QUASISTATIC INDENTATION OF A RUBBER COVERED ROLL BY A RIGID ROLL - THE BOUNDARY ELEMENT SOLUTION

Transcription

1 259 QUASISTATIC INDENTATION OF A RUBBER COVERED ROLL BY A RIGID ROLL - THE BOUNDARY ELEMENT SOLUTION Batra Department of Engineering Mechanics, Univ. of Missouri-Rolla ABSTRACT The linear elastic problem involving the indentation of a compressible rubberlike layer bonded to a rigid cylinder and indented by another rigid cylinder is analyzed by the boundary element method. For the same contact width. the pressure at the contact surface is found to depend noticeably upon the thickness of the layer and the Poisson ratio of the material of the layer. Results computed and presented graphically include the pressure distribution over the contact surface. the shape of the indented surface and the stress distribution at the bond surface. INTRODUCTION Tr.:.ction in vehiclef, nip acti'jn in cylindrical ro:ls in the paper-making process and in the textile industry, and friction drives are examples of the kind of problem studied herein. Each of these problems involves indentation, by a steel or granite cylinder, of a rubberlike layer bonded to a cylindrical core made also of steel or granite. Hertz's (1881) solution for determining stresses in two bodies in contact is applicable when both bodies are homogeneous and linear elastic and the dimensions of the contact region are small compared with the dimensions of the bodies. The contact problem involving two rollers, one of which is assumed perfectly hard, in contact with their axes parallel is a special case of Hertz's problem and has been investigated by Thomas and Hoersch (193) on the assumption that plane strain state of deformation prevails. Solutions of the problem when the deformable ro1.ler is not homogeneous but consists of a thin elastic layer bonded to a hard supporting core has been given by Hannah (1951), and Hahn and Levinson (1974). Whereas Hannah's approach involves the numerical solution of an integral equation, Hahn and Levinson

2 26 use an Airy stress function and obtain the solution in the form of a double series one of which converges rather slowly In the aforementioned studies the rubberlike layer is taken to be linear elastic and the deformations involved are presumed to be infinitesimal so that the linear theory applies Aiso for the case of infinitesimal deformations, Batra et al. (1976) solved the problem by taking the rubberlike layer to be made of a thermorheologically simple material. In order to explore the effects of material and geometric nonlinearities, Batra (198, 1981) recently studied the problem in which the rubberlike layer is modeled as a homogeneous and incompressible or nearly incompressible Mooney-Rivlin material. We note that Batra used the finite element method to solve the problem Experimental work involving varying thicknesses of the rubberlike layer and different combinations of the diameters of mating cylinders has been carried out by Spengos (1965). However, the only material property listed for the rubber is the durometer hardness. For the linear elastic problem involving incompressible material. the durometer hardness is sufficient to find Young' modulus and hence solve the problem. Since the length to diameter ratio for the rolls used by Spengos is of the order of one, the assumption of plane strain state of deformation made in the work referenced above is not valid. For such geometrical configurations one needs to solve the three dimensional problem. It seems that the Boundary Element Method has an advantage over the Finite Element Method for three dimensional problems since at least the data preparation for the former method is easier. With the ultimate aim of solving the three dimensional problem, in this paper, we solve the two dimensional (plane strain) problem by the Boundary Element Method. Results obtained from the use of the boundary element method are found to compare very favorably with those obtained by Hahn and Levinson (1974). iormulationof the problem The system to be studied and the location of rectangular Cartesian axes are shown in Fig. 1. Since the length of rolls perpendicular to the plane of paper is large as compared to their diameters we assume that plane strain state of deformation prevails. Also the deformations of the rubberlike layer are presumed to be infinitesimal so that a linear theory applies. The rubberlike layer is assumed to be made of a homogeneous elastic material. In the absence of friction at the contact surface, the deformations of the rubberlike layer are symmetrical about the line joining the centers of two rolls. Neglecting the effect of body forces such as gravity, equations governing the quasistatic deformations of the rubber-

3 261, x Elastic Layer Rigid Material Fig. 1 System to be studied like layer are P = Po (1 - u..), (1) J.,J. (LijkR. uk,r.)'j =, (i,j = 1,2) (2) E\I LijkR. = (1+\1)(1-2\1) ij kr. E (3) + 2(1+\1) (csik csjr. + csir. csjk). The pertinent boundary conditions are: at the inner surface SI (xixi = Rl)' ui =, (4) at the outer surface S2 (xixi= RO); eioijnj =, (5) ni o..n. =, Xl > b (6) J.J J - nioijnj = P(xl' x2)' x2 < b, (7) lim P(xl' x2) =, x2 b, (8) and at the plane surface S3 (x2 = ); where u2 =, (9) 12 =, (1) ij = LijkR.uk,R.. (11)

4 262 In these equations, P is the mass density of the material particle in the deformed state, Po is its mass density in the undeformed state, y is the displacement of a material particle, E and v are respectively Young's modulus and Poisson's ratio for the material of the rubberlike layer, a comma followed by an index i indicates partial differentiation with respect to xi' the summation convention is used, is a unit vector tangent to the surface, U is an outward unit normal to the surface, 6ij is the Kronecker delta, cr.. is the Cauchy stress tensor and the function p gives the prsure distribution at the contact surface. The condition (8) implies that the normal pressure at the boundary of the contact surface vanishes. This ensures that a contact problem rather than a punch problem is being solved. e note that the semicontact width b and the pressure p (xl,x2) at the contact surface are not known a priori but are to be determind as a part of the solution. As has been employed by Hahn and Levinson (1974), the boundary condition (7) can be replaced by a displacement type boundary condition. We find it convenient to se the boundary condition (7) rather than the equivalent displacement type boundary condition. BOUNDARY ELEMENT FORMULATION OF THE PROBLEM Taking the inner product of both sides of Equation (2) with wi' integrating the resulting equation over the domain G consisting of the region occupied by the layer above the plane x2 = and using the divergence theorem, we obtain J (LijkR. Wi,j)'R. uk dg aj (gkuk - fiwi) ds (12) where i: LijkR- uk,r- Dj' gk = LijkR, Wi,j nr," (14) (13) Details of deriving Equation (12) from Equation (2) are grrei' by lrebbia 1978). We now choose.. to be a solution of where m) is the Dirac delta function and represents a unit load at point m in the k direction. With this choice of wi' Equation (12) becomes ui (m) + f f. wi ds = f gkuk ds (16) an J. an Note that wi and gi are displacements and tractions due to a unit concentrated load at the point m in the k direction. Considering unit forces acting in the three directions, Equation (14) can be written as.-

5 263 where Wi and Gi represent the displacements and tractions in the direction due to a unit force acting in the i direction. Equation (17) is valid for the particular point m where these forces are applied. Expressions for Wi and Gi are given in Brebbia's book (1978). In Equation (17) the integrations are over the boundary of the domain. At points of the boundary, either surface tractions fi or displacements ui or a suitable combination of the two are known. Equation (17) enables us to solve for the unknown surface tractions or unknown surface displacements at each point of the boundary. COMPUTATION AND DISCUSSION OF RESULTS In order to solve the problem by the boundary element method, the boundary of the rubberlike layer in the first quadrant is divided into a large number of segments as shown in Fig. 2. Displacements and surface tractions within each segment (boundary element) are assumed to be constant. The mesh is finer within approximately twice the semicontact width. As has been shown in earlier studies (Batra, 198, 1981), the boundary conditions on the vertical plane have essentially no effect on the pressure profile at the contat surface and the deformations of the rubberlike layer within the vicinity of the contact region. In the results presented herein the vertical plane is taken to be traction free. The computer program given in Brebbia's (1978) book for constant boundary elements has been modified to solve the present contact problem. Half nip width b and the form of the pressure function p are presumed. With this additional data the problem is well defined and can be solved. Having solved the problem a check is made to insure that the deformed surface in the assumed contact zone matches, within a prescribed tolerance, with the circular profile of the indentor and that the nodal point just outside the contact area has not penetrated into the indentor. If the second conditi9n is not satisfied implying thereby that the nodal point just outside the presumed contact width has p,1netrated into the indent;')]:, either the value of b is increased or the total load is decreased. However, if the second condition is satisfied and the first is not, the values of p at various nodal points are adjusted so that the deformed shape of the assumed contact area conforms to the circular profile of the indentor. The deformed surface of the roll cover is taken to match with the profile of the indentor if the distance of each nodal point on the contact surface from the indentor is less than 1.5 percent. of the indentation Uo (see Fig. 1). Usually, with a little experience one can make pretty good est1mates of band p (x1,x2) so that the entire process converges in four or five iterations. The total load P is obtained by integrating p over the contact area.

6 264 Test cases To ensure that the discretization of the boundary as adequate, the contact problem with RO = R = rom, R1 = 441 rom, b = 22.9 rom, E = 8.96 x 15 N/m2, and v =.3 was solved. The pressure on the contact surface matched exactly with that obtained by Hahn and Levinson (1974). Stresses on the bond surface could not be compared since such information is not given in Hahn and Levinson's work for v =.3 and our formulation is not valid for v =.5. Next we solved the problem for a different set of material and geometric parameters, namely that RO = 61 mm, R = 76.2 min, Rl = 46.5 min, b = 12.7 mm, v =.45 and E = 1 x 16 N/m2. Batra (1981) has solved this problem by the finite element method. The pressure profile at the contact surface came out to be virtually the same by the two methods. For this case we plot, in Fig. 3, the variation of the normal stress rr and the shear stress ore at the bond surface with the angular distance from the center line of rollers. These stresses decay to essentially as e/eo approaches 3. Thus the assumption that the vertical plane is traction free is verified to be true. In this and all other figures eo = b/ro. Effect of thickness of the layer Keeping RO = 61 mm, R = 76.2 mm, b = 12.7 mm, v =.3 and E = 1 x 16 N/m2 fixed, the value of Rl was varied. The total load required and the!esultant indentation for various thicknesses of the layer are listed in Table 1. As expected, the load required to keep the contact width constant increases as the thickness of the layer decreases. For thinner layer the indentation is less even though the load is more. However, the indentation/thickness increases with the decrease in the thickness of the ].ayer. Thickness/Rn Load P!(Gb) Indentation/b Indentation Thickness CJ lQO ] Table 1. Load and indentation for different thicknesses of the layer Fig. 4 depicts the pressure profile at the contact surface for two different thicknesses of the layer. Note the change in the shape of the curve near the ends of the contact width as the thickness of the layer is reduced. With the decrease in the thickness of the layer it seems that the pressure at the center increases more than the decrease in the pressure near hp pnd so that the total load increases. Such a chane

7 265 in the curvature of the curve near the ends of the contact width is also apparent in Meijer's (1968) work which studied the indentation, by a rigid cylinder, of a layer fixed to a rigid plane surface. In Fig. 5 is shown the deformed surface of the rubberlike layer for two different thicknesses of the layer. Note that the scales along the vertical and horizontal axes are different. Therefore, the undeformed position of the rubberlike layer plots as an ellipse rather than as a circle. It seems that the change in the curvature of the deformed surface near the ends of the contact zone is heavily dependent upon the thickness of the rubberlike laver. Effect of Poisson's ratio To investigate the effect of Poisson's ratio on the load required to cause the same contact width the values of RO = 61 mm, R = 76.2 mm, Rl = 46.5 mm, b = 12.7 mm, and the shear modulus G = 3.45 x 15 N/m2 are kept fixed. The total load required and the resulting indentation for various values of v are listed in Table 2. These data show that Table 2. Load and indentation for various values of Poisson's ratio the total load increases with the increase in the value of Poisson's ratio upto a certain value of v. As v is increased beyond.45, the total load changes little but the pressure distribution on the contact surface differs in that it is more at the center and less near the ends. This becomes clear from Fig. 6 in which we plot the pressure distribution at the contact surface for two different values of v. Remarks We should emphasize that the results presented above and the conclusions drawn are applicable only to the geometric configuration studied herein. For other values of geometric parameters, one will expect similar qualitative and not necessarily quantitative results. For the present problem the CPU time and the core require-

8 266 ment for the finite element solution and the boundary element solution were virtually the same. In the boundary element solution of the problem the grid used had 1 elements across the thickness of the layer and 35 elements across the circumference. The boundary element solution perhaps gave slightly better values of tractions at the bond surface. The grid data in both caseswere generated internally in the computer program and required the same effort. Even though it is usually said that the finite element solution generates a lot of unneeded information about stresses and strains within a body, to us, it seems to be an advantage especially in a problem like the one studied here. In such a problem one does not know a priori where the maximum principal stress or the maximum principal strain will occur. It is more expensive to compute stresses and strains at internal points from the boundary element solution than it is from the finite element solution. REFERENCES Batra, R. C., Levin'3on. M and Be,:z, F. (19i6) Rubb"'r CCJve.red Rolls--The Thermoviscoelastic Problem. A Finite Element Solution, Int. J. Num. Meth. Engng. lq: Batra, R. C. (198) Rubber Covered Rol1s--The Nonlinear Elastic Problem. Trans. ASME - J. Appl. Mechs.. ]: Batra. R. C. (1981) Quasistatic Indentation of a Rubber Covered Roll by a Rigid Roll, Int. J. Num. Meth. Engng. li: (to appear). Brebbia. C. A. (1978) the Boundary Element Method for Engineers. John Wiley & Sons. New York. Hahn, H. T. and Levinson, M (1974) Indentation of an Elastic Layer Bonded to a Rigid Cylinder--I. Quasistatic Case Without Friction. II - Unidirectional Slipping with Coulomb Friction,.!..: Hannah, Margaret (1951) Contact Stress and Deformation in a Thin Elastic Layer, Quart. Journ. Mech. and Applied Math. i: Hertz, H. (1881) J. F. Math. 92: Meijers. P. (1968) The Contact Problem of a Rigid Cylinder on an Elastic Layer. Appl. Sci. Res. 1!: Spengos, A. C. (1965) Experimental Investigation of Rolling Contact, Trans. ASHE - J. Appl. Mechs., 11: Thomas, H. R. and Hoersch, V. A. (193) University of Ill.. Eng. Exp. Sta. Bull...'

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