CC BY
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УДК 539.612, 544.722.54
On the tensor of tangential stiffness in contact problems
Q. Li
Technische Universität Berlin, Berlin, 10623, Germany
In a nonsliding tangential contact, vectors of tangential force and tangential displacement are related by the tensor of tangential stiffness. Using the boundary element method, we calculate the principal values of the tensor of tangential stiffness for a number of contact shapes. Of special interest for applications is that the anisotropy of the tensor of tangential stiffness is generally relatively small, even for extremely anisotropic contact shapes.
Keywords: tangential contact, anisotropy, aspect ratio, boundary element method
О тензоре тангенциальной жесткости в контактных задачах
Q. Li
Берлинский технический университет, Берлин, 10623, Германия
При касательном контакте без скольжения векторы касательной силы и касательного смещения связаны тензором тангенциальной жесткости. С использованием метода граничных элементов найдены главные значения тензора тангенциальной жесткости для контактов различной формы. Особый интерес с прикладной точки зрения представляет тот факт, что анизотропия тензора тангенциальной жесткости, как правило, относительно мала даже для контактов с сильно выраженной анизотропией формы.
Ключевые слова: касательный контакт, анизотропия, соотношение сторон, метод граничных элементов
1. Introduction
Study of contact problems under a directed tangential traction is of great interest in such industrial fields as transmission of products on roller conveyor [1], wheel-rail contact [2] and many others. In the most practical cases, the contact patch is complicated (noncircular), so that a different value of tangential deformation will be generated if the tangential force is applied in varying directions in the contact plane, meaning that the tangential stiffness is orientation-dependent. A classic and most studied tangential contact area is the elliptic shape, such as in contact of the angled teeth of the helical gears [3], or two crossed wire cables [4]. It was firstly studied by Cattaneo [5] and Mindlin [6] independently. They gave a theoretical expression of tangential compliance between displacement-force relation along the principle axes of ellipse for the nonslip case. It was found that the tangential compliance along the minor axis is smaller than in the other principle direction, with exception of the case when the Possion's ratio is equal to zero (isotropic tangential stiffness). This problem was later
further studied for partial slip case under the assumption of Coulomb's law of friction, and also for some other directed tangential forces inclined to the principle axes [7-9]. Similar work was also carried out for dynamic tangential loading by Szalwinski [10] and for multicontact of elliptical shapes by Sevostianov [11]. In the most work of related studies, the surface deformation only in the direction of applied tangential force was considered, however, it is known that this force will also generate a component perpendicular to it, although it is very small in comparison with the other one. In 1989, Raoof presented a solution of tangential compliance of elliptical contact for arbitrarily directed tangential force, and then proposed a second order tensor of tangential contact compliance, in which the both parallel and perpendicular components of deformation are included [12]. However, the provided transformation is valid only for the case that the initial parallel and perpendicular components are the stiffness in the direction of principle axes. In 2000, Agartov gave an asymptotic solution of the stiffness tensor in the frictionless normal con-
© Li Q., 2017
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tact problem where a stamp indenter is loaded by the force and moments [13]. In the following recent decade, very rare literature can be found for the investigation of anisot-ropy of the tangential contact stiffness, especially for rough contact.
In the present paper, we give the complete expression of stiffness tensor valid for any complicated contact zone, and then numerically calculate the tangential stiffness for arbitrary contact shape in arbitrary direction using the boundary element method (BEM). This method was recently developed by Pohrt and Li for fast simulation of various three-dimensional contact problems including rough contact and partial contact [14], and was later further developed for adhesive contact [15] and viscoelastic contact [16]. The proposed tensor of tangential stiffness will be validated numerically for other forms of contact geometries. In this study, we make the following two assumptions: (i) no-slip (infinite large coefficient of friction): the indenters adhere to each other; (ii) elastic similarity of materials of contacting bodies:
1 - 2V1 = 1 - 2V 2, (1)
G1 G2
where G1 and G2 are shear moduli of both bodies, v1 and V 2 are Poisson's ratios. The latter ensures the independence of normal and tangential contact problems [17], so that only the geometry information of contact area is needed in the simulation of tangential contact.
2. Tensor of contact stiffness and its transformation
The deformation-force relation in tangential contact can be described by a second order tensor of tangential contact compliance
/ \ ux / + \ uxx + uyx ( J J xx J \ J xy ± X
uy \ y J K uyy + uxy , K Jyx Jyy , Fy \ y J
(2)
or
u = JF. (3)
Here uxx and uxy are the resulted deformations in x and y direction by the force Fx, and uyx and uyy by force Fy. The contact stiffness is the inverse matrix of the compliance
K = J-1 =
Ky
— Ky
xy
(4)
^yx yy ^
The transformation of the tensor in a new coordinate system (Fig. 1) is then
iv1
x
K =
—nn —
—5n —
n5
cos 0 - sin 0
Kx Ky
K
xy
K
v yx —yy J
sin 0 cos 0 cos 0 - sin
sin 0 cos 0
(5)
Fig. 1. Sketch of tangential contact with loading in a certain direction
— + — — - —
TyT Kxx yy . Kxx Kyy /OA\ I
—nn =-— +-— cos(20) +
nn 2 2
K + K
-^ sin (20),
Kn5 =
— — — — + —
—yx —xy + — yx —xy
cos(20)-
(6)
— — —
— xx —yy • /0A\
--— sin (20),
—5n = —yx —xy + —yx + —xy cos(20) -— - —
— xx —yy • s^fW
--— sin (20),
K + K K — K
TyT Kxx Kyy Kxx Kyy /OA\
% =---cos (20) -
K + K
--yy^_jxL sin (20).
This transformation rules can be found in the plane stress theory [18], therefore, the Mohr's circle can also be used to describe the Eq. (6) as well as the maximal and minimal values. In the following, we will numerically investigate the case for arbitrary forms.
3. Boundary element simulation of the tensor of contact stiffness
For pure tangential loading, the surface deflection of elastic half-space at any location (x, y) under a distributed tangential stress tx (x\ y) acting on an area A in x-direc-tion was given by Cerruti by use of potential theory [19]
uxx (x' y) 1
2nG
1 -v (x - x')2 - + v--T-^-
t x( x\ y/)dx/dy/,
1
and its components are
uxy ( x, y) =-
xyW 2nG xjjv( x - x/)( y - y)
A S3
tx ( x\ y)dxdy.
(7)
(8)
x
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We focus on the tangential displacements, so the deflection in normal z-direction is not presented here. The equations for the case of loading in y-direction can be obtained by analogy of Eqs. (7) and (8). In the boundary element simulation, Eqs. (7) and (8) are discretized into many rectangular elements (1024 x 1024 in this paper) and the influence matrix is obtained with the assumption of uniform stress on each element. The surface deflection can then be calculated for arbitrary stress on arbitrary contact area. The inverse problem, calculation of stress distribution necessary for generating a given deflection is also solved using conjugate gradient method. The details can be found in [14].
3.1. Anisotropic tensor of stiffness
For the nonslip contact, the stiffness is calculated as following. Let all elements of a given contact geometry A be displaced by a constant value unn = const in a certain tangential n-direction inclined with respect to the x-direc-tion by the angle 0, then we calculate the tangential stress distribution Tnn in this area. The tangential force is easily obtained by Fn = jA TnndA, and the contact stiffness is then Knn = Fn/unn. In the numerical simulation, due to the convenience of calculation of stiffness along the direction of meshing square grids, we rotate the shape of contact area by the angle -0 and keep the tangential displacement always in x-direction, instead of the above tangential movement in n-direction by an angle 0. In the following figures, the normalized contact stiffness KnnE* ¡(KZZG*) is used, where E and G are effective elastic and shear moduli
Kxx — Kz
(10)
E —-
and G —
4G
(9)
1 -V 2-v
Kzz is the normal stiffness. It is known that for a nonslip isotropic contact (circular contact area), the tangential stiffness has the following relation with the normal stiffness
Therefore, the values of normalized tangential stiffness in y-axis in the following figures will be around the value 1. It is noted that the calculation of tangential stiffness is independent of normal contact. The shape of contact area, for example the Fig. 2, a, could be generated from previous normal contact of a complicated rough indenter, or be a cross-section of a stamp indenter. Here normal stiffness Kzz is numerically calculated by indentation of a stamp with the same shape of cross-section.
Figure 2 shows the results of a few examples from numerical simulation. The geometry of contact areas is arbitrarily obtained: Figs. 2, a and b are some parts of "rough contact area", Figs. 2, c and d are two common geometries. The axis y is limited in the range from 0.98 to 1.03. It is clearly seen that the stiffness changes regularly with the direction of tangential loading. There exist principle axes for maximal and minimal values. A higher Poisson's ratio leads to a larger difference of stiffness. In Fig. 3, cases of regular polygonal shapes are presented with y-axis in the range from 0.99 to 1.01. It is known that the tensor of second rank is symmetric for any shapes having axis of symmetry of the third or higher order. Numerical simulations give in these cases still some angle dependence which however is all cases less than 0.5% (Fig. 3, b). This variation can only be explained by the finite accuracy of numerical simulations, which in this case obviously is on the order of magnitude of 0.5%.
3.2. Tensor of contact compliance
Now we consider the complete components of stiffness tensor in Eq. (6). For inverse tangential contact problem, the conjugate gradient method in the BEM simulation can
0° 45° 90° 135
Fig. 2. Tangential contact stiffness Knn changing with the direction for a few different shapes of contact zones. The angle and coordinate system are use as shown in Fig. 1
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1.01
g
p-
.00
0.99
1.01
Mr
P-
1.00
0° 45° 90° 135°
0.99
0° 45° 90° 135
Fig. 3. Tangential contact stiffness Knn in different directions for regular polygonal contact zones
provide an accurate result for calculation of the stress parallel to known constant deformation distribution, but not for the case of stress perpendicular to displacement. Unlike the simulation in Sect. 3.1 for given constant tangential displacement, here we study it in a simpler way. The contact zone A is loaded by two constant stress distributions Tnn = const and Tjj = const perpendicular to each other, which defines two tangential forces Fnn and Fjg.
Each force will generate two components of displacement, unn and unj by , and Ujj and Ujn by Fjj. The displacement distribution is not constant in the contact area (usually some are positive, some are negative), so we use its average value un and Uj to define the contact compliance
"n
unn + + un5
Jnn Jn5
(11)
Fig. 4. Components of stiffness tensor. Numerical simulation results are shown with points, the solid lines correspond the transformation equations (6)
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Fig. 5. Sketch of tangential contact of rectangle
as well as stiffness
K=J-1. (12)
This K is not the real contact stiffness, however it still reflects its property. In this way, we numerically calculate the components of K for the example of Fig. 2, c—shape of octopus.
The numerical results are shown in Fig. 4, plotted with points. The theoretical values are calculated according to the transformation rules (6), in which the stiffness Kxx, Kyy, Kxy and Kyx are selected at angle 0 = 0° (see solid lines in Fig. 4). It is seen that this stiffness tensor changes with the angle following the transformation rules. Figures 4, c and d show that the components K^, K^ obtained in this way are same Kn^ = K^n.
4. Anisotropy of tensor of contact stiffness for strongly anisotropic contact shapes
In Sect. 3 we studied the square- or circle-similar shapes. They have an isotropic tensor of stiffness. Now we consider strongly anisotropic contact shape, for example of rectangle whose length and width are a and b (a > b). Similar to Sect. 3.1, we focus only on the important component of tangential stiffness Kw The stiffness in the length direction (0 = 0°) is denoted as Ka, and Kb in width direction (0 = 90°) (sketched in Fig. 5).
A simulation case for b/a = 0.2 is shown in Fig. 6. Like the other shapes, the tangential stiffness Knn changes also
Fig. 6. Tangential contact stiffness Knn changing with the angle (aspect ratio bja = 0.2)
with the angle, and the deviation is higher for a larger Poisson's ratio. Moreover, the tangential stiffness in length direction is smaller than in width direction. Here we are interested in the influence of this aspect ratio b/a on the ratio of tangential stiffness Ka/Kb, because they are the maximum and the minimum. This result is shown in Fig. 7 (solid line) with aspect ratio b/a varying from 0.01 (strip) to 1 (square). For comparison, we added also the theoretical solution for elliptical contact given by Mindlin [6]:
Ja
Jb.
(2-v)(1 + v) 1 2Vy , 2 E 4A n '
1 ±-
1 + y2
2-v 1-
k + -
2 v
2—v 1 -
(13)
where k and 8 are the complete elliptic integral of first and second kind of argument >/l-y2, y is aspect ratio y = = b/a < 1 with major axis a and minor axis b. A is area of ellipse A = nab. Tangential stiffness Ka and Kb is simplified as Ka = J-1 and Kb = J-1.
In Fig. 7, it is found that the ratio of Ka/Kb decreases for a lower value of aspect ratio. A narrower shape and larger Poisson's ratio will lead to a stronger anisotropy of tangential stiffness. However, this difference has a limiting value when the shape is extremely narrow: Ka/Kb = 0.5390 for very very small aspect ratio of b/a = 10-8 and v = 0.5.
Poisson's ratio 0.1-0.5
— Rectangle
— Ellipse
0.0 0.2 0.4 0.6 0.8 1.0 Aspect ratio
Aspect ratio
Fig. 7. Influence of aspect ratio on the ratio of tangential stiffness Ka/Kb. Logarithmic aspect ratio is used in (b)
)%&*+&*%&%%&&%%$%%&*&&*$&&%&
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5. Conclusion
Using the boundary element method, we calculated the tensors of contact stiffness and contact compliance for a number of shapes. As predicted by the general theory, the components of the tensor of contact stiffness are transformed according to the usual transformation rules for any tensor of second rank. For shapes having a symmetry axis of third or higher order, the tensor is shown to be isotropic up to numerical error. In the case of "statistically isotropic" shapes, it is also almost isotropic. It is very important to note that even in the case of strongly anisotropic contact shapes as for example an elliptic shape with high aspect ratio, the principal values of the contact stiffness are different but still not as anisotropic as the shape itself. In particular the ratio of the largest and smallest principal values remains finite even if the aspect ratio tends to zero or to infinity. Therefore, for any shapes which are not very pronounce anisotropic, the use of isotropic tensor of stiffness is a good approximation. In this approximation, tangential contact problem can be solved in the usual way using the method of dimensionality reduction in the formulation for arbitrary contact shapes [20], in particular to contacts with isotropic contact stiffness as tapered indenters of various order [21] or line contacts [22].
The author is thankful to V.L. Popov for useful discussions.
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Поступила в редакцию 15.04.2017 г.
Сведения об авторе
Qiang Li, Dr.-Ing., Technische Universität Berlin, Germany, qiang.li@tu-berlin.de
