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УДК 531.43
Решение задачи фрикционного контакта на основе метода диаграмм памяти при произвольном трехмерном нагружении
V.V. Aleshin, O. Bou Matar
Институт электроники, микроэлектроники и нанотехнологий, Вильнёв-д'Аск, 59652, Франция
В статье обсуждается метод диаграмм памяти, разработанный для решения задачи фрикционного упругого контакта. Целью работы является установление связи между контактной силой и смещением для общего случая плоского контакта двух тел произвольной формы без учета качения и кручения. Описание механического взаимодействия двух контактирующих тел в присутствии трения является сложной задачей, поскольку необходимые отношения между силой и смещением носят гистерезисный характер и зависят от предыстории. Произвольное изменение действующего усилия (или смещения) вызывает сложное распределение напряжений сдвига в области контакта, которое требует адекватной параметризации и учета. В связи с этим предложено вместо сложного распределения напряжений сдвига рассматривать более простую функциональную форму, называемую диаграммой памяти, которая содержит ту же самую информацию о памяти в системе. Получены два интегральных соотношения, которые связывают вектора силы и смещения с данной внутренней функциональной зависимостью. Полученные интегральные соотношения дополнены двумя правилами эволюции диаграмм памяти, вытекающими из закона трения Кулона. Эти правила определяют обновление диаграммы памяти, происходящее в соответствии с обновлением истории силы контактного взаимодействия. Полученная таким образом диаграмма памяти используется для вычисления обновленной истории смещения, т.е. для получения искомого соотношения между силой и смещением.
Ключевые слова: трение, механика контактного взаимодействия, теория Герца-Миндлина, фрикционный контакт, произвольная история нагружения, метод диаграмм памяти
Solution to the frictional contact problem via the method of memory diagrams for general 3D loading histories
V.V. Aleshin and O. Bou Matar
Institute of Electronics, Microelectronics and Nanotechnologies, Villeneuve d'Ascq Cedex, 59652, France
This paper is concerned with the method of memory diagrams developed for solving a problem of frictional elastic contact. Our goal is to establish a link between contact force and displacement for a general plane contact between two arbitrarily-shaped bodies; rolling and torsion are not considered. Description of mechanical interactions between two solids in contact in the presence of friction is a nontrivial task since the desired force-displacement relationships have hysteretic (memory-dependent) character. Arbitrarily changing applied force (or displacement) creates a cumbersome shear stress distribution in the contact zone that has to be adequately parameterized and accounted for. In that regard, it is suggested to consider, instead of complex shear stress distributions, a simpler functional form called memory diagram that contains the same memory information. We have established two integral relationships that link the force and displacement vectors with that internal functional dependency. The integral relationships are supplemented with two other evolution rules for memory diagrams that eventually follow from the Coulomb friction law. The memory diagram is updated with the help of these rules following a given force history. Then the calculated memory diagram is used to update the history of displacement i.e. to produce the desired force-displacement relationship.
Keywords: friction, contact mechanics, Hertz-Mindlin theory, frictional contact, arbitrary loading history, method of memory diagrams
1. Introduction
The history of the contact problem started in 1880s when H. Hertz [1] published the classical solution for two elastic spheres compressed by a normal force. In absence of adhesion and plasticity, this solution is fully reversible. How-
ever, the addition of a tangential force and friction [2, 3] makes the problem hysteretic and memory-dependent. It was noted that even a small tangential force acting on two precompressed spheres results in appearance of a slip an-nulus at the periphery of the contact circle where the sur-
© Aleshin V.V., Bou Matar O., 2015
faces are compressed weakly. The coexistence of the stick (central) and slip (peripheral) zones actually means mixed-type boundary conditions that correspond to zero local tangential displacement in the central region and, in the slip annulus, to the Coulomb friction law written for local tangential t and normal a stresses, t = ^a (here ^ is friction coefficient considered as a constant for two contacting materials). The increase in the tangential force results in the slip propagation towards the contact centre. If now the tangential force starts decreasing, a new slip annulus develops at the contact periphery in which t = -^a. Hence, the same values of the normal N and tangential T forces can correspond to different distributions of stresses and displacements in the contact zone. This fact explains a complex hysteretic character of the solution.
Allowing the normal force to evolve [4] adds a new complexity factor to the problem. The matter is that the slip zone always arises at the contact border where a = 0 and propagates inward, but, if the normal force increases, the contact border itself propagates outward. The result depends on the value of the derivative dN/ d T.
Finally, the introduction of a general contact geometry instead of spherical profiles introduces even more complexity into the description. Indeed, for two rough surfaces, the contact zone consists of a multitude of contact spots having random geometry. For changing normal force, those contact spots can merge or split. Further, each of them supports slip and stick zones, and traction distribution in the stick zones can contain residual stresses left from the previous moments of evolution. With continuously varying normal and tangential loading, this complicated picture continuously evolves.
Here we propose a solution to such a general mechanical problem which is based on known works [4-9] as well as on original developments [10-14]. The geometric aspect of the problem can be successfully dealt with by using the reduced elastic friction principle (REFRP, see [5-9]) that, under some restrictions, makes it possible to replace an arbitrary contact geometry by a pair of axisymmetric profiles. Then, following the original method of memory diagrams (MMD) we introduce an internal functional dependency called memory diagram that replaces a cumbersome traction distribution in the contact zone but contains the same information. Further, the method of memory diagrams enables us to link the tangential force and tangential displacement via two integral equations each depending on the memory diagram. Finally, generalizing an incremental procedure established by Mindlin and Deresiewicz in 1953 [4], we develop an algorithm that produces a force-displacement relationship for an arbitrary loading history applied to a general contact geometry illustrated in Fig. 1, a.
The idea of the method of memory diagrams is close to the principle used by the method of dimensionality reduction (MDR) described in [15] and in a series of papers cited therein. The method of dimensionality reduction uses the
exact solution to frictional problem in axisymmetric geometry to build up a mapping of axisymmetric geometry into 1D geometry. Then the contact interaction is represented as the penetration of a one-dimensional profile into a series of infinitesimally spaced springs having normal and tangential stiffnesses. The detailed comparison of the method of memory diagrams and method of dimensionality reduction is planned in the nearest future.
2. Reduced elastic friction principle
The reduced elastic friction principle is an important theorem of contact mechanics which states that, for constant loading and for a wide range of contact geometries, the tangential force and displacement can be expressed through the normal force and displacement. This principle is illustrated in Fig. 1, b for axisymmetric bodies. Consider two situations: first one when the system is loaded only by normal force Q, and second one when both force components, N and T, are applied (N > Q). The force values are chosen in such a way that the stick zone in the second case coincides with the contact zone in the first case. Then the tangential force and displacement in the second situation are given by
jT = N -Q),
jb = e^(a ( N ) -a( N )| N=Q ), (1)
where the dependency of the normal displacement on the normal force a = a(N) is considered as known, and e is a
h
Arbitrary contact system
Txy E, V, ц
N
v<8>-
Tangential displacements
.......i
a\
Normal displacement
0
Arbitrary geometry
/ ,a
Contact
4zone
N
= Contact zone ) zone
o:
p
Contact radius
Equivalent axisymmetric system with the same a = a(N)
Fig. 1. Forces and displacements in a general contact system (case of rough surfaces is shown) (a); reduced elastic friction principle for axisymmetric bodies (b); equivalent axisymmetric system that has the same normal reaction as an original one, e.g. contact of rough surfaces (c)
material constant that depends only on Poisson's ratio v, e = (2-v)/(2(1 -v)).
An important feature of Eq. (1) is that it does not contain any geometry-related characteristics. Thus a simple consequence of the reduced elastic friction principle is a statement that, for two contact systems with the same normal response, the tangential responses are also identical (see Fig. 1, c). Consequently, a contact between surfaces of almost arbitrary topography can be replaced by an equivalent axisymmetric system. The related restrictions that limit the class of the considered contact types are discussed in [5-9]. In Fig. 1 the word "arbitrary" should be understood in that context.
3. Method of memory diagrams
The method of memory diagrams (for detailed proofs see [13] is formulated below for an equivalent axisymmetric system. In fact, it represents an extension of the solution Eq. (1) for arbitrary loading history, i.e. when forces N and T are not constant but change in 3D so that loading is actually described by vector (N, Tx, Ty), where Tx and Ty are coordinates of the tangential force vector, and x and y are coordinates on the contact plane. The method assumes that the normal solution to the contact problem is known, so that the functions N = N(c) and a = a(c) linking the contact radius c and normal force N and displacement a are given. The principal MMD statement asserts that there exists a function D(p) (p = \] x2 + y2 ) called memory function or memory diagram such as two relationships, dN -
- dp and (2)
c=p
T = ^J D(p)
b = 0(i| D(p)
dc da
dC
dp
(3)
c=p
hold simultaneously (here T = (Tx, Ty ), b = (bx, by)). This property (called here rule I) corresponds to the force balance equation; the contact friction force equilibrates the external tangential force. In addition, two other statements are valid. The second one (rule II) claims that | D(p)| < 1 that eventually follows from the Coulomb friction law.
The rule III reflects the fact that slip (if occurs) propagates from the contact boundary (weakly compressed zone for convex contact shapes) towards the contact centre (strongly compressed zone). As far as memory diagrams are concerned, this rule suggests that a possible adjustment of the memory function D(p) in the purpose of complying with the rule I is made by setting the norm of the memory function vector | D(p)| = 1 on a final segment s < p < c of the memory diagram. This final segment corresponds to the presence of slip that propagates inward until, by setting of s, the rule I is satisfied.
Equations (2) and (3) are written using the radial coordinate p. In order to get rid of any geometry-related characteristics, it can be rewritten in the form
T = ^J D( n)d n and
N da
b = 0ц[ D(n)—
dN
dn
(5)
N=n
particularly convenient for a force-driven contact system in which forces N and T are known and the displacements a and b are to be calculated. Here the substitution p ^ n = = N(c = p) has been made. For a system driven by displacement the solution is analogous with D(a), p ^ a = = a (c = p).
Equations (4) and (5) are integral equations whose approximate solution can be obtained for small but final force or displacement increments. Below we discuss an implementation of the method for an arbitrary 2D loading history for a force-driven system. In that case, Eq. (4) is used for updating the memory diagram while the former produces the unknown displacement via Eq. (5).
3.1. Method of memory diagrams: 2D implementation
The algorithm of the method of memory diagrams developed for final force increments for 2D loading, i.e. when normal and tangential forces always lay in one plane is shown in Fig. 2. The algorithm includes two binary choices. Firstly, if AN is positive (set (a)), the memory diagram extends by the respective interval N < n < N + AN on which the memory function has to be defined keeping the equality Eq. (4) (in 2D case vector signs in Eqs. (4) and (5) have to be omitted). Further, if |AT | < ^N, Eq. (4) can be satisfied by updating the memory function only on the new interval as indicated in set (b), so that the area under the newly de-
Apply AN, AT 1
0
1 0 -1
Yes
AN > 0?
Дл)
No
; t
AN > 0
Yes
|A71 < liA/V?
d
Remember \
1Дл)1 * i 0 -l
ДТ1) -дтуц
Л
bi
о -1
No
.....■. : ••••
i I j i А 77ц
0.1ОД A.
If % :
0 -1
Л
slip j,
ДТУц AT=ATX + AT2
oM-j-j—
-ll 5J U дтуц AT = AT л + A T2
■ДТУц
a = a(N(t)) b = b[T(t),N(t)]
Fig. 2. Algorithm of the method of memory diagrams for arbitrary loading history in 2D
D
Fig. 3. Numerical implementation of the 2D MMD algorithm: two kinds of nodes are introduced. Color of nodes correspond to various ways of function interpolation, black node means constant interpolation on its left indicated by black arrows, function values between white nodes are to be obtained by the linear (or of higher order) interpolation
ments in Fig. 3, only the limiting points are saved that makes the scheme much more efficient in comparison to, say, a simple equidistant grid of fixed points.
Compensating for AT at each step of the algorithm is possible via explicit computation. Indeed, adjustment of the length of the memory diagram according to a current increment AN and calculation of the related AT] is an explicit operation since at each interval n-i < n < n, function D(n) is given by an analytical expression. Similarly, in order to account for slip and to calculate Q (the boundary between stick and slip zones defined in terms of n-variable, see Fig. 3) we successively consider segments ni-1 < <n<ni with reducing i until the segment containing Q is found. Then the actual Q is analytically evaluated.
fined memory function equals AT/^. In contrast, if the latter inequality does not hold, even allocating the maximum possible area on the new interval is not enough to compensate for AT (set (c)). Indeed, in set (c), those maximum area is AT1 /^ = AN because of the rule II, |D(n)| ^ 1, and no more tangential force can be "absorbed". In order to take into account the remaining part AT2 = AT - AT1, we have nothing but to use rule III that explains how to update the memory function by assuming slip, i.e. segment on which |D(n)| = 1. In accordance to that rule, point A' moves to the left and sets |D(n)| = 1 at its right until the remaining force increment AT2 is equilibrated. In the case when the inequality AN > 0 is not fulfilled, the memory diagram shrinks by the value of | AN thus releasing some tangential force AT1 previously "saved" in the contact system (set (d)). The remaining value AT2 = AT - AT1 is equilibrated by assuming the slip propagation as before (set (e)).
The application of the algorithm shown in Fig. 2 allows us to update the memory diagram for known force increments AN and AT in such a way that the force balance Eq. (4) is kept. Then the updated memory diagram is used for calculating the tangential displacement according to Eq. (5).
The above reasoning shows that a memory diagram can contain both straight segments with |D(n)| = 1 corresponding to slip and curvilinear segments obtained in a situation (b) (in Fig. 2 the increment values are highly exaggerated). This suggests an economic way of representing a memory diagram for numerical implementation (see Fig. 3). We introduce two kinds of nodes, black and white, and define the memory function between them. Black point (ni, Dt) (i is the node number) indicates that on the interval ni-1 < n < ni function D equals Dt; white point (ni, Dt) indicates that on the interval ni-1 < n < ni function D has to be interpolated between points (ni-15 Dt-1) and (ni, Dt) using some interpolation method (linear or of higher order). Saving white memory points is a memory-consuming process since the distances between the points correspond to force increments AN and are small. As for the black seg-
3.2. Method of memory diagrams: 3D implementation
A 3D implementation of the method of memory diagrams is similar except the memory diagram consists of two functions Dx (n) and Dy (n) (Fig. 4). The algorithm first adjusts the length of the memory diagram and calculates the related AT1. If now the total increment AT is not fully equilibrated, slip starts propagating from the contact boundary n = N towards the centre n = 0. The remaining unbalanced force AT2 = AT - AT1 is known and we have to compensate for it by finding a proper Q. To do so, first an interval (black or white, see the interpolation rule above) that contains Q has to be found. Despite the number of equations now equals two but not one as in the 2D case, the interval is still possible to locate explicitly. In that purpose, we introduce the angle y of the slip direction and write the force balance equation in the form
ATX2 = (N - Q) cos у
■J £Xd(n)dn,
Q
N
(6)
ATy 2 = (N - Q)sin y-J D0ld(n)dn,
where D^ (n) is the previous (i.e. not yet updated at this step) memory diagram. By excluding y we obtain a single equation and then consider function
Fig. 4. MMD numerical implementation in 3D: memory diagram is a vector function (Dx (n), Dy (n)) of a scalar argument n; two kinds of nodes are introduced similarly to Fig. 3
-л old,
+
+
ATy 2 + N <d(n)dn
- (N - Q)2
(7)
v ^ ;
that has different signs at the ends of the interval containing Q. The final computation of actual Q on that segment is done analytically since the memory function D(n) is explicitly given.
4. Results and examples
Below we show a number of curves b(T) for some input dependencies N(t) and T(t) in the 2D loading case. Figure 5 illustrates a particular case when two arguments, N and T, are linked via a functional relation, so that actually there is
only one independent argument in the system. Curve b(T) in Fig. 5 is typical for one-parameter hysteresis (other dependencies of this kind as discussed by [10-12]. In particular, it exhibits closed loops for periodic T(t); partial increase in the argument T on the globally decreasing branch results in appearance of an inner loop, etc. Note that the inner loop has the property of the end-point memory which means that the curve exits the loop with the same tangent as just before entering it.
However, in a more general case when two arguments, N and T, are independent, the hysteretic behavior differs considerably. Since variations in N are not linked with T(t)-protocol, even for periodic T(t) the "loops" are not closed. Indeed, Fig. 6 shows that the same T(t)-history produces a curve in which all monotonous parts are shifted, bent, etc. Of course, generating such curves via the direct analysis of traction and without use of the method of memory diagrams is an extremely cumbersome task.
In the case of loading in 3D is more appropriate to show the solution (bx, by) for the excitation (Tx, Ty) on the same plot (Fig. 7). Here time dependences of all three force components (Tx, Ty, N) are harmonic with different periods. Again, the displacement curve has a complex behavior; its calculation by means of the method of memory diagrams is fully automated.
5. Discussion
Below there are some remarks that concern the solution to the contact problem for an arbitrary 3D loading presented here.
The method discussed here assumes that the normal solution a = a(N) is known. Such solutions for some regular contact geometries can be found in books by [15, 16] on contact mechanics. For randomly rough surfaces, we refer the reader to recent papers by [17-21] (see also [13] for a possible derivation of a = a(N)).
All geometry-related features are taken into account by the normal solution a = a(N); the rest of the theory is for-
Fig. 6. Typical force-displacement relationship b(T) in the case when N and T are independent (two-parameter hysteresis)
Fig. 7. Typical force-displacement relationship shown as curves by (bx) and Ty (Tx) in the case of 3D loading (three-parameter hysteresis)
mulated independently of contact geometry. The possibility of doing so follows from the reduced elastic friction principle.
This principle assumes some conditions discussed by Jäger [6] and Ciavarella [8, 9], such as neglect of the effects of dissimilarity between the bodies, etc.
Other assumptions are similar to those of the Hertz-Mindlin theory. They include purely elastic deformations, absence of adhesion, absence of torque, plane contact, aligned and non-rotating normals to all contact spots, etc. In fact, the present method can be considered as an extension of the Hertz-Mindlin theory for an arbitrary 3D loading and a wide range of geometries.
Although the solution is computer-assisted, we still can consider it as analytical for final force (or displacement) increments. The algorithm shown in Fig. 2 just selects an appropriate branch of the analytical solution in accordance to the relationships between the force or displacement increments.
As many other models in theoretical contact mechanics, our theory assumes the friction coefficient to be a universal constant for two materials in contact.
The theory can be applied as a basis for a description of materials with internal contacts (geomaterials or nondestructive testing of damaged pieces or components), as well as for designing new nonlinear metamaterials with desired properties.
Acknowledgements
The authors gratefully acknowledges the support of the European Commission (ALAMSA-FP7-AAT-2012-RTD-1, grant agreement No. 314768).
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Поступила в редакцию 19.06.2015 г.
Сведения об авторах
Vladislav V. Aleshin, PhD, Senior Researcher, Institute of Electronics, Microelectronics and Nanotechnologies, France, vladislav.aleshin@iemn.univ-lille1.fr, aleshinv@mail.ru
Olivier Bou Matar, hab. PhD, Prof., Institute of Electronics, Microelectronics and Nanotechnologies, France, olivier.boumatar@iemn.univ-lille1.fr
