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Heating Element

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Dryer Heating Element Test

Kinetic finite element model to optimize sulfur vulcanization: application to extruded EPDM weather-strips.


A weather-strip is typically an extruded elastomers bulb with
complex geometry and curved shape. Typically, weather-strips (1) are
installed in automotive industry and civil engineering, in order to
prevent water leakage, block exterior noises, minimize body and window
vibration, and provide some shock absorbing capacity. Initial use of
weather-strip seals in automotive applications was aimed at
accommodating for manufacturing variations. Lately, the isolation issue
became important, and the key design was to better isolate the passenger
compartment from dust, air, and water leakage. Nowadays, the general
trend in seal design is the isolation of the passenger compartment from
noise and vibration. Isolation is again the main role played by
weather-strips installed in windows and doors. Here, stiffness,
strength, isolation capacity, and aging resistance are key issues to be
considered, since the average life of a building is typically longer
than that of a car.

Weather-strips are realized by elastomers, which typically are
viscoelastic materials. Their mechanical properties are strain,
frequency, and temperature dependent. In addition, being the geometry of
weather-strip often rather complex, this generally implies that a
numerical model has to be performed with a high level of detail. Unlike
plastic and other materials, elastomers have characteristics such as
high flexibility, high elasticity, and high elongation. Their major
drawback is that they require vulcanization during the manufacturing
process, and (2) they cannot be reshaped after curing.
Ethylene-propylene-diene monomers (EPDM) are elastomers utilized in a
wide range of applications, including weather-strips. The advantage of
EPDM is its outstanding resistance to heat, ozone, and weather. Usually,
EPDM vulcanizates are composed of rubber, filler (carbon black and
calcium carbonate), curatives (fatty acids, zinc oxide, accelerators,
and sulfur), antidegradants, and processing aids.

As a matter of fact, weather-strip producers are interested in the
improvement of cured EPDM final mechanical performance, at the same time
limiting the production costs. For instance, sulfur is normally
preferred to peroxides merely for economic reasons, despite the fact
that the performance of rubber cured with peroxides--in terms of final
mechanical properties--may be sensibly higher than that of rubber
vulcanized with sulfur. Indeed, from a chemical point of view, sulfur
vulcanization determines transversal chains constituted by more than one
sulfur atom (link energy 270-272 kJ/mol) whereas for peroxides the link
is created between two back-bone carbons belonging to contiguous chains
(energy 346 kJ/mol). In addition, it has to be emphasized that, when
peroxides cure method is used, the rubber base could exhibit a peculiar
smell, which is obviously inacceptable for the production of

Despite the wide diffusion of sulfur vulcanization and the fact
that its discovery and utilization go back to Goodyear (3-6), the
chemistry of vulcanization remains an open issue. In this field, among
the others, historical contributions by Ding and Leonov (7) and Ding et
al. (8) are worth noting. Globally, it can be stated that the majority
of the available approaches are models enforced to resemble to peroxidic

Another important issue to consider is reversion, which occurs
quite frequently in practice. From a macroscopic point of view, it
consists of remarkable decrease of rubber vulcanized properties at the
end of the curing process. Chen et al. (9) have shown that this
phenomenon seems to appear when two reactions are competing during
vulcanization. Reversion is often associated with high-temperature
curing. For instance, Loo (10) demonstrated that, as the cure
temperature rises, the crosslink density drops, thus increasing the
degree of reversion. Morrison and Porter (11) confirmed that the
observed reduction in vulcani-zate properties is caused by two reactions
proceeding in parallel, i.e., desulfuration and decomposition, see Table
1. Generally speaking, this could give rise to the items with
considerable thickness and undergoing different temperatures gradients
during curing a strongly inhomogeneous final level of vulcanization.

TABLE 1. Products and schematic reaction mechanisms of accelerated
sulfur vulcanization of polydiene and EPDM elastomers.

Reaction  Compounds        Process/reaction    Kinetic    Model
ID                                             constant   constants

NA        [S.sub.8] +      Mechanical mixing   NA         NA
          accelerators +   by open roll mill,
          ZnO + stearic    internal mixers,
          acid [right      and/or extruders
          arrow] soluble   ai T <
          sulphurate zinc  100[degrees] C
          complex (A) +
          elastomers (P)

(a)       [MATHEMATICAL    Allylic             [K.sub.1]  [K.sub.1]
          EXPRESSION NOT   substitution

(b)       [MATHEMATICAL    Disproportionate    [K.sub.2]  [K.sub.2]

(c)       [MATHEMATICAL    Oxidation           [K.sub.3]  [~.K]

(d)       [MATHEMATICAL    Desulfuration       [K.sub.4]

(e)       [MATHEMATICAL    De-vulcanization    [K.sub.5]

Very recently Milani and Milani (12) have proposed a simple kinetic
numerical model to predict EPDM reticulation level, which may also take
into account reversion. The model is a simplified one and relies into
the derivation of a single second order nonhomogeneous differential
equation, representing the degree of reticulation (or conversely the
torque resistance) of rubber in dependence of curing time. Kinetic
parameters to set in the kinetic model are only three and they can be
evaluated by at least two torque curves performed on the same blend at
two different vulcanization temperatures. Cure tests have to be
maintained at fixed vulcanization temperature and may be performed by
means of both traditional oscillating disc (ODR) and rotor-less (13),
(14) (RPA2000) cure-meters.

In the present article a RPA2000 cure-meter is utilized to perform
the experimentation. Such device has a test chamber with controlled
stable vulcanization temperature allowing the storage of 5 gm of
product, with diameter 20 mm and height 12.5 mm (total neat volume 8
[cm.sup.3]). Basically, there are no perceivable geometric effects
related to the torque measure because (1) the test is fully standardized
and (2) in case of RPA2000 devices the ODR is missing. Quite small
secondary torques may be present as a consequence of material viscosity
and plates friction, which are obviously negligible in practice. Also in
case of experimentations conducted with ODRs, it is worth remembering
that the dimension of the disc is relatively small, allowing to
disregard inertia forces, always present when mechanical elements move
and responsible of secondary geometric effects.

After experimental data reduction, the aforementioned kinetic model
is adopted to predict rubber degree of vulcanization during the
industrial curing process of a thick weather-strip used in civil
engineering applications. Once evaluated the kinetic constants involved
in the reticulation process, the second phase relies in implementing
kinetic model parameters within a nonstandard finite element (FEM)
software for a thermal analysis of the item. Such approach follows a
relatively long tradition regarding FEMs applied to then-no--mechanical
problems of weather-strips installed within devices subjected, after
thermal curing, for static and dynamic loads (15-19). FEM is, indeed,
recognized as the most suitable technique to interpret the
thermo--mechanical behavior of vulcanized rubber items with complex
geometries, giving the possibility to quickly study combined nonlinear
3D problems, with error estimates and error reduction upon mesh
refinements. The software developed allows obtaining, element by
element, temperature profiles at increasing curing times. In addition,
it is possible to evaluate output mechanical properties (tensile
strength, tear resistance, and elongation) increase as a function of
curing time. In this case, the numerical database collected in the first
phase (reticulation kinetic model) is used, allowing a point by point
estimation of any output mechanical property.

The blend studied to realize the weather-strip is a mix of two
different EPDMs (Dutral TER 4049 and 9046) with a medium amount of
propylene content (ca. 31% or 40% in weight) and 9% or 4.5% in weight on
ENB (5-ethylidene-2-norbornene), vulcanized through accelerated sulfur,
as described in detail next. Once evaluated the final mechanical
properties of the item point by point, a compression test is numerically
simulated, assuming that the rubber behaves as a Mooney-Rivlin material
under large deformations and using contact elements between the
compression device and the item. From an industrial point of view, the
numerical approach may be useful to optimize (especially in economic
terms) (i) vulcanization time, (ii) energy utilization, (iii)
temperature of vulcanization, and (iv) accelerators quantities. The
procedure is quite general and can be used in presence of any rubber
blend, provided that suitable experimental data are at disposal to
characterize crosslinking reactions at different temperatures.


The recently presented kinetic model (20) is utilized to evaluate
the degree of vulcanization reached by a rubber specimen subject to
thermal predefined conditions (constant temperature) and vulcanized with
sulfur. The model relies into a second order differential equation with
solution evaluable in closed form, having only three kinetic constants
to be determined. Constants are usually evaluated by means of rheometer
experimental tests realized following the ASTM D 2084 and D 5289 methods

Focusing exclusively on EPDM rubber, the commonly accepted basic
reactions involved-see also Refs. [5. 2124 and Table 1, are:





where P and A are the polymer (EPDM) and soluble sulfureted zinc
complex ([S.sub.8] + accelerators + ZnO + stearic acid) respectively,
[P*.sub.1] is the pendent sulfur (crosslink precursor), [P.sub.v] is the
reticulated EPDM. [K.sub.1,...,5] are kinetic reaction constants, which
depend only on reaction temperature, [MATHEMATICAL EXPRESSION NOT
REPRODUCIBLE IN ASCII] [Q.sub.x]and [D.sub.e] are the matured crosslink,
the oxidation product and diaryl-disultide, respectively. Reaction (a)
in Eq. / represents the allylic substitution in Table 1, reaction (b) is
the disproportionation, whereas reactions (c-e) occurring in parallel
are respectively the oxidation, the de-sulfuration and the

Chemical reactions occurring during sulfur vulcanization reported
in Eq. I obey the following rate equations:


The set of differential Eq. 2 may be obviously solved numerically
by means of standard Runge--Kutta procedure (25), (26). However, this
approach may become tedious both from a numerical and practical point of
view. Closed form solutions are obviously preferable.

In (20) it is shown how the concentration of vulcanized polymer
[P.sub.v](t) within the material during the vulcanization time range is
ruled by the following single second order non-homogeneous differential
equation with constant coefficients:

[d.sup.2][P.sub.v]/d[t.sup.2] + [K.sub.2]d[P.sub.v]/dt +
[[~.K].sup.2][P.sub.v] =
[K.sub.1][K.sub.2][P.sub.0.sup.2]/[([P.sub.0][K.sub.1]t + 1).sup.2] (3)

Having indicated with [[~.K].sub.2] the following constant:

[[~.K].sub.2] = [K.sub.2]([K.sub.3] + [K.sub.4] + [K.sub.5]) +
[K.sub.3.sup.2] + [K.sub.4.sup.2] + [K.sub.5.sup.2] (4)

It can be shown that Eq. 3, after some reasonable simplifications
on the nonhomogeneous term fully explained in Ref. (20), may be solved
in closed form and the solution is:


As it is possible to notice, in Eq. 5 kinetic constants to be
determined are only three, namely [K.sub.1], [K.sub.2], and
[[bar.K].sub.2]. The most straightforward method to provide a numerical
estimation of kinetic constants is to fit Eq. 5 on experimental
cure-curves, normalized scaling the peak value to Po (for instance equal
to 1) and translating the initial torque to zero, as suggested by Ding
and Leonov (7). As a rule, variables [K.sub.1], [K.sub.2], and
[[~.K].sup.2] are estimated through a standard nonlinear least square
routine. The initial part of the curve, before scorch point, which is
typically linked to viscosity exhibited by a fluid, cannot enter into
the optimization process, because it is obviously ruled by other
physical mechanisms.

EPDM Blends Under Consideration

Two different EPDMs (Dutral TER 4049 and Dutral TER 9046)
experimentally tested in [20.1 are reanalyzed in this work. The
characteristics of such EPDM blends, in terms of Mooney viscosity and
compositions are summarized in Table 2. In the same table, a hypothetic
product derived ad hoc mixing the two experimentally tested blends is
also indicated. Generally speaking, from a practical point of view, an
elastomeric blend is an interesting method commonly used in the rubber
industry in order to increase final properties of rubber items. It has
been shown in many applications that, while two polymers may be
virtually mutually insoluble, blends may be industrially produced, which
are macroscopically homogeneous and have improved properties, provided
that mechanical mixing is suitable and viscosities after mixing are
sufficiently high to prevent gross phase separation. In this situation a
master-batch process would be desirable, i.e., the component polymers
should be precompounded with vulcanizing agents and additives, and then
individual stocks should be blended in desired proportions. In our
specific case, the two rubber types are quite similar in terms of
molecular weight, molecular weight distribution, chemical properties,
and structure. They exhibit different cure rates because different
amounts of ENB are present, namely 1% and 2% in terms of % moles in the
polymers. The mix under consideration deserves to be studied for the
following reasons: (1) it may be produced by means of the same catalyst
system and the same process used for Dutra! TER 4049 and 9046; (2) when
dealing with Mooney viscosity number, differences do not exceed 30
points between the two products, meaning that the mix between them do
not modify the MWD extensively; (3) the composition in terms of ethylene
of both principal polymers is about the same and in any case in the
range in which the polymers are perfectly amorphous.

TABLE 2. Composition of Dutral 9046, Dulral 4049 and Duiral
9046-4049 blend.

                                                         Mix 70%
Product type             Dutral 9046  Dutral 4049  4049-30% 9046

Propylene (wt%)                   31           40           37.3
ENB (wt%)                        9.0          4.5           5.85
Ethylene (wt%)                    60         55.5          56.85
ML(1 + 4) [Degrees]C              67           93           85.2
ML(1 + 4) 125[Degrees]C           49           76           67.9


For all types of EPDM and the hypothetical mix considered, the same
formulation was used, with the aim of comparing the experimental data
with the data derived numerically from the solution of the ordinary
differential equation system Eq. 2--hereafter labeled as ODEs system
model for the sake of clearness--and the single second order
differential Eq. 5--hereafter labeled as single EQ-DIFF model. Rheometer
curves at 160 and 180[degrees]C are shown in Fig. 1. It is interesting
to notice that Dutral TER 9046 exhibits more visible reversion at
180[degrees]C, whereas the behavior of Dutral TER 4049 is less critical.
Being the mix composed by 70% Dutral TER 4049 and only 30% Dutral TER
9046, its reversion is expected to be quite little and in any case
limited to the final vulcanization range at 180[degrees]C. Reversion is
totally absent at 160[degrees]C. In Table 3, the compounds formulation
adopted (in parts per hundred resins) is summarized. The experimental
compounds were generally prepared in an internal mixer and by adding
both curing and accelerating agents on the roll mixer, maintaining the
temperature of mixing lower than 100[degrees]C.

TABLE 3. Compounds formulation adopted (in phr).

Ingredients   Description                   phr

Polymer       Dutral 4049, 9046             100
Zinc oxide    Activator                       5
Stearic acid  Coagent                         1
HAFN 330      Carbon black                   80
Paraffin oil  Wax                            50
Sulfur        Vulcanization agent           1.5
TMTD          Tetramethylthiuram disulfide  1.0
MBT           Mercaptobenzothiazole         0.5

A comparison among experimental data, ODEs system and single
EQ-DIFF at temperatures equal to 180 and 160[degrees]C is represented in
Fig. 2 for Dutral TER 4049, in Fig. 3 for Dutral TER 9046 and in Fig. 4
for the mix between 4049 and 9046, respectively. Convergence curves are
also represented for both models. Each curve represents the difference
between experimental and numerical normalized torque at a given
iteration number. The mix of products here presented will be adopted for
studying the production phase of whether-strips analyzed in the
following sections. Here it is worth noting that, for these particular
items, it is crucial to have at disposal a rubber compound with a
rheometer test curve, which reaches asymptotically the maximum Mooney
viscosity, without exhibiting reversion. As it has been demonstrated in
Ref. (20) in a different technical problem, the overall vulcanization
level of an item may be numerically predicted by means of the
determination of temperature profiles and the knowledge of rheometer
curves at different temperatures. In general, vulcanization density
depends on curing temperature and time as well as on thickness of the
items, obviously at given recipe used. Compounds exhibiting reversion at
temperatures lower than 220[degrees]C should be avoided, since may
result into items over-vulcanized near the skin. Conversely, from an
economical point of view, it is important to reduce vulcanization time
and hence to use high temperatures, blends that vulcanize more quickly
(as Dutral TER 9046), but without reversion (as Dutral TER 4049): for
these reasons, the polymer mix here proposed appears to be a good
compromise between quality of final reticulation and reduction of
production time needed.




Cured Rubber Mechanical Properties

To evaluate numerically the degree of vulcanization of a given
rubber item, it is necessary to link degree of vulcanization with output
mechanical properties. There are many important parameters that
producers are interested to maximize at the end of the vulcanization
process, as for instance tear resistance, tensile strength, tension set,
etc. (27), (28). Unfortunately, in almost the totality of the cases,
such parameters cannot be maximized at the same time. In this article,
we will focus, for the sake of simplicity, only on a single output
parameter (tensile strength).

As experimental evidences show, there is a precise relationship
between torque values obtained in the vulcanization curve and output
mechanical properties of a real cured item (see for instance Refs. (27),
(29-40). In Fig. 5, several experimental data available in the
literature concerning torque values M and tensile strength exhibited by
vulcanized commercial products are represented, along with a possible
numerical experimental data fitting. The numerical relation between
torque and tensile strength found for Dutral blends and represented in
Fig. 5 will be used in the following section to perform some simulations
on a real weather-strip. An interesting aspect of the experimental
relation existing from torque and tensile strength is that it is not
necessarily mono-tonic, i.e., tensile strength may slightly decrease
increasing torque resistance. In the same figure, the relation between
hardness and percentage vulcanization degree is also depicted. The curve
is taken from Coran (21) and depends on the blend used. Having at
disposal such relationship, from the thermal model proposed hereafter it
is possible to evaluate point by point the vulcanization degree at the
end of the thermal process, and hence the local hardness of the item.
From well-known numerical relationships (usually based on empirical
formulas or equations based on theoretical formulations, e.g., Gent, BS
903, ASTM 1415, Qi et al. (41-43) it is finally possible to estimate
resultant Young Modulus E of the rubber after vulcanization, to be used
within the mechanical compression simulations reported at the end of the

Another important issue to underline is that, obviously, the
representation of Fig. 5 is able to give correct information on output
parameters only at constant temperatures (note that T [not equal to]
[T.sub.n] being T rubber temperature and [T.sub.n] curing agent
temperature), i.e., results cannot be applied directly to a rubber
infinitesimal element subjected to curing. Indeed, the actual
temperature of a point inside an item subjected to heating is rather
variable and such variability can be estimated only resorting to
numerical methods. In general, each point P of the item has its own
temperature profile T = T(P,t). At a fixed time during the curing
process, each point has also its own torque value, which can be
calculated from the database collected previously. For each value of the
torque reached, a corresponding value of the output parameter can be
identically evaluated. In this way, torque-exposure time and output
property-exposure time (or alternatively output property-temperature)
profiles can be numerically estimated. Nonetheless. Fig. 5 gives a
precise (although approximate) idea of the complex behavior of rubber
during vulcanization, addressing that a strong variability of output
mechanical properties is possible when changing curing time and
vulcanization temperature.

Vulcanization of Thick Weather-Strips with Accelerated Sulfur

In a weather-strip subjected to any vulcanization process, an
inhomogeneous distribution of temperatures between internal (cool) zone
and (hot) skin occurs. With the aim of optimizing a production line,
many parameters have to be chosen carefully but, among the others, the
following variables play a crucial role: exposure time and temperature
of the heating phase. The software presented here--fully developed in
Matlab (44) language--is basically an assemblage of two elemental
blocks, further subdivided into sub-blocks.

BLOCK 1: for each node of the item (already discretized by means of
FEMs) at fixed values of the input parameters [T.sup.n] (curing agent
temperature) and [t.sup.c] (curing time), temperature profiles are
evaluated by solving numerically a Fourier's heat transmission
problem in two-dimensions (45-51). Since, closed form solutions are
rarely available and usually refer to simple geometries, a FEM (52)
discretization is needed in the most general case. For each node of the
item and for each value of exposure time and curing temperature
inspected numerically, local final mechanical properties are evaluated.
The average uniaxial tensile strength is considered as objective
function. To maximize this quantity, lour sub-block instructions are
repeated node by node.

SUB-BLOCK 1.1: at increasing times, the temperature of each node
varies (increasing until the end of the vulcanization phase and then
quickly decreasing during cooling). Each point is thus characterized by
a couple of values representing temperature and time.

SUB-BLOCK 1.2: for each temperature value collected in the previous
sub-block, the corresponding cure curve (evaluated numerically with the
model described previously and having at disposal kinetic constants of
the partial reactions) is uploaded. Here, it is worth noting that each
node of the item to vulcanize undergoes continuously different
temperatures, thus passing from a cure curve to a contiguous one (more
precisely to a contiguous curve associated to a higher temperature in
the heating phase).

SUB-BLOCK 1.3: for each point of the item, the torque-time diagram
is evaluated making use of all the numerical cure curves provided by the
single differential equation model described previously.

SUB-BLOCK 1.4: tensile strength and hardness (or elastic modulus)
to be used in mechanical simulations may be finally evaluated (e.g.,
tensile strength--time and tensile strength--temperature profiles) for
each node or point as a function of the torque reached by the point at
successive time steps. This is possible only having at disposal a
relation between final torque and tensile strength (or hardness) as
those provided in Fig, 5. Usually experimental data may be useful in
this step.


BLOCK 2: the procedure summarized in BLOCK 1 should be repeated
changing input parameters (i.e., total curing time and vulcanization
agent temperature). A very direct way to evaluate the best input couple
([t.sup.c] [T.sup.n]) is probably to subdivide the input domain in a
regular grid of points, but for practical purposes random attempts and
the experience of the user should be sufficient. Very advanced
algorithms based on Genetic Algorithms or bisection has been already
presented in Refs. (51), (53), (54). They are not used here, because the
knowledge of the blend behavior drives almost directly to the choice of
the most indicated production parameters to adopt.

The Modified Heat Transmission Problem

A modified heat transmission problem has to be solved to evaluate
item temperature profiles, accounting for the chemical reactions
occurring during vulcanization. We schematically subdivide the curing
process into two phases: heating and cooling. In the first phase,
elastomers are exposed to high temperatures in order to activate
crosslinking and thus vulcanization, whereas in the second phase rubber
is kept to ambient temperature through air and/or water. In the most
general case of 2D items, temperature profiles for each point of the
element are obtained solving numerically Fourier's heat equation
law (45-51):

[[rho].sub.p][c.sub.p.sup.p]([partial derivative]T/[partial
derivative]t) - [[lambda].sub.p][[gradient].sup.2]T -
[r.sub.p][DELTA][H.sub.t] = 0 (5)

where [[rho].sub.p], [c.sub.p.sup.p] and [[lambda].sub.p] are EPDM
density, specific heat capacity, and heat conductivity, respectively;
[DELTA][H.sub.r](KJ/mol mol) is rubber specific heat (enthalpy) of
reaction and [r.sub.p] [mol/([m.sup.3] sec)] is the rate of

When heat transmission at the external boundary is due to
convection and radiation (extrusion process), the following boundary
conditions must be applied in combination with the field problem:

[[lambda].sub.p][partial derivative]T(P, t)/[partial
derivative]n(P) + h(T(P, t) - [T.sub.n]) + [q.sub.rad] = 0 (7)

where h is the heat transfer coefficient between EPDM and
vulcanizing agent at fixed temperature T, [T.sub.n] is vulcanizing agent
(e.g., nitrogen) temperature, P is a point on the object surface, n is
the outward unitary vector on P. and [q.sub.rad] is the heat flux
transferred by radiation. Radiation contribution for the vulcanization
of complex 2D geometries may not be determined precisely, but the
following formula may be utilized without excessive inaccuracies:

[q.sub.rad] = [sigma]([T.sub.n.sup.4] - T[([R.sub.p],
t).sup.4])/[1/[[epsilon].sub.p] +
[A.sub.p]/[A.sub.n](1/[[epsilon].sub.n] - 1)]

where [sigma] = 5.67 x [10.sup.8] W/([m.sup.2][K.sup.4]) is the
Stefan--Boltz-mann constant, s are emissivity coefficients, [A.sub.p,n]
are the areas of heat exchange (p: rubber item; n: curing agent).

The same considerations hold for the cooling phase, with the only
differences that (a) the actual cooling agent temperature has to be used
and (b) that heat exchange for radiation is negligible, i.e.:

[[lambda].sub.p][partial derivative]T(P, t)/[partial
derivative]n(P) + [h.sub.w](T(P, t) - [T.sub.w]) = 0 (9)

where [h.sub.w], is the water (air) heat transfer coefficient,
[T.sub.w] is the water (air) cooling temperature, and all the other
symbols have been already introduced.

Initial conditions on temperatures at each point at the beginning
of the curing process are identically equal to the room temperature.
Initial conditions at the beginning of the cooling phase are obtained
from the temperature profiles evaluated at the last step of the cooling
zone, i.e., at T(P,[alpha][t.sub.c]), where ate is the heating interval,
and P is a generic point belonging to f2. To solve partial differential
equations system (6)-(9), standard elements (46), (47), (52) are used.
The procedure has been completely implemented in Matlab (44) language
and interfaced with the first phase of the procedure. In this way,
resultant FEM temperature profiles at each time step are directly
collected from the numerical analysis and utilized for the evaluation of
output rubber mechanical properties by means of an integrated tool.

For the discretization of the weather-strip, which is schematically
represented by its cross-section, 4-and 3-noded 2D elements are
utilized. Temperature field interpolation is assumed linear inside each
element, i.e., = [N.sup.e][T.sup.e], where, for 4-noded elements,
[T.sup.e] = [[[T.sup.1], [T.sup.2], ... [T.sup.4]].sup.T] is the vector
of nodal temperatures, [N.sup.e] = [[N.sup.1], [N.sup.2], ... [N.sup.4]]
is the vector of so-called shape functions [N.sup.i]( i = 1, ... 4) and
P is a point of coordinates [x.sub.p], [y.sub.p], and [z.sub.p] inside
the element. Following consolidated literature in this field, e.g., see
Ref. (55), in the numerical simulations reported, the following
parameters have been used: EPM/EPDM density [[rho].sub.p] = 922
kg/[m.sup.3], rubber specific heat capacity [c.sub.p.sup.p], = 2700
J/(kg [degrees]C), [[lambda].sub.p] = 0.335 W/(m [degrees]C),
[DELTA][H.sub.r] = 180 kJ/mol, water heat transfer coefficient [h.sub.w]
= 1490.70 W/([m.sup.2] [degrees]C), curing agent heat transfer
coefficient h = 900 W/([m.sup.2] [degrees]C) (for air heat transfer
coefficient we assume h = 5 W/([m.sup.2] [degrees]C), [[epsilon].sub.p]
= 0.60, [[epsilon].sub.c] = 0.70, water cooling temperature [T.sub.w] =

The FEM discretization and the geometry of weatherstrip considered
are sketched in Fig. 6 (top subfigure). In the same Fig. 6 (bottom
subfigure), temperature contours obtained during a FEM simulation at
increasing instants are represented (nitrogen temperature [T.sub.n] =
160[degrees]C, curing interval [alpha][t.sub.c], = 300 sec). As it is
possible to notice, internal points heating is sensibly slower with
respect to the skin, as also schematically represented in Fig. 7a, where
temperature profiles of three different points of the item are depicted.
This may influence the final quality of the vulcanized rubber. Points
are labeled as A, B, and C. B and C are near the external boundary of
the item. whereas A is an internal point. It is therefore expected that
B and C thermal behavior is similar and at the same time quite different
to point A behavior. Data adopted in the thermal simulations are in
agreement with literature indications (56), (57). For the sake of
simplicity, specific heat capacity [c.sub.p.sup.p] and conductivity
[[lambda].sub.p] are assumed constants in the simulations. In reality,
such parameters reasonably exhibit certain variability in the
temperature range inspected. In particular heat capacity [56] ranges
from 1700 J/(kg [degrees]C) at 80[degrees]C to 2900 J/(kg [degrees]C) at
200[degrees]C, with an almost linear behavior in such temperature range,
whereas conductivity decreases asymptotically from 0.355 to 0.335 W/(m
[degrees]C). In Fig. 7b and c, part of the huge amount of results
obtained from a sensitivity analysis conducted varying material heat
capacity and conductivity is summarized. In particular, temperature
profiles for points A (b) and B (c) are represented, assuming three
different constant values for the heat capacity ([c.sub.p.sup.p], equal
to 1700, 2200, and 2700 J/ (kg [degrees]C), respectively) at fixed
conductivity [0.335 W/(m [degrees]C)], a linear behavior of
[c.sub.p.sup.p] with either [[lambda].sub.p] constant or 'P
asymptotically variable following (58). As it is possible to notice, the
variability of the temperature profiles is quite small and in any case
the error introduced assuming [c.sub.p.sup.p] equal to 2700 J/(kg
[degrees]C) is in the most unfavorable case within 10%, fully acceptable
from an engineering standpoint. The greatest variability is obviously
experienced for internal points, whereas for points near the core,
temperature profiles almost coincide. Finally, authors experienced very
little variability with [[lambda].sub.p].



The thermal behavior during vulcanization of the core and the skin,
i.e., points A and B, subjected to the vulcanization conditions
previously discussed and also with a different [T.sub.n], =
200[degrees]C, are represented from Figs. 8-11. The relationship used to
evaluate the final tensile strength of the point is depicted in Fig. 5a,
which is a reasonable hypothesis of the blend behavior under mechanical
tests. As already pointed out, there is very little or absent reversion
even at high temperatures when a Dutral TER 4049 70% and Dutral TER 9046
30% mix is utilized. In particular, the temperature profile of point A
for both [T.sub.n] = 160[degrees]C and [T.sub.n] = 200[degrees]C, Fig. 8
(2nd and 4th row, 2nd column sub figures), shows a gradual initial
temperature increase, meaning that Point A passes very slowly from
rheometric curves at low temperatures to rheometric curves at high
temperatures. The vulcanization phase stops at 300 sec and in both cases
([T.sub.n] = 160[degrees]C and 200[degrees]C) the point is not able to
reach the maximum possible temperature, i.e., external nitrogen
temperature. This was expected, because A is positioned in the core of
the item. Such curves are numerically interpolated, as discussed
previously, from experimental data using the kinetic model proposed, see
Fig. 9, once that constants [K.sub.1], [K.sub.2], and [[~.K].sup.2] are
at disposal. When the maximum temperature is reached, point A follows
the rheometric curve associated to such temperature, stabilizing its
behavior at increasing times on that rheometer curve. In other words,
point A follows almost completely the rheometer curve at around
150[degrees]C and 190[degrees]C, respectively. When curing finishes,
temperature decreases quickly. Here, two paths should be followed
numerically for point A, represented in Fig. 9 with circles and crosses.
The numerical behavior represented by crosses is not real, because it
would be referred to a material without any memory of the vulcanization.
Therefore, it relies on a material that ideally jumps from a rheometer
curve at high temperature to the successive at low temperature
(reversible process). On the contrary, the real behavior of the material
under consideration is represented by the circles, indicating that the
level of vulcanization reached at the end of the curing process cannot
be inverted reducing temperature, but can only be stopped.



Once that the curing time-torque diagram of the rheo-metric curve
is known numerically (2nd and 4th row, 1st column diagrams) for the
point under consideration, tensile strength reached increasing exposure
times can be easily determined through the relation sketched in Fig. 5.
For point A, the exact vulcanization history in terms of curing
time-[[sigma].sub.t], and temperature-[[sigma].sub.t] diagrams is
represented in Fig. 8 (1st and 3rd row, 1st sub-figures). The same
considerations can be repeated for points B (or similarly C) of the
item, thus giving the possibility to evaluate the average final tensile
strength. In particular, in Figs. 10 and 11 the curing behavior of point
B is represented, whereas point C data are omitted for the sake of
brevity. From a comparative analysis, it is rather evident that points B
and C reach maximum vulcanization temperature more quickly with respect
to point A. since the cure curves followed by such points are very near
with each other. Also in this case, the reversion range is very little
even at 160[degrees] and, as expected, it occurs after that the peak
strength has been reached.



From a detailed analysis of points A, B, and C behavior, it can be
argued that at 160[degrees] the item is slightly undervulcanized,
because the thickness of the core would require either a higher
vulcanization external temperature or a longer exposure time. The
technical usefulness of the approach proposed seems rather clear, since
manufacturers could calibrate (with a few numerical attempts) both
curing time and vulcanization temperature of the production phase, to
avoid reversion and under-vulcanization, thus minimizing the industrial
process cost.

Mechanical Compression Test on the Weather-Strip

After a detailed numerical analysis of the industrial vulcanization
process, a FE mechanical analysis of weather-strip installed in the body
of a door (civil engineering application) is finally conducted using
Strand7 commercial software package. The mechanical simulations
reproduce a typical experimental compression test, where a part of the
boundary (shown in Fig. 6) is supposed clamped and a rigid punch
compresses the opposed boundary. Displacements in the x- and
y-directions are restricted only in such region, to simulate the actual
constraints acting where the weather-strip is installed in a door.
Contact elements between the punch (Fig. 6) and the rubber material are
utilized to simulate mono-lateral contact. Zero gap elements available
in Strand 7 are used to model contact.

A two constants Mooney--Rivlin model under large deformation
hypothesis is used for rubber, with parameters [C.sub.10] and
[C.sub.20], respectively equal to 0.270 and 0.160 MPa. Such constants
are suitably chosen in order to fit as close as possible, the
experimental uniaxial stretch--strain curve available for the blend
used. Here it is worth noting that, from the vulcanization model
proposed in the previous section, only the initial Young Modulus is
indirectly known, once that the average vulcanization level is evaluated
and hence the final hardness S is at disposal. For a vulcanization
executed at [T.sub.n], = 185[degrees]C and [t.sub.e] = 600 sec a very
good vulcanization level with average hardness equal to 56 is obtained.
The corresponding Young modulus is evaluated by means of BS 903 as 2.58
MPa. The elastic shear modulus is always E/3 and is linked with
Mooney--Rivlin constants by means of the following formula: G =
2([C.sub.10] + [C.sub.20]). A further condition is necessary to set
univocally [C.sub.10] and [C.sub.20], meaning that the uniaxial
stretch--stress curve test is needed. In absence of specific
experimental data available, from authors experience, the typical
uniaxial behavior of the compound is well approximated, at least for
stretches lower than 3, assuming [C.sub.20] roughly equal to 0.6

In such a situation, defined the stretch as the ration between the
length in the deformed configuration divided by the length in the
undeformed state, let [[lambda].sub.1] = [lambda] be the stretch ratio
in the direction of elongation and [[sigma].sub.l] = [sigma] the
corresponding stress. In uniaxial stretching, the other two principal
stresses are zero, since no lateral forces are applied ([[sigma].sub.2]
= [[sigma].sub.3] = 0). For constancy of volume, the incompressibility
condition [[lambda].sub.1] [[lambda].sub.2] [[lambda].sub.3] = 1 gives:

[[lambda].sub.2] = [[lambda].sub.3] = 1/[square root of [lambda]].

In the most general case, the strain energy function for a
two-constant Mooney--Riv lin model is given by:

W = [C.sub.1]([I.sub.1] - 3) + [C.sub.2]([I.sub.2] - 3) (6)

where [I.sub.1.sup.2] + [[[lambda].sub.2.sup.2] +
[[lambda].sub.2.sup.3] and [t.sub.2] = [[lambda].sub.1.sup.-2] +
[[[lambda]sub.2.sup.-2] + [[[lambda].sub.3.sup.-2].

The FE model, as mentioned earlier, is constituted by
two-dimensional elements, which are supposed here to undergo a plane
strain mechanical condition under large displacement and contact
elements. Quadrilateral 4-noded elements with bubble-shaped
extra-functions and triangular linear elements are utilized to mesh the
rubber weather-strip. To speed up computations, only half of the device
is meshed for symmetry, applying suitable boundary conditions on
displacements on the symmetry axis. The compression device is assumed to
be a rigid body, since it is supposed to exhibit very little deformation
when compared to rubber. The compression device is loaded by means of an
increasing horizontal displacement up to 4 mm from its unloaded
position. Displacement sub-steps equal to 0.2 mm are imposed on the FE
model to facilitate convergence.

Contact forces distributions at successive time-steps are
represented in Fig. 12a. It is particularly evident that the compressed
contact zone, as expected, spreads at successively increased
displacements of the compression device. A direct integration of contact
forces allows the evaluation of the force to apply to the compression
device at successive displacements, as represented in Fig. 12b. The
loading condition obviously justifies the increase of first derivative
of the load at increased displacements, since the contact surface
becomes larger. Figure 12b shows the overall behavior is very important
for producers, because it indicates the overall expected stiffness of
the device under design loads. Such stiffness should fall within some
lower and upper bounds to be acceptable. The global behavior of the
device depends not only on the rubber compound used but also on the
vulcanization process used to cure it, since elastic properties are
Shore A dependent. It is therefore crucial to have at disposal of an
integrated thermo--chemo--me-chanical software as that used here to
predict such behavior.

For the sake of completeness in Fig. 12b compression curves
obtained assuming a hardness S equal to 65 and 45, respectively are
represented. For the blend and the specific item under consideration,
such hardness values are obtained respectively when the following
vulcanization conditions are assumed: [T.sub.n] = 195[degrees]C with
[t.sub.e] = 500 and [T.sub.n] = 175[degrees]C with [t.sub.e] = 650. The
same hypotheses assumed previously are adopted for Mooney--Rivlin
constants [C.sub.10] and [C.sub.20]. From the sensitivity analysis
conducted, it is very straightforward to conclude that the evaluation of
the overall stiffness of the item in presence of different vulcanization
conditions is crucial, especially when the item is installed as
weather-strip in a real door. As a matter of fact, from an engineering
point of view, the knowledge of such stiffness may be very useful,
especially when a maximum displacement threshold must be respected, as
for instance for large windows subjected to wind.



In the present article, the possibility to utilize, in
weather-strip for industrial production processes, two different
ter-polymers has been investigated by means of a comprehensive numerical
model. The first ter-polymer has a low amount of ENB, whereas the second
exhibits the maximum amount of ENB nowadays possible in EPDM production
plants. For both polymers. experimental rheometer curves were at
disposal at two different temperatures. The behavior of a mix of both
polymers, 70-30% in weight, has also been investigated numerically. In
particular, rheometer curves of the mix at two different temperatures
have been numerically extrapolated from components experimental data
available, taking into account their presence in the mix in terms of
relative percentage in weight. The vulcanization behavior of the mix has
been then numerically studied within the production line of a real
weather-strip. Generally, weather-strips are produced from ter-polymers
with an ENB content equal to around 6. but this custom is almost always
empirical. In this framework, this article gives a numerical insight
into the most suitable blend to be used in the production of such kind
of items.

From a production point of view, it is well known that a huge
amount of different EPM--EPDM commercial products are available.
Differences are mainly due to the ENB content, the distribution of
molecular weights, and the Mooney viscosity. The choices to mix two
different products with low and high ENB content, respectively, appears
interesting, since blends with an intermediate behavior could
potentially increase the final mechanical quality of the vulcanized
item, as well as reduce production time and hence costs. From the
producer point of view, costs reduction is derived not only from the
reduction of materials to stock, but also from the possibility to
modulate time and temperature used to obtain an optimal vulcanization of
the final item. It has been shown, indeed, that a ter-polymer with low
ENB content exhibits a rheo-metric curve monotonically increasing, but
with low rate of vulcanization. Conversely, a ter-polymer with high ENB
content shows a very good vulcanization rate but exhibiting also
reversion, especially at high temperatures. The present article shows
how an intermediate blend, obtained ad hoc mixing the aforementioned
products, may have both good vulcanization rate (intermediate between
the products with low and high ENB content) and no or very little
reversion. Furthermore, it has to be emphasized that the possibility to
mix ter-polymers with different molecular weights and Mooney viscosity
higher than 60, should modify the molecular weight distribution of the
mix, with the important advantage to obtain an ENB uniform distribution
in the back bone, which should allow a higher crosslinking homogeneity.

From the numerical model proposed, it seems that, at least
theoretically, it is possible to obtain mixtures whose vulcanization can
be tuned as a function for industrial production requirements, with an
obvious increase of the flexibility. A comparison with experimental data
on the production line, unavailable at this stage, will be obviously
rather useful to have an insight into the actual predictive capabilities
of the comprehensive numerical approach proposed.


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G. Milani, (1) F. Milani (2)

(1.) Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133
Milano, Italy

(2.) CHEM.00 Consultant, Via J.F. Kennedy 2, 45030 Occhiobello,
Rovigo, Italy

Correspondence to: G. Milani; e-mail: gabriele.milani@polimi.it

Published online in Wiley Online Library .(wileyonlinelibrary.com).

[c] 2012 Society of Plastics Engineers

DOI 10.1002/pen .23270

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