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The tearing of aluminium foils
Journalof the Mechanlw and Physiu of Solids, 1a110,Vol. 4, pp. 17% to 181.
THE TEARING By
Prem Ltd., London.
Pemon
OF ALUMINIUM J.
FOILS
HENNEPHCW
Laboratorium voor Technische Physics, Dell%
(Z&c&d
5th Dccmobtr;
1955)
SUMMARY
To investigate the tearing of foils, thin hard aluminium foils (thickness 0*0188cm) were provided with two parallel cuts. The part of the foil between these two cuts ~84 torn out while the rest was clamped against a flat surface. The base angle of the triangular part that ia tom out was measured, and also the tearing-force. The angle under which the tearing take-a plaee was variable and appeared to be of great importance. The energy for tearing a unit length out of the foil is the essential quantity (and not the tearing force). This energy is made up of two parts : the energy for bending the foil and the energy
for the plastic deformation of the borders of the tear. The first quantity was determined aepa~~tely; it was also calculated in good agreement with the measurements ; it proved to be the greater part of the total tearing energy. The energy necessary for forming the fresh surfaces is thought to be negligible. 1.
IN the theories
on the fracture
cracks exist in the material. through
the material.
propagation
for II brittle
INTRODUCTION
of materials
At a certain
it is generally
GRIFFITIXand others have calculated material.
For a ductile
Fig.
material
that micro-
the stress for crack
the fracture
mechanism
1.
is more complicated, as plastic. llow ran ocacur in the We thought it interesting to study experimentally propagation of a crack in a cluc*tile material. Our wcrc with hard alumilriunl foils. The c4reumstanees 172
supposed
stress these cracks open and propagate
neighhourhoocl of on a macroscopic cxperinients were chosen in suc4i a
the crack. scale the executed way that
The
tearing
of aluminium
Fig. 2.
Fig. 3.
Fig. 4.
foils
173
174
J. J~ENNEPEOP
the essehtial parameters of the tearing mechanism could be measured in a reproducible way. 2.
THE
TEARING
OF THE
Forrs
(i) Method. To get reproducible tearing, foils are provided with two parallel cuts. A lip is set free which can be torn out of the foil (Fig. 1). For this purpose
Lou
3sol
r
WI .
I&!-
306
2so
2oa
130
60
70
80
90
loo
110
120
130
a;’ Fig. 5. The tearing
force FT (gwt) us u function of y, the angle uuder of the foil.
which the lip is torn out
the foil is forced against a flat wooden surface with the help of two mebat strips Irarallel to the above-mentioned cuts. Fig. 4 shows the mounting of the foil in more detail. The distance between the metal strips and the parallel cuts is of secondary importance, and was of the order of magnitude of 0.05 rm. Fig. 2 out-
The
tearing
foils
of aluminium
175
lines the situation seen frbm the side. The force F,. necessary to tear the lip out of the material is obtained by the use of weights. The angle y, under which the lip is torn out of the material, can be varied by varying the elevation angle a of the foil. Fig. 3 reproduces a (fiattened) torn out lip, the torn out piece of metal being always triangular.
x x
x
. ISU
r
P
X
x
J
x X
X
r
x
0
X X
bL ml
xx
X
Fig. 6. The tearing force FT (gwt) (w :I function of the distance d (cm.) Mween the psrallel cuts. (ii) Heszll~~. Fig. 5 shows that the force k’, necessary to tear-the lip out of the material depedds largely on the angle y under which the lip is torn out. The distance d between the two parallel cuts was held constant (0.92 cm). As the experiment approaches the tensile test (y = 0), I’, rises fast. For y > 130“, F, increases again. Fig. 6 shows the dependence of F, on the distance d between the two parallel cuts. Here y was constant (103’). The smallest distance d was 0.07 cm. The base angle E of the torn out triangle (Fig. 3) was also determined and appeared to depend on the angle y. As the angle y changed from 60“ to 130”, E increased from 70’ to 80°. It proved, however, that the base angle c.was independent of the distance d between the parallel cuts (d > 0-07 cm.). Thus Fig. 6 demonstrates the decrease of the force F, during a single test. As a matter of fact the used quantity F, is the maximum load which occurs when
176
J. HENNEPHOF
a lip is torn out. AS we start loading, small but already visible cracks appear at both sides of the lip. The situation however rests in stable equilibrium until the load F, is reached. Then the lip is torn out all at once. The 0.0188
material cm.
used was fine-grained
Coarse-grained
of reproducibility.
There
the torn
rents were no longer straight each time they crossed their direction
hard
soft aluminium
aluminium
out pieces of metal
lines but broken
a crystal
border.
foil with a thickness
of the same thickness
had fanciful
lines which altered
Howe\.er,
sometimes
of
showed a lack forms.
The
their direction
they also changed
in the middle of a crystal.
3.
THE
BENDIM:
EFFECT
In our case the energy necessary to tear the lip out of the foil is used partly to bend the lip to a certain radius of curvature and straighten it again. We will now
describe
the
determination
of this
bending
effect
both
by calculation
and
by
experiment.
Fig. 7.
Fig. 8.
(i) Calculation. When the deformation of a foil is known, the deformation of a single layer can be determined (e.g. the hatched layer in Fig. 7). The forces necessary to elongate this layer can be calculated from the stress-strain curve of the material. So the deformation energy of the single layers can be calculated. And the energy necessary to deform the whole foil is found by simple addition. The determined stress-strain curve of the material could easily be schematized to the one represented iti Fig. 8 with an ultimate stress u,, = 11.9 kg/mm* and a Young’s modulus E = 5500 kg/mm 2. The simplification will introduce an error to the extent of a few per cent. In describing the mainly plastic behaviour of the single layers, we .assumed that the neutral zone would remain in the middle, of the foil. This limits the
The tearing of ehrminium foils
177
exactness of the calculation only to about 1% in our case (WOLTER 1952). Now if the foil is bent in such a way that the neutral zone has a radius of curvature of r, the strain in a layer at a distance y of the neutral zone is given by c = y/r. We can determine the energy used to get this result with the help of the surfaces SE + Is, = +uo co + u. [C (y) - ~a] (Fig. 9). When the foil is straightened again
il
Fig. D. our layer is pressed back to its original length. This needs an energy represented by - S, $ SE -I- S,, = S,, = S, - 2S,. So the energy kV (y) per unit of volume by necessary to elongate our layer and push it back is represented W(y)==2S,--SE or U’(Y) = 2% [e (Y) -
fol - !POq)
o. represents the ultimate stress of the material, E (y) the strain of a layer at a distance y from the neutral zone co the elastic strain ; here so/E by definition. Hence the total energy IV, per unit length of the foil necessary to bend the foil till the neutral zone has a radius of curvature of r and straighten it again is where :
where :
b represents the width of the foil, t the thickness of the foil, y. (= co T) the distance from the neutral zone of the layer with a strain co, E Young’s modulus.
The layers with a distance y to the neutral zone given by y. < Iy 1 < 2 go3in which Y. = co r, are not taken into account, as their total contribution here is less than 0.2%. The layers with a distance IyI < Y. do not contribute at all to the energy WI39 as their deformation is purely elastic.
J. HENNEPAOF
178
It may be remarked that with E = 00 our expression change3 into W, = b . g
:
a first approximation
of the problem.
Some data are given for a foil 2.00 cm wide, 0.0188cm thick. Data with E = 7400 kg/nuns, the Young’s modulus for aluminium, are also listed. As the material was not 100% hard, these data give certainly too high values for JVB. It is remembered that E = 5500 has been introduced by scshematizing the stressstrain curve.
r cm. O*loo
0*1a4 O*lSO o*m
F, Fig. lo. (ii) Experiment. The energy per unit of length of the foil necessary to bend the foil and straighten it again is determined by forcing a foil to roll over a (revolving) axis. Fig. 10 outlines the situation seen from the side. To realize this situation two forces are needed : N to keep the foil in its place and F, to draw it over the axis. At 1 (Fig. 10) the foil is bent, at 2 straightened again. When the foil is pressed against the axis the radius of curvature is known to be half the sum of the diameter of the axis and the thickness of the foil. And, as F, is the only force that supplies energy, the numerical value of F, must be equal to that of W,, the energy necessary to bend the unit of length of our foil to the above-mentioned radius of curvature and straighten it again. So
The tearing of aluminium fo& The (e
experiment
can
4 . L cos 8, which
which
is supported
axle mentioned
bc
carried
is always
out
about
as designed
in Fig.
600 gm) is obtained
by two axles revolving
before (Fig.
179 11.
in ball-bearings.
force
N
load L
The top axle is the
The load L is held in equilibrium
10).
The
by a variable
by a tare-weight
T = I, sin 6. In principle there is no force acting on the foil in a direction tlirular
to hr.
A correction
has to be made for the friction
perpenof the ball-bearings.
The force I’, is obtained by weights attac,hed to one end of the foil. In measuring is observed which can be suppressed by bending the foil F H, a starting-effect round
the axle
<*banged.
at the commencement
In principle
F,
of the experiment.
is independent
The angle
6 can
be
of w (W = 90” + S) and thus of 6. pulley
A----_.--__ Fig. 11. (iii) Results. and a thickness curvature
Measurements of 0.0188
were carried
cm.
The results
r acts as a parameter.
are given.
the
lowest
small crosses are mean values. the highest
with
The vertical
The discrepancy
Thus, for our purpose,
We return
and
the
a foil can be given
to our starting-point an arc of a circle
now
: the tearing
highest
of 6.
The
these crosses gi\-e
for 6 = 15” may be due
calculated
a fair approximation
of
values of IV*
in this case the foil does not follow the asle surface.
to bend and straighten
not describe
of 2.00 cm
12. The radius
values are independent
lines drawn through
negative angles S such an effect was remarked. On the whole a reasonable agreement between is obtained.
in Fig.
E = 5500 kg/mm2,
These calculated
and lowest measurements.
to the fact. that
are plotted
For each value of r two calculated
one calculated
one with B = 7400 kg,/mm2.
out on foils with a breadth
Indeed for
and measured
of the energy
values
necessary
by calculation.
of lips out of foils.
Such lips do
being torn out but this is unimportant,
for if a
180
J. HENNEPHW
f
X
250
rm
50
Fig. 1%. The energy
W,
neeeasmy
to bend and straighten a unit length angle 6 (Fig. 11).
of the
of the foil w a function
181
The tearing of aluminiurn foil
minimum radius of curvature does exist this radius determines the bending energy. When this radius is known we can calculate the energy necessary to bend and straighten the lip and compare it with the energy necessary to tear the lip out of the foil. To determine the minimum radius of curvature r, the load F, is taken away as soon as tearing takes place. This, however, causes the lip to spring back elastically somewhat, with the effect that the radius of curvature increa.ses by a few per cent. For this effect a correction has to be made. The radii of curvature were measured with the aid of a set of bars with decreasing diameters. The energy IVB necessary to bend and straighten the unit length of a lip can now be calculated with the aid of the formula : IV,=b
-%)(l -*y)] [(O??._d f
r
where E -
5500 kg/mma, a,, = II.9 kg/nuns, t = 0.0188
cm, and b = d c
0.90 cm.
A t.rin a
Fig. 18.
The energy IV, necessary to tear out a lip, per unit length of that lip, can be calculated with the help of the elevation angle Q of the foil and the tearing force F,.. The work the force F, performs when a length Al of the foil is torn out amounts so IV, = F, (1 + sin a) per to F,. . Al . (1 + sin a) as Fig. 13 illustrates. unit of length of the lip. Measurements were executed on foils with a thickness t = 0.0188 cm and a distance between the parallel cuts d = O-94 cm. It appeared that the radius of curvature decreased with increasing angle y under which the lip was torn out of the material. The lower curve of Fig. 14 gi\res the energy IV, per unit length of the lip necessary to bend and straighten this lip while being torn out. The upper-curve in Fig. 14 represents the tearing energy IV,. per unit lip length as a function of y. By comparing this curve with that of Fig. ci the influence of the factor (1 + sin a) on W, can be noticed. Low \.alues for the tearing force F, correspond here with high values for the tearing energy IJ’,, and vice versa. It is obvious that the iucrease of the tearing energy II’,. with increasing angle y is (*aused by the increase in energy necessary to bend and straighten the lip. C’oncerning the difference M’,, - v’, between the two curves little can be said. ‘l(llere is. however, no evidence to shop that in first approximation this difference
1x2
J. HENNILPUOF
lOOO-
O-
M
70
80
100
90
110
1M
130
b;” Fig. 14.
The tearing
function
energy WT and the bending energy W B, per unit length of the lip, as a of the angle y under which the lip is torn out of the foil.
The tearing
of aluruiniuu~
fdr
183
should not be constant. In our range of measurements the base angle E of the torn out triangle (Fig. 8) changes only from 70” to 80“ with increasing angle y. So here the length of the rents, per unit torn-out length of the lip, can be regarded as being nearly constant (2.19 cm/cm lip) compared to the error of some 10% in the difference IV, - W,. This very low accuracy of IV, - WB is chiefly caused by the error (10%) in the determination of the radii of curvature of the lip. As the energy necessary to enlarge the edge surfaces of the newly formed rents is negligible compared to the energy W, - W,, (some 800 gm cm/cm lip), this latter energy may be regarded as the deformation energy per unit torn-out lip length along the newly formed rents. So we may conclude with the following remarks. (1) The tearing force, and consequently the tearing energy, depends largely on the angle y under which the lip is torn out. (2) In our case the tearing force is less essential than the tearing energy. (9) The tearing energy consists of two parts : one part for bending the foil and straightening it again, and the other (here almost constant) part for deforming the material at the borders of the tears. The latter is the smallest part. These conclusions are only valid in our circumstances. ACKNOWLEDGMENT The author is grateful to Prof. M. J. DRUYVESTEYNfor his advice and encouragement throughout the work. REFERENCE WOLTEE, K. H.
1952
V. D.I. Fonchun@eft,
485.
THE TEARING By
Prem Ltd., London.
Pemon
OF ALUMINIUM J.
FOILS
HENNEPHCW
Laboratorium voor Technische Physics, Dell%
(Z&c&d
5th Dccmobtr;
1955)
SUMMARY
To investigate the tearing of foils, thin hard aluminium foils (thickness 0*0188cm) were provided with two parallel cuts. The part of the foil between these two cuts ~84 torn out while the rest was clamped against a flat surface. The base angle of the triangular part that ia tom out was measured, and also the tearing-force. The angle under which the tearing take-a plaee was variable and appeared to be of great importance. The energy for tearing a unit length out of the foil is the essential quantity (and not the tearing force). This energy is made up of two parts : the energy for bending the foil and the energy
for the plastic deformation of the borders of the tear. The first quantity was determined aepa~~tely; it was also calculated in good agreement with the measurements ; it proved to be the greater part of the total tearing energy. The energy necessary for forming the fresh surfaces is thought to be negligible. 1.
IN the theories
on the fracture
cracks exist in the material. through
the material.
propagation
for II brittle
INTRODUCTION
of materials
At a certain
it is generally
GRIFFITIXand others have calculated material.
For a ductile
Fig.
material
that micro-
the stress for crack
the fracture
mechanism
1.
is more complicated, as plastic. llow ran ocacur in the We thought it interesting to study experimentally propagation of a crack in a cluc*tile material. Our wcrc with hard alumilriunl foils. The c4reumstanees 172
supposed
stress these cracks open and propagate
neighhourhoocl of on a macroscopic cxperinients were chosen in suc4i a
the crack. scale the executed way that
The
tearing
of aluminium
Fig. 2.
Fig. 3.
Fig. 4.
foils
173
174
J. J~ENNEPEOP
the essehtial parameters of the tearing mechanism could be measured in a reproducible way. 2.
THE
TEARING
OF THE
Forrs
(i) Method. To get reproducible tearing, foils are provided with two parallel cuts. A lip is set free which can be torn out of the foil (Fig. 1). For this purpose
Lou
3sol
r
WI .
I&!-
306
2so
2oa
130
60
70
80
90
loo
110
120
130
a;’ Fig. 5. The tearing
force FT (gwt) us u function of y, the angle uuder of the foil.
which the lip is torn out
the foil is forced against a flat wooden surface with the help of two mebat strips Irarallel to the above-mentioned cuts. Fig. 4 shows the mounting of the foil in more detail. The distance between the metal strips and the parallel cuts is of secondary importance, and was of the order of magnitude of 0.05 rm. Fig. 2 out-
The
tearing
foils
of aluminium
175
lines the situation seen frbm the side. The force F,. necessary to tear the lip out of the material is obtained by the use of weights. The angle y, under which the lip is torn out of the material, can be varied by varying the elevation angle a of the foil. Fig. 3 reproduces a (fiattened) torn out lip, the torn out piece of metal being always triangular.
x x
x
. ISU
r
P
X
x
J
x X
X
r
x
0
X X
bL ml
xx
X
Fig. 6. The tearing force FT (gwt) (w :I function of the distance d (cm.) Mween the psrallel cuts. (ii) Heszll~~. Fig. 5 shows that the force k’, necessary to tear-the lip out of the material depedds largely on the angle y under which the lip is torn out. The distance d between the two parallel cuts was held constant (0.92 cm). As the experiment approaches the tensile test (y = 0), I’, rises fast. For y > 130“, F, increases again. Fig. 6 shows the dependence of F, on the distance d between the two parallel cuts. Here y was constant (103’). The smallest distance d was 0.07 cm. The base angle E of the torn out triangle (Fig. 3) was also determined and appeared to depend on the angle y. As the angle y changed from 60“ to 130”, E increased from 70’ to 80°. It proved, however, that the base angle c.was independent of the distance d between the parallel cuts (d > 0-07 cm.). Thus Fig. 6 demonstrates the decrease of the force F, during a single test. As a matter of fact the used quantity F, is the maximum load which occurs when
176
J. HENNEPHOF
a lip is torn out. AS we start loading, small but already visible cracks appear at both sides of the lip. The situation however rests in stable equilibrium until the load F, is reached. Then the lip is torn out all at once. The 0.0188
material cm.
used was fine-grained
Coarse-grained
of reproducibility.
There
the torn
rents were no longer straight each time they crossed their direction
hard
soft aluminium
aluminium
out pieces of metal
lines but broken
a crystal
border.
foil with a thickness
of the same thickness
had fanciful
lines which altered
Howe\.er,
sometimes
of
showed a lack forms.
The
their direction
they also changed
in the middle of a crystal.
3.
THE
BENDIM:
EFFECT
In our case the energy necessary to tear the lip out of the foil is used partly to bend the lip to a certain radius of curvature and straighten it again. We will now
describe
the
determination
of this
bending
effect
both
by calculation
and
by
experiment.
Fig. 7.
Fig. 8.
(i) Calculation. When the deformation of a foil is known, the deformation of a single layer can be determined (e.g. the hatched layer in Fig. 7). The forces necessary to elongate this layer can be calculated from the stress-strain curve of the material. So the deformation energy of the single layers can be calculated. And the energy necessary to deform the whole foil is found by simple addition. The determined stress-strain curve of the material could easily be schematized to the one represented iti Fig. 8 with an ultimate stress u,, = 11.9 kg/mm* and a Young’s modulus E = 5500 kg/mm 2. The simplification will introduce an error to the extent of a few per cent. In describing the mainly plastic behaviour of the single layers, we .assumed that the neutral zone would remain in the middle, of the foil. This limits the
The tearing of ehrminium foils
177
exactness of the calculation only to about 1% in our case (WOLTER 1952). Now if the foil is bent in such a way that the neutral zone has a radius of curvature of r, the strain in a layer at a distance y of the neutral zone is given by c = y/r. We can determine the energy used to get this result with the help of the surfaces SE + Is, = +uo co + u. [C (y) - ~a] (Fig. 9). When the foil is straightened again
il
Fig. D. our layer is pressed back to its original length. This needs an energy represented by - S, $ SE -I- S,, = S,, = S, - 2S,. So the energy kV (y) per unit of volume by necessary to elongate our layer and push it back is represented W(y)==2S,--SE or U’(Y) = 2% [e (Y) -
fol - !POq)
o. represents the ultimate stress of the material, E (y) the strain of a layer at a distance y from the neutral zone co the elastic strain ; here so/E by definition. Hence the total energy IV, per unit length of the foil necessary to bend the foil till the neutral zone has a radius of curvature of r and straighten it again is where :
where :
b represents the width of the foil, t the thickness of the foil, y. (= co T) the distance from the neutral zone of the layer with a strain co, E Young’s modulus.
The layers with a distance y to the neutral zone given by y. < Iy 1 < 2 go3in which Y. = co r, are not taken into account, as their total contribution here is less than 0.2%. The layers with a distance IyI < Y. do not contribute at all to the energy WI39 as their deformation is purely elastic.
J. HENNEPAOF
178
It may be remarked that with E = 00 our expression change3 into W, = b . g
:
a first approximation
of the problem.
Some data are given for a foil 2.00 cm wide, 0.0188cm thick. Data with E = 7400 kg/nuns, the Young’s modulus for aluminium, are also listed. As the material was not 100% hard, these data give certainly too high values for JVB. It is remembered that E = 5500 has been introduced by scshematizing the stressstrain curve.
r cm. O*loo
0*1a4 O*lSO o*m
F, Fig. lo. (ii) Experiment. The energy per unit of length of the foil necessary to bend the foil and straighten it again is determined by forcing a foil to roll over a (revolving) axis. Fig. 10 outlines the situation seen from the side. To realize this situation two forces are needed : N to keep the foil in its place and F, to draw it over the axis. At 1 (Fig. 10) the foil is bent, at 2 straightened again. When the foil is pressed against the axis the radius of curvature is known to be half the sum of the diameter of the axis and the thickness of the foil. And, as F, is the only force that supplies energy, the numerical value of F, must be equal to that of W,, the energy necessary to bend the unit of length of our foil to the above-mentioned radius of curvature and straighten it again. So
The tearing of aluminium fo& The (e
experiment
can
4 . L cos 8, which
which
is supported
axle mentioned
bc
carried
is always
out
about
as designed
in Fig.
600 gm) is obtained
by two axles revolving
before (Fig.
179 11.
in ball-bearings.
force
N
load L
The top axle is the
The load L is held in equilibrium
10).
The
by a variable
by a tare-weight
T = I, sin 6. In principle there is no force acting on the foil in a direction tlirular
to hr.
A correction
has to be made for the friction
perpenof the ball-bearings.
The force I’, is obtained by weights attac,hed to one end of the foil. In measuring is observed which can be suppressed by bending the foil F H, a starting-effect round
the axle
<*banged.
at the commencement
In principle
F,
of the experiment.
is independent
The angle
6 can
be
of w (W = 90” + S) and thus of 6. pulley
A----_.--__ Fig. 11. (iii) Results. and a thickness curvature
Measurements of 0.0188
were carried
cm.
The results
r acts as a parameter.
are given.
the
lowest
small crosses are mean values. the highest
with
The vertical
The discrepancy
Thus, for our purpose,
We return
and
the
a foil can be given
to our starting-point an arc of a circle
now
: the tearing
highest
of 6.
The
these crosses gi\-e
for 6 = 15” may be due
calculated
a fair approximation
of
values of IV*
in this case the foil does not follow the asle surface.
to bend and straighten
not describe
of 2.00 cm
12. The radius
values are independent
lines drawn through
negative angles S such an effect was remarked. On the whole a reasonable agreement between is obtained.
in Fig.
E = 5500 kg/mm2,
These calculated
and lowest measurements.
to the fact. that
are plotted
For each value of r two calculated
one calculated
one with B = 7400 kg,/mm2.
out on foils with a breadth
Indeed for
and measured
of the energy
values
necessary
by calculation.
of lips out of foils.
Such lips do
being torn out but this is unimportant,
for if a
180
J. HENNEPHW
f
X
250
rm
50
Fig. 1%. The energy
W,
neeeasmy
to bend and straighten a unit length angle 6 (Fig. 11).
of the
of the foil w a function
181
The tearing of aluminiurn foil
minimum radius of curvature does exist this radius determines the bending energy. When this radius is known we can calculate the energy necessary to bend and straighten the lip and compare it with the energy necessary to tear the lip out of the foil. To determine the minimum radius of curvature r, the load F, is taken away as soon as tearing takes place. This, however, causes the lip to spring back elastically somewhat, with the effect that the radius of curvature increa.ses by a few per cent. For this effect a correction has to be made. The radii of curvature were measured with the aid of a set of bars with decreasing diameters. The energy IVB necessary to bend and straighten the unit length of a lip can now be calculated with the aid of the formula : IV,=b
-%)(l -*y)] [(O??._d f
r
where E -
5500 kg/mma, a,, = II.9 kg/nuns, t = 0.0188
cm, and b = d c
0.90 cm.
A t.rin a
Fig. 18.
The energy IV, necessary to tear out a lip, per unit length of that lip, can be calculated with the help of the elevation angle Q of the foil and the tearing force F,.. The work the force F, performs when a length Al of the foil is torn out amounts so IV, = F, (1 + sin a) per to F,. . Al . (1 + sin a) as Fig. 13 illustrates. unit of length of the lip. Measurements were executed on foils with a thickness t = 0.0188 cm and a distance between the parallel cuts d = O-94 cm. It appeared that the radius of curvature decreased with increasing angle y under which the lip was torn out of the material. The lower curve of Fig. 14 gi\res the energy IV, per unit length of the lip necessary to bend and straighten this lip while being torn out. The upper-curve in Fig. 14 represents the tearing energy IV,. per unit lip length as a function of y. By comparing this curve with that of Fig. ci the influence of the factor (1 + sin a) on W, can be noticed. Low \.alues for the tearing force F, correspond here with high values for the tearing energy IJ’,, and vice versa. It is obvious that the iucrease of the tearing energy II’,. with increasing angle y is (*aused by the increase in energy necessary to bend and straighten the lip. C’oncerning the difference M’,, - v’, between the two curves little can be said. ‘l(llere is. however, no evidence to shop that in first approximation this difference
1x2
J. HENNILPUOF
lOOO-
O-
M
70
80
100
90
110
1M
130
b;” Fig. 14.
The tearing
function
energy WT and the bending energy W B, per unit length of the lip, as a of the angle y under which the lip is torn out of the foil.
The tearing
of aluruiniuu~
fdr
183
should not be constant. In our range of measurements the base angle E of the torn out triangle (Fig. 8) changes only from 70” to 80“ with increasing angle y. So here the length of the rents, per unit torn-out length of the lip, can be regarded as being nearly constant (2.19 cm/cm lip) compared to the error of some 10% in the difference IV, - W,. This very low accuracy of IV, - WB is chiefly caused by the error (10%) in the determination of the radii of curvature of the lip. As the energy necessary to enlarge the edge surfaces of the newly formed rents is negligible compared to the energy W, - W,, (some 800 gm cm/cm lip), this latter energy may be regarded as the deformation energy per unit torn-out lip length along the newly formed rents. So we may conclude with the following remarks. (1) The tearing force, and consequently the tearing energy, depends largely on the angle y under which the lip is torn out. (2) In our case the tearing force is less essential than the tearing energy. (9) The tearing energy consists of two parts : one part for bending the foil and straightening it again, and the other (here almost constant) part for deforming the material at the borders of the tears. The latter is the smallest part. These conclusions are only valid in our circumstances. ACKNOWLEDGMENT The author is grateful to Prof. M. J. DRUYVESTEYNfor his advice and encouragement throughout the work. REFERENCE WOLTEE, K. H.
1952
V. D.I. Fonchun@eft,
485.