Steady state rolling contact of cylinders with viscoelastic layers M. Tervonen Department of Mechanical Engineering, University

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1 Steady state rolling contact of cylinders with viscoelastic layers M. Tervonen Department of Mechanical Engineering, University ABSTRACT The finite element method is used to solve the steady state rolling contact problem of two layered cylinders. The layer thicknesses, radii of the rolls and the relaxation spectra of the linearly viscoelastic materials can be arbitrary. The contact formulation presented here uses the numerical fundamental solution for a moving surface force. The full Newton-Raphson iteration is used when solving the spatial equilibrium equations. Test and convergence results show the validity of the method. Other, more practical, cases are also considered. INTRODUCTION Slightly filled engineering polymers are frequently used, for instance, as roll covers in paper industry. At normal operating temperature, these materials have an equilibrium extensional modulus in the order of 1 GPa and can withstand a maximum strain of about one to two percent. They have a glass-transition temperature of about 100 C or more and are used in the stiff state. The viscoelastic ity of these materials has been demonstrated in rolling contact experiments, in which remarkable heat generation (power consumption) and speed dependent stiffness were observed, Koriseva et al. [10]. Many studies have investigated the rolling contact of viscoelastic bodies. Various cases based on the half-space solution can be found in Golden and Graham [5] and the references therein. Engel [4] has proposed a solution to the rolling of a viscoelastic slab between rigid cylinders using viscoelastic influence coefficients which were determined analytically. Lynch [11] seems to be one of the first to use the FEM for a rolling contact problem involving viscoelasticity when he studied the rolling of a viscoelastic sheet between rigid cylinders. Similarly, FEM has been used by Batra et al. [3] to solve the steady state thermoviscoelastic rolling of a covered cylinder with a rigid one. Babat and Batra [1, 2] have studied the problem of one resilient cylinder when the cover is modelled as a special incompressible material exhibiting finite strains.

2 94 Contact Mechanics In a variational formulation of the same problem Telega and Bielski [15] took friction into account. All of the formulations in [1, 2, 3 and 15] rely on the rigidity of the facing surface. Kalker [9] has presented a method based on the influence functions of layered half-space for the rolling contact of layered cylinders but admits that only the elastic case was implemented on the date of issue. Rodal [13] has offered practical hints and an elastic solution to the same practical problem as dealt with in this paper. This paper proposes a method that can be effectively used to solve the steady state, small strain viscoelastic rolling contact problem when both rolls are covered with layers having arbitrary thickness and relaxation spectra. The methodology can be expanded to those cases when special consideration is given to the thin viscoelastic sheet, Tervonen [16]. The proposed formulation and method can be used as part of the thermomechanical solution under the practical assumptions given in the next chapter. GROUNDS, ASSUMPTIONS, STATEMENT OF THE PROBLEM Since the paper sheet is porous and has a very small Poisson's ratios in the plane under study, it can be thought to act as a lubricating film between the cylinders. On the other hand, the effect of the tangential traction on the normal pressure and displacements is small in the non-conformal, small-strain contact (Johnson [8]). Contrary to the elastomeric materials, the cover materials in this study are compressible. Measurements have shown the Poisson's ratio of a definite material to be almost constant (independent of frequency) at a value of about This value is also reported by Rodal [13]. The steady state requires that no considerable deformations remain from the previous loading cycles, since the cover would otherwise soon cease to exist. The material, thus, has a short memory compared to the time of revolution. The principle of fading memory is also valid. Measurements by Koriseva et al. [10] indicate that the rise in temperature is slow compared to the rotation speed. The temperature distribution is thus practically axisymmetric and hence the thermal strains have no effect on the contact deformations and forces. The thermal and mechanical problems are coupled only through the temperature dependent material parameters which produce the temperature dependence of the heating power. Here, we use the material parameters determined at the average temperature in the radial direction so that the time dependence of these parameters is independent of location. The error produced is small, if all the temperatures are below that of the glasstransition, so that the temperature dependence of the parameters remains low. Based on these assumptions, it is then possible to separately formulate the mechanical and thermal problems. In this study, I will concentrate only on the mechanical problem. The rolling speed in real situations is about one per cent of the longitudinal wave speed in the materials (greater than 10^ m/s).

3 Contact Mechanics 95 Furthermore, practical measurements have shown no vibrations, Koriseva et al. [10]. On account of the given arguments, the quasi-static, steady state isothermal analysis is performed for covered cylinders having rigid (steel) cores. It is assumed that a specific cover material has the same time-dependence in all terms and that microslipping and other frictional effects are insignificant. FEM- FORMULATION USING STIELTJES CONVOLUTIONS The field equations used in the linear theory of elasticity are given by Gurtin and Sternberg [7]. The boundary is here assumed to be conventionally divided into the traction and displacement parts. The material law in the form of Stieltjes convolution is expressed as <T=C*dE, (1) where C is the time dependent material response matrix. For the definition and properties of the convolutions, see [7]. Next, we use the principle of virtual work, assuming that the displacement boundary conditions are exactly satisfied. The body is discretized using the conventional displacement formulation of FEM. Using the properties of Stieltjes convolutions, we then obtain the time dependent equilibrium equations for a typical element in the form of Stieltjes convolutions, Tervonen [17]. After assembling we obtain The displacement vector D(t) can, in principle, be solved from this equation. Here, we use first the properties of the materials at hand. If the common time dependent function of the material stiffness matrix is G(i), then the solution reads (Tervonen [17]) where Lf(t) is the solution to the problem in the case G(t) = 1. The function J(t) is the Stieltjes inverse of the real G(t) such that where h(t) is the unit step function. For the existence of the Stieltjes inverse, see Gurtin and Sternberg [7]. Equation (3) includes both the substitution of the initial conditions and the possibility of discontinuous functions and can be used for selected degrees of freedom. It is worth emphasizing that the time-independent part of the material response matrix can be orthotropic. (2) (3) (4)

4 96 Contact Mechanics SPATIAL FLEXIBILITY MATRICES Here, I describe briefly how to determine the spatial flexibility matrix for a steadily moving surface using the numerical fundamental solutions for a viscoelastic body whose thickness and surface curvature normal to the moving direction are constant. The material properties are independent of the moving direction. Elements having only corner nodes are used on the line along which the force moves. The nodal spacing is constant on that line. Detailed reasons for these choices, a complete formulation and basic characteristics of the fundamental solutions are given by Tervonen [17]. Let us now assume, that the static displacements D* of the nodes 1... Ware known when a unit force is acting (fig. 1) and G(t) = 1. The spatial domain is so large that outside it these displacements are zero for sufficient accuracy. N _^/wla-» jo... j... N' v Figure 1. Steady flow of the material nodes D through the displaced spatial nodes A is equivalent to a load (here a unit force) moving with a constant speed. The situation in figure 1 can be taken as the zero initial conditions for the material nodes 1... N due to the conditions presented in the problem statements. Because of the steady state and non-retroactivity, we only need the time dependent solution D± (t) for the material node 1 to obtain the solution for all material nodes. From this solution we obtain the displacements of the spatial nodal positions 1... N'... M, where M is, for the needs of a flexibility matrix to be inverted, so large that the viscoelastic deformation there is negligibly small. If the relaxation modulus G(t) is in the form of arbitrary discrete spectrum (Gross [6]) having G( 4 # 0, then the inverse can be found using equation (4). The relaxation times in creep are t^ and the proportional creeping factors are Q. Taking into account the facts given above, equation (3) yields, after quite extensive derivation, the fundamental displacement *=1

5 Contact Mechanics 97 where (6) and ( -a(l-e^")) /;( ) ; <1 : \<t<2 (7) The sum of the factors Q is equal to one, and thus 7(0)"' = (l+f3)g( ) = G(0). Displacements at spatial nodes are obtained from (5) by assigning the integer values 0... M to the non-dimensional time (spatial placement). Displacements for simultaneous nodal forces are calculated by shifting and summing. Using the definition for the flexibility coefficients yields the elements A(i,j); i,j Na ( NZ > M), of the asymmetric spatial matrix A [0 if k > /o or k < -(M-/Q), where Z> - - D^(j-l). For computational reasons, it is better to first expand D* with NZ -JQ leading zeros and with N^+JQ-M-l trailing zeros. THE ROLLING CONTACT PROBLEM OF TWO CYLINDERS Formulation We shall now consider the rolling contact problem of two cylinders whose core is assumed to be rigid with cover materials that behave viscoelastically according to the laws described in previous sections. We shall also deal with a cross- section which is far from the cylinder ends compared to the layer thicknesses such that a plane strain condition can be assumed. A further assumption is that there is no gross sliding between the contacting surfaces. Because the strains are small, the contact angles (c/ / #,, GI /#, ; leading and trailing contact widths divided by the initial radii of curvatures) must also be small compared to unity. Consistent linearization demands that we use first order approximation in the contact angles for all quantities. Let us now suppose that the spatial flexibility matrices of the surface nodes have been determined with the method described in the previous section. The segments of the surfaces are of equal length and must have the same nodal spacing. The surface having the longer deforming length determines the segments needed. The dimension of the matrices A% and Ag and the point of the

6 98 Contact Mechanics first contact are such that there is no coupling between the contacting and cutoff nodes. At the candidate contact surfaces the no-penetration and the notension conditions must be satisfied. These conditions are realized by the penalty method using simple gap-elements between the node pairs. Neglecting all the second and higher order terms in strains or, equivalently, in the contact angles, the equilibrium equations in the direction of the line loading (load per unit axial length) PI are d, -4, / = AiFi = -Ai*,<rfi -d2 -g) <?2 = ^2^2 = &2kg(di ~d2~g} <rf i (/) - d2 (i) -g(d} = PI, 1=1 1=1 where /, g, kg and N^ are a unit column vector, initial gap vector, penalty number and the dimension of the flexibility matrices, respectively. The primary unknowns are the displacements d± and ^ of the surfaces and therigidbody motion 6^ of the loaded cylinder. The operator < > is described as part of the equations (6). The first two equation groups can be combined so that we obtain NC + 1 equations instead of 2A^ + 1. The new matrix operator is A* + ^ and the primary unknowns are d - ^ - ^i + dj and d. The original unknowns are solved using equations (9) after the non-linear solution. Solution and computer implementation The non-linear equations are solved using full Newton-Raphson iteration. The tangential operator is the constant inverse K of the flexibility matrix supplemented with the varying part derived by differentiating the functions kg<-> which gives the coupling terms. The method is implemented using the MATLAB 3.5-system [12], which includes effective matrix and vector operations. Important benefits are the short and easy coding and good graphical facilities. A value kg max{k(i,i)} is used for the penalty number based on the study done by Sepponen and Tervonen [14]. Variation of the gap-status is used as the convergence criterion. When no status is changed, the system is linear and one more iteration cycle is enough. The Euclidean norm of the force unbalance was always smaller than 10'^ PI at the end of the iteration. Eight to fourteen cycles are needed depending on the asymmetry of the problem. In [14], where the corresponding elastic system is considered, eight iteration cycles were on the average sufficient to achieve convergence.

7 Contact Mechanics 99 RESULTS AND DISCUSSION The first example acts partially as the test case. One cylinder is rigid and the other has a cover of different thicknesses and the material behaves as a standard linear solid. The reference results were obtained by programming in MATLAB the equations given on pages and in the appendices of Golden and Graham [5]. The values of the parameters used in all cases are: PI = 40 kn/m, #i=#2= m, v= 0.42, E( *)= 1 GPa, (3= 4,vt= mm. The numerical analyses use the values 20 and 2 mm for the layer thickness b. The cover was discretized with a nodal spacing mm on the surface; 40 and 10 elements were used in the thickness direction for the two cases, respectively. The half-space solution yielded a contact width c, = mm. The pressure distributions of the half-space solution and the numerical solution for the thick cover are quite similar (fig. 2) as they should be. The results indicate that the layer thickness has a significant effect on the contact parameters Figure 2. Pressure distributions (left) for different layer thicknesses b when the upper cylinder is rigid and the material flows from left to right; solid line: solid cylinder (classical solution), o : thick layer (blc^ ~ 6), + : thinner layer (blc,. «0.6, blc «0.8). Contact configuration (right) for the thicker layer; the x- symbols show the nodal positions on the lower surface for the coarser mesh used in the convergence study (notice the scales). Pressures are scaled with the maximum pressure /*//<,( (<*>)) of the Hertzian contact (slow rolling). Coordinates y and x are measured from the first point of contact. The convergence of the results was tested using different mesh densities. For the physical situations given above, the maximum change of the results was about 0.2 % when the nodal spacings a mm and a mm were used. The contact widths were within the spatial discretization accuracy. From the contact configuration seen in figure 2 (right) we can see among other things the fulfilment of the no-penetration condition.

8 100 Contact Mechanics The second example shows the effects of the rolling speed. Both cylinders are covered (table 1). Some of the results are drawn in figure 3. Both cylinders have the same asymptotic stiffnesses. In case a) the upper cylinder behaves almost elastically with E = E( ) and only the last creeping term has a significant effect on the behaviour of the lower one. In case b) the viscoelasticity has a great effect especially on the deformations of the upper cylinder because the relaxation time is almost the same as the contact time. In case c) both cylinders are virtually elastic with the instantaneous modulus 0.2 a) 0.2 b) { ::;/ ::::.,.-.;:/ ;:" ;..._ /f i ^^K x [mm] c) x [mm] d) /~ t^x o -10 o 10 x [mm] x [mm] Figure 3. Contact configurations for: a) low, b) intermediate and c) high speed rolling (notice scales), d) Contact pressure in the three cases; slow: dashed (--); intermediate: solid line; high speed: dotted (... ). The upper cylinder is loaded in the y-direction and the rigid body motion of the lower one is prevented. PH 30 b 20 80) 10 Table 1. Spectra (C(i), t(i)) of the cover materials and shared parameters used in the example illustrated in figure 3. Upper cylinder A 1 and lower A 2. Case a b c vti /mm vtgt i )/mm 0.006, 0.06, 0.6, 6 0.6, 6, 60, , 600, 6E3, 6E4 d = 1, Q(i) = 0.25; i = =62 = 10 mm, RI=KZ = 400 mm EiM=EzM = 1 GPa,Vj=V2 =.42 ft = & = 3, PI = 300 kn/m CL ~~ 125 mm 30 elms in thickness

9 Contact Mechanics 101 CONCLUSIONS The steady state rolling contact problem of two viscoelastically covered cylinders is formulated using asymmetric flexibility matrices. The derivation of these matrices is based on numerical fundamental solutions. The tests and other results demonstrate the effectiveness of the method. The effects of layer thickness and rolling speed on the contact parameters are demonstrated. REFERENCES 1. Babat, C.N. and Batra, R.C. 'Finite Plane Strain Deformations of Nonlinear Viscoelastic Rubber-Covered Rolls' Int. J. Num. Meth. Eng., Vol.20, pp , Babat, C.N. and Batra, R.C. 'Finite Deformations of a Viscoelastic Roll Cover contacting a rigid Plane Surface' Commun. Appl. Num. Meth., Vol.1, pp , Batra, R.C., Levinson, M. and Betz, E. 'Rubber Covered Rolls- the Thermoviscoelastic Problem, a Finite Element Solution' Int. J. Num. Meth. Eng., Vol.10, pp , Engel, P.A. 'Rolling and Impact on a Linearly Viscoelastic Slab, by Point Matching' Int. J. Num. Meth. Eng., Vol.5, pp , Golden, J.M. and Graham, G.A.C. Boundary Value Problems in Linear Viscoelasticity Springer-Verlag, Berlin Heidelberg, Gross, B. Mathematical Structure of the Theories of Viscoelasticity Hermann, Paris, Gurtin, M.E. and Sternberg, E. 'On the linear theory of elasticity' Arch. Rat. Mech. Anal, Vol. 11, pp , Johnson, K.L. Contact Mechanics Cambridge University Press, Cambridge, Kalker, J J. Elastic and Viscoelastic Analysis of two Multibly Layered Cylinders Rolling over each other with Coulomb Friction Delft University of Technology, Report 89-50, Koriseva, J., Kiema, T. and Tervonen, M. 'Soft Calender Nip: an Interesting Subject for Research and Measurement' Paperija Puu - Paper and Timber, Vol. 73, pp , Lynch, F. de S. 'A Finite Element Method of Viscoelastic Stress Analysis with Application to Rolling Contact Problems' Int. J. Num. Meth. Eng., Vol.1, pp , PRO-MATLAB User's Guide, The MathWorks, Inc, Rodal, J J.A. 'Modelling the State of Stress and Strain in Soft-Nip Calendering', in Fundamentals of Papermaking, Vol. 2 (Ed. Baker, C.F), pp , Transactions of the ninth Fundamental Research Symbosium on Papermaking, Cambridge, UK, Mechanical Engineering Publications, London, Sepponen, A. and Tervonen, M. 'Practical Modelling and Solution Strategies in Normal Contact of Multilayered Anisotropic Cylinders' Submitted to Conta93, Computational Methods in Contact Mechanics, First International Conference, July 1993, Southampton, UK. 15 Telega, JJ. and Bielski, W.R. 'Variational Formulation of a Rolling Cylinder Undergoing Plane Finite Deformations' Mech. Res. Commun., Vol.17, pp , Tervonen, M. 'Rolling of a Thin Viscoelastic Sheet between Viscoelastically Covered Cylinders' Unpublished manuscript, Tervonen, M. 'Numerical Fundamental Solutions for Moving Surface Forces on Viscoelastic Bodies Having Constant Surface Compliance' Unpublished manuscript, 1993.