 Figure 1: Two dimensiona contact as a specia case of three dimensiona contact ρ ξ and the unit norma vector ν. Then a save contact point S is found as r s (ξ, ζ) = ρ(ξ) + ζν(ξ). (1) The norma unit vector ν in the case of arbitrary Lagrangian parameterizations with ξ can be defined via a cross product in a Cartesian coordinate system as: ρ ξ = ρ ξ ; = ν = [k ρ ξ] ρξ ρ ξ, (2) where k is the third unit vector in this Cartesian coordinate system. According to this definition the traction vector R s is defined via the contravariant basis vectors: R s = T ρ 1 + Nρ 2 ρ ξ = T + Nν. (3) (ρ ξ ρ ξ ) The decomposition of the traction in eqn. (3) eads to the foowing contact integra δw c = (Nδζ + T δξ)d. (4) A coser ook reveas that the contact integra (4) contains the work of the contact tractions T and N defined on the master contact curve and is computed aong the save curve s. 2

4 2 Reguarization of the contact tractions For the norma traction N, the foowing reguarized equation in the cosed form is taken N = ɛ N ζ, if ζ 0, (5) where ɛ N is a penaty parameter for the norma interaction. As a reasonabe equation for the reguarization of the tangent traction T we choose a proportiona reation between the fu time derivative dt and the dt reative veocity vector expressed in covariant form on the tangent ine ζ = 0: D 1 T dt ρ1 = ɛ T ξρξ. (6) Expressing the covariant derivative in eqn. (6) the foowing equation is obtained dt dt = ɛ T (ρ ξ ρ ξ ) ξ + ρ ξξ ρ ξ (ρ ξ ρ ξ ) ξ h 11 a 11 ζ, (7) where ɛ T is a penaty parameter for the tangentia interaction, a 11 and h 11 are components of the metric tensor resp. of the curvature tensor for the cyindrica surface. If a path ength s = ξ is used for parameterization, then eqn. (7) is transformed as dt dt = ɛ T ṡ κ ζ, (8) where κ is a curvature of the master contact pane curve. 3 Linearization for a Newton type soution scheme The derivation of the contact matrices can be performed either according to the cyindrica geometry, or according to the geometry of the pane curves. The atter provides a straightforward geometrica expanation of each part of the contact matrix and can be used for the judgment of their necessity within the soution process. Thus, the inearization of the norma part in eqn. (4) given by the foowing integra: δwc N = Nδζd, (9) eads to the foowing. D(δW N c ) = ( dn dt ) dδζ δζ + N d = dt = ɛ N (δr s δρ) (ν ν)(v s v)d (10) 3

5 ( ɛ N ζ δτ (ν τ )(v s v) + (δr s δρ) (τ ν) τ t ) d (10 a) ɛ N ζκ(δr s δρ) (τ τ )(v s v)d. (10 b) The form written via the path ength aows a simpe geometrica interpretation of each part in eqn. (10) and even aows to estabish situations where some of the parts are zero. The first part eqn. (10) is caed main part and defines the constitutive reation for norma contact conditions. The second part eqn. (10 a) is caed rotationa part and defines the geometrica stiffness due to the rotation of the tangent vector of the master curve. It disappears when a master segment is moving in parae, because ony in this case the derivative of a unit vector τ becomes zero. The third part eqn. (10 b) is caed curvature part. This part disappears when the curvature κ of a master segment is zero, i.e. in the case of inear approximations of the master segment. The structure of the tangentia part for the sticking case is as foows: D v (δwc T ) = ( dt dδξ δξ + T dt dt ) d = ε T (δr s δρ) (τ τ )(v s v)d (11) [ T (δr s δρ) (τ τ ) τ ] t + δτ (τ τ ) (v s v) d (11 a) + κ(δr s δρ) (τ ν + ν τ ) (v s v)d. (11 b) Here T is a tria tangent traction computed from the discrete evoution equation (6) at each oad step, e.g. under the assumption that the save point obeys the eastic deformation aw. One can see that the symmetry is preserved for the fu sticking case for any curviinear geometry of the contact bodies. If siding is detected, i.e. if T > µ N, then the siding force is computed according to Couomb s friction aw within the return-mapping scheme, see [2] and [3]. We aso keep a covariant form: T s = µ N T tr T tr = µ N (ρ ξ ρ ξ ) 1/2 sgn(t tr ). (12) The inearized contact integra for the tangentia part in the case of siding has the same geometrica structure, but contains non-symmetric parts due to the non-associativity of the Couomb friction aw. 4

6 Detais for the impementation of the inear contact eement as we as for the computation according to the return-mapping scheme can be found in Konyukhov and Schweizerhof [7]. 4 Treatment of specia cases Some specia cases can appear when a direct appication of the return-mapping scheme can ead to improper resuts. The first probem is arising when the appied oad is not simpy modified proportionay. In this situation a tria oad can not be computed ony via the evoution equation, because the attraction point ξ 0 must be updated. Thus we have to extend the agorithm as is shown in the foowing. The second probem is arising when the projection point is crossing an eement boundary during the incrementa oading. In this case, the computation according to the incrementa evoution equation wi produce a jump, because the convective coordinate ξ beongs to different eements. 4.1 Update of the siding dispacements in the case of reversibe oading A geometrica interpretation of the tria step in the return-mapping scheme eads to the definition of the eastic region with an attraction point ξ 0 T (n) tr < µ N (n) a 11 = ɛ T ξ (n) ξ 0 < µ N (n) (13) If a point ξ (n+1) appears to be outside of the domain at oad step (n + 1), then its ony admissibe position is on the boundary of the domain. A siding force is appied then at the contact point. As ong as we have a motion of the contact point ony in one direction the sign function for the siding force sgn(t (n+1) tr ) = sgn( ξ (n+1) ) does not change and the computation wi be correct. However, when a reversibe oad is appied and it forces the contact point to move forward or backward, the attraction point ξ 0 must be updated in order to define the sign function for the siding force correcty. This update can be defined geometricay from Fig. 2: ξ (n+1) = ξ (n+1) µ N (n). (14) ɛ T Thus, computation of the tria force at the next oad step (n + 1) has to be made in accordance to the update procedure, see more in Konyukhov and Schweizerhof [7]. 5

7 Figure 2: Couomb friction. Updating of siding dispacements in convective coordinates. Motion of friction cone and center of attraction. 4.2 Crossing an eement boundary continuous integration scheme Consider two adjacent eements AB and BC, see Fig. 3. If we foow the computation of the tria force according to the formua expressed in convective coordinates as T (n+1) tr = ɛ T a 11 (ξ (n+1) ξ (n) ), (15) then this eads to a jump in the contact tractions. The maximum of the jump is computed as ( ) T jump = ɛ T a 11 im (ξ) ξ BC im (ξ) ξ AB = 2ɛ T a 11. (16) ξ 1+0 ξ +1 0 This jump appears ony due to the different approximation of the adjacent eements. In order to overcome this, we can compute the force in geometrica form. Foowing this procedure the incrementa coordinate ξ can be expressed as ξ = ( ρ(ξ (n+1) ) ξ BC ( ρ(ξ (n) ) + u(ξ (n) ) ) ) ξ AB, ρ (n+1) ξ a (n+1) 11 (17) where u is an incrementa dispacement vector. The evoution equation becomes then T (n+1) tr = T (n) ɛ T a (n+1) 11 ξ. (18) In the 2D case, the computation can be made via the ength parameter s eading to the continuous scheme as we, see Wriggers [2]. 5 Numerica exampes 5.1 Siding of a bock. Linear approximation of the contact surfaces. Reversibe oading process. We consider the siding of an eastic bock on the rigid base oaded with horizontay prescribed reversibe dispacements, see Fig. 4(a). The geometrica 6

8 Figure 3: Crossing an eement boundary within a oad increment. Typica case for the continuous integration scheme. and mechanica parameters are easticity moduus E = , Poisson ratio ν = 0.3, ength a = 20, height b = 5, Couomb friction coefficient µ = 0.3. The penaty parameters are chosen as ε N = ε T = The main point is to show the update procedure for siding dispacements. The oading is appied as prescribed dispacements at the top side of the deformabe bock. For contact the node-to-segment approach is taken. The hysteresis curve representing the computed horizonta dispacement at point D vs. the appied dispacement at point C is given in Fig. 4(b) G F6 F F4 F3 F2 F 5 1 F h u 2 u 1 B A v C B 1 C 1 D b Observed dispacement (pt. D) H O Appied dispacement (pt. C) O 1 O2 (a) Pane deformation of a bock. Appied dispacement oading at top of the bock. (b) Hysteresis curve. Observed horizonta dispacement at point D vs. appied horizonta dispacement at point C. 5.2 Drawing of an eastic strip into a channe with sharp corners. A particuar exampe in which appication of the continuous integration is absoutey necessary is a deep drawing of an eastic strip into a channe with sharp corners, see Fig. 4. The crucia point during the anaysis is the siding of a sharp corner C over the eement boundaries 1, 2, 3. A oad-dispacement curve computed for the oading point is chosen as the representative parameter to compare various contact approaches. We obtain, see aso [7], that the appication of the segment-to-segment approach, as described in [6], without the 7

9 continuous integration scheme aows to compute the force-dispacement curve ony for non-frictiona cases. For frictiona cases a more carefu transport of the history variabes is necessary as it is suggested here. 6 Concusions Figure 4: Drawing of an eastic strip into a channe with sharp corners. In this contribution a convective description is reconsidered for the 2D quasistatica frictiona contact probem. Specia attention is paid to the derivation of the necessary equations either as a reduction of the known 3D covariant formuation, or directy from the specia 2D cyindrica geometry of the contact surfaces. The agorithmic inearization in the covariant form aows to obtain the tangent matrices before the inearization process. Thus, an impementation can be easiy carried without providing any specia attention to the approximation of the contact surfaces. Further it is shown that specia agorithmic techniques have to be taken to provide robust answerers for oad reversion and for the crossing of eement boundaries. References [1] Wriggers P., Vu Van, Stein E. Finite eement formuation of arge deformation impact-contact probem with friction. Computers and Structures, 37, pp , [2] Wriggers P. Computationa Contact Mechanics. John Wiey & Sons, [3] Laursen T. A. Computationa contact and impact mechanics. Fundamentas of modeing interfacia phenomena in noninear finite eement anaysis. Springer, [4] Konyukhov A., Schweizerhof K. Contact formuation via a veocity description aowing efficiency improvements in frictioness contact anaysis. Computationa Mechanics, 33, pp ,

10 [5] Konyukhov A., Schweizerhof K. Covariant description for frictiona contact probems. Computationa mechanics, 35, 3, pp , [6] Harnau M, Konyukhov A, Schweizerhof K. Agorithmic aspects in arge deformation contact anaysis using Soid-She eements. Computers and Structures, 83, , [7] Konyukhov A., Schweizerhof K. A specia focus on 2D formuations for contact probems using a covariant description. submitted to Internationa Journa for Numerica Methods in Engineering. 9