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Experiments on inextensional collapse of circular paper disks including the influence of holes
ELSEVIER
Thin-WalledStructures Vol. 25, No. I, pp. 47 60. 1996 Copyright ~c) 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0263-8231/96 $15.00 0263-8231(95)00045-3
Experiments on Inextensional Collapse of Circular Paper Disks Including the Influence of Holes J. A m i r b a y a t & J. W. S. Hearle Department of Textiles, University of Manchester, Institute of Science& Technology, PO Box 88, Manchester M60 1QD, UK (Received 18 November 1994; accepted 6 July 1995)
ABSTRA CT Buckling behaviour of circular filter papers, as isotropic plates with high aspect ratio, negligible weight-st,]hess ratio and little ability to undergo inplane strains prior to delamination, has been studied. The study covers the results of deforming circular papers under three equally spaced radial loads and the effect of holes punched within the samples to help formation o[ single curvature modes without points of discontinuous slope. The results o[ a.fen, experiments on eccentricity of the holes are also presented.
INTRODUCTION A circular plate can be in equilibrium under an infinite variety of in-plane external loadings each causing a different continuous three-dimensional (3-D) shape, provided the material can withstand the bending and the membrane stresses developed. Figure 1 shows examples of external loads including the extremes of a sample under two concentrated loads along a chord (which may be a diameter) and of loads distributed unequally around the circumference. Except for some simple cases of the above-mentioned loadings such as a pair of loads along a diameter, the analytical solutions of load deformations relationships, including the application of energy methods are far from being simple. Some 2-D materials such as rubber sheets and textile fabrics buckle into smooth surfaces due to their ability to undergo high levels of in-plane 47
J. Arnirbayat, J. W. S. Hearle
48
p
Q
p
p
A
P
Fig. 1. Some possible types of in-plane loading of a disk.
strains, while the majority of materials, including paper and the sheet metals, develop sharp corners during 3-D deformations when deflections reach beyond the level for which the compressive membrane strains are negligible as shown in Fig. 2. The classical theory of plates and shells shows that the membrane strains become significant when the deflections equal the plate thickness ~ or exceed beyond a few percent of the diameter. 2 By analysing the order of magnitude of energies involved in complex deformations, the authors showed that 3"4 the ratio between the membrane strain energy, Urn, and bending energy, Ub, can be expressed as
Um
--~b Ub
y¢2
-a
(1)
where Y B t
= membrane modulus of the sheet based on the force per unit width = the bending stiffness = a characteristic length defining the double curvature zone (the radius in the case of a circular sample) = a geometrical measure of the externally imposed deformation.
If the material is a homogeneous continuum, the bending stiffness can be expressed in terms of the thickness, t, the Young's modulus, E = Y/t, and the Poisson's ratio,/4 and eqn (1) can be rewritten as
lnextensional collapse of circular paper disks
49
Fig. 2. Circular sheets under large deformations. (a) Textile fabric; (b) sheet rubber.
Um ~ (1
~2 - ~2)
(2)
This shows that for a given material, the square of the aspect ratio, t/t, determines the level of deformation for which the membrane strain energy becomes significant. The present paper is concerned with an experimental study of the way in which filter papers, which cannot accommodate appreciable in-plane strain and have high aspect ratio, respond to displacements which cause buckling.
50
J. Amirbayat, J. 14I. S. Hearle
Fig. 2. Contd. (c) writing paper.
EXPERIMENTAL PROCEDURE For three-point loading the samples were deformed by a rig made of a lathe chuck and three connections supporting loading pins, Fig. 3. The samples were standard filter papers o f l l 0 m m diameter and the holes were cut by hard steel punches.
Fig. 3. Simple rig to deform the filter papers.
Inextensional collapse of circular paper disks
51
In the first set of experiments, the displacements were imposed in such a way that there were no bending moments at the points of application of the forces, but only the main inward radial forces and small frictional forces which resist the sample sliding out of place. In later experiments, where the specimens were held within the boundary, the conditions at the loading points can be regarded as built-in, i.e. zero slopes at the loading points. It should be noted that the experiments reported are obtained by controlled radial displacements, and not by controlled or measured forces.
EXPERIMENTAL RESULTS
Parallel displacement Under sets of parallel displacement of the types shown in Fig. l(a) and (b) the samples deform in a single curvature surface, behaving as a thin column with variable width. There is no need of double curvature zones to satisfy the requirement of geometry. With no membrane stresses developed, the curvature can be increased until the material delaminates, with micro-buckling due to compressive stresses on the inside of the bend. This limit was not reached for the standard size filter papers, even when the sample was bent round so that opposite edges touched as shown in Fig. 4(a). This shows that the single curvatures are not large enough to produce bending stresses higher than the tolerable amount, and that it would be necessary to push the two surfaces of the fold into closer contact before delamination occurs.
Radial displacements For radial displacements under the loads shown in Fig. l(c) and (d), it is not geometrically possible for the entire surface to deform in single curvature modes, except (up to certain levels of displacements) by means of separate lips at each loading-point, with the inner part remaining at zero curvature. For evenly spaced and equal radial forces the initial deformation of the samples approaches this form. Figure 4(b) shows the continuous 3-D surfaces, with no sharp discontinuities but a very slight double curvature central zone, virtually no transverse curvature in the edges which bend over like a cantilever, and possibly very small regions of higher double curvature where the lips join, with membrane strains tending to zero as the size of the double curvature region tends to zero.
52
J. Amirbayat, J. W. S. Hearle
Fig. 4. Buckling under different types of load. (a) Under parallel loads; (b) initial deformation under radial loads; and (c) large deformation under radial loads.
As the radial displacement increases, the difference between the low curvatures of the middle portions, ABC, Fig. 1(c) (being a triangular plate under distributed moments and forces around external edges) and the high curvature of the rest of the sample (being cantilevers with variable width), becomes more apparent. Upon further displacement, the high stresses at the regions where the lips join cause the paper to collapse into sharp corners, as shown in Fig. 4(c). The sample under large deformations can be regarded as divided into four different regions as shown in Fig. 5: the central roof F (which would have a very small degree of double curvature); the outer segments O with slight single curvatures, the zones S between F and O with higher single curvature acting as a 'fillet,' region, and the sharp discontinuities at the lines L. With no need of any membrane strain, the failure is entirely due to the bending stresses at the singularity points where the slopes become discontinuous. If the number of points of externally imposed displacements is increased, the central zone F increases in number of sides and in size,
53
lnextensional collapse of circular paper disks
" - - Z'~_- ~griginaXl circumference Fig. 5. Different zones of a buckled sample.
f
) Fig. 6. Buckling under large number of external loads.
as shown schematically in Fig. 6. This leads to an increase in the number of sharp corners but also makes the limiting displacement PP' smaller. Consequently, the difference between the directions of single curvatures of the adjoining lips, shown by double headed arrows in
J. Amirbayat, J. W. S. Hearle
54
Fig. 6, decreases. When the number of points increases above a certain value, the intermediate mode of buckling at the edges will cease to occur; under these conditions, once the slight permissible double curvature is exceeded, there will be a buckling in a sharp discontinuity somewhere within the central zone. There is no simple geometric way in which deformation can occur in the limiting case, indicated in Fig. l(e), with a uniform inward displacement all round the circumference of the circle.
The influence of holes
If there is a hole in the centre of the sheet, an alternative form of deformation is possible. The paper can buckle circumferentially in folds of single curvature, since a connecting dome of double curvature is not present and the membrane strains can be relieved by other movements of the paper around the hole. If the hole is smaller than a critical size for a given paper diameter, a crease forms after a certain amount of radial movement, with the sample buckling in the mode found without a hole, namely radial buckling of lips at the edges. Figure 7(a) and (b) shows identical paper samples with different sizes of central openings. A sample of 110 mm diameter deforms in 3-fold buckling without any sharp corners with a central hole of 21 mm diameter, while one with a 16 mm diameter hole cannot change the buckling shape. The critical diameter is somewhere between 16 and 21 mm, namely between 0.15 and 0.19 of the total diameter.
Fig. 7.
Different buckling modes of a paper disk under three external loads.
Inextensional collapse of circular paper disks
55
Influence of nature of forces causing displacement The experiments so far described are for the simplest form of buckling at the edges. Other methods of holding the sample and different sequences of application of displacement lead to different patterns of buckling with point and line discontinuities at various locations. If instead of applying the displacements on the circumference of a sample, they are applied by pins within the sample, creases within the sample form after the slightest radial movement of the pins. The sample deforms in single curvature portions bent around radii and a central flat portion with sharp corners at the interface. This is because the zero slopes at the loading points prevent the formation of single curvature zones around the sectors such as AB, BC and so on, Fig. l(c). The double curvature required to join the regions bent around the radii and the flat centre, delaminates the paper similar to the case mentioned before. Figure 8 shows the sample under three radial loads which are applied at 5 mm from the outer edge. In this case also a central hole of a proper size makes it geometrically possible for the sample to deform without any double curvature and therefore no sharp corners from, Fig. 9. For a central opening of 13.5 mm diameter namely 0.12 of the sample diameter, the sample deforms without any crease, Fig. 9(a), while a 12 mm diameter hole cannot stop creasing at this level of radial displacement, Fig. 9(b). Curve A in Fig. 10 shows the effect of central hole size on the amount of deformation to form the crease. The test results show that for papers with central openings whose radii are larger than 12% of the sample radius. there is no limit of deformation to cause crease.
Fig. 8. Bucklingof paper disks when the loads are applied from within the samples.
56
J. Amirbayat, J. W. S. Hearle
Fig. 9. Buckling of paper disks with central holes when the loads are applied from within the samples. (a) Hole diameter 13.5 mm, no crease formation; and (b) hole diameter 12 mm, crease forms.
If the displacements are not symmetrical or one is applied before or after the other two (which is the case of deforming a plate with initial single curvature already generated by the other two displacements) the crease develops after the radial displacement exceeds a limit lower than the corresponding value for the symmetrical case, depending on the amount of the offset and the size of the central hole. Curve B in Fig. 10 belongs to an offset loading of 1 mm, i.e. one of the loading pins is set 1 m m closer than the other two to the centre of the sample.
Eccentric holes
Further experiments show that if the holes are not concentric with the sample, the crease forms at lower radial displacements. According to the limited test results, there is a definite relationship between the critical radial deformation, the opening size and the a m o u n t of offset for a given sample size. Figure 11 shows the effect of relative size of the hole and its relative offset from the centre on the external displacement for crease formation. Figure 12 (a)-(d) shows the effect of the size and the location of the opening. The hole diameter of the first three samples is 16 m m and their offsets are 35, 20, and 15 mm, respectively. The crease formation is stopped when the offset is reduced to 15 m m in Fig. 12(c). The corresponding diameter of the hole to stop the crease when punched centrally is 13.5 m m as previously mentioned and shown in Fig. 9(a). The offset of the hole in Fig. 12(d) is 20 m m but the larger diameter of the hole (40 mm)
Inextensional collapse of circular paper disks
57
30 - -
m
20 - -
<
L.
,7 j"
0
/
I
I
I
I
5
I0
15
20
Percentage of opening
ro Ro
x 100
Fig. 10. Effect of eccentricity of the central opening on the radial displacement to cause crease.
makes it possible to deform in combination of single curvature modes. Figure 12(e) shows a sample identical to Fig. 12(d) with one of the loading pins positioned 1 mm closer to the centre before the experiment. There is again no crease formation due to the large diameter of the eccentric hole. According to our observations, with the ratio ro/Ro approaching 0.3 any asymmetry of loading or offset of the hole can be tolerated without any limitation of the radial displacement, Fig. 12(f). When the opening is large enough, the radial movements of the elements become merely translatory and no membrane strain develops.
CONCLUSION The present paper is part of a series concerned with the complex buckling of sheet materials. The principal area of interest in the whole research programme is the way in which textile fabrics, which can easily
J. Amirbayat, J. W. S. Hearle
58
25I 0
2O ~
~o =10%
2
0
I
I
I
t
I
I
I
10
20
30
40
50
6(I
70
Eccentricity of the opening e
Ro
x 100
Fig. 11. Effect of eccentricity of the central opening on the radial displacement to cause crease.
accept large membrane strains due to their low resistance to certain forms of in-plane deformation, buckle into rounded folds of double curvature. In contrast to this, a material like paper which has a high resistance to in-plane deformation relative to its ease of bending will not accommodate double curvature, and this led us to the subsidiary investigation reported here of the forms of buckling which do occur in such a material. The experimental results show that the samples avoid formation of any substantial double curvature zones by the occurrence of point or line singularities, even if this requires delamination in order to release membrane energy. Furthermore, it has been shown that a hole of proper size and location within the samples changes the mode of buckling, and allows accommodation of the deformation by different forms with a lower energy for a given externally imposed deformation. It is worth noting that the softness of some nonwoven fabrics is improved by making apertured sheets containing holes.
Inextensional collapse of circular paper disks
59
O
e-, O
.=_ e~ G
~D
~d
60
J. Amirbayat, J. I4I. S. Hearle
REFERENCES 1. Den Hartog, J. P., Advanced Strength of Materials. McGraw-Hill, New York. 2. Timoshenko, S. & Woinosky-Krieger, S., Theory of Plates and Shells. McGraw-Hill, New York. 3. Amirbayat, J. & Hearle, J. W. S., The complex buckling of sheet material, part II. Int. J. Mech. Sci. 7,8 (1986) 359. 4. Amirbayat, J. & Hearle, J. W. S., The anatomy of buckling of textile fabrics. J. Text. Inst. 80 (1989) 51.
Thin-WalledStructures Vol. 25, No. I, pp. 47 60. 1996 Copyright ~c) 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0263-8231/96 $15.00 0263-8231(95)00045-3
Experiments on Inextensional Collapse of Circular Paper Disks Including the Influence of Holes J. A m i r b a y a t & J. W. S. Hearle Department of Textiles, University of Manchester, Institute of Science& Technology, PO Box 88, Manchester M60 1QD, UK (Received 18 November 1994; accepted 6 July 1995)
ABSTRA CT Buckling behaviour of circular filter papers, as isotropic plates with high aspect ratio, negligible weight-st,]hess ratio and little ability to undergo inplane strains prior to delamination, has been studied. The study covers the results of deforming circular papers under three equally spaced radial loads and the effect of holes punched within the samples to help formation o[ single curvature modes without points of discontinuous slope. The results o[ a.fen, experiments on eccentricity of the holes are also presented.
INTRODUCTION A circular plate can be in equilibrium under an infinite variety of in-plane external loadings each causing a different continuous three-dimensional (3-D) shape, provided the material can withstand the bending and the membrane stresses developed. Figure 1 shows examples of external loads including the extremes of a sample under two concentrated loads along a chord (which may be a diameter) and of loads distributed unequally around the circumference. Except for some simple cases of the above-mentioned loadings such as a pair of loads along a diameter, the analytical solutions of load deformations relationships, including the application of energy methods are far from being simple. Some 2-D materials such as rubber sheets and textile fabrics buckle into smooth surfaces due to their ability to undergo high levels of in-plane 47
J. Arnirbayat, J. W. S. Hearle
48
p
Q
p
p
A
P
Fig. 1. Some possible types of in-plane loading of a disk.
strains, while the majority of materials, including paper and the sheet metals, develop sharp corners during 3-D deformations when deflections reach beyond the level for which the compressive membrane strains are negligible as shown in Fig. 2. The classical theory of plates and shells shows that the membrane strains become significant when the deflections equal the plate thickness ~ or exceed beyond a few percent of the diameter. 2 By analysing the order of magnitude of energies involved in complex deformations, the authors showed that 3"4 the ratio between the membrane strain energy, Urn, and bending energy, Ub, can be expressed as
Um
--~b Ub
y¢2
-a
(1)
where Y B t
= membrane modulus of the sheet based on the force per unit width = the bending stiffness = a characteristic length defining the double curvature zone (the radius in the case of a circular sample) = a geometrical measure of the externally imposed deformation.
If the material is a homogeneous continuum, the bending stiffness can be expressed in terms of the thickness, t, the Young's modulus, E = Y/t, and the Poisson's ratio,/4 and eqn (1) can be rewritten as
lnextensional collapse of circular paper disks
49
Fig. 2. Circular sheets under large deformations. (a) Textile fabric; (b) sheet rubber.
Um ~ (1
~2 - ~2)
(2)
This shows that for a given material, the square of the aspect ratio, t/t, determines the level of deformation for which the membrane strain energy becomes significant. The present paper is concerned with an experimental study of the way in which filter papers, which cannot accommodate appreciable in-plane strain and have high aspect ratio, respond to displacements which cause buckling.
50
J. Amirbayat, J. 14I. S. Hearle
Fig. 2. Contd. (c) writing paper.
EXPERIMENTAL PROCEDURE For three-point loading the samples were deformed by a rig made of a lathe chuck and three connections supporting loading pins, Fig. 3. The samples were standard filter papers o f l l 0 m m diameter and the holes were cut by hard steel punches.
Fig. 3. Simple rig to deform the filter papers.
Inextensional collapse of circular paper disks
51
In the first set of experiments, the displacements were imposed in such a way that there were no bending moments at the points of application of the forces, but only the main inward radial forces and small frictional forces which resist the sample sliding out of place. In later experiments, where the specimens were held within the boundary, the conditions at the loading points can be regarded as built-in, i.e. zero slopes at the loading points. It should be noted that the experiments reported are obtained by controlled radial displacements, and not by controlled or measured forces.
EXPERIMENTAL RESULTS
Parallel displacement Under sets of parallel displacement of the types shown in Fig. l(a) and (b) the samples deform in a single curvature surface, behaving as a thin column with variable width. There is no need of double curvature zones to satisfy the requirement of geometry. With no membrane stresses developed, the curvature can be increased until the material delaminates, with micro-buckling due to compressive stresses on the inside of the bend. This limit was not reached for the standard size filter papers, even when the sample was bent round so that opposite edges touched as shown in Fig. 4(a). This shows that the single curvatures are not large enough to produce bending stresses higher than the tolerable amount, and that it would be necessary to push the two surfaces of the fold into closer contact before delamination occurs.
Radial displacements For radial displacements under the loads shown in Fig. l(c) and (d), it is not geometrically possible for the entire surface to deform in single curvature modes, except (up to certain levels of displacements) by means of separate lips at each loading-point, with the inner part remaining at zero curvature. For evenly spaced and equal radial forces the initial deformation of the samples approaches this form. Figure 4(b) shows the continuous 3-D surfaces, with no sharp discontinuities but a very slight double curvature central zone, virtually no transverse curvature in the edges which bend over like a cantilever, and possibly very small regions of higher double curvature where the lips join, with membrane strains tending to zero as the size of the double curvature region tends to zero.
52
J. Amirbayat, J. W. S. Hearle
Fig. 4. Buckling under different types of load. (a) Under parallel loads; (b) initial deformation under radial loads; and (c) large deformation under radial loads.
As the radial displacement increases, the difference between the low curvatures of the middle portions, ABC, Fig. 1(c) (being a triangular plate under distributed moments and forces around external edges) and the high curvature of the rest of the sample (being cantilevers with variable width), becomes more apparent. Upon further displacement, the high stresses at the regions where the lips join cause the paper to collapse into sharp corners, as shown in Fig. 4(c). The sample under large deformations can be regarded as divided into four different regions as shown in Fig. 5: the central roof F (which would have a very small degree of double curvature); the outer segments O with slight single curvatures, the zones S between F and O with higher single curvature acting as a 'fillet,' region, and the sharp discontinuities at the lines L. With no need of any membrane strain, the failure is entirely due to the bending stresses at the singularity points where the slopes become discontinuous. If the number of points of externally imposed displacements is increased, the central zone F increases in number of sides and in size,
53
lnextensional collapse of circular paper disks
" - - Z'~_- ~griginaXl circumference Fig. 5. Different zones of a buckled sample.
f
) Fig. 6. Buckling under large number of external loads.
as shown schematically in Fig. 6. This leads to an increase in the number of sharp corners but also makes the limiting displacement PP' smaller. Consequently, the difference between the directions of single curvatures of the adjoining lips, shown by double headed arrows in
J. Amirbayat, J. W. S. Hearle
54
Fig. 6, decreases. When the number of points increases above a certain value, the intermediate mode of buckling at the edges will cease to occur; under these conditions, once the slight permissible double curvature is exceeded, there will be a buckling in a sharp discontinuity somewhere within the central zone. There is no simple geometric way in which deformation can occur in the limiting case, indicated in Fig. l(e), with a uniform inward displacement all round the circumference of the circle.
The influence of holes
If there is a hole in the centre of the sheet, an alternative form of deformation is possible. The paper can buckle circumferentially in folds of single curvature, since a connecting dome of double curvature is not present and the membrane strains can be relieved by other movements of the paper around the hole. If the hole is smaller than a critical size for a given paper diameter, a crease forms after a certain amount of radial movement, with the sample buckling in the mode found without a hole, namely radial buckling of lips at the edges. Figure 7(a) and (b) shows identical paper samples with different sizes of central openings. A sample of 110 mm diameter deforms in 3-fold buckling without any sharp corners with a central hole of 21 mm diameter, while one with a 16 mm diameter hole cannot change the buckling shape. The critical diameter is somewhere between 16 and 21 mm, namely between 0.15 and 0.19 of the total diameter.
Fig. 7.
Different buckling modes of a paper disk under three external loads.
Inextensional collapse of circular paper disks
55
Influence of nature of forces causing displacement The experiments so far described are for the simplest form of buckling at the edges. Other methods of holding the sample and different sequences of application of displacement lead to different patterns of buckling with point and line discontinuities at various locations. If instead of applying the displacements on the circumference of a sample, they are applied by pins within the sample, creases within the sample form after the slightest radial movement of the pins. The sample deforms in single curvature portions bent around radii and a central flat portion with sharp corners at the interface. This is because the zero slopes at the loading points prevent the formation of single curvature zones around the sectors such as AB, BC and so on, Fig. l(c). The double curvature required to join the regions bent around the radii and the flat centre, delaminates the paper similar to the case mentioned before. Figure 8 shows the sample under three radial loads which are applied at 5 mm from the outer edge. In this case also a central hole of a proper size makes it geometrically possible for the sample to deform without any double curvature and therefore no sharp corners from, Fig. 9. For a central opening of 13.5 mm diameter namely 0.12 of the sample diameter, the sample deforms without any crease, Fig. 9(a), while a 12 mm diameter hole cannot stop creasing at this level of radial displacement, Fig. 9(b). Curve A in Fig. 10 shows the effect of central hole size on the amount of deformation to form the crease. The test results show that for papers with central openings whose radii are larger than 12% of the sample radius. there is no limit of deformation to cause crease.
Fig. 8. Bucklingof paper disks when the loads are applied from within the samples.
56
J. Amirbayat, J. W. S. Hearle
Fig. 9. Buckling of paper disks with central holes when the loads are applied from within the samples. (a) Hole diameter 13.5 mm, no crease formation; and (b) hole diameter 12 mm, crease forms.
If the displacements are not symmetrical or one is applied before or after the other two (which is the case of deforming a plate with initial single curvature already generated by the other two displacements) the crease develops after the radial displacement exceeds a limit lower than the corresponding value for the symmetrical case, depending on the amount of the offset and the size of the central hole. Curve B in Fig. 10 belongs to an offset loading of 1 mm, i.e. one of the loading pins is set 1 m m closer than the other two to the centre of the sample.
Eccentric holes
Further experiments show that if the holes are not concentric with the sample, the crease forms at lower radial displacements. According to the limited test results, there is a definite relationship between the critical radial deformation, the opening size and the a m o u n t of offset for a given sample size. Figure 11 shows the effect of relative size of the hole and its relative offset from the centre on the external displacement for crease formation. Figure 12 (a)-(d) shows the effect of the size and the location of the opening. The hole diameter of the first three samples is 16 m m and their offsets are 35, 20, and 15 mm, respectively. The crease formation is stopped when the offset is reduced to 15 m m in Fig. 12(c). The corresponding diameter of the hole to stop the crease when punched centrally is 13.5 m m as previously mentioned and shown in Fig. 9(a). The offset of the hole in Fig. 12(d) is 20 m m but the larger diameter of the hole (40 mm)
Inextensional collapse of circular paper disks
57
30 - -
m
20 - -
<
L.
,7 j"
0
/
I
I
I
I
5
I0
15
20
Percentage of opening
ro Ro
x 100
Fig. 10. Effect of eccentricity of the central opening on the radial displacement to cause crease.
makes it possible to deform in combination of single curvature modes. Figure 12(e) shows a sample identical to Fig. 12(d) with one of the loading pins positioned 1 mm closer to the centre before the experiment. There is again no crease formation due to the large diameter of the eccentric hole. According to our observations, with the ratio ro/Ro approaching 0.3 any asymmetry of loading or offset of the hole can be tolerated without any limitation of the radial displacement, Fig. 12(f). When the opening is large enough, the radial movements of the elements become merely translatory and no membrane strain develops.
CONCLUSION The present paper is part of a series concerned with the complex buckling of sheet materials. The principal area of interest in the whole research programme is the way in which textile fabrics, which can easily
J. Amirbayat, J. W. S. Hearle
58
25I 0
2O ~
~o =10%
2
0
I
I
I
t
I
I
I
10
20
30
40
50
6(I
70
Eccentricity of the opening e
Ro
x 100
Fig. 11. Effect of eccentricity of the central opening on the radial displacement to cause crease.
accept large membrane strains due to their low resistance to certain forms of in-plane deformation, buckle into rounded folds of double curvature. In contrast to this, a material like paper which has a high resistance to in-plane deformation relative to its ease of bending will not accommodate double curvature, and this led us to the subsidiary investigation reported here of the forms of buckling which do occur in such a material. The experimental results show that the samples avoid formation of any substantial double curvature zones by the occurrence of point or line singularities, even if this requires delamination in order to release membrane energy. Furthermore, it has been shown that a hole of proper size and location within the samples changes the mode of buckling, and allows accommodation of the deformation by different forms with a lower energy for a given externally imposed deformation. It is worth noting that the softness of some nonwoven fabrics is improved by making apertured sheets containing holes.
Inextensional collapse of circular paper disks
59
O
e-, O
.=_ e~ G
~D
~d
60
J. Amirbayat, J. I4I. S. Hearle
REFERENCES 1. Den Hartog, J. P., Advanced Strength of Materials. McGraw-Hill, New York. 2. Timoshenko, S. & Woinosky-Krieger, S., Theory of Plates and Shells. McGraw-Hill, New York. 3. Amirbayat, J. & Hearle, J. W. S., The complex buckling of sheet material, part II. Int. J. Mech. Sci. 7,8 (1986) 359. 4. Amirbayat, J. & Hearle, J. W. S., The anatomy of buckling of textile fabrics. J. Text. Inst. 80 (1989) 51.