# Fiber draw using draw pins under isothermal conditions.

INTRODUCTIONIn this paper we model the draw process of as-spun, undrawn fiber with elastic-plastic constitutive behavior. In an isothermal draw process involving a series of rollers with increasing surface speeds, the tension in the elastic-plastic filament monotonically increases in the absence of draw-pins as shown by Bechtel et al. (1-3). Even though the as-spun fibers undergo morphological changes while the tension and stretch are monotonically increasing in the tow, resulting in permanent plastic deformation, one can develop a one-to-one relation between filament tension and stretch that is valid until the filament unloads downstream of the draw process. This results directly from plasticity as a special case: In uniaxial tension with no unloading the incremental elastic-plastic theory relating strain, increment of strain, and increment of stress integrates to a function relating stress to strain, i.e. the equations behave as equations of nonlinear elasticity, even though there is permanent deformation. Generally this result holds for loading in which the 3-D stress components monotonically increase during the experiment in proportion to one another, called 'simple,' 'radial,' or 'proportional' loading, cf. Kachanov (4) and Lubliner (5).

The insertion of a stationary draw pin between rollers, however, introduces the possibility of unloading of the filament during the draw process, i.e. decreasing tension along the drawline. This unloading adds the complication of inelastic behavior of spun fiber into the analysis of the draw process. In this paper, we restrict the cause of unloading to draw pins, rather than radiant heaters or heated rollers. By restricting to isothermal conditions we are able to isolate the effect of unloading due to draw pins. Figure 1 shows a schematic diagram of the general draw process with a draw pin.

BASIC EQUATIONS

The governing equations in a steady draw process for fibers with a general constitutive model are given by Bechtel et al. (1, 4):

Conservation of Mass and Momentum, and Friction

Denoting mass per unit volume and cross-sectional area of the fibers as [rho] and A, respectively, conservation of mass requires that the mass flow rate G is constant,

[FIGURE 1 OMITTED]

G = [rho]A(s)v(s) = constant. (1)

where s is the arclength for a fixed position in space from a reference point. Conservation of momentum in a free span demands an increment dT of fiber tension T to be related to an increment dv of fiber speed v through

dT = Gdv. (2)

Ignoring aerodynamic forces, conservation of momentum projected in the tangential and normal directions to the roller surface is

dT - fds = Gdv. (3a)

n = [T - Gv/r]. (3b)

where f and n denote the projections of the force per unit length from a roller on the fibers in the tangential and normal directions, respectively, with positive n taken as compressive and positive f in the direction of decreasing s. Assuming that there is no adhesion to the roller, the normal force per unit length n on the fibers must be compressive, which combined with Eq 3b demands

T - Gv [greater than or equal to] 0; (4)

implying that an insufficiently tensioned tow will fly off the roller.

In a region where there is no slip between the roller and fiber, the sign and magnitude of friction is determined by the momentum equation in order to maintain the velocity constraint v = r[omega], where r is the radius of the roller and [omega] is its angular velocity. Where there is slip, we assume the magnitude of friction force f is given by the familiar equation

|f| = [mu]n, (5)

where [mu] is the coefficient of friction.

The fixed draw pin is a special case of a roller with zero surface speed. On the pin there is always slip, as the fiber is moving faster than the pin surface.

Constitutive Behavior

We assume that in the draw process the fibers of the tow form a layer resembling a belt, and every fiber within the tow experiences the same tension and stretch. The axial strain [epsilon] at a point s of the tow is

[epsilon](s) = [[dl(s)]/[d[l.sub.ref]]] - 1, (6)

with dl(s) the length of an infinitesimal segment of the fiber at location s and d[l.sub.ref] the length of that segment in the initial reference state in which the strain is assigned the value zero. Without loss of generality we identify this reference state with zero strain to be where the fiber tow attaches to the first roller. In this analysis we assume that the tow attaches to the first roller without any slip. The tensile force in the tow as it attaches to the first roller is denoted by [T.sub.0]. In the draw process [T.sub.0] is constrained; from Eq 4 the tension [T.sub.0] must be greater than Gr[[omega].sub.1], where r[[omega].sub.1] is the surface speed of the first roller.

In this work we adopt an elastic-plastic constitutive equation for fiber behavior which captures the behavior of a soft draw plateau between two stiffer regions encountered for as-spun, undrawn polymeric fibers. We assume that the processing is done at high speeds, allowing little time for viscoelastic effects to occur, so that viscoelasticity can be neglected without resulting in significant errors. In our model, when strains are small the tow is elastic and stiff with a large modulus [K.sub.1]. At a threshold strain value [[epsilon].sub.a] during loading the modulus abruptly reduces to a value [K.sub.2] less than [K.sub.1]. Beyond a second transition strain [[epsilon].sub.b] the modulus in loading again increases, to a value [K.sub.3] greater than the plateau modulus [K.sub.2]; ultimately, at critical strain [[epsilon].sub.c] the filament breaks. If the filament unloads, the decrease of tension is with the initial modulus [K.sub.1], i.e. dT = [K.sub.1]d[epsilon]. Therefore the first transition strain [[epsilon].sub.a] is the elastic limit: If the filament has been loaded to less than [[epsilon].sub.a] then it unloads along the same path that it was loaded on, since the filament is elastic. If, however, the filament has been loaded beyond [[epsilon].sub.a] and then unloaded, there is permanent deformation, since [K.sub.2] < [K.sub.1]. A schematic representation exhibiting three possible paths in load/strain space for the filament is given in Fig. 2. In each path depicted on this figure the filament stretches monotonically to some strain [[epsilon].sub.max] and then unloads. If the maximum stretch of the filament is less than the elastic limit [[epsilon].sub.a] (path a) then the loading and unloading are along the same line. If the maximum stretch exceeds the elastic limit, either into the soft plateau (path b) or through the plateau into the second stiff region (path c), the filament unloads with the initial elastic stiffness [K.sub.1], revealing the permanent deformation.

In the isothermal case, our elastic-plastic constitutive model can be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where [K.sub.1], [K.sub.2], and [K.sub.3] are in general specified functions of strain [epsilon], the transition strains [[epsilon].sub.a], [[epsilon].sub.b], and [[epsilon].sub.c] are specified constants, and [[epsilon].sub.max] is the maximum value of strain attained in history of deformation since the state at s = 0.

[FIGURE 2 OMITTED]

In an anisothermal drawline, the increment of tension depends on temperature. Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where d[THETA] is the increment of temperature; [K.sub.i] and [G.sub.i] are in general functions of the current strain [epsilon] and temperature [THETA], and [[epsilon].sub.a], [[epsilon].sub.b], [[epsilon].sub.c] are functions of [THETA]. In this paper we restrict to the isothermal case, given by Eq 7.

For the moving fiber the strain [epsilon](s) at any point s is related to the fiber speed v(s) at that point. We have selected the attachment point of the fiber to the first roller as its reference state where [epsilon] is assigned the value zero, hence

[epsilon](s) = [[v(s)]/[r[[omega].sub.1]]] - 1, (9)

where r[[omega].sub.1] is the surface speed of the first roller. Using Eq 9. the incremental constitutive equation (Eq 7) between fiber tension and fiber strain can be rewritten as an incremental relation between tension and fiber speed.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where [K.sub.i] are now functions of v. The transition speeds [v.sub.a], [v.sub.b], and [v.sub.c] follow from the transition strains [[epsilon].sub.a], [[epsilon].sub.b], and [[epsilon].sub.c], i.e. [v.sub.a] = r[[omega].sub.1] (1 + [[epsilon].sub.a]), [v.sub.b] = r[[omega].sub.1] (1 + [[epsilon].sub.b]), and [v.sub.c] = r[[omega].sub.1] (1 + [[epsilon].sub.c]); [v.sub.max] is the maximum speed achieved by the fiber since it attached to the first roller at s = 0.

In the free span the relevant equations are 1, 2, and 10. The only way in which both Eqs 2 and 10 are jointly satisfied at all times for an isothermal process is when

dv = 0. (11)

Therefore the speed v and, from Eq 10, the tension T are constant in the free spans. We emphasize that this result is a consequence of the assumptions made in this paper of elastic-plastic fiber characterization and isothermal, steady conditions. Draw in the free span is predicted by other models if the conditions are not isothermal [see (2) for an example], or if effects other than elasticity and plasticity are incorporated in the model.

When the fiber is on a roller and moving faster than the roller surface, friction f acting on the fiber is kinetic due to the slip, and positive according to our sign convention. Hence Eq 5 becomes

f = [mu]n (12)

The relevant equations when the fiber is moving faster than the underlying roller surface are Eqs 3 and 12; combining them produces

[FIGURE 3 OMITTED]

dT - [[mu]/r] (T - Gv)ds = Gdv, (13)

where T and v are the values of tension and speed, respectively, at the upstream boundary of the increment ds in arclength, and the particular dT depends on v and the sign of dv as described in Eq 10. When the fiber is moving slower than the roller surface, friction is kinetic, and negative according to our sign convention. Hence Eq 5 becomes

f = -[mu]n (14)

and Eqs 1, 3, 10, and 14 combine to produce

dT + [[mu]/r] (T - Gv)ds = Gdv, (15)

Over a no-slip zone on a roller, the fiber speed matches that of the underlying roller surface, and hence is constant. Therefore in these ranges of s.

dv = 0. (16)

When passing over a draw pin, the fiber is always moving faster than than the stationary draw pin surface, and hence the governing equation for the fiber on a draw pin is Eq 13.

A TWO-STAGE DRAW PROCESS

We model the isothermal two-stage process with an intermediate draw pin in the second free span shown in Fig. 1. For simplicity we assume all three rollers have the same radius, labeled r. The angular velocities of the first, second, and third rollers are specified as [[omega].sub.1], [[omega].sub.2], and [[omega].sub.3], respectively, with [[omega].sub.1] < [[omega].sub.2] < [[omega].sub.3].

In Fig. 3 we indicate conceivable slip and no-slip zones on the rollers and pin for this process, and introduce our labels for important locations along the drawline. In general, transition locations which are known before we solve our boundary value problem modeling the drawline (namely the locations where the fibers attach to and depart from the rollers and pin) and denoted by [x.sub.i], i = 1, 2.... Locations which we deduce as part of the solution (i.e. the boundaries between slip and no-slip on the rollers) are denoted by [y.sub.i]. Speeds that are determined as part of the solution are denoted by [v.sub.i].

The fiber tow attaches to the first roller at location [x.sub.0] with the surface speed r[[omega].sub.1] of the roller, and continues at that speed to an as-yet-undetermined location [y.sub.1] where the fiber begins to slip. From [y.sub.1] to the point of departure from the first roller at [x.sub.1], the fiber speed increases from r[[omega].sub.1] to the to-be-determined departure speed [v.sub.1] > r[[omega].sub.1]. We label this first conceivable draw zone by 1 in Fig. 3. The first free span extends from [x.sub.1] to [x.sub.2], where the fiber attaches to the second roller with a speed [v.sub.2]. Since the draw process is isothermal, there is no change in speed (i.e. no draw) in the free span. Hence the fiber speed [v.sub.1] at the exit from the first roller is the same as the speed [v.sub.2] of the fiber at entry to the second roller.

The fiber is in contact with the second roller from the attachment location [x.sub.2] to the detachment location [x.sub.3]. Between [x.sub.2] and [x.sub.3] there are two conceivable draw zones. The first draw zone (labeled 2 in Fig. 3) occurs if the attachment speed [v.sub.2] is less than the roller surface speed ([v.sub.2] < r[[omega].sub.2]) and extends from [x.sub.2] to the asyet-undetermined location [y.sub.2] where v reaches r[[omega].sub.2]. The second draw zone (labeled 3 in Fig. 3) occurs if the departure speed is different than the roller surface speed ([v.sub.3] [not equal to] r[[omega].sub.2]). This zone extends from the as-yet-undetermined location [y.sub.3], where the fiber speed first departs from the roller speed, to [x.sub.3], where it leaves the roller. It should be noted that because of the presence of the draw pin, both [v.sub.3] < r[[omega].sub.2] and [v.sub.3] > r[[omega].sub.2] are possible.

The second free span is divided into two sections by the draw pin. In the first section the fiber has the speed [v.sub.3]. Since the pin is fixed, there is necessarily slip between fiber and pin throughout the contact region, and, as shown in the next section, slip is always accompanied by draw. Hence there is draw over the entire contact with the pin, from [x.sub.4] to [x.sub.5]; this draw zone is labeled 4 in Fig. 3. The fiber therefore departs the pin with a speed [v.sub.5] different than the speed [v.sub.4] = [v.sub.3] with which it attaches to the pin. The fiber now enters the second section of the second free span; in this section the fiber stays at a speed [v.sub.5].

Draw takes place on the third roller marked as zone 5 extending from [x.sub.6] to [y.sub.4] if the speed of the fiber [v.sub.6] = [v.sub.5] at entry to the roller is less than the speed r[[omega].sub.3] of the roller. The fiber reaches the speed r[[omega].sub.3] at [y.sub.4], and stays at that speed until it exits the final roller at [x.sub.7].

DRAWLINE MODEL AND SOLUTIONS

In this section we integrate the basic equations to obtain relations between the process conditions and unknown slip boundaries [y.sub.1], [y.sub.2], [y.sub.3], [y.sub.4], unknown fiber departure speeds [v.sub.1], [v.sub.3], [v.sub.5], and unknown attachment speeds [v.sub.2], [v.sub.4], [v.sub.6], and to obtain relations that determine the speed and tension profiles v(s) and T(s) everywhere along the drawline in terms of these ten quantities. Introducing a draw pin creates many new conceivable paths of fiber behavior, and hence this section necessarily contains a large number of equations and conditional statements. However, in the next section, it will be seen that the solution of these equations and conditions is straightforward and easy to understand.

To reduce the number of possibilities and equations to consider, we consider a case where the roller speeds and fiber properties are such that in the no-slip zone on the second roller the fiber has been drawn into its soft plateau (r[[omega].sub.1] < [v.sub.a] < r[[omega].sub.2] < [v.sub.b]), and in the no-slip zone on the third roller the fiber has been drawn into its second stiff state but below its breaking strain ([v.sub.b] < r[[omega].sub.3] < [v.sub.c]). Although this is invariably the situation in industrial practice, we note that our solution in this case be readily extended to the other cases.

Since there is no draw pin in the first free span, the departure speed [v.sub.1] of the fiber from the first roller must be in the range between the surface speeds of the first and second rollers, r[[omega].sub.1] [less than or equal to] [v.sub.1] [less than or equal to] r[[omega].sub.2]. If there is no draw on the first roller then [v.sub.1] = r[[omega].sub.1], and if there is a draw zone on the first roller then [v.sub.1] > r[[omega].sub.1]; part of the solution procedure is to identify which of these two situations occurs for specific drawline conditions. If there is a draw zone on the first roller then Eq 13 applies, and in this draw zone the fiber speed is monotonically increasing, so that dv > 0 and v = [v.sub.max] everywhere in the zone. For such monotonically increasing loading, the incremental constitutive Eq 10 integrates to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

Equation 17 combined with Eq 13 can be integrated from the beginning ([y.sub.1]) to the end ([x.sub.1]) of the draw zone, yielding a relation between the unknowns [v.sub.1] and [y.sub.1]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

The solution to Eq 18 obviously must satisfy the condition [y.sub.1] [less than or equal to] [x.sub.1], since the slip zone [y.sub.1] [less than or equal to] s [less than or equal to] [x.sub.1] must be on the pulley (see Fig. 3). If in a specific process condition [y.sub.1] [less than or equal to] [x.sub.1] is satisfied by the solution of Eq 18 and if the fiber departs the first roller still in the initial stiff state (i.e. [v.sub.1] [less than or equal to] [v.sub.a]), then integrating Eqs 13 and 17 from [y.sub.1] to arbitrary s [less than or equal to] [x.sub.1] gives the speed profile

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (19)

If the condition [y.sub.1] [less than or equal to] [x.sub.1] is satisfied and the fiber departs the first roller in the soft plateau (i.e. [v.sub.1] [greater than or equal to] [v.sub.a]), then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

where [y.sub.a] is the location with the draw zone where the fiber draws into the soft plateau,

[y.sub.a] = [x.sub.1] - [r/[mu]] ln [[[T.sub.0] + [K.sub.1] ([[v.sub.a]/[r[[omega].sub.1]]] - 1) + [K.sub.2] ([[v.sub.1]/[r[[omega].sub.1]]] - [[v.sub.a]/[r[[omega].sub.1]]]) - G[v.sub.1]]/[[T.sub.0] + [K.sub.1] ([[v.sub.a]/[r[[omega].sub.1]]] - 1) - G[v.sub.a]]] (21)

The tension profile T(s) is computed by inserting the pertinent equation. Eq 19 or 20, into Eq 17.

If the solution to Eq 18 predicts the inadmissible result [y.sub.1] > [x.sub.1] for all possible [v.sub.1], then there is no draw zone on the first roller, and [y.sub.1] = [x.sub.1] and [v.sub.1] = r[[omega].sub.1]. The fiber leaves the first roller at [x.sub.1] with a speed [v.sub.1]. After crossing the free span the fiber attaches to the second roller at [x.sub.2] with a speed of [v.sub.2]; for the isothermal free span there is no draw in the free span and [v.sub.2] = [v.sub.1].

Since there is no pin in the first free span, the attachment speed [v.sub.2] of the fiber to the second roller must be in the range r[[omega].sub.1] [less than or equal to] [v.sub.2] [less than or equal to] r[[omega].sub.2]. If [v.sub.2] < r[[omega].sub.2] then there is a take-up draw zone on the second roller with the fiber moving slower than the roller surface, so that Eq 15 applies. In this draw zone, the fiber speed is monotonically increasing, with dv > 0 and v = [v.sub.max] everywhere, so that the incremental constitutive Eq 10 integrates to Eq 17. Combining Eqs 17 and 15, and integrating from [x.sub.2] to the point [y.sub.2] of slip termination where v first reaches r[[omega].sub.2] produces

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

The solution of Eq 22 must satisfy the condition [y.sub.2] [greater than or equal to] [x.sub.2]. If this condition is satisfied and the fiber attaches to the second roller still in its initial elastic state ([v.sub.2] < [v.sub.a]), then the speed profile is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

where the transition location [y.sub.a] into the soft plateau of fiber behavior is now given by

[y.sub.a] = [x.sub.2] - [r/[mu]] ln [[[T.sub.0] + [K.sub.1] ([[v.sub.a]/[r[[omega].sub.1]]] - 1) - G[v.sub.a]]/[[T.sub.0] + [K.sub.1] ([[v.sub.2]/[r[[omega].sub.1]]] - 1) - G[v.sub.2]]]. (24)

If the condition [y.sub.2] [greater than or equal to] [x.sub.2] is satisfied and the fiber attaches to the second roller already drawn into its soft plateau ([v.sub.2] [greater than or equal to] [v.sub.a]), then the speed profile is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (25)

If Eq 22 nonsensically predicts [y.sub.2] < [x.sub.2] (i.e. the slip zone on the roller is not on the roller) for all possible [v.sub.2], then there is no take-up draw on the first roller, and [x.sub.2] = [y.sub.2] and [v.sub.2] = r[[omega].sub.2]. In the no-slip zone on the second roller fiber speed is given by v(s) = r[[omega].sub.2]. It is conceivable that at some location before departure from the second roller the fiber will begin to slip and hence the departure speed [v.sub.3] will be different from the roller speed r[[omega].sub.2]. If there is no draw pin present in the second free span then the departure speed must be some-where between the surface speeds of the second and third rollers, r[[omega].sub.2] [less than or equal to] [v.sub.3] [less than or equal to] r[[omega].sub.3] (for a model of two-stage draw without draw pins, see Bechtel (2)). If there is a pin, however, then [v.sub.3] can be either greater than or equal to the second roller speed r[[omega].sub.2] (qualitatively the same as if there were no pin) or less than r[[omega].sub.2] (possible because of the pin). We handle each of the three cases ([v.sub.3] = r[[omega].sub.2], [v.sub.3] > r[[omega].sub.3], and [v.sub.3] < r[[omega].sub.2]) separately; which of these cases happens depends on the draw process conditions and is part of the solution to the model equations.

If [v.sub.3] > r[[omega].sub.2] then there is a feed draw zone before departure from the second roller in which the fiber is moving faster than the roller surface, for which Eq 13 applies. The fiber speed in this draw zone would be monotonically increasing (dv > 0 and v = [v.sub.max] everywhere), so that Eq 17 is valid (i.e. the incremental constitutive Eq 10 integrates to T as a function of v). We combine Eqs 13 and 17 and integrate from the beginning ([y.sub.3]) to the end ([x.sub.3]) of the draw zone to obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

It is useful to recall that we are modeling draw processes with the transition speed [v.sub.b] of the fiber from soft plateau to stiff behavior between the second and third roller speeds, r[[omega].sub.2] < [v.sub.b] < r[[omega].sub.3]. The solution of Eq 26 obviously must satisfy [y.sub.3] [less than or equal to] [x.sub.3]. If this condition is satisfied and the fiber departs the second roller still in its soft plateau ([v.sub.3] < [v.sub.b]), then the speed profile is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (27)

If the condition [y.sub.3] [less than or equal to] [x.sub.3] is satisfied and the fiber departs the first roller already drawn into its second stiff phase ([v.sub.3] [greater than or equal to] [v.sub.b]) then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (28)

where the location [y.sub.b] of transition into the final stiff region of the fiber behavior is given by

[y.sub.b] = [x.sub.3] - [r/[mu]] ln [[[T.sub.0] + [K.sub.1] ([[v.sub.a]/[r[[omega].sub.1]]] - 1) + [K.sub.2] ([[v.sub.b]/[r[[omega].sub.1]]] - [[v.sub.a]/[r[[omega].sub.1]]]) - G[v.sub.b]]/[[T.sub.0] + [K.sub.1] ([[v.sub.a]/[r[[omega].sub.1]]] - 1) + [K.sub.2] ([[r[[omega].sub.2]]/[r[[omega].sub.1]]] - [[v.sub.a]/[r[[omega].sub.1]]]) - Gr[[omega].sub.2]]]. (29)

If the solution to Eq 26 predicts the nonsensical result [y.sub.3] > [x.sub.3] for all possible [v.sub.3], then there is no feed draw zone on the second roller and [y.sub.3] = [x.sub.3] and [v.sub.3] = r[[omega].sub.2].

If the fiber departs the second roller with a speed slower than the underlying roller surface ([v.sub.3] < r[[omega].sub.2], a situation possible if and only if there is a draw pin present between the second and third rollers) then dv < 0 and v < [v.sub.max] = r[[omega].sub.2] for some length of fiber on the second roller. In this case we integrate the incremental constitutive Eq 10 to obtain

T(v) = [T.sub.0] + [K.sub.1] ([[v.sub.a]/[r[[omega].sub.1]]] - [[v.sub.max]/[r[[omega].sub.1]]] - 1) + [K.sub.2] ([[v.sub.max]/[r[[omega].sub.1]]] - [[v.sub.a]/[r[[omega].sub.1]]]) + [K.sub.1] ([v/[r[[omega].sub.1]]]). (30)

In the departing slip zone of the second roller the fibers speed and tension decrease, according to Eq 30, from their maximum values.

[v.sub.max] = r[[omega].sub.2],

[T.sub.max] = [T.sub.0] + [K.sub.1] ([[v.sub.a]/[r[[omega].sub.1]]] - 1) + [K.sub.2] ([[r[[omega].sub.2]]/[r[[omega].sub.1]]] - [[v.sub.a]/[r[[omega].sub.1]]]), (31)

(attained in the no-slip zone on the second roller) to v = [v.sub.3] < r[[omega].sub.2] and T = T([v.sub.3]) = [T.sub.0] + [K.sub.1] ([[v.sub.a]/[r[[omega].sub.1]]] - [[r[[omega].sub.2]]/[r[[omega].sub.1]]] - 1) + [K.sub.2] ([r[[omega].sub.2]/r[[omega].sub.1]] - [[v.sub.a]/[r[[omega].sub.1]]]) + [K.sub.1] ([[v.sub.3]/[r[[omega].sub.1]]]). Integrating the coupled Eqs 30 and 15 from beginning ([y.sub.3]) to end ([x.sub.3]) of this unloading zone produces

[y.sub.3] = [x.sub.3] + [r/[mu]] ln [[[T.sub.0] + [K.sub.1] ([[v.sub.a]/[r[[omega].sub.1]]] - [[r[[omega].sub.2]]/[r[[omega].sub.1]]] - 1)]/[[T.sub.0] + [K.sub.1] ([[v.sub.a]/[r[[omega].sub.1]]] - 1) + [K.sub.2] ([[r[[omega].sub.2]]/[r[[omega].sub.1]]] - [[v.sub.a]/[r[[omega].sub.1]]]) - Gr[[omega].sub.2]]] + [[[K.sub.2] ([[r[[omega].sub.2]]/[r[[omega].sub.1]]] - [[v.sub.a]/[r[[omega].sub.1]]]) + [K.sub.1] ([[v.sub.3]/[r[[omega].sub.1]]]) - G[v.sub.3]]/[[T.sub.0] + [K.sub.1] ([[v.sub.a]/[r[[omega].sub.1]]] - 1) + [K.sub.2] ([[r[[omega].sub.2]]/[r[[omega].sub.1]]] - [[v.sub.a]/[r[[omega].sub.1]]]) - Gr[[omega].sub.2]]]. (32)

The corresponding speed profile is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

which decreases from r[[omega].sub.2] at [y.sub.3] to the departure speed [v.sub.3] from the second roller at [x.sub.3]. The fiber then passes through the first section of the second isothermal free span and attaches to the draw pin at [x.sub.4] with speed [v.sub.4] = [v.sub.3].

On the draw pin the speed of the fiber is always greater than the speed of the stationary pin surface, so that momentum Eq 13 applies. Throughout contact with the pin the fiber speed is monotonically increasing (dv > 0), and hence the fiber exits the pin with a speed greater than that with which it entered. Two different cases and (and five subcases) of fiber behavior on the draw pin are possible, depending on whether or not the fiber has unloaded on the immediately preceding second roller (see Figs. 4 to 6 and 7 and 8).

We first investigate the more interesting Case 1, sketched in Fig. 4, where the fiber has unloaded on the second roller (indicated by the condition [v.sub.3] = [v.sub.4] < r[[omega].sub.2]). In this case there is always at least a portion of the contact interval with the pin where the fiber's speed and tension are increasing, but are not yet up to the maximum values [v.sub.max] and [T.sub.max] that they had on the second roller before the unloading, given by Eq 31. In this portion, the incremental constitutive Eq 10 integrates to Eq 30; until it returns to its previous maximum, the fiber loads along the path it just unloaded. Three subcases are possible, depending on if the fiber leaves the pin still below its previous maximum tension (Subcase 1a), the fiber fully reloads on the pin and leaves with a speed greater than r[[omega].sub.2] but less than the the transition speed [v.sub.b] (Subcase 1b), or it fully reloads and in addition draws into the final stiff region (Subcase 1c). For each subcase we have different equations for fiber speed and tension, and different relations between locations of attachment and departure:

Subcase 1a: The entry speed [v.sub.4] to the pin and exit speed [v.sub.5] are related through

[x.sub.5] = [x.sub.4] + [r/[mu]] ln {[[T.sub.0] + [K.sub.1] ([[v.sub.a]/[r[[omega].sub.1]]] - [[r[[omega].sub.2]]/[r[[omega].sub.1]]] - 1) + [K.sub.2] ([[r[[omega].sub.2]]/[r[[omega].sub.1]]] - [[v.sub.a]/[r[[omega].sub.1]]]) + [K.sub.1] ([v.sub.5]/[r[[omega].sub.1]]) - G[v.sub.5]] / [[T.sub.0] + [K.sub.1] ([[v.sub.a]/[r[[omega].sub.1]]] - [[r[[omega].sub.2]]/[r[[omega].sub.1]]] - 1) + [K.sub.2] ([[r[[omega].sub.2]]/[r[[omega].sub.1]]] - [[v.sub.a]/[r[[omega].sub.1]]]) + [K.sub.1] ([v.sub.4]/[r[[omega].sub.1]]) - G[v.sub.4]]} (34)

with speed profile on the pin given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)

[FIGURE 4 OMITTED]

Subcase 1b:

[x.sub.5] = [x.sub.4] + [r/[mu]] ln [[[T.sub.0] + [K.sub.1] ([[v.sub.a]/[r[[omega].sub.1]]] - 1) + [K.sub.2] ([[v.sub.5]/[r[[omega].sub.1]]] - [[v.sub.a]/[r[[omega].sub.1]]]) - G[v.sub.5]]/[[T.sub.0] + [K.sub.1] ([[v.sub.a]/[r[[omega].sub.1]]] - [[r[[omega].sub.2]]/[r[[omega].sub.1]]] - 1) + [K.sub.2] ([[r[[omega].sub.2]]/[r[[omega].sub.1]]] - [[v.sub.a]/[r[[omega].sub.1]]]) + [K.sub.1] ([[v.sub.4]/[r[[omega].sub.1]]]) - G[v.sub.4]]]. (36)

and

[FIGURE 5 OMITTED]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (37)

where [y.sub.p] is the intermediate location on the pin where drawing in the soft plateau resumes.

[y.sub.p] = [x.sub.4] + [r/[mu]] ln [[[T.sub.0] + [K.sub.1] ([[v.sub.a]/[r[[omega].sub.1]]] - 1) + [K.sub.2] ([[r[[omega].sub.2]]/[r[[omega].sub.1]]] - [[v.sub.a]/[r[[omega].sub.1]]]) - Gr[[omega].sub.2]]/[[T.sub.0] + [K.sub.1] ([[v.sub.a]/[r[[omega].sub.1]]] - [[r[[omega].sub.2]]/[r[[omega].sub.1]]] - 1) + [K.sub.2] ([[r[[omega].sub.2]]/[r[[omega].sub.1]]] - [[v.sub.a]/[r[[omega].sub.1]]]) + [K.sub.1] ([[v.sub.4]/[r[[omega].sub.1]]]) + G[v.sub.4]]]. (38)

Subcase 1c:

[x.sub.5] = [x.sub.4] + [r/[mu]] ln [[[T.sub.0] + [K.sub.1] ([[v.sub.a]/[r[[omega].sub.1]]] - 1) + [K.sub.2] ([[v.sub.b]/[r[[omega].sub.1]]] - [[v.sub.a]/[r[[omega].sub.1]]]) + [K.sub.3] ([[v.sub.5]/[r[[omega].sub.1]]] - [[v.sub.b]/[r[[omega].sub.1]]]) - G[v.sub.5]]/[[T.sub.0] + [K.sub.1] ([[v.sub.a]/[r[[omega].sub.1]]] - [[r[[omega].sub.2]]/[r[[omega].sub.1]]] - 1) + [K.sub.2] ([[r[[omega].sub.2]]/[r[[omega].sub.1]]] - [[v.sub.a]/[r[[omega].sub.1]]]) + [K.sub.1] ([[v.sub.4]/[r[[omega].sub.1]]]) - G[v.sub.4]]]. (39)

and

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (40)

where the intermediate location [y.sub.p] on the pin where drawing in the soft plateau resumes is again given by Eq 38, and [y.sub.b] is the intermediate location on the pin where fiber draws out of its soft plateau and into its final stiff state,

[y.sub.b] = [x.sub.5] - [r/[mu]] ln [[[T.sub.0] + [K.sub.1] ([[v.sub.a]/[r[[omega].sub.1]]] - 1) + [K.sub.2] ([[v.sub.b]/[r[[omega].sub.1]]] - [[v.sub.a]/[r[[omega].sub.1]]])]/[[T.sub.0] + [K.sub.1] ([[v.sub.a]/[r[[omega].sub.1]]] - 1) + [K.sub.2] ([[v.sub.b]/[r[[omega].sub.1]]] - [[v.sub.a]/[r[[omega].sub.1]]]) - G[v.sub.b]]] + [[[K.sub.3] ([[v.sub.5]/[r[[omega].sub.1]]] - [[v.sub.b]/[r[[omega].sub.1]]]) - G[v.sub.5]]/[[T.sub.0] + [K.sub.1] ([[v.sub.a]/[r[[omega].sub.1]]] - 1) + [K.sub.2] ([[v.sub.b]/[r[[omega].sub.1]]] - [[v.sub.a]/[r[[omega].sub.1]]]) - G[v.sub.b]]]. (41)

We now investigate Case 2, sketched in Fig. 5, in which the fiber has not unloaded on the second roller (indicated by the condition [v.sub.3] = [v.sub.4] [greater than or equal to] r[[omega].sub.2]). Two subcases are possible, depending on whether the fiber leaves the pin at a speed less than the transition speed [v.sub.b] (Subcase 2a), or greater than [v.sub.b] so that it draws on the pin into the final stiff region (Subcase 2b). As with the previous case we obtain different equations for fiber speed and tension:

Subcase 2a: The entry and exit speeds to the pin, [v.sub.4] and [v.sub.5] respectively, are related through

[x.sub.5] = [x.sub.4] + [r/[mu]] ln ([[T.sub.0] + [K.sub.1] ([[v.sub.a]/[r[[omega].sub.1]]] - 1) + [K.sub.2] ([[v.sub.5]/[r[[omega].sub.1]]] - [[v.sub.a]/[r[[omega].sub.1]]]) - G[v.sub.5]]/[[T.sub.0] + [K.sub.1] ([[v.sub.a]/[r[[omega].sub.1]]] - 1) + [K.sub.2] ([[v.sub.4]/[r[[omega].sub.1]]] - [[v.sub.a]/[r[[omega].sub.1]]]) - G[v.sub.4]]). (42)

and the speed profile on the pin is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (43)

Subcase 2b:

[x.sub.5] = [x.sub.4] + [r/[mu]] ln ([[[T.sub.0] + [K.sub.1] ([[v.sub.a]/[r[[omega].sub.1]]] - 1) + [K.sub.2] ([[v.sub.b]/[r[[omega].sub.1]]] - [[v.sub.a]/[r[[omega].sub.1]]])]/[[T.sub.0] + [K.sub.1] ([[v.sub.a]/[r[[omega].sub.1]]] - 1) + [K.sub.2] ([[v.sub.4]/[r[[omega].sub.1]]] - [[v.sub.a]/[r[[omega].sub.1]]]) - G[v.sub.4]]] + [[[K.sub.3] ([[v.sub.5]/[r[[omega].sub.1]]] - [[v.sub.b]/[r[[omega].sub.1]]]) - G[v.sub.5]]/[[T.sub.0] + [K.sub.1] ([[v.sub.a]/[r[[omega].sub.1]]] - 1) + [K.sub.2] ([[v.sub.4]/[r[[omega].sub.1]]] - [[v.sub.a]/[r[[omega].sub.1]]]) - G[v.sub.4]]]). (44)

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (45)

with the location [y.sub.b] on the pin where the fiber draws out of its soft plateau is given by Eq 41.

After exiting the draw pin the fiber enters the second section of the second free span. The fiber does not undergo a change in speed in this isothermal free span, and hence attaches to the third roller at [x.sub.6] with a speed [v.sub.6] = [v.sub.5]. If the attachment speed to the third roller is less than the roller speed ([v.sub.6] < r[[omega].sub.3]) then draw takes place on the roller, with the draw zone extending from [x.sub.6] to some [y.sub.4], where the fiber speed reaches r[[omega].sub.3]. Integrating Eq 15 from [x.sub.6] to [y.sub.4] produces

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (46)

and integrating to some intermediate location s yields the speed profile,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (47)

If Eq 46 nonsensically predicts [y.sub.4] < [x.sub.6] for all possible [v.sub.6], then there is no draw on the final roller, i.e. [v.sub.6] = r[[omega].sub.3] and [x.sub.6] = [y.sub.4]. Beyond the point of termination of draw at [y.sub.4] the fiber stays at the surface speed of the roller of r[[omega].sub.3], marking the end of the two stage draw.

SOLUTION PROCEDURE

Recall that the slip boundaries [y.sub.1], [y.sub.2], [y.sub.3], [y.sub.4] and the fiber departure and attachment speeds [v.sub.1], [v.sub.2], [v.sub.3], [v.sub.4], [v.sub.5], [v.sub.6] are unknown. These ten unknowns are related by Eqs 18, 22, 26, 46, and any one of Eqs 34, 36, 39, 42, and 44, and the three constraints [v.sub.1] = [v.sub.2], [v.sub.3] = [v.sub.4], [v.sub.5] = [v.sub.6] due to the free spans being isothermal. Hence there are eight equations for ten unknowns, and the problem is underdetermined. We identify [v.sub.2] and [v.sub.6] as the free parameters, noting that they must satisfy r[[omega].sub.1] [less than or equal to] [v.sub.2] [less than or equal to] r[[omega].sub.2] and [v.sub.6] [less than or equal to] r[[omega].sub.3], and solve the set of equations mentioned earlier (i.e. Eqs 18, 22, 26, 46, and any one of Eqs 34, 36, 39, 42, and 44, and the three constraints [v.sub.1] = [v.sub.2], [v.sub.3] = [v.sub.4], and [v.sub.5] = [v.sub.6]) for [y.sub.1], [y.sub.2], [y.sub.3], [y.sub.4], [v.sub.1], [v.sub.3], [v.sub.4], and [v.sub.5] in terms of [v.sub.2] and [v.sub.6]. In the isothermal simulations of this paper we find that the free parameters [v.sub.2] and [v.sub.6] are uniquely determined by the conditions that the solutions of Eqs 22 and 46 must satisfy [y.sub.2] [greater than or equal to] [x.sub.2] and [y.sub.4] [greater than or equal to] [x.sub.6], respectively. We comment that in simulations of anisothermal drawlines this is not the case, and the values of [v.sub.2] and [v.sub.6] are selected so as to minimize energy, as demonstrated by Bechtel et al. (2).

Once the transition locations [y.sub.1], [y.sub.2], [y.sub.3], [y.sub.4] and fiber departure and attachment speeds [v.sub.1], [v.sub.2], [v.sub.3], [v.sub.4], [v.sub.5], [v.sub.6] are determined, we construct the speed profile v(s) along the entire drawline from Eqs 19, 20, 23, 25, 27, 28, 33, 35, 37, 40, 43, 45, and 47 and the tension evolution T(s) by inserting this speed profile into Eqs 17 and 30.

SIMULATIONS

In this section we simulate four draw processes. To highlight the effect of the addition of a draw pin, we first simulate a draw process without a draw pin (Simulation 1). We then simulate three processes with a draw pin in the second free span that demonstrate the three qualitatively different drawline behaviors that are possible with a pin (Simulations 2, 3, and 4). To facilitate study of the effect of the pin, the same elastic moduli [K.sub.1], [K.sub.2], [K.sub.3], transition strains [[epsilon].sub.a], [[epsilon].sub.b], and breaking strain [[epsilon].sub.c] are employed in the four simulations (listed in Table 1), as well as the same process conditions of upstream fiber tension [T.sub.0], friction coefficient [mu], fiber linear density [rho], roller radius r, and roller speeds r[[omega].sub.1], r[[omega].sub.2], r[[omega].sub.3] (listed in Table 2). Simulations 2, 3, and 4 differ only in the offset h of the draw pin, labeled in Fig. 1. Note in Table 2 that because of geometrical constraints the length of contact of the fiber with roller 2, roller 3, and the pin vary with pin offset h.

Simulation 1: No Draw Pin

In Bechtel et al. (2) the solution of Simulation 1 is obtained using the isothermal special case of our model for two-stage draw without draw pin. This solution is shown in Fig. 6. In all plots we have used crosses (X) to denote points where the fiber either enters or departs from a roller, circles ([circle]) to indicate the locations of initiation or cessation of slip, and triangles ([??]) to indicate the transitions between the initial stiff zone and soft plateau of the fiber, and the soft plateau and final stiff zone. On first two plots (fiber speed vs. arclength and fiber tension vs. arclength) these points are connected with either solid lines (----) that indicate zones of draw on rollers, dashed-dotted lines (-.-.-) that indicate zones of no slip, or dotted lines (........) that indicate free spans. In Fig. 6c (tension vs. speed), the arrows show the loading path of the fiber, and the numbers indicate important points on this loading path.

As represented by the solution depicted in Fig. 6, in an isothermal two-stage draw without pins, draw occurs only as feed draw on the first roller and feed draw on the second roller (zones 1 and 3 in Fig. 3). For the process in Simulation 1, the fiber attaches to the first roller at the first X (s = 0 cm) with speed r[[omega].sub.1] = 396 cm [s.sup.-1]. The fiber then proceeds without slip zone on the first roller until the first [circle] at s = 67 cm; in this 67 cm of no slip the speed and tension are constant. This no-draw portion of the process occurs at point 1 in the speed vs. tension plot in Fig. 6c.

The draw zone on the first roller is the 33 cm from the first [circle] ([y.sub.1] = 67 cm) to the second X ([x.sub.1] = 100 cm), where the fiber departs the roller. In this draw zone, lying between points 1 and 3 of the speed vs. tension plot, the fiber speed increases from the surface speed 396 cm [s.sup.-1] of the first roller (at point 1 on the speed vs. tension plot) to the surface speed 476 cm [s.sup.-1] of the second roller (at point 3). The corresponding tension increases from 4.00 X [10.sup.5] dyn to 6.76 X [10.sup.5] dyn. Note by kink at s = 99 cm (marked by a [??] and numerial 2) that the fiber is drawn on the first feed roller from its initial stiff response into its soft plateau: For 67 cm < s < 99 cm the draw is in the initial stiff portion of the filament response, where a large increase in tension accompanies a small increase in strain and hence speed. For 99 cm < s [less than or equal to] 100 cm the draw is in the soft portion of the filament response, where a small increase in tension accompanies a large increase in strain and speed.

The fiber departs the first roller already at the surface speed of the second roller for two reasons: The free span between the first and second rollers is an isothermal free span without draw pins, and hence speed and tension are constant; the 50 cm in the top two plots between the second X ([x.sub.1] = 100 cm) and third X ([x.sub.2] = 150 cm) is the free span. Then, for all possible attachment speeds [v.sub.2] to the second roller (i.e. all r[[omega].sub.1] [less than or equal to] [v.sub.2] < r[[omega].sub.2]). Eq 22 predicts [y.sub.2] < [x.sub.2]; hence, as discussed after Eq 25, for this process there is no takeup draw on the second roller (the second [circle] and third X on the plots coincide).

The fiber attaches to the second roller at [x.sub.2] = [y.sub.2] = 150 cm already at its surface speed r[[omega].sub.2] = 476 cm [s.sup.-1] and proceeds without slip until the third [circle] at [y.sub.3] = 216 cm; from its departure from the first roller until this location the fiber is at point 3 on the tension vs. speed diagram. Beyond [y.sub.3] = 216 cm the fiber begins to slip; this draw zone on the second roller is 14 cm from the third [circle] to the fourth X at [x.sub.3] = 230 cm. where the fiber departs from the second roller already at the surface speed of the third roller. This draw on the second roller extends from point 3 to 8 of the speed vs. tension plot. Note from the kink marked by the second triangle at 217 cm (numeral 7) that the fiber is drawn out of its soft plateau during this feed draw.

Between the fourth X and fifth X (at [x.sub.4] = 280 cm) is the second free span, again isothermal with no draw pins and hence without draw. In this process Eq 46 spuriously predicts [y.sub.4] < [x.sub.6] for all possible [v.sub.6], and hence there is no takeup draw at the end of the second draw stage, indicated in Fig. 6 by the fourth [circle] coinciding with the fifth X. The fiber proceeds without slip on the third roller.

Simulation 2: Pin Adjusted so That There Is Draw on Both Rollers and the Pin

When a draw pin is present in the second free span, three different types of behavior are possible in the draw line, depending on the pin offset h (all other process conditions being fixed): There can be (i) draw on the both the first and second rollers and the pin, (ii) draw on the first roller and pin, and not on the second roller, with no unloading of the fiber on the second roller, and (iii) draw on the first roller and pin with unloading on the second roller. Note that case (ii) is the boundary between cases (i) and (iii).

In Simulation 2 we select h so that behavior (i) is obtained (specifically h = 8.71 cm). In this simulation, shown in Fig. 7, we again find that for all [v.sub.2] and [v.sub.6] in the ranges r[[omega].sub.1] < [v.sub.2] [less than or equal to] r[[omega].sub.2] and [v.sub.6] [less than or equal to] r[[omega].sub.3] other than [v.sub.2] = r[[omega].sub.2] and [v.sub.6] = r[[omega].sub.3]. Eqs 22 and 46 produce the spurious results [y.sub.2] < [x.sub.2] and [y.sub.4] < [x.sub.6], respectively. Hence, as in Simulation 1 without a draw pin, there is no takeup draw on either the second or third roller. The solution in Simulation 2 is identical to that of Simulation 1 for the first stage of the draw process, since the process conditions are identical and there is no draw pin in that stage. The first sign of the presence of the draw pin in Simulation 2 is the migration of the onset of slip on the second roller (the third [circle]) downstream in the draw process, closer to the fiber exit [x.sub.3] from the second roller (the fourth X). The effect of the pin is to transfer draw from the second roller to the pin. Referring to the speed vs. tension plots for the two simulations, the draw on the second roller between points 3 and 8 in Fig. 6c has been split in Fig. 7c into 3 to 6 on the second roller and 6 to 8 on the pin. In this simulation the transition from the soft plateau to final stiff zone (point 7) occurs on the pin (Subcase 2b shown in Fig. 5).

Quantitatively, on the feed zone of the second roller, starting at the third [circle], the fiber has been drawn to a speed [v.sub.3] = 550 cm [s.sup.-1] when it exits the roller at [x.sub.3] = 236 cm, marked by the fifth X. The fiber stays at 550 cm [s.sup.-1] in the first section of the free span before attaching to the draw pin at [x.sub.4] = 256 cm. The fiber is at the surface speed of the third roller when it leaves the pin ([v.sub.5] = r[[omega].sub.3] = 740 cm [s.sup.-1] at [x.sub.5] = 259 cm). There is no additional draw in the second part of the second free span and on the third roller.

Simulation 3: Pin Adjusted so That There Is Neither Draw nor Unloading on the Second Roller

All other process conditions held constant, as the pin offset h increases the location of the onset of slip on the second roller migrates downstream toward the exit from the roller into the free span, until at a critical value h = [h.sub.c] the two points coincide. In our family of simulations, [h.sub.c] = 9.17 cm. For h < [h.sub.c] (such as h = 8.71 cm in the Simulation 2 above) there is a feed draw zone on the second roller immediately upstream of the pin. For h > [h.sub.c] (e.g. Simulation 4 to follow, with h = 20.17 cm) the fiber unloads on the second roller. exiting the second roller with a speed less than the surface speed of the roller.

In Simulation 3, as shown in Fig. 8, we set h = [h.sub.c] = 9.17 cm. At this critical value there is no draw on the second roller (it all having been transferred to the pin), but also no unloading. The point [y.sub.3] of initiation of draw on the second roller (the third [circle]) coincides with the departure location [x.sub.3] (the fourth X). At this point the fiber enters the second free span with a speed of 476 cm [s.sup.-1]. On the draw pin the fiber is drawn from 476 cm [s.sup.-1] to 740 cm [s.sup.-1] (the final roller speed) over a length extending from [x.sub.4] = 256 cm to [x.sub.5] = 259 cm. Referring to Fig. 8c, the fiber draws from 1 to 3 on the first roller, and 3 to 8 on the pin.

[FIGURE 9 OMITTED]

Simulation 4: Pin Adjusted so That There Is Unloading on the Second Roller

In Simulation 4, as depicted in Fig. 9, the draw pin is lowered past the critical location to h = 20.17 cm. In this supercritical case [y.sub.3] = 231 cm (the third [circle]) marks the start of the unloading zone on the second roller. The fiber unloads to an exit speed 472 cm [s.sub.-1] less than the surface speed r[[omega].sub.2] = 476 cm [s.sup.-1]. The fiber maintains this exit speed up to the point of attachment [x.sub.4] = 263 cm to the draw pin. On the pin the fiber first stretches back to the second roller speed and then further draws to the final roller speed 740 cm [s.sup.-1] when it departs the pin at [x.sub.5] = 272 cm.

In Fig. 9c, the fiber loads on the first roller from 1 to 3. From 3 to 4 the fiber unloads on the second roller and then from 4 to 5 reloads on the draw pin. The direction of the arrows indicates the unloading and reloading. After returning on the draw pin to the second roller speed the fiber undergoes new draw on the pin to the third roller speed. This new draw, from 5 to 8 on the speed vs. tension plot, passes out of the soft plateau and into the final stiff zone at point 7.

CONCLUSION

Using the governing equations developed in our earlier work, incorporating full radial and tangential inertial effects we describe the analysis of draw for a commercially relevant draw process where a draw pin is included for localizing and enhancing fiber draw. In the absence of the draw pin, axial stress in the fiber increases monotonically. However, the presence of a non-rotating draw pin introduces the possibility of unloading in the fiber and additional complexity in the analysis, since the modulus in the unloading region is not the same as the modulus realized in continuous loading for the soft plateau and the second stiff region.

The methodology we present is general, and can be applied to any isothermal drawing process with one or more draw pins. We give the results of simulations of two-stage isothermal draw processes in which the fiber is at the soft plateau when moving over the draw pin, thereby limiting the number of possible combinations in the analysis. Within this class of processes we show that the position of the draw pin (i.e., the offset from the feed/take-up roller) has a significant effect on the distribution of draw. For small offsets of the draw pin, no unloading occurs before the draw pin. Hence, the fiber stress monotonically increases, with draw present on the first two rollers as in a two stage draw process without the presence of a draw pin: a portion of the draw is transferred from the second roller to the pin, however. At a critical offset of the draw pin, neither draw nor unloading occurs on the second roller. Further increase in offset results in unloading in the fiber on the second roller followed by a significant draw on the pin.

Table 1. Values of Material Constants Used in Eq 7 To Characterize the Fiber. Constant Value Units [K.sub.1] 1.030050E + 07 dyn [K.sub.2] 0.002945E + 07 dyn [K.sub.3] 0.087309E + 07 dyn [[epsilon].sub.a] 0.0262 [[epsilon].sub.b] 0.6990 [[epsilon].sub.c] 2.3204 Table 2. Process Conditions for the Four Simulations. Process Condition Sim.1 Sim.2 Sim.3 Sim.4 upstream tension [T.sub.0] (dyn) 40000.0 40000.0 40000.0 40000.0 coefficient of friction [mu] 0.2 0.2 0.2 0.2 linear density of fiber [rho] (denier) 450 450 450 450 roller radius r (cm) 12.7 12.7 12.7 12.7 draw pin radius [r.sub.p] (cm) 3.18 3.18 3.18 3.18 wrap on roller 1 (cm) 99.75 99.75 99.75 99.75 wrap on roller 2 (cm) 79.80 86.39 86.65 91.96 wrap on draw pin (cm) NA 3.3 3.43 6.08 wrap on roller 3 (cm) 79.80 106.34 106.6 111.91 free span, roller 1 to 2 (cm) 50.0 50.0 50.0 50.0 free span, roller 2 to pin (cm) NA 20.0 20.0 20.7 free span, pin to roller 3 (cm) NA 20.0 20.0 20.7 surface speed r[[omega].sub.1] (cm [s.sup.-1]) 396 396 396 396 surface speed r[[omega].sub.2] (cm [s.sup.-1]) 476 476 476 476 surface speed r[[omega].sub.3] (cm [s.sup.-1]) 740 740 740 740 draw pin offset h (cm) NA 8.71 9.17 20.17

ACKNOWLEDGMENT

This work was sponsored in part by the National Textile Center and the U.S. Department of Commerce under Grant E27B51, the National Science Foundation under Grant CTS-9711109, and NSF/CAEFE under grant 2706M65.

REFERENCES

1. S. E. Bechtel, S. Vohra, and K. I. Jacob, "Stretching and slipping of fibers in isothermal draw processes," Textile Research Journal. 72 (9): 769-776 (2002).

2. S. E. Bechtel, S. Vohra, and K. I. Jacob, "Modeling of a two-stage draw process." Polymer, 42 (5): 2045-2059 (2001).

3. S. E. Bechtel, S. Vohra, K. I. Jacob, and C. D. Carlson, "Stretching and slipping of belts on pulleys," Journal of Applied Mechanics, 67(1): 197-206 (2000).

4. L. M. Kachanov, Fundamentals of the Theory of Plasticity, MIR Publishers, Moscow (1974).

5. J. Lubliner, Plasticity Theory, Macmillan Publishing Company, London (1990).

S. E. BECHTEL

Department of Mechanical Engineering Ohio State University Columbus, Ohio 43210

S. VOHRA and K. I. JACOB*

School of Textile and Fiber Engineering Georgia Institute of Technology Atlanta, Georgia 30332

*Corresponding author.

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Author: | Bechtel, S.E.; Vohra, S.; Jacob, K.I. |
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Publication: | Polymer Engineering and Science |

Date: | Feb 1, 2004 |

Words: | 9874 |

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