When either or both of the surfaces are polymers, Equation 3.1
no longer applies. Polymers deform viscoelastically: the deformation
depends not only on the normal load N but also on the geometry and time
of loading. With fixed geometry and duration of loading, the area of true
contact is proportional to where . For a
truly elastic solid (for example rubber), [Lincoln,
1952].
From Figure 3.2 for
(typical for a hard sphere on a Plexiglas surface),
a 16-fold increase in
load from 1 to 16 pounds produces an 8-fold increase in frictional
force, hence the
coefficient of friction is halved. This is a general feature of polymers;
the effective coefficient of friction reduces at higher loads.
Howell and Mazur [1953] performed some of the first experiments to study
the effect that the elastic behaviour of the asperity-to-asperity contact
has on the nature of sliding for polymeric materials.
The stress-strain curves of three
hypothetical materials are shown in Figure 3.3.
The dashed lines represent the stress range in the contacting asperities.
The asperities of material (a)
deform plastically, so Amontons' law applies. The asperity-to-asperity
contact in material (b) is
elastic; so the true asperity-on-asperity contact area
will be of the form where C
involves the modulus and dimensions of the asperities, hence
(where S is the shear breaking strength).
Increasing the normal stress would cause plastic
deformations as for material (a).
Figure 3.3: Stress-strain curves for various hypothetical asperity models
(Howell, 1953). The dashed lines represent the stress range in
the contacting asperities. The models are:
(a) plastic deformation of asperities, so Amontons' law applies,
(b) elastic deformation of asperities, so ,
(c) elastic and plastic deformation of asperities, so .
For material (c) the deformation of the asperity contact starts with
a linear elastic region followed by a gradual
yield; the true asperity-on-asperity contact area is thus
likely to lie between the bounds of
materials
(a) (plastic ) and (b) (elastic ). Therefore a possible equation for the friction could be the
empirical one, , where . Most
synthetic polymers used in ropes correspond most closely to material (c)
where the actual mechanism of asperity deformation is indeterminate.
The relationship fits Howell and Mazur's test
results very well; some of their results are shown in
Table 3.2 and Figure 3.4.
Material | K | |
cellulose acetate | 0.60 | 0.96 |
viscose rayon | 0.49 | 0.91 |
drawn nylon | 0.92 | 0.80 |
undrawn nylon | 0.85 | 0.90 |
Howell's equation for friction can be re-written as
, where is the ``equivalent
coefficient of friction''. For use in the finite element analysis
it was decided that expressing this relationship in terms of
stresses would make modelling possible. Therefore, it is proposed
here that Howell's equation can be re-expressed as; ,
where is the frictional stress, is
the contact pressure, and A and are frictional constants.
Due to the viscoelastic properties of polymers, when a polymer is in
contact with another surface for a period of time, the asperities start to
creep under the normal load. This increases the actual area of contact,
thereby increasing the frictional limit. This creep explains why
contact involving polymers
typically has a greater difference between static and dynamic
coefficients of friction, than for metal-to-metal contact.
An alternative view of friction is put forward by Kragelsky et al. [1982].
They support the hypothesis that the coefficient of friction is
the sum of two terms; molecular and mechanical, i.e.;
Molecular interaction processes take place in the surface `film' and
affect the surface layers to a depth of a few hundredths of a
micron. Mechanical interaction takes place in layers with a thickness of
a few tenths of a micron. As these processes occur at different levels
they are largely uncorrelated and hence can be separated.
This equation suggests a very complex relationship between the normal
load and the coefficient of friction; both components of
include a pressure
term, as well as extra terms for the hysteresis loss during sliding, the
surface roughness, and the strength of the molecular bond, amongst
others.
The additional complexity of Kragelsky's model does not seem justified;
Howell's equation
has been shown to adequately represent experimental data.
Therefore a modified Howell's equation, , is used.
Spin finish is a surface coating, added to the fibres during
manufacture, for the purpose of gluing together the fibres during
the processing of the yarn [Piller, 1973]. This prevents
them being snagged by the guides and thus being broken or drawn
out of the yarn. The finish also evens out, and so improves, the
running properties and friction coefficients, as well as removing
any electrostatic charge; these keep the draw-off conditions as
even as possible.
Yarn sizes are also used; these coat the yarn with a protective film to
reduce the abrasion damage during processing and service.
By selecting the right finish, the abrasion resistance of the yarn
can be increased [Beers and Ramirez, 1990; Crawford and McTernan,
1988; Schick, 1975, 1977].
The yarns used in the tests referred to in
this chapter are from spools used to make Parafil
ropes and so carry the finishes present in actual ropes.