K. Ärölä* and R. von Hertzen Department of Mechanical Engineering Helsinki University of Technology, Espoo, Finland
e−mail: kilwa.arola@hut.fi, rvhertz@csc.fi M. Jorkama
Metso Paper Inc., Järvenpää, Finland e−mail: marko.jorkama@metso.com
ABSTRACT
Summary A wound roll of paper is loaded against a nip roller and the measured values of the nip width and the roll indentation are compared with the corresponding calculated values. The numerical problem is solved using the Finite Element Method in conjunction with the Panagiotopoulos process. A suitable form of the incremental stress-strain relation will be discussed. A least squares fit to the experimental results is used to determine values for the paper roll elastic moduli.
INTRODUCTION
Winding a flexible web into a compact roll is a widely used process for materials such as paper, thin films and magnetic tapes. By the aid of winding models [1,2] one can analyse the internal stress state of the roll, which is an important part of the defect free roll structure design.
Winding models can predict several roll defects directly, and will help to understand many others.
Every winding model needs a proper constitutive law for the material wound and numerical values for the material parameters. The earliest models were based on the assumption that the wound roll could be treated as a linear, plane strain and orthotropic medium. Later the radial modulus of a wound roll was modeled as a nonlinear function of the radial stress or strain [3]
and also viscoelastic effects were accounted for [4]. A common feature of modern winding is the presence of a nip in the winding path. It should be noted that the nip-induced stresses destroy the rotational symmetry of the stress field of the wound roll. Therefore, shear stresses are generated into the roll. Since the conventional models concentrate solely on cases with rotational symmetry, the information on elastic constants in the literature is restricted to the tangential and radial moduli. However, there is an increasing interest and need for models of winding with a nip so that the values of the shear moduli and Poisson’s ratios also become important. It has been shown recently, in particular, that the value of the shear modulus of the roll may have a marked effect on the nip width [2].
The aim of the present work is to present a method for the determination of the elastic moduli of the roll by comparing the results of roll compression tests with those of a nonlinear FE-model in the sense of a least squares fit.
THEORY
The basic contact configuration of a paper roll and a nip roller is shown in Figure 1, which shows the wound roll pressed against the steel drum.
Figure 1. Nip contact configuration and the coordinate system used.
Let us consider first a general stress-strain relation of an orthotropic nonlinearly elastic material.
In a plane strain state, some distance away from the roll ends, the incremental constitutive equation in the principal directions of the material can be written as
úú the case of a paper roll? No general answer exists so far. Pfeiffer, for example, suggests [5] the relations
Instead of equation (2) Hakiel prefers the relation [1]
2 equation (2) for the radial modulus.
Winding drum
Our numerical solution of the compression experiment consists of two main steps as described below:
• The elastic solution, i.e. the flexural matrix for the discretized contact surface is calculated using FEM.
• The contact stress distributions are calculated using the Panagiotopoulos iteration utilizing the flexural matrix.
The winding steel drum can be considered as rigid when compared to the paper roll. The finite element mesh of the roll cross section is shown in Figure 2. The calculated contact pressures are used to update the stress and strain state of the material and a new elastic solution is sought.
This process is repeated until a prescribed accuracy in the contact force distribution is obtained.
−600 −400 −200 0 200 400 600
Figure 2. Cross section of the paper roll meshed with four node quadrilateral elements.
Typically six or seven material iterations are needed to reach a relative change ≤ 10-6 of forces in successive iterations. Note that a wound roll represents a strongly pre-stressed body of material. Therefore, it is important to account for the initial strains in the solution procedure. If the wound-in-tension history of a roll is known, one can account for the initial strains of a finished axisymmetric roll using Hakiel’s model; otherwise, experimentally determined (pull tab) values must be used.
Calculations show that the value of Erθ has practically no effect, whereas Grθ has a significant effect on the contact width and indentation depth. Hence, it is evident that the value of Grθ can be determined by a least squares fitting to measured data. The residual error to be minimized is
[ ] [ ]
Here ai and δi denote the measured values of the nip half-width and the roll indentation, respectively, for the line loads Pi (i = 1,…, N ) used in the compression experiment, whereas
a(Grθ; Pi) and δ(Grθ; Pi) denote the corresponding numerically computed values. Note that in place of Grθ in eqs (5) and ( 6 ) there may be several unknown elastic parameters. In practice, however, the computational cost may be reduced by taking the values of K1 and K2 from a stack test and the value of Eθθ from a machine direction tensile test.
EXPERIMENTAL SETUP
The experimental setup resembles the configuration shown in Figure 1. A paper roll of radius R1 rests on a nip roller of radius R2. A compressive load is applied from the core chucks to the roll.
The resulting nip load P is calculated from the roll weight and the core load. The distance x of the roll top from a fixed point is measured with a high precision laser sensor (resolution < 10 µm). It is assumed that the nip and core loads do not cause any significant deformation at the roll top. Hence, the nip compression can be calculated from the measured value of x. The nip width was measured using the Pressurex Micro Film. The smallest measurable pressure with this film is 13.8 kPa and, hence, also the nip edges could be quite accurately detected. The evenness of the nip distribution was checked by measuring the nip width from three locations of the roll: both edges and the center. The initial internal pressure distribution of the roll was estimated by pull tab measurements. The total number of pull tabs was 10. Near the roll periphery the internal pressure was measured at 50 mm spacings. The nip roller diameter was 0.425 m and the roll diameter 1 m.
CONCLUSIONS
An indirect method for determining the values of elastic moduli of a paper roll is presented. In particular, the value of the shear moduli Grθ is rarely reported in existing literature, and yet it is an important parameter in modern winding models including the nip action. The present method can in principle be utilized in an automated on-line measurement system.
REFERENCES
[1] Z. Hakiel. Nonlinear Model for Wound Roll Stress. TAPPI Paper Finishing and Converting Conference Proceedings, pages 9-15, (1986).
[2] M. Jorkama. Contact Mechanical Model for Winding Nip. Acta Polytechnica Scandinavica, Me 146, (2001).
[3] J.D. Pfeiffer. Prediction of Roll Defects from Roll Structure Formulas. TAPPI Journal, 62, No. 10, pages 83-88, (1979).
[4] W.R. Qualls, and J.K. Good. An Orthotropic Viscoelastic Winding Model Including a Nonlinear Radial Stiffness. Journal of Applied Mechanics, 64, pages 201-208, (1997).
[5] J.D. Pfeiffer. Measurement of the K2 Factor for Paper. TAPPI Journal, 64, No. 4, pages 105-106, (1981).