Numerical simulation of film casting using an updated lagrangian finite element algorithm

Numerical Simulation of Film Casting Using an Updated Lagrangian Finite Element Algorithm

SPENCER SMITH

Department of Computing and Software McMaster University

1280 Main Street West Hamilton, Ontario, Canada, L8S 4K1

DIETER STOLLE

Department of Civil Engineering McMaster University

1280 Main Street West Hamilton, Ontario, Canada, L8S 4L7

This paper presents a new numerical algorithm for 2D nonisothermal time-stepping simulations of a nonlinear viscoelastic cast film process. A significant contr- bution of the algorithm is that an updated Lagrangian description of motion is employed, as opposed to the more conventional Eulerian description generally used for continuous polymer processing simulations. Furthermore, use is made of a Perzyna- type constitutive equation, which is different from what is usually employed for molten polymers. The constitutive equation accommodates viscoelasticity, extensional thinning/thickening, and strain-hardening. This new numerical algorithm can find the steady-state film properties, and it can predict the onset of instability by observing draw resonance. The critical draw ratio is determined from the response problem, which means that the mathematical complications of the more common linear stability analysis are avoided. In terms of the stability of the film, it was observed that stability is decreased by extensional thinning, strain-hardening, and higher relaxation times, and stability is increased by higher heat transfer coefficients and higher ratios of air-gap length to die width.

INTRODUCTION

wo main approaches exist for describing the mo- T tion of a body (1): the material formulation, where the conservation equations are applied to a control mass: and the spatial formulation, where the conservation equations are written for a control volume. Al- though the different formulations should yield identical results for any smooth motion of a body, for any given type of motion there is often an obvious choice as to which formulation is preferable (2). For instance, fluid mechanics problems lend themselves nicely to spatial, often termed Eulerian (E), formulations of motion. On the other hand, solid and structural mechanics problems are usually best treated using material, or the so-called Lagrangian (L) and updated La- grangian (UL), descriptions of motion. Although the choice for description of motion is well established for fluid and solid mechanics problems, it is not so clear

which choice is best when the material model in question exhibits properties of both a fluid and a solid; that is, when a viscoelastic model is appropriate, as is often the case for polymer processing.

In the literature on continuous polymer processing, such as for the film casting process illustrated in Fig. 1 , the usual choice to describe motion is an E formulation. However, the UL approach also holds promise for simulating the cast film process (3), since it provides a simple framework for handling viscoelasticity and it naturally captures the important draw resonance phenomenon. Draw resonance consists of peri- odic oscillations in the film’s geometry that occur once the draw ratio (Dr ) exceeds some critical value [Dr,--). The draw ratio is defined as the ratio of the speed at the roll (urou) relative to that at the die (udte). A time- stepping algorithm, which is necessary for the UL framework, can predict draw resonance via a response

POLYMER ENGINEERING AND SCIENCE, MAY 2003, Vol. 43, No. 5 1105

Spencer Smith and Dieter Stolle

Table 1. Summary of the Kinematic Assumptions.

Direction

Dim Machine (x ) Transverse (y ) Out-of-Plane (2) Neck-in Edge Bead

1D u = U ( X ) v = 0 w = C(x, z)' No No 1.5D u = U ( X ) v = V(x, y) w = C(x, 2)' Yes No 2D u = U ( X , y) v = v(x, y) w = W(x, y, z)* Yes Yes 3D u = U(x, y, z) v = V(x, y, 2) w = C(x, y, z)* Yes Yes

* W varies linearly in the z-direction

problem: that is, when the time-dependent governing equations are solved, resonance is observed directly as oscillations in the simulated film's geometry. It should be noted that the E framework can also be used to predict draw resonance as a response problem, if a time-stepping algorithm is adopted. However, this is usually not done since the addition of time- stepping removes the speed advantage inherent in solving steady-state casting problems within an E framework. Given the advantages of the time-stepping UL approach, the UL framework has been adopted for the current numerical study of the film casting process.

A number of film casting studies using an E description of motion have been published (4-35). This literature can be categorized according to three key decisions: i) whether the material is assumed viscous or viscoelastic: ii) whether the conditions are assumed isothermal or nonisothermal: and, iii) the assumed

dimensional dependence of the velocity field. With respect to the dimensional dependence, this paper clas- sifies a film casting model as either lD, 1.5D. 2D or 3D. Table 1 clarifies this terminology by summarizing the kinematic constraints on the velocity components (u, u, d) for each type of analysis. The corresponding coordinate directions (x, y. z ) used in the table are the same as those shown in Rg. I. The dimensionality of the film problem determines, as shown in Table 1 , whether two important phenomena can be predicted by a given model: i) neck-in, a downstream reduction in the film's width; and, ii) edge bead, an increase in the thickness of the film at its edges.

Table 2 provides a summary of the references that address the steady-state film casting problem: that is, the literature that does not consider transient behavior or the related phenomenon of draw resonance. The table compares the research according to the three key decisions mentioned previously. As Table 2

+--Die--+( 2h, I Y

f Roll

FRONT VIEW RIGHT VIEW Flg. 1. TwcFdimenswnal fh casting setup.

1106 POLYMER ENGINEERING AND SCIENCE, MAY 2003, Vol. 43, No. 5

Numerical Simulation of Film Casting

Table 2. Summary of Research Simulating Steady-State Film Casting.

Viscous Fluids Viscoelastic Fluids ~

Isothermal Nonisothermal Isothermal Nonisothermal

( 4 , 5 ) ( 6 7 ) (8) (9) 1.5D (10,11) (1 2- 1 5*) (-) (1 6) 2D (17) (1 8) (19, 2P*) (-1 3D (21) (-) (-) (-)

I D

* Acierno eta/ is 1D for the temperature predictions ** Rajagopalan is 20 in the sense that the thickness vanes in the transverse direction but edge-bead and neck-in are not included because the computational domain IS restricted to the

central oortion of the film

shows, most of the steady-state film casting studies are 1D or 1.5D. As a consequence of this, the influence of the edge bead defect on the film has received only limited attention. The table also shows that the majority of film casting studies assume a viscous fluid.

Table 3 has the same format as Table 2, except that the focus is on the stability of the film casting problem. This summary shows that only a small number of papers consider the stability of 2D films. The table also demonstrates that nonisothemal problems have received only limited attention and that the majority of studies consider viscous fluids. With respect to the determination of draw resonance, the most popular approach is linear stability analysis, which is used for almost all of the studies of Table 3. The approach of observing draw resonance via a response problem is only employed in a few studies (24, 27, 34, 35).

Following the headings used in Tables 2 and 3, the current study can be classified as 2D and nonisothermal, with a viscoelastic constitutive equation. The constitutive equation adopted here, which accommodates Newtonian fluids, power-law viscosity, elastic effects and strain-hardening, is new in the sense that it has not previously been applied to molten polymers. The influence of inertia, self-weight, film sag, die-swell, surface tension and air-drag are assumed negligible. Furthermore, only uniform boundary conditions at the die and roll are used, the influence of an air-knife is neglected, and only a single layer of film is modeled. The algorithm described in this paper can be used for both steady-state and draw resonance studies.

The first section presents the governing equations and boundary conditions for the cast film process. Thereafter, the finite element algorithm is summarized and the results of the simulations are presented and

discussed. Results are presented for lD, 1.5D and 2D film casting, with the 1D simulations focusing on the influence of the material and heat transfer properties on Dr,, the 1.5D results addressing the influence of the aspect ratio on stability, and the 2D simulations investigating the effect of the material and processing conditions on both steady-state and unstable conditions. Concluding remarks are given in the final section.

GOVERNING EgUATIONS

Figure 1 shows the Cartesian coordinate system used for the film casting problem. The origin is located at the center of the die with the coordinates running in the machine direction x, the transverse direction y. and the out-of-plane direction z. The dimensions of the film problem are defined by the air-gap length L, the die width 2W,, and the die thickness 2bie A fac- tor of 2 is used in labeling the dimensions of the die to facilitate the introduction of the symmetry constraints.

Equilibrium Equation

At every instant in time the film must satisfy the equilibrium equation. If inertia, self-weight, air-drag and surface tension are neglected, then the equilibrium equation can be written as

LTo = 0 (11

where L is the linear differential operator that relates incremental strains A& to incremental displacements u, such that AE = Lu, and a is the Cauchy stress tensor for the deformed configuration. The incremental displacements (u) within a time step (A t ) are related to the velocities (u) of a material point via u = At u. The

Table 3. Summary of Research on the Stability of Film Casting.

Viscous Fluids Viscoelastic Fluids

Isothermal Nonisothermal Isothermal Nonisothermal

1D 1.5D 2D 3D

(22-24) (29,30)

(33*, 34) (-)

'Lee models the film differently than the other studies, by considering the film as a parallel composition of numerous fibre filaments spun simultaneously.

POLYMER ENGINEERING AND SCIENCE, MAY 2003, Val. 43, NO. 5 1107


notation used here is similar to that of Zienkiewicz (36). in which symmetric 2nd order tensors, such as stress and strain, are represented as vectors. The film is assumed to behave as a membrane, which reduces the number of components in the equilibrium equation to only those necessary for a plane stress problem: i.e. u = [a, uyu a,IT. This reduction is possible because for a membrane, the normal to the film's surface is approximately in the z-direction, and the m a g nitude of the out-of-plane shear is negligible when compared to that of the other components. This assumption applies when the film is thin and the thickness gradient is small. Strictly speaking the membrane approximation does not hold at the edges of the film (4).

Incompressibility Equation

The equilibrium calculation using the membrane approximation only predicts the in-plane incremental displacements u and v. To determine the out-of-plane incremental displacement w, an explicit calculation invoking continuity must be introduced. If the melt is assumed to be incompressible, the out-of-plane strain can be related to the two in-plane components via

(2)

This equation is used to calculate w, which allows the film's thickness to be updated.

A&,, = - (AExx + b , y )

Conservation of Thermal Energy Equation

In the case of nonisothermal film casting, with New- ton's law of cooling assumed to apply on the film's upper surface, the transient 2D temperature field T(x, y, t) of the membrane can be determined using

dT a t

kVT(VT) - u(T- Tair) - pCh- = 0 (3)

where VT = [a/& a/ay], (Y is the heat transfer coefficient from the film's surface, T& is the surrounding air's temperature, h is the film's thickness, and k, p and C are material properties for the thermal conductivity, density and specific heat capacity, respectively. Implicit in this equation are the assumptions that the temperature varies little through the thickness of the film and that the effects of viscous dissipation are negligible.

Constitutive Equation

Adopting a constitutive equation for an elastic material that is creeping and invoking the additivity pos- tulate, the incremental form of Hooke's law may be written as

A U = D ( A E - A e c ) = G 2 4 0 ( A E - Aec) (4) [e 2 :I where A u is the stress increment, D represents the elasticity matrix, G is the shear modulus and AE and

A E ~ are the total strain and creep strain increments, respectively. The components of D are derived by starting with the usual linear elastic constitutive equation with an unknown pressure, and then eliminating the pressure by using the plane stress and incompressibility assumptions, in the same manner as for film casting of a viscous fluid (1 7).

The creep strain increment may conveniently be expressed using the approach adopted by Perzyna (371, which in a modified form can be written as

with A t being the time step, EG the effective creep strain rate and + the creep potential function defined as

IJJ = q, with q = v3J2 (6)

The parameter q is the effective stress (38). and J , represents the second invariant of the deviatoric stress tensor.

If the creep strain rate is assumed to follow a time- hardening creep law, then

E: = Aqm t n (7)

where A, rn and n are constants and t is the total time. This law can be transformed into the strain- hardening form, by holding q constant and taking the time derivative of Eq 7. After some manipulations, the strain-hardening relationship for the creep strain rate is obtained by eliminating time t in the rate equation through the use of Eq 7; i.e.,

A close examination of the above equation indicates that it is an equation of state, in which .5: = f ( ~ ; , q) . The special case of a Von Mises material is obtained for m = n = 1 and A = 1/(2qc), with qc being the creep viscosity. The effect of temperature on viscosity may be modeled by introducing the Arrhenius relation for the A parameter via

where Q is the activation energy, R is the gas constant (8.314 J mol-I Kpl) and A, and To are the reference values for the A parameter and for the temperature, respectively.

It should be emphasized here that the approach followed by the authors takes a solid mechanics perspective, rather than the usual fluid mechanics one, owing to the use of the UL framework. The constitutive equation can be described as a nonlinear, nonisothermal, Maxwell body, with a relaxation time of A = qS/G = 2qC/(3G), where qs is the shear viscosity. One should be aware that the Maxwell body given by Eq 4 uses a strain measure that is different from that usually adopted in an E framework. Thus, except for the case of small strains, its viscoelastic behavior differs from



that of the nonlinear Maxwell equations generally adopted in the polymer processing literature. On the other hand, an examination of the constitutive equation shows that it provides a good approximation of a viscous fluid, when the relaxation time is low.

For a viscous power-law fluid n = 1.0 and rn is the reciprocal of the power-law index generally cited in the chemical engineering literature. The constitutive equation can describe Newtonian behavior (rn = 1). extensional thinning (m > 1) and extensional thickening (rn < 1). A n additional feature of the constitutive equation described here is that it allows for strain- hardening and softening, which respectively mean an increasing and decreasing resistance to deformation as the creep strain accumulates. Given that the softening behavior (n > 1) is unstable, the materials considered in this study require that R 5 1.

Boundary Conditions and Initial Conditions

Taking into account symmetry about the centerline, the boundary conditions for the half width of the film casting problem are shown in Fig. 2. The mechanical boundary conditions specified inside the die and on the roll imply that the film moves downstream (in the machine direction) as a rigid body at these locations. On the roll, the transverse uand out-of-plane ~ velocities are set to zero to simulate the sudden freezing of the melt when it contacts the chill roll. At the free surface and along the line of symmetry, the natural

boundary conditions are assumed to be zero normal stress (a,) and zero shear, respectively. The thermal boundary conditions for the film consist of prescribing the temperature inside the die as TdC and setting the normal thermal flux q ,, to zero on all other surfaces. A zero thermal flux approximation along the edges is reasonable given the thinness of the film and its poor thermal conductivity.

The transient analyses of this study typically start with the film inside the die with the temperature set to Tdie The initial stresses and accumulated creep strain for the material inside the die are assumed to be zero. Strictly speaking, the initial stresses and creep strain are not likely to be zero, as flowing through the die deforms the material. However, the initial stresses often relax quickly because of the high temperatures at the die. Thus, in the absence of upstream information, the stresses and accumulated strains are assumed to be zero. This assumption is the same as that used elsewhere (19) for simulating the film casting of a viscoelastic fluid. A consequence of the zero stress condition, along with use of the membrane approximation, is that the die-swell phenomena cannot be accommodated by the present study.

FINITE ELEMENT ALGORITHM

Figure 3 provides a flowchart for the time-stepping UL finite element algorithm. To simplify the presenta- tion, the flowchart shows only those variables that are

Flg. 2. Thennomechanical boundary conditions for a typicaljhite element mesh.


Spencer Smith and Dieter StoZZe

Initialize Variables:

Update Constraints h 4 + V + W + - O >

Solve: Au,Av7Aw

F- Simulation Complete

Update time, mesh and temperatures: t + t + A t X + - X + U

Y + Y + V Z + Z + W T + T + A T Output results Renumber mesh

Update dof, stress, strain: U + U + A U v + - v + A v w + - w + A w a + a + A a E,C + E: + AE:

Solve: 4 AT

Fig. 3. Flowchart for the solution algorithm

most important, which include the current time ( t ) , the final time ( tfinal), the nodal coordinates (x, y, z) , the displacement degrees of freedom (u, u, w) , the stress tensors at the integration points (a), the total effective creep strain at the integration points (E:) and the temperature degrees of freedom (T). As usual, the symbol A is used to denote a change in a variable.

Each time step of the UL algorithm starts from a known configuration and solves a materially and geo- metrically nonlinear finite element problem for the nodal displacements and temperatures. At the end of each time step the mesh, stresses, effective creep strains and temperatures are all updated. The updated values provide a new reference configuration and state, which allows one to repeat the prediction process for the following time step. After many time

1110

steps, a given finite element will experience the following sequence of events: i) exits the die: ii) travels through the air-gap: iii) moves onto the chill roll: iv) is removed from the roll and is reinserted and reinitial- ized inside the die to begin the sequence again. In principle, the time-stepping process can be continued indefinitely: however, this is unnecessary for a stable film problem as the transient behavior eventually damps out and a spatially steady-state solution is reached.

Initialize Variables and Update Constraints At the beginning of each time step the constraints

are updated for the displacement degrees of freedom. To determine the constraints, each node of the finite element mesh is classified according to its spatial

POLYMER ENGINEERING AND SCIENCE, MAY 2003, Vol. 43, No. 5


location, which as shown in Fig. 2 are inside the die, on the chill roll, in the air-gap, on the line of symmetry, and at the free surface. In addition to constraints required by a node's spatial location, constraints are also required if the problem is assumed to be either 1D or 1.5D. For the 1.5D problem, all of the nodes for each column of nodes in the air-gap share the same degree of freedom for each 0f.u and w. The 1D problem uses the same constraints, with the additional requirement that u = 0 for all of the nodes. The constraints on the 1.5D and 1D problems mean that all of the columns of nodes remain parallel to the die and that the thickness across a given column of nodes is constant. In the case of the 1D problems, the width of the film also remains unchanged.

Solve for Nodal Displacements

To estimate the displacements for the ( i + l)th time step, the process starts with the fact that the residual load (u' + 1) for that time step should be zero; that is,

'Pi+, B T ~ , + , d V i - R i = O (10) i V,

where R, is the load vector, V, is the volume of the current domain, B is a matrix such that he = Ba, + 1,

with a, + the desired degree of freedom (dof) vector for the in-plane displacement increments (u and u). The finite element equations for an implicit time-stepping algorithm can be derived (39, 40) using Eq 10 as a starting point. On the first pass, the finite element equations for equilibrium can be expressed as

jBTD"B dV, a,, =

v

R i - BTU,dv+ B T A u C d X (1 1) J V,

! V,

with

c, = [ l + (He + HJ-1, He =

where F = At . &;, A u C is the creep stress increment, and Due is the viscoelastic constitutive matrix. At this stage of the calculations ai + represents a first estimate for the change in displacements. For subsequent passes within an equilibrium iteration loop, the finite element equations, which provide a correction for ai +

simplify to

B T D B dVi A a = R i - BToi dv (15)

with ai+l t ai+ + A a , where the right arrow notation represents the assignment of a value to the variable a, + (41). In Eqs 11 and 15, the so-called initial stress matrix W,, which is usually included for an updated reference configuration formulation (42), is not introduced, as it was found to have little influence on the final results for the class of problems addressed in this study. The integrals in Eqs 11 and 15 are over the current geometry, so equilibrium will no longer be satisfied when the geometry is updated. This small error is not a problem, since it is accounted for in the next time step via the integral over BTu.

Owing to the membrane approximation, A a does not contain the out-of-plane component of the incremental displacement A w . To find A w , the continuity equation (Eq 2) is multiplied by the trace of the virtual strain tensor S(Ae) , which is consistent with the virtual displacements, and is then integrated over the film's domain

i v,

I V,

c

(16) ( A E ~ + AE,, + Aczz) d V = 0

Equation 16 has the advantage that it preserves the units of work done, as the change in the virtual vol- umetric work done can be expressed as S(AWJ = tr(6Ae) - A p and A p = a . tr(AE), where A p is the change in pressure and tr (A&) --f 0 as a -+ x, for an incompressible material. I t can be simplified by observing that Sagxw = S A E ~ ~ = 0, since both ALL and Au are specified by A a , which is known:

S ( A E z ) A E z d V = - S ( A E ~ ) ( A E ~ + AE,,)dV (17) [ The above equation can be expressed in the usual finite element notation, with the use of B,,, which contains the components of the B matrix that are used to calculate AE,; that is,

V V

in which A w contains the nodal values for A w and (Asxu + AE,,) is calculated for each integration volume after ai+ or A a is determined from the equilibrium considerations.

For each time-step, the changes in the dof vectors are calculated repeatedly, until the convergence crite- ria satisfies a given tolerance (tokr), as follows:

(19)

where 11 . I/ implies the Euclidean norm of the vector in question.


Spencer Smith and Dieter StoIIe

The Radial Return Algorithm and Average Strain Elements

After solving for the displacement increments, the local stresses and strains are updated using a radial return algorithm (43, 44). In the radial return algorithm, the stress invariants p and q. which correspond to the pressure and the effective stress, respectively, are first updated and then used to scale the stress tensor u and to update the effective creep strain E;. Since the creep response of the constitutive equation (Eq 8) does not depend on the pressure p, q may be updated in an uncoupled manner. The details of the application of this procedure to the film casting problem are provided elsewhere (40).

To construct the finite element stiffness matrices an average strain approach (45, 46) is employed. The average strain approach, which is an alternative to Gauss quadrature, has the advantage of defining gradients for a region, rather than for a point. This suits the assumption that the film properties,are averaged through its thickness. Another advantage for the average strain approach. which is beneficial for the de- forming mesh of a UL formulation, is that the integra- tions are exact, even for distorted elements.

As Fg. 2 shows, the nodes of the film elements that are crossing the die or roll rarely coincide with the start location of these spatial boundary conditions. To address the problem of differences between the mesh and the spatial boundary conditions, two special elements are used in the 1D and 1.5D simulations: a die element and a roll element. For an element that is crossing the die or roll, the average shape function gradients are calculated using a redefined geometry. For these elements the x-coordinates are redefined to coincide with the x-coordinate of the die or roll. In the case of the die element, no further redefinition of the local coordinates is required, whereas for the roll element the y and z values must be interpolated to their values where the element crosses onto the roll. In the case of the 2D simulations the nodes cannot be redefined in this simple manner because the 2D elements may rotate as they cross onto the roll resulting in an odd number of nodes on the roll. This makes it impos- sible for a simple redefinition of the nodes to form a “new” quadrilateral element in the air-gap. Therefore, for the 2D analyses the die and roll elements are not employed and no special measures are taken to cor- rect the elements at the die or roll. This implies that the boundary is allowed to wander a little at these locations. Although this may introduce a small error with respect to solving the mathematical problem, it likely more realistically reflects the physical problem.

Solution of the Nodal Temperatures

The finite element equations for predicting the temperature are fairly straightforward. A fully implicit algorithm, which is linear and uncoupled from the mechanical analysis, is used. Details on this type of analysis can be found in many sources (36, 40, 47,

48). In summary, however, the change in the temperature for the ith time step (Ti +, +- Ti i AT,), is predicted via the following finite element equations:

[IT+:] h T i = F i - H T i (20)

with

F = N ; a T a i r d A J (23) A

where Ti is the temperature dof vector, A is the area of the film, N , is a shape function matrix such that T = NTTi , and B, is a matrix such that VT = B,T,. A noteworthy characteristic of the above equations is that the advective term, which appears in the Euler- ian analysis, is not required since the finite element mesh moves with the material.

The following sections present results of simulations using the previously described numerical model. The program was complied using Borland Delphi 5.0 and simulations were completed using a 400 MHz Pentium 111 computer with 128 MBytes of RAM. Com- pletion time ranged from roughly an hour, for the 1D simulations, to several days, for the 2D simulations. The greatest burden on the computations, which is not present in non-transient algorithms, was the sometimes significant number of time steps required to reach steady-state. Further discussion of efficiency is provided in Reference (49). where Eulerian and updated Lagrangian algorithms for 1D film casting are compared.

INSTABILITY IN 1D FILM CASTING

The critical draw ratio (Dr,,) may be identified by monitoring the history of the rate of energy dissipation Wc, which is defined as

V V

(24)

where the subscript i refers to the values for the ith integration volume and ninteg is the total number of integration volumes for the film. In the case of 1D and 1.5D film casting, the calculation of Wc uses the die and roll elements discussed previously. By definition, an upper bound for stability corresponds with the sit- uation where large oscillations in the rate of energy dissipated do not damp out over time. This definition

1112 POLYMER ENGINEERING AND SCIENCE, MAY2003, Vol. 43, No. 5


Table 4. Film Casting Parameters for the 1D Simulations.

Geometry h,jle=o.ool m; W ~ l e = 0 . 5 m ; L = 0 . f m ( l D ) ; L = WdIe.Af(l.5Dand2D)

Boundary Conditions OdIe = 0.01 m/s: ore,/ = o& Df

Material Parameters

rn = 1 .O; n = 1 .O; x = 0.002 s; qs = 2000 Pa. s; A = 1/(3qs) = 1.6667 x (Pa. s)-'; E = l/(XA) = 3.0 X l o 6 Pa

Numerical Parameters

At = s ; foler = nelW* = 1 (1 D); nelW = 30 (1.5D and 2D)

nelP = 200 (1 D and 20); nelL = 150 (1.5D);

"ne/L and nelW refer to the number of elements in the machine and transverse directions, respectively.

of stability does not require a constant Wc history, only one that stays reasonably bounded. An altema- tive approach is to define the processing conditions as stable if the time rate of change of the film geometry at the roll approaches zero as time progresses (24, 27, 34, 35, 49). However, the use of the Wc history has the advantage that the rate of energy dissipated is a scalar measure that depends on the changing configuration of the entire body, not just on the thickness and width at the chill roll. When completing an analysis, only integer values of the draw ratio are considered because of the uncertainties inherent in the approach. This approach was used to investigate the influence of the constitutive parameters and processing conditions on the stability of the film, employing

the processing conditions and numerical parameters of Table 4, except where noted otherwise.

The Stability of a Viscous Material Figure 4 shows the Wc histories at Dr values of 20

and 21. An examination of Fig. 4 shows that, for Dr = 20, after an initial increase in Wc the oscillations decrease. According to the proposed definition, this implies that the process is stable when Dr = 20. A draw ratio of 21, on the other hand, shows oscillations that do not decrease over time. Therefore, the film is considered to be unstable for Dr = 2 1 . This allows one to conclude that the critical draw ratio lies between 20 and 2 1, which agrees well with the theoretical prediction for a viscous fluid of Dr,, = 20.2 (22). It is also

Fg. 4. Rate of energy dissipation historiesfor Dr = 20 and Dr = 21.

POLYMER ENGINEERING AND SCIENCE, MAY 2003, Yo/. 43, No. 5 1113


noteworthy that the plots in Fig. 4 show more than one frequency. The higher frequency is likely associ- ated with the imperfect satisfaction of the boundary conditions over each time step. Although this higher frequency makes for a noisy plot, it approximates the actual physics of the process where the boundary conditions likely migrate slightly over time, as indi- cated previously. This conclusion is supported by ex- perimental data for film casting (24), which shows noise, even for stable operations.

Influence of Extensional ThinninglThickening and Strain-Hardening

Figure 5 shows the dependence of the critical draw ratio on the extensional thinning/thickening parameter for fluids with and without strain-hardening. This figure also plots the closed-form solution (23) for the critical draw ratio of a fluid without strain-hardening. The curve in Flg. 5 for n = 1 .O agrees very well with theoretical expectations, although as the extensional thickening nature increases (rn < 1.0). the numerical model underpredicts the closed-form solution. This is most likely a consequence of the numerical algorithm, which does not perform as well at the higher draw ratios, because the downstream elements are considerably

n = 1.0 \

elongated. To partially compensate for the effect of the elongation of the downstream elements at higher draw ratios, the simulations used to produce the m 5 1 points of Fig. 5 employed 1600 elements, instead of the usual 200. When strain-hardening is added, the shape of the curve is similar. but it is shifted downwards.

Influence of the Relaxation Time

The influence of the relaxation time on the critical draw ratio is shown in Fig. 6. This figure shows that a higher relaxation time decreases the stability of the system. The decrease in Dr,, could be related to the increase in the elastic energy, which is available to do work when the energy is released. The trend shown in Fig. 6 differs from that predicted by linear stability analysis of viscoelastic fluids, which suggests that increasing relaxation time has a stabilizing effect (26, 35). Another reference (32) observes that viscoelasticity enhances stability for extensional-thickening fluids, but has the opposite effect on extensional-thinning fluids. Unfortunately, as previously mentioned, the Max- well body used in this study fundamentally differs from those used in previous stability studies. Thus a direct comparison to previously reported results is, strictly speaking, not applicable.

0.50 1 .oo 1.50 2.00 2.50

Extensional Thinningmhickening Parameter m Rg. 5. Dependence of Dr,, on the extensional thinninglthickening parameter, with and without strain-hardening for the cbsed-fom (-1 and numerical solutions (0 and 0).



8 1 I I I I I

0.00 0.20 0.40 0.60 0.80

Relaxation Time A (s)

Fg. 6. Critical draw ratio Dr,, versus the relaxation time A [s).

Influence of Nonisothermal Conditions the polymer was modeled as a viscous fluid. The ap-

To investigate the influence of increasing heat transfer on a viscous fluid the parameters of Table 5 were introduced into the analysis. The parameters selected for Table 5 are typical parameters for polypropylene (PP) (50, 51). The results of the analysis are summarized in Fig. 7. As the figure shows, increasing heat transfer has a significant benefit for the stability of the film. The same stabilizing influence is observed in fiber spinning (52), which has essentially the same governing equations as 1D film casting. An increase in the critical draw ratio with increasing heat transfer provides an explanation with regard to why industrial film lines operate a t much higher draw ratios than should be possible according to the isothermal theory.

proach-to identifymg instability for the 1.5D problems is the same as that employed for the 1D simulations discussed in the previous section, except that increments of 0.5 are allowed in Dr to accommodate varia- tions that are too small to observe with integer values alone. The findings are summarized in Fig. 8, which shows how the critical draw ratio varies with the aspect ratio. The first point (Ar = 0) in FQ. 8 was found using the 1D film model, which is mathematically the same as an infinite width 1.5D film, since both cases have the same requirement of zero straining in the y- direction. As the aspect ratio increases, a stabilizing influence is observed, although the benefits seem to level off at Ar F= 1 .O. A stabilizing influence from an increasing aspect ratio confirms what has been demon- stratedin other studies (29, 30, 53). INFLUENCE OF ASPECT RATION ON

STABILITY FOR 1.5D FILM CASTING PROBLEMS WITH THE 1.5D KINEMATIC ASSUMPTION

Although the 1.5D approximation has been popular, it is not a good approximation of the film casting process because edge beads, which are observed in the real film casting process, cannot be accommodated

The parameters provided in Table 4 were used to conduct the simulations presented in this section, but with various values of the aspect ratio Ar = L/ Wde To avoid the complications of extensional thinning/thickening, strain-hardening or effects of material elasticity,

Table 5. Typical Nonisothermal Film Casting Parameters for PP.

Boundary Conditions

Material Parameters

Tdie = 215°C; T,, = 30°C; a = 20 W/(m2 K)

p = 910 kg/m3; k = 0.15 W/(m K); C = 2100 J/(kg K); QIR = 5100 K; To = 190°C; -r$ = 1/(3A0) = 3200 Pa. s

POLYMER ENGlNEERlNG AND SCIENCE, MAY 2003, Yo/. 43, No. 5 1115

1116


8o 1

0.00 2.00 4.00 6.00

Heat Transfer Coefficient a (W/(m2K)) Rg. 7. Critical draw ratio Dr,, versus the heat transfer coefficient a.

"1

am a40 0.80 1.B 1.63

Aspect Ratio Ar

FQ. 8. Critical draw ratio Dr,, versus the aspect ratio Ar.



within the 1.5D assumption. Not being able to predict the edge bead may not seem important, as it represents only a small fraction of the overall width and is typically trimmed from the film. However, the edge bead does have a strong influence on the overall film geometry as it provides a restraining influence on the film, with the result that the middle of the film is essentially in a state of plane strain. When a state of plane strain is combined with the membrane approximation, the 1D model discussed in the previous section is obtained. One could therefore argue that the 1D model is more consistent with the behavior across the majority of film’s width, than is the 1.5D model.

The idea that the middle of the film is in a state of plane strain, and that the edge bead is in a state of uniaxial extension is the usual explanation for the edge bead phenomenon (54). This theory is supported by the results of Smith and Stolle (18), where the restraining influence of the edge bead is illustrated for nonisothermal film casting, film falling under its own weight, and for the case when the edge bead is partially removed by employing a nonuniform thickness across the die. However, for viscous polymers the assumption of 1D behavior in the middle of the film does not seem to hold for larger aspect ratios. For instance, simulation results with A r = 1.0 (19) do not show a zone of 1D behavior. Therefore, when higher aspect ratios are used for viscous polymers, or when there is interest in the behavior of the edge bead, it is necessary to adopt at least a fully 2D formulation. The next section presents steady-state and stability results for the 2D formulation.

2D SIMULATIONS All of the simulations conducted for this part of the

study were based on the parameters from Table 4. The first subsection, which compares simulation results to those of other published studies, uses an aspect ratio of A r = L/ W,, = 1.0. Subsequent subsec- tions present a parametric study and employ A r = 1.4. The simulations were continued until Wc con- verged to a n approximately constant steady-state value. Although most film lines operate at lower aspect ratios, this value was chosen for two reasons. Firstly, the 2D nature of the film is more apparent at higher aspect ratios and secondly, the higher aspect ratio simulations amplify the influence of changes in the constitutive parameters or operating conditions, since the larger air-gap allows more time for an ad- justment in the flow characteristics. The second rea- son for using a large aspect ratio is important, since the film geometry is relatively insensitive to changes in the material properties for the simulations shown in this section.

All of the simulations of this section have steady- state finite element meshes similar to that shown in Fig. 2, except that aspect ratio and the mesh density are higher for the simulations discussed here. The mesh, which is a “snapshot” at one instant of time, provides a clear picture of the deformation history for

the material, as one can easily identify the distortions that develop as the material is stretched through the air-gap. Along the centerline of the film the accumulated strain is mostly extensional, whereas nearer the edge, the material experiences significant shear strain. The shape of the finite element mesh highlights the fact that the elements near the edge have a longer path and residence time within the air-gap. Moreover, the pronounced differences between the shapes of the finite elements at the chill roll clearly indicate that the desired uniformity of the film properties across the film’s width is difficult to achieve, because the residence times of the different material particles within the air gap are quite dissimilar. From a practical point of view, the shape of the mesh and its elements provides useful information on the behavior of the film.

Comparison to Published Studies

The thickness profile at the chill roll for Dr.= 9 and 20 as predicted by Debbaut et al. (19), the E algorithm of Smith and Stolle (7, 18) and by the current UL formulation are shown in Fig. 9. The different algorithms lead to different predictions, with the most pronounced difference for the UL simulation versus the two E algorithms at a draw ratio of 20. Several possible explanations for the differences between the E and UL results were investigated (40). including the element shapes, the approximation of the boundary conditions, and the initial mesh used for the UL analysis. None of these factors seemed to explain the differences between the solutions of the UL and E algorithms. An examination of the dissipation energy suggests that the differences might be explained by the nonlinear nature of the problem and the fact that the algorithms cannot easily distinguish between dissimilar solutions because they are so close in an energy sense. Further- more, unlike the E algorithms in question, the UL algorithm accommodates transient behavior, so the observed increase in differences between the solutions as the draw ratio increases might also be related to the approaching instability.

Influence of Extensional Thinninflhickening

To investigate the influence of extensional thickening/thinning, simulations were conducted with rn values of 1.0 (Newtonian fluid), 0.75 (thickening fluid) and 1.5 (thinning fluid). Figure 10 illustrates how the normalized thickness contours are affected by the power-law nature of the viscosity. A close examination reveals that the extensional thickening fluid under- goes a higher thickness gradient in the machine direction at the die, as well as a more rapid neck-in at the free surface, when compared with the extensional thinning fluid. However, the final width of the film is almost identical for all three fluids under considera- tion. In fact, the differences in geometry between the different fluids are fairly small. This fmding is in keep- ing with the results presented by Debbaut et al. (191, who also present steady-state simulation results for



0.05

0.35 0.3 0.25

-- --

8 0.2 r, ~ 0 . 1 5

0.

the 2D film casting of an extensional thinning fluid. For small degrees of extensional thinning (m = 1.251, only a small change from the thickness predictions for a Newtonian fluid is shown by them. On the other hand, for larger degrees of extensional thinning (m = 2 and m = 3). they show a dramatic change in the thickness distribution. These high values of M could not be reproduced with the UL algorithm because the stiffness matrix became ill-conditioned, which likely reflects the increased physical instability at the higher m values. As demonstrated in Rg. 5 for the 1D formulation, higher rnvalues lead to a dramatic decrease in the critical draw ratio. The algorithm presented by Debbaut e t al. (19) does not account for instability, so the possibility exists that the high extensional thinning value simulations they present may correspond to physically unattainable states.

Influence of Strain-Hardening The influence of the n parameter on normalized

thickness contours is demonstrated in Fig. 1 I , for n = 1.0, 0.5 and 0.25. One observes that strain-hardening leads to a higher thickness gradient at the die. In fact,

the initial decrease in thickness is so rapid for n = 0.25 that the h/&@ = 0.9 and 1.0 contours are almost coincident. Although the jump in thickness reduction at the die may be partly due to the imprecise satisfaction of the boundary conditions at the die, the high gradients point to a potential problem area in the physical process that an engineer should be aware of.

Strain-hardening also has an influence on the shape of the free surface. For every film casting simulation the free surface must be normal to the die and to the roll to ensure that the free surface is tangent to the velocity vector. At these locations the boundary conditions spec@ that the only non-zero velocity component is in the machine direction. On the other hand, as the film moves away from the die the free surface adjusts so that it is no longer perpendicular to the die. In the case of a strain-hardening fluid this ad- justment in the free surface occurs more rapidly than for a Newtonian fluid. Another influence of strain- hardening is a slightly wider final film width. A likely explanation for this is related to the longer particle path along the free surface. The longer path provides more time for the creep strain to accumulate and this

h/h& = 1 .o 1 .o 1 .o

m = 1.0 m = 0.75 m = 1.5 Fig. 10. Thickness contours for various values of the extensional thinning/thickening parameter m

D r = 9

Dr = 20



Ml&= 1.0 1 .o 1 -0

n= 1.0 n = 0.5 n = 0.25 Fig. 1 1 . Thickness contours for various values of the strain-hardening parameter n.

results in a stiffer material. The stiffer material at the edge provides a greater restraining influence, which in turn leads to a larger final film width. As a final point, the differences between the contour plots in Fig. 11 are most noticeable near the die. Once the film reaches the roll, the changes in the constitutive behavior do not have a dramatic influence on the film’s width and thickness.

Influence of Relaxation Time

Simulations were completed with relaxation times corresponding to 0.002, 2 and 7s. Figure 12 illustrates the influence of the relaxation time on the normalized thickness contours. Although the influence is not dramatic, higher relaxation times appear to lead to a more sharply defined edge bead and a larger final film width. A similar result is observed by Debbaut et aL (1 9), although they predict a much more significant influence. They also observed a much wider region of uniform thickness at the chill roll and a higher thickness gradient a t the die with increasing relaxation time. The differences between the current study and Debbaut et al. are not surprising, given that their Maxwell equation is, strictly speaking, not equivalent to the Maxwell equation used by the current formulation.

Influence of Nonisothermal Conditions

The normalized thickness contours shown in Fig. 13 were found by using the parameters of Tables 4 and 5, with heat transfer coefficients a of 0, 20 and 40

h/h,;.= 1.0 1 .o

W/(m2 . K). For the material properties and processing conditions chosen, the nonisothermal conditions are seen to increase the thickness gradient at the die. This finding was also observed using a 2D Eulerian algorithm (18). Another observation from using an Eulerian algorithm (18) that is reproduced by the simulations presented here is a decrease in the neck-in with increasing heat transfer. Smith and Stolle (18) suggest that the decrease in neck-in is a consequence of the longer particle path at the edge of the film, as this leads to increased cooling and a greater restraining influence of the edge, owing to the increase in the temperature-dependent viscosity.

Instability for 2D Film Casting

It is not possible to reliably investigate instability for the 2D simulations in the same way as was done for the 1D and 1.5D simulations, owing to the distortion of the mesh at higher draw ratios. The approach for the 2D problem was to track +e number of oscillations in the indicator variable ( Wc) at low draw ratios. If a change in operating conditions led to more oscillations, then this was considered to be a likely indica- tion that the change in question had a destabilizing influence. Using this idea, the following observations were made:

trends in instability for the 2D simulations with changing constitutive parameters and nonisothermal conditions are the same as for the 1D simulations:

1 .o

h = 0.002 s h = 2 s h = 7 s FUJ. 12. Thickness contours for various values of the retavation time X (s).



hih,,. = 1 .o 1 .o 1 .o

a=O a = 2 0 a=40 Fg. 13. Thickness contours for various values ofthe heat transfer coeflicient 01 (W/(m2 . K)).

- Dr,, is increased by extensional thickening

- Dr,, is decreased by extensional thinning,

higher aspect ratios have a stabilizing effect, similar to that observed for the 1.5D simulations.

All of the Wc histories used to make the above conclu- sions are not reproduced here, but an example is provided to illustrate the approach. The example histories, which are shown in Fig. 14, correspond to m values used for Flg. 10; that is, rn = 0.75, 1.0 and 1.5. One observes that the number of oscillations in the +, before steady-state, increases with increasing

and an increased heat transfer coefficient

higher relaxation times, and strain-hardening

values of rn, as shown in Fig. 14. Furthermore, with Ar = 1.4, m = 1.5 the film is stable, but for the 1D simulations, where A r = 0, rn = 1.5 it is not stable, as shown in Fig. 5. Therefore, in this case, a higher aspect ratio appears to have a stabilizing influence on the 2D simulation. The same approach was followed to examine the impact of the other constitutive parameters and nonisothermal conditions on stability.

CONCLUDING REMARKS This paper presented an updated Lagrangian algo-

rithm for lD , 1.5D and 2D nonisothermal simulations of a nonlinear viscoelastic cast film process. The algorithm is new with respect to how it is implemented

40 r

m = 0.75 20 - F W

I I I

5 10 15 20 .H oo

n

.c1 a n 1.5-

1 -

& 0.5 -

m = 1.0 .H cn m .H

&

I I I

5 10 15 20 w 2 0 L O

0 1 I I I

0 5 I 0 15 20 Time (s)

Fig. 14. Rate of energy dissipation history for the three d@erent rn values.

1120 POLYMER ENGINEERING AND SCIENCE, MAY2003, Vol. 43, No. 5


and with respect to the physical phenomena included in the governing equations. With respect to the imple- mentation, the algorithm described in this paper is the first to model film casting using an updated La- grangian description of the motion. Furthermore, the algorithm provides one of the few examples of deter- mining draw resonance via a response problem, as opposed to the more common approach of linear stability analysis. When taking into account the existing literature, the algorithm described in this paper also contributes to the body of knowledge on film casting by removing some of the simplifying assumptions used in previous studies. The new features of the model include observing the transient behavior of a 2D nonisothermal film and employing a constitutive equation that adopts a solid mechanics perspective rather than the fluid mechanics one generally employed in polymer processing studies. Since the same algorithm accommodates lD, 1.5D and 2D film casting, it provides an opportunity to contrast the different kinematic assumptions. One conclusion from this comparison is that the 1.5D assumption is a worse approximation of the film casting process than the 1D model, because the plane strain conditions induced by the restraining influence of the edge bead are not included in the 1.5D case.

Some of the UL algorithm results for steady-state film casting of a viscous polymer were found to be different when compared to two E studies. These differences are attributed to the nonlinear nature of the problem and the fact that different solutions were found to still be close in an energy sense. The parametric study for 2D steady-state simulations shows that, for the draw ratios considered, changes to the material properties do not have a dramatic influence on the film’s geometry. This observation suggests that, at least for the case of lower draw ratios, the solution for the film casting problem is continuity dri- ven. The instability of 2D films was also discussed, and it was concluded that the trends with changing constitutive parameters and thermal conditions are the same as for the 1D simulations. I t was also found that higher aspect ratios have a stabilizing effect, which is consistent with the observations corresponding to the 1.5D simulations.

ACKNOWLEDGMENTS

The financial support provided by the Natural Sci- ences and Engineering Research Council (NSERC) of Canada is gratefully acknowledged.

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Numerical simulation of film casting using an updated lagrangian finite element algorithm