In the present method 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 of the non-linear problem. The nip width is measured by a sensitive sensor film and the roll indentation by a laser displacement sensor. The nonlinear numerical problem is solved using the Finite Element Method with four-node isoparametric quadrilateral elements and Newton-Raphson-type iteration. A suitable form of the constitutive equation and the stress state dependence of the moduli of the incremental stress-strain law will be discussed. A least squares fit to the experimental results determines the values of the paper roll elastic moduli

Arola, K.

Jorkama, M.

von Hertzen, R.

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STUDY OF STRESS-STRAIN RELATION FOR PAPER ROLL by ABSTRACT K. Arola1, R. von Hertzen1 and M. Jorkama2 1Helsinki University of Technology 2Metso Paper, Inc. FINLAND In the present method 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 corre-sponding calculated values of the nonlinear problem. The nip width is measured by a sensitive sensor film and the roll indentation by a laser displacement sensor. The non-linear numerical problem is solved using the Finite Element Method with four-node isoparametric quadrilateral elements and Newton-Raphson-type iteration. A suitable form of the constitutive equation and the stress state dependence of the moduli of the incremental stress-strain law will be discussed. A least squares fit to the experimental results determines the values of the paper roll elastic moduli. NOMENCLATURE a e E Err• Er0• E00, Gr0 fr, f0, fr0 K1,K2 p R1 Rz nip half-width measured nip half-width roll diameter residual error modulus of elasticity elastic moduli of the orthotropic roll constitutive relations Pfeiffer' s constants compressive load between the cylinders paper roll radius nip roller radius radial and tangential strains of the roll 135 Yr& () O; Llo-0 Lio-& LI Tr& V INTRODUCTION shear strain of the roll indentation of the roll measured indentation of the roll incremental radial and tangential stresses of the roll incremental shear stress of the roll Poisson's ratio radial and tangential stresses of the roll shear stress of the roll 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 modem 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 sym-metry, 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 be-come 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 When determining the material parameter values from experiments one must have measured data which is diversified enough. For example, in the case of an isotropic homogeneous roll of diameter d, pressed against a rigid half-space by a line load P, one obtains for the contact half-width [5] a= (1) and for the indentation of the roll 136 <5=P-- 2ln(2dla)-- , 1-v2[ VJ ,rE 1-v (2) where Eis the modulus of elasticity and v Poisson's ratio of the roll. It can be seen from equation (1) that measuring the contact width 2a as a function of the line load P fixes the value of E I ( 1- v2) only but not the values of E and v separately. On the other hand, measuring the indentation t5 for different line loads P can in principle separate the values of E and v . In practice, however, this works purely since the term 2ln(2dla) (idO for d=lm and a=lcm) dominates over the term v/(1-v)in equation (2) so that some noise in the experimental data may annihilate the effect of the latter term. Consequently, one must in practice use the measured values of both a and t5 in order to determine the values of E and v . This is on the other hand sufficient since a fixes the value of EI (1- v 2) and then t5 fixes the value of v /(1- v) . In the case of an orthotropic nonlinear material the situation is somewhat more complex. Let us consider first a general stress-strain relation. In a plane strain state, some distance away from the roll ends, one can write the general constitutive relations a,= f,.(&,,&o,Y,o) ao = fo(6 ,, 6o,Y,o) -z-,o = f,o(&,,&o,Y,0) • (3) Of what form are the functions f,, fo and fro in the case of a paper roll? No general answer exists so far. The incremented form of equations (3) is also frequently needed. One obtains (4) and for an orthotropic material, in particular, (5) For a hyperelastic material one can derive the stresses from an elastic potential function so that E,0 = E0, • In general, all the coefficients Err, E,0 , E0,, E00 and G,o may depend on the strain components &, , &0 , and y,0 . Due to the lack of more complete material models, one may postulate simpler models such as 137 (6) (7) (8) For example, Pfeiffer suggests [6] (9) (10) Instead of equation (9) Hakiel prefers the relation [ 1] (11) In the present work we consider the parameters E&&, Er& and Gr& as constants and employ equation (9) for the radial modulus. Our numerical solution of the compression experiment is performed by using a nonlinear FE-model of the continuum combined with a modified Panagiotopoulos-iteration for the contact equations. Since in this case the contact is static the Panagioto-poulos process is much simplified. The FE-discretization of the virtual work equation yields the standard FE-formulation of small strain 2D-elasticity 'I'= fBratdA-F=O, A (12) where B is the strain-displacement matrix containing the derivatives of the shape func-tions, vector a contains the stress components and F= jNTttd[+ fNTbtdA. (13) r A Above t contains the boundary tractions and b the body forces. The symbols A and r denote the area and boundary of the plane cross section, respectively. The thickness of the considered specimen in the direction orthogonal to the plane is t. It is emphasized that no assumptions on the relation between stresses and strains have been made during the derivation of equation (12). The introduction of a nonlinear constitutive relation renders the problem nonlinear and an iterative solution is required. Using the Newton-Rhapson method leads to the following solution algorithm for the elastic problem: • Seti= 0 and specify the starting point. Usually u0 = 0. • Compute the strains due to displacements e; = Bu; • Compute the corresponding unbalance 'Iii= fBT a(e;)tdA-F A 138 • Compute the tangential stiffness matrix O\JI froa froaoe fT Kri=-·-= B -tdA= B --tdA= B Er(ei)BtdA ' OU OU 0800 A A A • Compute the increment of the displacement vector Liu.= -Kr-I· u,. l ,I Tl • Compute the new total displacement vector • Set i = i + I and return to step two. Using the obtained elastic solution of the wound roll the contact equations are solved. The 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 distribution of contact forces is obtained. Typically six or seven material iterations are needed to reach a relative error ::;; 1 o-6 in our calculations. Note that a wound roll represents a strongly prestressed body of material. Therefore, it is important to account for the initial strains in the solution pro-cedure. If the wound-in-tension history of a roll is known, one can account for the ini-tial strains of a finished axisymmetric roll using Hakiel's model; otherwise, experi-mentally determined (pull tab) values must be used. The values of the material parameters are detemined by a least squares fitting pro-cedure. The residual error to be minimized is e(K1,K2,Eoo,Ero,Gro) = N L {[ai -a(K1,Kz,E(J(J,Ero,Gro;P;)]2 +[0;-o(K1,K2,Eoo,Ero,Gro;P;)]2} i=l (14) Here ai and bi denote the measured values of the nip half-width and the roll indenta-tion, respectively, for the line loads P; (i = 1, ... , N) used in the compression experi-ment, and a(K1, ••• ;P;) and o(K1, ... ;P;) denote the corresponding numerically com-puted values. In practice, the computational cost may be reduced by taking the values of K1 and K2 from a stack test and the value of E88 from a machine direction tensile test. Figure 1 shows the dependence of the nip width on the nip load for different values of the roll shear modulus G,8 • It can be seen that, especially at higher nip loads, G,0 is an important elastic parameter in the non-axisymmetric nip contact models. 139 EXPERIMENTAL SETUP The experimental setup is shown in Figure 2. 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 weigth and the core load. The dis-tance 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 signifi-cant 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. An estimation based on the Hertzian pressure distri-bution shows that approximately a width of 0.2 - 0.35 mm at the nip edges will fall be-low the sensing range of the film. 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 lm. 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 G,0 is rarely reported in existing literature, and yet it is an important parameter in modem winding models including the nip action. Our preliminary test results indicate that a strain deficiency [7] seems to exist in the top layers of the wound roll. This means that the measured nip widths must be subtracted by some value before the fitting procedure to the nonlinear solid roll model can be performed. REFERENCES 1. Hakiel, Z., "Nonlinear Model for Wound Roll Stress", TAPPI Paper Finishina and Convertini Conference Proceedin~s, pp. 9-15, 1986. 2. Jorkama, M., "Contact Mechanical Model for Winding Nip", Acta Polytechnica Scandinavica. Me 146, 2001. 3. Pfeiffer, J.D., "Prediction of Roll Defects from Roll Structure Formulas", IAffl .Io!Jm.al, Vol. 62, No. 10, 1979. 4. Qualls, W.R. and Good, J.K., "An Orthotropic Viscoelastic Winding Model In-cluding a Nonlinear Radial Stiffness", Journal of Ap_plied Mechanics. Vol. 64, 1997. 5. Johnson, K.L., Contact Mechanics, Cambridge University Press, 1984. 6. Pfeiffer, J.D., "Measurement of the K2 Factor for Paper", TAPPI Journal. Vol. 64, No. 4, 1981. 7. Pfeiffer, J.D., "Surface Winding to Overcome Strain Deficiency", Proceedings of TAPPI Finishin~ and Convertina Conference, pp. 233-236, 1990. 140 29 27 25 I ,s 23 1 r 21 19 17 15 +------,-----,----.-------,-----,----.-------l 1.5 2 2.5 3 3.5 4 4.5 Nip load [kN/m] -G=5MPa ---G=IOMPa ····· G=20MPa -··-G=40 MPa Figure 1. Effect of the shear modulus G,8 on the nip width by a nonlinear FEM-calculation (K1 = 0.275 MPa, K2 = 12.8, EfJfJ= 5.1 GPa and E,o= 1.0 MPa). Figure 2. Setup of the compression experiment. 141 Study of Stress-Strain Relation for Paper Roll K.Arola1, R. von Hertzen1, and M. Jorkama2 - Helsinki University ofTechnology1 and Metso Paper, Inc.2, Finland Name & Affiliation Question D. Pfeiffer- JDP Pfeiffer wants to defend his Kl-K2 model and explain why Innovations you might have trouble with it to explain the compression of a roll. The Kl-K2 model applies to a stack under uniform applied pressure. In a wound roll a pressure gradient exists starting from very low pressure between the outer layers of paper or so into an increasing pressure gradient. In your model somehow you would need to incorporate that ramp up of pressure between the layers and the impact on the radial modulus. My second comment is that you have a very high value of Kl - much higher than stack measurement would normally predict in the metric system. You would normally find a Kl about 40 KPa instead of 1.9 MPa. Name & Affiliation Answer R. von Hertzen - Helsinki Yes, thank you for your comments. As to the first University of Technology comment, we thought the same thing -- this was not an attack against your model. In this case a more complex material model should have been used. I think that the main reason for the pretty high values of the coefficients is that the material model in this case was not describing the real behavior of the roll. Name & Affiliation Reply M. Jorkama - Metso Paper We did measure the internal stress pressure distribution inside the wound roll and that was accounted for in Mr. Arola's finite element calculation. Name & Affiliation Answer R. von Hertzen - Helsinki The pre-stresses of the roll were measured by pull tabs and University of Technology were included in the simulation. Name & Affiliation Question K. Good-OSU The new parameter that you introduced, a0• How did you come up with a0? Name & Affiliation Answer R. von Hertzen - Helsinki We added an additional parameter there to test the University of Technology optimization strategy to see if it could find values for the parameters so that the calculated results would fit the measured data. Name & Affiliation Question K. Good-OSU So it was arbitrary - the setting of a0? 142 Name & Affiliation Answer R. von Hertzen - Helsinki It was optimized by the optimization strategy. It was University of Technology calculated. Name & Affiliation Question Y. Wang - Sonoco I noticed that you used a two-parameter model and actually used the 2-D model and then reduced to the model with only a diagonal element in that constitutive question. So it's totally decoupled, then you used the elastic model for the stress-strain relationship. Name & Affiliation Answer R. von Hertzen - Helsinki The equations were coupled. A full orthotropic material University of Technology model was used. Of course, the linear orthotropic material parameters include three elastic moduli, three Poisson's ratios and three shear moduli. But we are not actually interested in calculating each one of these separately. What we are interested in are values of the combinations of these parameters in the elastic coefficient matrix. These combinations can be calculated using the introduced method. This way we have only four material parameters to determine for the linear orthotropic case. 143 

Study of stress-strain relation for paper roll

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Study of stress-strain relation for paper roll

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