L Sc; 1NSTRUM., 1965, VOL. 42
`
`The measurement of flexural rigidity of thin polymeric films
`
`D. H. MORTON and A. MARKS
`British Cellophane Ltd., Bridgwater, Somerset
`MS. received 25th March 1965, in revisedform 15!}: April 1965
`
`Abstract. The ‘Handle-o-Meter’ method of determining flexural rigidity of thin (10 to
`200 pm thick) polymeric films is discussed. This method determines the force which has
`to be applied along the centre line of a film specimen laid over a slot, so that the film is
`forced into the slot. The relationships between the ‘Handle-o-Meter’ stiffness as measured
`by this force, the flexural rigidity and the tensile moduli of a variety of polymeric films
`have been investigated. An unexpected relation obtained by previous workers was shown
`to be due to failure to allow for the effects of gravitation. Computation showed that
`frictional forces have also to be taken into account. Within the limits of experimental
`error, the results for ‘Handle-o-Meter’ stiffness were shown to be in agreement with
`flexural rigidities calculated from film dimensions and tensile moduli. Considerable un-
`certainty is introduced by the limited accuracy of measurement of coefiicient of friction.
`
`1. Introduction
`It is well known that the flexural rigidity of a thin sheet or
`rod, composed of a homogeneous isotropic material, can
`be calculated simply from a knowledge of its dimensions
`and of the Young's modulus of the material (e.g. Champion
`and Davey 1948), provided that only moderate curvatures
`are considered. Polymeric films for packaging purposes are
`often coated with layers of other polymeric materials or
`are laminated to one another to provide some desirable
`combination of barrier properties or heat~sealability. The
`flexinal rigidity of such a structure is calculable, in principle,
`provided the dimensions of the component layers are known
`and provided that the Young‘s modulus of each layer is
`constant and known. However, virtually all processes of
`fabricating polymeric films result
`in a product which is
`inhomogeneous and is frequently markedly anisotropic.
`The calculation of
`the flexural
`rigidity from tensile
`properties is therefore bound to be both imprecise and
`laborious and in many cases will be impossible. There is
`Ll_1us_a need for a direct method of measuring the flexural
`1'1_E1d1ty, which is an important property of a packaging film
`since it has a great influence on the behaviour of the film
`whenit is used on packaging machines. Surprisingly few
`methods are available.
`H511 (1964) has applied the vibrating reed method to obtain
`the dynamic flexural
`rigidity. This sufl'ers from several
`drawbacks. The curvatures involved are infinitesimal, and
`“mi polymeric materials have non-Hookean elasticity, so
`ma‘ rigidities determined in this way may well differ signi-
`fi°'3DflY from those obtaining at larger curvatures. A more
`Important practical difliculty is that it is extremely difficult
`l0 ensure that reeds of these thin films (usually from 10 to
`SOP“ thlckl are free from transverse curvature. Such
`Wifature results in the sample adopting the form of an axial
`:°°"."“ °f_3- Cylinder, flattened at the clamping end. Such a
`olficlmen is far stiffer than a fiat reed, and Hsu's method can
`Y_ be used with specially flattened films, whose pro-
`“mes may not be representative of those it is desired to
`measure.
`method of Hansen et al. (1963) using the so-called
`‘1 3-0-Meter‘i' (Brownlec 1955) is thus of considerable
`1'
`-
`.
`.
`Albe;’t*“é°;I;rl13‘tI:‘i:"cVia‘l!_s‘;e5(s,i¢i3i.this instrument is made by Thwlllg.
`
`interest, since it does not suffer from the disadvantages of the
`vibrating reed method. The apparatus has been described
`in detail by previous authors and the description need not be
`repeated here.
`In principle, this method consists of measur-
`ing the force which has to be applied along the centre line
`of a film specimen laid over a slot, so that the film is forced
`into the slot. The piece of filrn is laid over the slot, which
`may be typically one centimetre wide, in a horizontal position.
`A horizontal bar which is parallel to the slotdescends so that
`it passes accurately through the centre of the slot, pushing
`the film with it. The instrument registers the maximum
`force of resistance to the movement of the bar, which is
`related to the flexural rigidity of the film.
`There were some surprising features in the results reported
`by Hansen and his co-workers.
`In particular, they found
`the relationship between ‘Handle-0-Meter’ stiflness S per
`unit specimen width and film thickness 2 to be of the form
`
`S = kl Ef2‘s
`
`where k, is a constant of the apparatus and E is the tensile
`elastic modulus.
`(see
`Inspection of the mechanics of the measurement
`below) indicates that the ‘Handle-o-Meter‘ stiffness S should
`be proportional to the flexural rigidity R of the film so that we
`are left with the relationship
`
`R 7': kl EIZIS.
`
`This is at variance with the theoretical relationship for a
`homogeneous, isotropic material, which predicts a dependence
`of R on the cube of the thickness.
`It also contrasts with the
`results of work by Guthrie et al. (1954) on the flexing of
`single rayon fibres. Despite the irregularity, inhomogeneity
`and anisotropy of such fibres,
`there was shown to be a
`close overall dependence of the mean flexural rigidity on the
`mean square of the cross-sectional area of the fibres, as
`predicted by theory.
`It appeared unlikely that the com-
`paratively homogeneous and isotropic materials used by
`Hansen would produce such a marked divergence from the
`expected relationship.
`A further source of uncertainty about these results arises
`from the fact that in the ‘Handle-o-Meter’ type of test, which
`was designed in the first place to measure the ‘hand’ of
`clothing fabrics, frictional properties of the film surface play
`591
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`D. H. Morton and A. Marks
`
`It cannot necessarily be assumed that such forces
`a part.
`can be ignored in the case of films unless special precautions
`are taken to reduce friction to a negligible value.
`In view of these considerations it was decided to carry
`out an investigation of the mechanics of this type of bending
`measurement, and in particular to detennine whether a more
`rational form of relationship could be found between the
`Young’s modulus of films and fiexural rigidity.
`
`2. Experimental: apparatus and procedures
`2.1. Standard ‘Handle-0-Meter’ measurements
`‘Handle-o-Meter’ stiffness was measured using specimens
`measuring from 2-5 to 20cm wide (parallel to the slot),
`depending on the rigidity of the specimen. The results
`were adjusted to correspond to a one centimetre wide speci-
`men and quoted in gwt per cm width. Measurements
`were made on six specimens of each film in both machine and
`transverse directions of the films, the specimen widths being
`chosen to give an instrumental reading in a convenient range.
`The reproducibility of the results varied greatly from film to
`film, depending on the uniformity of thickness of the film
`being measured, but for most films the range of the six
`measurements was within i 15% of the mean.
`
`2.2. Measurement of film thickness
`A ‘Magna-Gage’,'j which incorporates a variable-in-
`ductance displacement transducer, was used to measure the
`thickness of the specimens. Contact pressure of the probe-
`head in this instrument is 50 g, acting through a spherical
`ball point of 0-3 cm radius of curvature on to a flat anvil.
`Experience has shown that such measurements give good
`agreement with the average value of thickness obtained
`from weight per unit area and density measurements,
`Measurements between flat anvils which are frequently used
`give a high value for the thickness because of the presence of
`‘high-spots’ on the film.
`Measurements were made at five different points on each
`specimen of film, the precision of each individual measure-
`ment being 1-10 pin.
`( 10-25 am). The extent of the
`variation of thickness from specimen to specimen depended
`on the type of film being tested. The range of average
`specimen thicknesses about
`the mean varied from :1”/,
`for regenerated cellulose films up to 15% or more for many
`of the plastic films.
`
`2.3. Measurement of tensile modulus
`Because of the non-linearity of the load—e1ongation curves
`of polymeric films, even at extensions of below 1%, it is very
`dificult to obtain a value for the ‘initial’ Young’s modulus
`of the material. For all the materials involved in the present
`work however, extreme curvature of the load—elongation
`relation does not commence below about 2% extension.
`It
`is therefore possible to use the nominal stress at 1 % extension
`q as an indication of the Young’s modulus of the film.
`Effects arising from the non-linearity of
`the load-
`elongation relationship are discussed further below.
`The measurements were made using an Instron tensile
`testing machine. Specimens measuring Sin. long by 0-4 in.
`wide (12-5 cm X 1cm) were stretched at the rate of 10%
`per minute and the nominal stress at 1 % extension calculated
`from the load—elongation graphs.
`Six measurements were
`made in both machine and transverse directions. The
`
`1’ Southern Instruments Ltd.
`592
`
`reproducibility of the results was such that in most cases the
`range of the six measurements lay within i 5 % of the mean
`
`2.4. Automatic film-flexing apparatus
`
`For other work in these laboratories it became desirable
`to construct an instrument which would overcome two of the
`drawbacks of the ‘Handle-o-Meter’. One of these dmw_
`backs is that, using the standard instrument, it is inconvenient
`to measure the stiffness in both senses on the same Sample
`since this involves recovery of the sample from beneath the
`slot and re-mounting the other way up. The other in.
`convenience is that the size and nature of the instmmem
`makes it diflicult to enclose in a controlled environmental
`chamber.
`The apparatus which was developed and which has been
`used for part of this work is built almost entirely from
`FAC constructional kit.I To avoid the gravitationaj
`problems involved in the ‘Handle-o-Meter’ discussed below,
`the film is arranged to lie in a vertical plane and is clamped
`between and inside a vertical array of 2 mm diameter rods.
`Bending of the film takes place as illustrated in figure 1,
`
`
`
`Figure 1.
`
`Method of flexing film in automatic film flexing
`apparatus.
`
`The pair of rods which clamp the film specimen reciprocate
`so that the film is flexed in alternating directions by the
`pairs of rods at each end of its travel. The amplitude of
`the movement is adjusted to be suflicient for the film to
`pass just beyond the position at which maximum forces
`exerted on the outer cage of rods. Movement of the film 15
`
`
`
`Figure 2. Swinging-arm suspension of rod-cage and link
`to dynamometer.
`
`achieved by the use of a 2rev min“ synchronous e1=0t_“°
`motor, an eccentric and a simple slide mechanism. Wh’
`maintains parallelism of the system of rods.
`So that the forces exerted by the bending of the film 93:
`be measured, as shown in figure 2 the outer cage Of 1'0“
`
`I Mark Sylwan Ltd., Stockholm.
`
`PAGE 000002
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`PAGE 000002
`
`

`

`Flexural rigidity of polymeric films
`
`fact that this line passes very nearly through the origin of
`coordinates is evidence that friction in the bearings of the
`swing-arm has negligible eflect. Reproducibility of results
`is similar to that achieved with the standard ‘Handbo-
`Meter’.
`Measurement was made of the ‘Handle-o-Meter’ s'tifi'ness
`SH of a small series of films using shortspecimenscut in both
`
`I-0
`
`20
`4‘) (cm)
`
`5-0
`
`4-0
`
`I 2
`
`I 0
`
`8
`
`'3-
`"; 6W)
`
`4 2
`
`0
`
`Figure 4. Mean advancing flexing force 5'8 as a function of
`sample width b.
`
`the machine and transverse directions of manufacture. The
`series was chosen to include a. wide range of flexural rigidities.
`When these stiffnesses were compared with results of Sa
`from the automatic flexing apparatus, a good correlation was
`established (figure 5).
`It should be noted that although the
`
`
`
`4
`3
`S“ (q cm")
`
`5
`
`6
`
`7
`
`Figure 5. Relationship between flexing force 3, from the
`automatic film flexing apparatus and that from the ‘Ha.ndle-o-
`Meter’ SH for a range of polymeric film types and thicknesses.
`
`slot width in both cases is 0-5 cm, the two instruments do
`not give numerically identical
`results. This is probably
`because of the difference in the geometry of the slot, which is
`between rounded uprights in one case and sharp cornered
`in the other.
`
`593
`
`is mounted on a horizontal swinging arm, to which is coupled
`3 Smimgauge dynamometer D with a load capacity of
`+100 g. This arrangement increases the sensitivity of the
`Shem by virtue of the leverage ratio. Good quality ball-
`bearings B were found to be adequate for elimination of
`significant friction at the bearings and adjustment is provided
`to ensure that the axis of swinging of the arm is accurately
`'
`l.
`vegllgcfoutput of the dynamometer is displayed on a recording
`potentiometric rnillivoltmeter. The whole apparatus, exclud-
`mg power supplies and recorder, can be fitted into a 20 cm
`cubical box of rigid transparent plastic for environmental
`control purposes. The lateral dimension of the specimen
`(1,, figure 1) can be from 1 m up to 30 mm. A convenient
`width to use is 15 mm for films of about 20 mm thickness, and
`the specimen needs to be about 2-3 cm long. The actual
`sample width chosen is unimportant, since it is found that
`me rigidity reading obtained is accurately proportional
`to
`sample width (see below). Calibration is achieved by dead-
`weight loading. The form of the record obtained is illu-
`slrated by the two complete cycles shown in figure 3. Each
`
`
`
`Figure 3. Tracing from a typical record of the automatic
`film-flexing apparatus.
`
`cycle has two distinct parts, corresponding to bending in the
`two opposing directions. Both parts of the trace have two
`Peaks, each corresponding to the force recorded when the
`film is in the most favourable position to exert lateral pressure
`011 the outer cage.
`It will be noted that the flexing force
`ft‘-corded when the film is advancing between the rods (SI)
`Ismarlcedly greater than that found when the film is receding
`E~§5,g.2)'I'he reasons for this difference are discussed below
`_ An initial longitudinal curvature in the film sample causes
`Wluality of s, and s,',
`but
`it has been found that
`Hsa + 5;) = S,
`remains constant and independent of
`moderate curvatures. Clamping pressure and pressure
`afiilnst the cage rods appear to eliminate any efl‘ects attri-
`hutable to lateral curvature of the specimen, except in Severe
`C3565.
`
`‘It was found for all the films investigated that there was a
`slight decrease in the value of S, during the first 5-10 com-
`plete flexing cycles, the decrease being usually about 10%.
`. ‘fnd this
`the recorded values
`remained constant
`to
`mm’ 14% for some hundreds of cycles. Experiment
`sh.°w°d that the recorded output S, is proportional to the
`“dth Of the film specimen b, as shown in figure 4. The
`
`PAGE 000003
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`

`

`D. H. Morton and A. Marks
`
`3. Results and discussion
`
`3.1. Efects of sample length correction for gravity
`When the ‘Handle-o-Meter’ is operated in the standard
`way, a 20 cm long specimen is used, which takes up the form
`indicated in figure 6. As the specimen is pushed into the
`
`An experiment was carried out to find the relation between
`‘Handle-o-Meter’ stifihess SH, the stress q at 1% cxtension
`modulus and thickness t allowing for gravity in this my
`Using a 2-5 cm long specimen and a 1 cm slot width, value;
`of these properties were determined for a series of twe1v¢
`I-O0
`
`Figure 6.
`
`lllustration of profile adopted by film during
`course of measurement.
`
`slot by the penetrator beam the resistance encountered by
`the beam will depend on the following factors:
`(i) The
`fiexural rigidity of the film. This is the property which is
`required to be measured. (ii) Frictional resistance between
`the film and the plates of the instrument. This was found
`to be particularly important for films carrying a high charge
`of static electricity which tend to cling to the plate instead
`of lifting, as illustrated in figure 6. This clinging can be
`reduced by dusting the plates with talc and was eliminated
`when using shorter specimens, as discussed below.
`(iii)
`Friction against the edge of the slot. Dusting the film with
`talc reduces the friction, but the extent of this efl°ect has yet
`to be discussed. (iv) The leverage exerted by the weight of
`overlapping film on each side of the slot. The magnitude
`of this efi'ect depends on the length of film overlapping the
`slot and the weight per unit area of the film.
`
`
`
`Figure 8. Variation of ‘Handle-o-Meter’ stiffness S3 with
`specimen length L forpolyethylene films covering a range of
`weights per unit area.
`
`polymeric films of widely varying types and thicknesses
`These
`included
`polyethylene,
`polypropylene,
`biaxially
`orientated polypropylene and regenerated cellulose films, and
`measurements were made on specimens cut in both machine
`and transverse directions.
`
`
`
`loq
`
`I‘
`
`to stress q 31 1%
`Figure 9. Relation of the stifiness Sp;
`extension ratio, to the thickness 2, for a variety of p01Ym°‘;‘§
`film types and thicknesses. The solid line is that calculal
`theoretically (see text).
`
`The results of this work are displayed in figure_9, WW:
`log (S/q) is plotted against
`log t. The best straigh‘ 1’?!
`about which these points are scattered have s1oP°5 if
`log (S/q) on log I of 2-97 and for log t on 108 (5/4)
`
`PAGE 000004
`
` 4
`
`B
`
`[2
`L (cm)
`
`I6
`
`20
`
`Figure 7. Variation of ‘Handle-o-Meter‘ stiffness SH with
`length ofspecimen L, for a regenerated cellulose film of 60 gm -2.
`
`Infigure 7, ‘Handle-o-Meter‘ stillness SH is plotted against
`Specimen length L. As the specimen length is increased
`SH increases until the specimen is long enough to bend over
`and touch the plate, after which the value of SH becomes
`constant. The rate of change of Sn with L increases as the
`weight per unit area (‘unit weight’) of the film increases
`(figure 8).
`Thus when long specimens are used the ‘Handle-o-Meter’
`reading depends not only on the flexural rigidity of the film
`but also on an extra term which is related to the unit weight
`of the film. The leverage effect may be approximately
`°°mDenSated by using a short specimen which overlaps the
`slot on each side by an amount which will just balance the
`weight of the film within the slot, taking the slot edge as
`the fulcnim. The gravitational forces within the slot are
`then balanced out by those acting outside the slot.
`594
`
`PAGE 000004
`
`

`

`Flexural rigidity of polymeric films
`
`It is therefore
`1/3.02, with a correlation coeflicient of.0-992.
`safe to assert that ‘Handle-o-Meter’ stiffness when measured
`in thjs way does have the expected power-law dependence on
`hickness
`film‘
`S = k q 15.
`A comparable set of measurements, using long (20 cm)
`film specimens gave results somewhat similar to those of
`Hansen et al. (1963) except that the exponent appeared to be
`about 2-3, compared with their 2-5. The appearance of the
`unexpected power-law in the results of the previous workers
`therefore is due to the failure to take into account the need
`to correct for the effects of gravity.
`
`3.2. Relationship of ‘Handle-a-Meter’ stiflness and flexural
`rigidity theory
`Although values from the ‘Handle-o-Meter’ are useful for
`comparative purposes, a more fundamental value for flexural
`rigidity is desirable. The standard engineering formulae for
`bending of beams are of no use in this application, since
`they are all restricted to small-angle approximations. An
`approximate solution can be obtained by assuming that the
`profile the film adopts is that of a circular arc, whose curvature
`isthe mean of that at the centre of the slot (where the bending
`moment has a maximum value) and that at the edge of the
`slot (which is zero). This approach gives a result which
`agrees roughly with the observed data but takes no account
`of frictional elfects.
`It was therefore decided to attempt to obtain an exact
`solution by computing the film path, using a method of
`successive approximation. The procedure adopted was as
`follows.
`
`
`
`Figure 10. Segmented film profile assumed for computer
`calculation of approximate true film profile.
`
`the
`Considering one half of the film profile (figure 10),
`its
`V§1l1e‘of the normal reaction force P at the slot edge,
`‘fi”°°i1°{1 (given by 9) and the depression h at the slot centre,
`°F maximum normal force S are calculated from the circular
`glc aPP1'0XlIl‘latlOn, in temis of the slot width W and the
`gm‘?-1 rigidity R of the film. Now assuming a value for
`f8 coefiicient of friction we have to introduce an additional
`Ewe fl-If acting at the slot edge, in the direction shown, if
`i 9
`is advancing into the slot.
`theuslflg 0, the apex of the film, as the origin of coordinates
`t
`actual values of P and 6 needed to cause the film path
`° Pass through the point G‘W, h) have to be found. The
`E‘ Path is divided into 72 segments along the x axis. Taking
`as‘ V3-1l_1<‘-S 0f_P and 9 from the circular arc approximation
`_ Starting points, the curvature in the film at the base of the
`moniemient at point
`(x,-, y,-) can be calculated from the
`mg fients of the forces 1’ and p.P and the flexural rigidity of
`11'“ R. Assuming the curvature to be constant along
`
`the ith segment, the slope cl,-+1 of the (i + 1)th segment can
`be calculated from
`
`<11,“ = 49; + {i-W/n x (curvature of ith segment)}.
`
`The coordinates of point x,+,, y,+, can then be found
`from x,, y,, ¢, and 5-W/n. Thus by working outward from
`the origin, the point on the line x = i- W which the film
`would intercept with the assumed values of P and 9 can be
`found. This intercept (/i’) will not, in general, be equal to h,
`so that at present the assumed forces do not lie in the plane
`of the film.
`The next step is to alter the assumed value of P to Ph/h’
`and to re-calculate the film path. At the end of this stage
`the assumed direction of P is altered to be perpendicular to
`the newly calculated film path, and the path re-calculated
`again. Thus by successive setting and resetting of P and 9
`we obtain the values needed to cause the segmented film
`path to pass through the point («}W, h) with any desired
`degree of accuracy.
`Then by progressively increasing the number of segments
`into which the film path is divided, a closer and closer
`approach is made to the true film path. The number of
`segments needed can be limited to the number beyond which
`any increase does not cause a significant change in the value
`of S, to within the desired degree of accuracy. S is of course
`given by vertical resolution of the forces P and p.P.
`The procedure so far provides a means of calculating S
`as a function of h.
`It remains to find the value of h for
`which S is a maximum. This is achieved by increasing or
`decreasing h, using a bracketing technique, until the approxi-
`mate value of h at which S(h) is a maximum is obtained.
`Then by progressively reducing the intervals by which h
`is changed, the value of SW can be found to any desired
`degree of accuracy.
`the
`the calculation,
`throughout
`By setting ii. = — 11.
`direction of the frictional force can be reversed, and the
`result obtained for the situation in which the film is receding
`from the slot.
`The procedure described was programmed in ‘Auto-Code’
`for an ICT 1300 computer. The computation proved to
`be extremely lengthy and it was possible to obtain results
`for only a limited range of variables with an accuracy to
`:§%. These results are given in the table.
`It is interesting
`
`Friction
`coeff./i
`0
`0- I
`0-1
`0-3
`0-3
`
`S3/Ra
`(cm '2)
`26-50
`28-38
`—
`32-15
`——
`
`S,/R,
`(cm -2)
`26-50
`—
`24-90
`—
`21-69
`
`hm,
`(cm)
`0-118
`0-124
`0-114
`0-135
`0-104
`
`Circular arc approximation gives S/R = 32-00;
`hm, = 0- 104; slot width = 0-5 cm.
`
`to note that the crude circular arc approximation gives very
`nearly the right value for S/R if the coefiicient of friction of
`the film on the slot edge is 0-3.
`It is clear that friction accounts for most of the difierence
`between S, and S,, such as that shown in figure 3. An
`attempt was made to check on the results of the computations
`by studying films of similar tensile moduli and thicknesses
`but of differing surface properties, taking into account the
`stress relaxation which takes place during the period that the
`film is bent inside the slot. This confirmed, approximately,
`the figures given in the table, but it was not possible to make
`precise comparisons, owing to the extreme difliculty of
`595
`
`PAGE 000005
`
`PAGE 000005
`
`

`

`obtaining reliable results for the coefiicient of meta1—film
`friction at the very light loads involved here.
`
`3.4. Comparison of calculated ‘Handle-o-Meter’ szif}‘ne,_, with
`observed results
`
`D. H. Morton and A. Marks
`
`3.3. Relation of nominal stress at 1% extension to Young’s
`modulus
`The effective mean linear strain involved in bending thin
`polymeric films to a diameter of 0-5 cm is of the order of
`0-1%, and generally does not exceed 0-5 "/0.
`It is diflicult
`experimentally to obtain values for the Young’s modulus
`of such films in the region of small strains. Any slight
`irregularity in alignment of the testing machine, or wrinkling
`of the sample itself results in a ‘toe’ on the load-elongation
`curve which completely obscures the region of interest.
`However, for relatively rigid polymers such as regenerated
`cellulose it
`is possible to obtain reasonably satisfactory
`curves of load as a function of extension up to 1%.
`This has been done for a regenerated cellulose film con-
`taining a proportion of glycerol as a plasticizer. Five
`specimens were measured and the load read off at intervals
`of 0-1% extension. The mean nominal stress on at each
`extension was then calculated, and when this stress is divided
`by the extension E a value for the apparent Young’s modulus
`E, = «rule is obtained. When E. is plotted against applied
`extension e, a smooth curve results (figure 11) showing that
`
`63
`
`6-0
`
`4-5
`
`4-0
`0
`0'2
`0'4.
`0'6
`0'8
`1'0
`6 (°/0)
`
`11._ Apparent Young’s modulus E.= on/0-Ole,
`Figure
`as a function of strain e at small strains, for a glycerol
`plasticized regenerated cellulose film.
`
`It
`apparent modulus decreases with increasing strain.
`should be noted, however, that the line relating E, and e is
`not straight, as has been suggested by Wellisch et al. (1961).
`For this material, therefore, it can be seen that the relation
`between apparent Young's modulus E, at very small strains,
`and the stress q at 1% extension is approximately
`E, = 141 q.
`
`It has not been possible to obtain a reliable estimate of the
`relation for the other polymers studied in this work. Cer-
`tainly the factor is greater than 100 for all the films for which
`results are included in figure 9, but it appears in general to
`be somewhat lower than the value obtained above for regener-
`ated cellulose. The best estimate which can be made is
`'.hat for thin thermoplastic polymeric films the factor lies
`between 115 and 140.
`
`596
`
`If an average factor relating stress at 1% extension
`to apparent Young’s modulus E, at infinitesimal stresses ‘(I
`assumed as 130, from the preceding section, a valueol?
`flexural rigidity can be calculated using classical mecham
`assuming the film to be homogeneous and isotropic
`R3 = Eat3/12 2 10-8 qt3
`where t is the film thickness.
`
`’
`(1)
`
`A representative value for the coeficient of friction of
`film on metal is about 0-3 for the polymer types included
`in the results of figure 9. From the table therefore, we have
`S, 2 32R, for a0«5crnslot
`
`or S,: SR, foralcmslot
`S, 2 86 qt3
`
`(2)
`
`(3)
`
`or log (S,/q) = 2-93 + 3 log t
`where S, is in g wt per cm, q in dyn cm‘2 and tin cm.
`This is the ‘theoretical’ line drawn in figure 9.
`It will be
`seen that a fair fit is obtained to the experimental points,
`although because of the form of the plot the intercepts on the
`axes are insensitive to the coefficient of equation (2).
`Residual deviations may be attributed to a combination of
`errors in the assumed values for coeflicient of friction, in
`the factor relating E, and q for each film, and to the departures
`of the films from the assumed homogeneous, isotropic state.
`
`4. Conclusion
`There are several factors which have to be allowed for in
`measuring flexural
`rigidity of polymeric fihns using the
`principle of the ‘Handle-o-Meter’. The most important of these
`is the effect of the weight of film overlapping the slot edges.
`When allowance is made for this gravitational pull the expected
`dependence of film stiffness on the cube of thickness is found.
`By making estimates of the ‘initial’ Young’s modulus from
`stress at 1% extension and allowing for the effects of friction
`on the edge of the slot it has been possible to relate the
`observed value of stiffness approximately to the tensile
`properties of a variety of polymeric films.
`It may be deduted
`by inference that
`the ‘Handle-o-Meter’
`stiffness gives a
`direct and reasonably accurate measurement of flexural
`ridigity of such films at large curvatures, provided that allow-
`ance is made for the effects of friction. The magnitude of
`the frictional effect is, however, such that small ditferences
`of flexural rigidity may be over-ridden by differences ‘in
`coefficient of friction which are difficult to determine with
`adequate accuracy.
`
`Acknowledgments
`Smith»
`Thanks are due to the Research Manager, Dr. J.
`and to the Directors of British Cellophane Limited for
`permission to publish this paper.
`References
`
`BROWNLEE, R. N., 1955, Pulp Pap., 29, 130.
`CHAMPION, F. C., and DAVEY, N., 1948, Properties of Mam’
`(Glasgow: Blackie), p. 53.
`Gurmun, J. C., MORTON, D. H., and OLIVER, P. H., 1954'
`J. Text. Inst, 45, T912-29.
`Hmsm, o. c., MARKER, L., Nmmnm, K. w., and Swim
`mo, 0. J., 1963, J. Appl. Polym. Sci., 7, 817-32.
`HSU, B., 1964, J. Sci. Instrum., 41, 153-6.
`WELLISCH, E., MARKER, L., and SWEETING, O.
`J. App. Polym. Sci., 5, 647-54.
`
`I
`J-. 196'
`
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