WO2017067799A1

WO2017067799A1 - A resilient body comprising a corrugated surface

Info

Publication number: WO2017067799A1
Application number: PCT/EP2016/074077
Authority: WO
Inventors: Robert Stanley Farr
Original assignee: Unilever NV; Conopco Inc
Current assignee: Unilever NV; Conopco Inc
Priority date: 2015-10-23
Filing date: 2016-10-07
Publication date: 2017-04-27
Anticipated expiration: 2018-04-23
Also published as: AR106434A1; WO2017067799A8

Abstract

A resilient body comprising a gradually undulating surface, the undulating surface defined by a notional two-dimensional surface having a plurality of undulations each having a positive or negative amplitude normal to the notional two-dimensional surface and randomly distributed thereon, such that in any and all profiles of the undulating surface, the mean amplitude is statistically zero.

Description

A Resilient Body Comprising a Corrugated Surface

Field of the invention The present invention relates to a resilient body comprising a gradually undulating surface.

Background of the invention Packaging for consumer products is widely seen as a waste by-product of modern society. In general packaging is disposed of by an end user and the packaging needs to be subsequently handled in some way, e.g. by going to landfill or by being recycled.

Therefore to reduce the quantity of packaging disposed of by consumers would provide great advantages.

One requirement of packaging is to provide structural integrity, so that the product can be handled through the supply chain and in use by the consumer. Therefore, if it were possible to modify packaging to improve its structural properties, it becomes possible to use less material to achieve the same structural effect, and thus reduce the amount of disposed packaging.

Recent work has shown that complex hierarchical designs can be used to make very lightweight structures for supporting compressive loads (without buckling for example). However, most of these designs require complex topologies that are difficult to manufacture, being multi-layer structures, or having multiple hollow regions within them. They are therefore not suited to making single-layer packaging, for example via blow moulding.

Therefore, improvements in this area would be highly desirable. WO02/016122 discloses flat panels which contain positive and negative deviations in the surface. There is no disclosure of improving structural properties such as bending and stretching. US 5,965,235 discloses a flat sheet onto which has been introduced deformations on one side only in order to provide separation between multiple stacked sheets. The surface contains sudden changes in the profile of the sheet.

WO99/07273 discloses a wipe article having a random series of peaks and valleys thereon.

JP3141022 discloses a bottle which comprises a region of decoration which resembles a cluster of ice cubes. WO02/13916 discloses a golf ball with randomly spaced dimples on an otherwise flat land between, e.g. in a Voronoi tessellation.

Summary of the invention In a first aspect, the invention relates to a resilient body comprising a gradually undulating surface, the undulating surface defined by a notional two-dimensional surface having a plurality of positive and negative undulations normal to the notional two-dimensional surface and randomly distributed thereon, such that in any and all profiles of the undulating surface, the mean undulation is statistically zero.

Thus, the present invention utilises an optimised geometry to provide improved linear mechanical properties such as stretching and bending. Improvements in crushing resistance have also been found. This increases the resistance to elastic buckling and crumpling under applied loads.

The present invention provides a method for applying random, statistically isotropic corrugations to a (possibly curved) panel, such as the wall of a packaging container, which increase the bending stiffness. The panel may initially be flat, but the method is applicable to panels that have any shape or topology - for example including holes or handles - with a radius of curvature larger than that of the corrugations. The method ensures that locally, the properties of the corrugated panel are statistically isotropic and on average are the same across the curved panel, with no seams or discontinuities, no matter what geometry or topology the initially curved panel may have.

In an embodiment the method consists of:

• Creating a statistically isotropic pattern of real numbers at an array of points in 3D space (for example using the Cahn-Hilliard theory of phase separation).

• Interpolating between these points and applying periodic boundary conditions so that one can assign a real value to every point in space.

• Using these values to move the surface of the shape parallel to its local normal, in order to create a corrugated surface.

The topology of the final surface should be the same as that of the initial surface:

provided the amplitude of the corrugations is not too large, no new handles, surface- surface intersections or enclosed volumes are created by this process. Note that the corrugations are applied to a panel, but this initial panel may itself optionally be foamed or topologically complex or a laminate or a sandwich composite. The random corrugated sheet may also be used as a layer in a sandwich composite.

In another step, it is possible to apply this procedure more than once at different scales, creating a hierarchical corrugation pattern.

By "gradually undulating" is meant that the surface profile is smooth and contains no sudden changes of direction. In mathematical terms the profile of the surface is therefore mathematically differentiate to first order. For example, a sudden change in direction would imply an infinite rate of change of direction and the surface would therefore not be differentiate. By "profiles of the undulating surface" is meant a portion of the surface, such as a slice through the surface or an extended region of the surface. Because the surface is undulating the profile must extend over a representative distance of the surface in order for the mean undulation in the profile to be statistically zero. A profile can thus be a perpendicular slice through the surface giving a trace in one dimension or could be an area of the surface giving a two dimensional sample of the surface. Indeed the profile could even be the entire surface.

By "statistically zero" is meant that the measurement of the mean undulation in any profile will not necessarily be exactly zero but will be zero as testable by statistical methods. For example taking a number of profiles of the surface, no matter their extent, will together provide a mean undulation that approaches zero the more samples of the profile are taken. As an example, if the head side of a coin is represented by +1 and the tails side of a coin are represented by -1 , then the average of a number of coin tosses is also statistically zero, even if the actual value measured for a limited number of coin tosses might be non-zero. By "resilient" is meant that the body returns or is able to recoil or spring back into shape after being bent, compressed or stretched in any direction up to an elastic limit.

By "statistically isotropic" is meant that the surface has the same linear material properties regardless of the direction they are measured in, to a high level of statistical confidence.

By "mean undulation" is meant the average height of the surface.

Preferably the surface has statistically isotropic material properties. This means that the material properties are independent of the alignment of the surface.

A significant advantage of the present invention is that the undulations can be applied to complex geometries without breaking or disturbing the pattern in any way. This, preferably the notional surface is curved, and more preferably undulating surface forms a three-dimensional shape.

Thus the body may be a container or a mould for an object such as a container, preferably for containing a consumer product. Additionally the body may be secondary or tertiary packaging such as a box or pallet. As the undulating surface provides improvements in material strength and can result in less packaging material being used, it is preferable that the body is made up of at least 50wt% of the undulating surface, more preferably at least 75 wt%, most preferably at least 85% wt%.

As discussed above the undulating surface does not contain any sudden changes of direction, as these can introduce a source of structural weakness, especially to crushing. As such, preferably the radius of curvature at any point on the undulating surface is not less than 0.1 times the mean magnitude of the amplitudes of undulation, preferably is not less than 0.2 times, more preferably not less than 0.5 times. In a preferred embodiment it is about 1.0 times the mean magnitude of the amplitudes of undulation. Additionally, as one of the main advantages of the undulating surface is to make a stronger, more crush-resistant structure, it is preferable that no part of the undulating surface exhibits a slope relative to the notional surface of greater than 60 degrees, preferably greater than 50 degrees. Additionally, although the present invention can in principle apply to materials of any thickness, preferably the undulating surface has a material thickness of from 0.1 to 3mm, more preferably less than 2mm, most preferably less than 1 mm.

The undulating surface may be made from a variety of materials; however a plastic, cardboard or metal material is preferred.

It has also been found that the benefits of the invention are more significant when the magnitude of the undulations is greater than the thickness of the surface. Thus, preferably the ratio of the mean amplitude of the undulations to the thickness of the surface is greater than 2:1 , preferably greater than 5:1 , more preferably greater than 10:1 .

The undulations preferably all have the same or similar magnitudes of amplitude, so that they project the same or similar distance from the notional surface. Thus, preferably the magnitude of the amplitude of the undulations for a plurality of positive and negative undulations extending across the undulating surface is essentially the same across the surface. In a highly preferred embodiment, all of the undulations have the same magnitude of amplitude.

It is also preferred that the length scale of the undulations is consistent throughout the surface. As such, preferably the distance, parallel to the notional surface, between the "peak" of a positive undulation and its nearest neighbouring "peak" of a positive undulation is substantially the same for all such "peaks" on the undulating surface.

Additionally, it is preferred that the distance, parallel to the notional surface, between the "peak" of a positive undulation and its nearest "trough" of a negative undulation is substantially the same for all such "peaks" on the undulating surface. It is also preferred that the size and spacings of the positive undulations are essentially the same as the negative undulations. Thus, preferably the mean distance, parallel to the notional surface, between the "peaks" of the positive undulations are essentially equal to the mean of the distance between the "troughs" of the negative undulations. It has also been found that a continuously undulating surface is preferable to one which has only a few scattered undulations separated by intervening flat surface. Thus preferably one peak flows into a trough and vice versa throughout the surface. Thus, preferably the ratio of (1 ) the mean distance, parallel to the notional surface, between the "peak" of a positive undulation and its nearest "trough" of a negative undulation, to (2) the mean amplitude of the undulations, is from 10:1 to 1 :3, preferably 5:1 to 1 :1 , more preferably from 3:1 to 1.5:1.

The undulations are preferably smooth and contain no sudden changes of direction. In a preferred embodiment, the profile of the undulating surface between a "peak" and its neighbouring "trough" is S-shaped.

In order for the undulations to have a sufficient effect on the strength of the surface there must be a sufficient number of them. Thus, preferably the surface comprises at least 10 positive undulations and at least 10 negative undulations. Additionally, if the surface forms a container, it is preferred that the ratio of (1 ) the minimum lateral dimension of the surface, to (2) the mean distance, parallel to the notional surface, between the "peak" of a positive undulation and its nearest "trough" of a negative undulation, is greater than 5:1 , preferably greater than 10:1 , more preferably greater than 20:1 .

By "minimum lateral dimension of the surface" is meant the closest separation of two parallel planes between which the surface will fit without touching either plane.

In a preferred embodiment the starting surface may be an undulating surface as produced according to the present invention. Thus, a second set of undulations can be superimposed onto such an undulating surface, e.g. to produce a hierarchical structure. If the second set of undulations is smaller than the first set then such a hierarchical structure is provided. Therefore preferably the ratio of the magnitude of (1 ) the amplitude of the undulations on the notional two-dimensional surface, to (2) the amplitude of the positive and negative undulations normal to the undulating surface is greater than 3:1 . In a second aspect, the invention relates to a process for the generation of a body as described herein, the process involving (1 ) generating a mathematical three- dimensional scalar field, (2) mathematically placing the notional two-dimensional surface into the three-dimensional scalar field, (3) recording the value of the scalar field over the two-dimensional surface, (4) creating a mathematical representation the undulating surface by applying undulations normal to the two-dimensional surface in proportion to the value of the scalar field at that point on the surface, (5) generating the body from the representation so obtained in step (4).

Thus, the shape of the surface can be generated as a mathematical object in steps (1 ) to (4) and step (5) is the production of the object according to the mathematical description so obtained.

Preferably the scalar field is generated by mathematical analogy with Cahn-Hilliard theory of incomplete polymer phase separation. Detailed description of the invention

The invention will now be illustrated, by way of example, and with reference to the following figures, in which:

Figure 1 is a schematic representation of a model surface illustrating how various parameters are defined.

Figure 2a is a periodic image of a simulation of incomplete Cahn-Hilliard phase separation.

Figure 2b is the simulation shown in figure 2a but with a corner of the illustrated cube missing to illustrate that the pattern does not vary according to orientation. Figure 3a is an image of a mathematical representation of a cylinder which has been treated to the application of undulations according to the present invention.

Figure 3b is an image of a mathematical representation of the cylinder shown in figure 3a which has been further treated to the application of smaller undulations to create a hierarchical structure.

Figure 4 shows images of mathematical descriptions of various patterns applied to a planar square surface (a) 'flat', (b) 'ribs', (c) 'ribs rotated 90°', (d) 'Cahn-Hilliard', (e) 'cubes', (f) 'cubes rotated 90°', (g) 'p31 m' and (h) 'p31 m rotated 90°.

Figure 5a is a chart showing the finite element results for stretching stiffness, S, divided by the stretching stiffness of a flat sheet, for the surfaces shown in figure 4.

Figure 5b is a chart showing the finite element results for bending stiffness, B, divided by the bending stiffness of a flat sheet, for the surfaces shown in figure 4.

Figure 6 is images of a variety of 3D printed test cylinders made from acrylonitrile butadiene styrene (ABS), the corrugation patterns are: (a) ^"p31 m', (b) ^"cubes' (p3m1 symmetry), (c) ^"helix' (a ribbed geometry, but rotated by 30° away from alignment with the cylinder axis, (d) ^"ribs', (e) ^"flat' and (f) ^"ribs rotated 90°'.

Figure 7 is a chart showing part of the same chart as figure 5b, with the results from mechanical tests also included as filled diamond symbols.

Figure 8 is an image of the mechanical testing arrangement showing a cylinder under test and a Stable Micro Systems Texture Analyser'. Figure 9 is an image of a solid object resembling the shape of a consumer product for containing liquid detergent made from ABS and printed by a 3D-printer.

Figure 10 is an image of a solid object based on the object shown in figure 9 but wherein part of the surface has been treated to the mathematical addition of undulations to the surface before being printed. A portion of the surface has not been treated and left flat for the application of a label.

Examples Consider a plate with a thickness f, and a corrugation that is characterized by three properties: First the pattern itself, which may be ridges in some orientation, or an egg- box like motif, or something more complicated. Second there is a lateral length scale λ, and third there is the amplitude a. These are illustrated for a simple pattern of ridges in figure 1. The plate thickness itself is defined as the volume of material per unit projected area of the plate. Therefore introducing a corrugation pattern by these definitions does not change the amount of material used in the plate or packaging.

In the examples the pattern and lateral scale λ change, but ^α/ remains fixed. In that sense, we regard different values of ^α/ as defining entirely different ^"types' of corrugation. Most of the corrugation patterns studied are based on a regular triangular lattice, with the heights of the vertices raised or lowered by various amounts, and the surface being a linear interpolation between these vertex positions. In this case, we take ^α/ to be the length of each triangular edge. For the case of a flat plate, clearly λ is arbitrary (or is undefined), so it is chosen to be the same as for the patterns based on triangular lattices.

A corrugation design which is defined statistically was also generated. The route chosen is to base the design on the Cahn-Hilliard theory of polymer phase separation. This pattern is statistically isotropic, defined in three dimensions and has a single length scale.

The technique is as follows: solve the Cahn-Hilliard equation for concentration c of one phase:

— = V² (c³ - c - γψ c)

at which has an equilibrium domain wall width of γ^1/2. Then simulate this equation in a cell with periodic boundary conditions, of side length 200 elements, using unit dimensions for each element, a timestep of 0.001 , and 7=0.5. The simulation is carried out for 6 time units, and store the 8 x 10⁶ values of c. By scaling, interpolation and use of periodic boundary conditions, a real value of c can then be assigned to any point in space. Figure 2(a) shows values of c for the boundary of one periodic image of the simulation, and also in 2(b) for some of the interior points.

For the irregular ^"Cahn-Hilliard' pattern, there is some latitude to choose the meaning of λ, and so because the scale of this pattern has been chosen to be visually similar to the corrugations based on the triangular lattice, we again use the same value for λ. An alternative definition would be to start from a peak or ridge in the structure, and measure how far one needs to go to reach the nearest neighbouring peak or ridge. For a random corrugation pattern, this length scale may differ from place to place, but for a statistically uniform and isotropic corrugation pattern, it will not vary systematically from place to place, nor will there be a preferred direction in which the nearest neighbouring peak will be found. The value of the amplitude is more easily defined: it is difference in ^"height' (normal to the notional surface bearing the corrugations) between the highest and lowest points on one side of the surface. For a flat plate, a=0, and for the other patterns, the value is specified by the code which generates the geometry. In general, we are interested in patterns which will be elastically isotropic, so that the bending and stretching stiffness does not depend on the direction of the deformation. For this reason, the regular patters which are of most interest are those with 3-fold or 6- fold symmetry. The ridge-type geometries are likely to be highly anisotropic, with bending stiffness across the ridges being much higher than when the deformation is aligned with the ridges, and similarly for stretching. These are included because some of the scaling properties of these structures can be calculated, and because they make useful test cases.

The irregular, random corrugation pattern can be designed to be statistically isotropic, and also have other favourable properties (in a statistical sense).

Although the regular patterns investigated can be applied to flat surfaces (or those with no intrinsic/Gaussian curvature, such as a cylinder, or, more generally, developable surfaces), it is not clear how they could be applied to the complex curved surfaces encountered in packaging designs. Possible approaches would

be to scale the corrugations so that, for example, they would be small near a narrow neck and large at the equator of a vase-shaped bottle. However this breaks the effective translational symmetry over the surface, leading to a design with different effective properties at different points on the packaging. It would also mean that the mechanical properties are tied to the bottle shape rather than being chosen

independently for reasons of strength. Another approach would be to cover the surface in a patchwork of corrugated regions; however, this could lead to weaknesses at the seams where the patches meet. However the random corrugated surface can be readily applied to any curved surface. In order to apply the corrugation pattern to a surface, the above method is used to assign a value of to every point of the surface of interest, and then translate these points by an amount proportional to the assigned value, and in a direction normal to the original surface.

An example is shown for a cylinder in figure 3a. It is also possible to use the random corrugated surface generated as a new starting surface and superimpose a further level of corrugation. Such a design is shown in Figure 3b.

Finite Element Simulations

For the finite element simulations, the Calculix FE solver was used. The unit system is Newtons and millimetres. A Poisson ratio v=0.35 and Young's modulus Y=1000Nmnr² (or equivalently 1 GPa) was chosen.

For the simulations, a 20mm by 25mm plate is used, cantilevered along one of the long edges (see figure 4a), comprising 140 by 175 quadratic hexahedral elements with full integration (element type C3D20 in Calculix or Abaqus), which is considered to give reasonably accurate answers for a plate simulation, and not be particularly prone to locking, soft modes or anomalously high stiffness in certain deformation modes. The relevant values of λ and a (defined above) are λ =2 mm and a=1 mm (except for the flat plate where a=0 mm). Figure 4 shows square portions 20mm by 20mm of the patterns used in the finite element simulations. The actual geometries are rectangular, with one side 20% longer than shown (20mm by 25mm). The fixed support in the simulations lies along the y-axis (the line x=z=0), as shown for the flat plate. The geometries are (a) to (h): ^"flat', ^"ribs', ^"ribs rotated 90°', ^"Cahn-Hilliard', ^"cubes', ^"cubes rotated 90°', ^"p31 m' and ^"p31 m rotated 90°'.

Figure 5a shows finite element results for the stretching and Figure 5b for bending of corrugated plates, scaled by the theoretical results for a flat plate, and plotted as a function of plate thickness divided by lateral scale of the corrugation. All results are on a log-log scale. It can be seen that in general, introducing corrugations reduces the stretching stiffness (except for the case where the corrugations are ribs parallel to the stretching direction) and increases bending stiffness (except for the case where the corrugations are ribs parallel to the cantilevered edge).

The Cahn-Hilliard random corrugation pattern is amongst the highest in stiffness for both bending and stretching. It should also be noted that its mechanical behaviour will be statistically isotropic (independent of orientation). Mechanical Testing

To corroborate the predictions of the finite element analysis, a variety of 3D printed test cylinders were made from ABS as shown in figure 6. From left to right, the corrugation patterns are: (a) ^"p31 m', (b) ^"cubes' (p3m1 symmetry), (c) ^"helix' (a ribbed geometry, but rotated by 30° away from alignment with the cylinder axis, (d) ^"ribs', (e) ^"flat' and (f) ^"ribs rotated 90°'.

Corrugated cylindrical geometries were generated by computer. These were then used to 3D print the shapes using a commercial 3D printer, out of ABS (acrylonitrile butadiene styrene). This is a glassy polymer at room temperature (Tg¾ 105°C), with a quoted density in the range 1060 to 1080kgnr³ and Young's modulus in the range 1 .4 to 3.1 GPa.

In order to test the density of the samples, the printed pieces were floated in salt solution, and found the concentration needed to ensure neutral buoyancy. The density measured in this way was found to be in the range 1060 to 1070kgnr³

The scale of the printing was adjusted to ensure the shapes all fit within the available sample volume, so we do not know in advance how large each of the pieces will be. The radii (inner and outer) and heights were therefore measured for each of the pieces, and the thickness of the plates (in terms of mass per unit normal projected area) were then determined by measuring the mass, converting to volume by the density of ABS, and dividing by the area of a cylinder with the measured average (mean of inner and outer) radius. See table 1 for results where the heights and radii were measured triplicate with vernier callipers.

Table 1

Name Mass/g Height/mm Outer diameter/mm Inner diameter/mm

Flat 15.70 77.97 77.92 77.95 68.0 68.0 67.8 65.0 65.7 66.0

Ribs 17.91 77.93 77.91 78.07 72.5 72.5 72.5 66.0 65.5 65.5

Ribs 16.34 77.88 77.99 77.89 72.5 72.5 72.7 65.5 66.0 66.0 rotated 90

Cubes 16.99 77.99 77.91 77.95 72.5 72.5 72.7 66.5 66.0 66.0

P31 m 17.30 77.98 77.78 77.97 72.5 72.5 72.7 66.5 66.0 66.0

In relative terms, the cylinder height L, circumference C, lateral scale λ and corrugation amplitude a are designed to be in the ratio L : C : λ : a :: 9Λ/3 :42 : 1 : 0.5, and corrugations are produced by moving points outwards from the basic uncorrugated cylinder, so the heights based on mean radius, λ and C might not match the prediction exactly. With these data, we can work out the ratios α/λ and t/λ, as shown in table 2.

Table 2

The pieces were subjected to bending and compressional tests, as illustrated in figure 8, by means of a Stable Micro Systems "Texture Analyser". The results of Table 2 are plotted on Figure 7 together with a magnified region of the finite element results shown in figure 5b. It can be seen that there is very good agreement between the two data sets, although the experimental results for the two orientations of 'ribs' corrugations lie a little below the finite element results.

This shows that the finite element results are a very accurate predictor of the elastic properties of the bodies analysed.

Discussion

For thin plates, bending is easier than stretching (as can be seen dramatically for paper, which is easy to bend and fold, but rather stiff under stretching stresses).

Corrugations offer the prospect of increasing the bending stiffness dramatically, at the expense of stretching stiffness; and therefore where bending stiffness is the limiting design constraint, such as in large panels with little or no curvature, this is a worthwhile compromise. Simple ridge-type patterns however lead to very anisotropic mechanical properties, and the advantage of some of the other patterns studied here (such as ^"p31 m' and ^"Cahn-Hilliard') is that it is possible to increase the bending stiffness more or less equally for any orientation of the bending stress.

Suppose a corrugation pattern were introduced to a previously flat region with corrugation size t/λ =0.05, then figures 5a and 5b indicate that an approximate 10 fold increase in bending stiffness can result. To achieve the same by increasing the thickness of the flat region would require more than twice as much material.

Additionally, the problem of a cylindrical shell, axially loaded under compression offers a very simple example that is relevant to primary packaging of products.

It is known that resistance to crushing is proportional to the s , where S is the stretching stiffness and B is the bending stiffness.

However upon investigation it turns out that the value of VSZ? for the ^"p31 m' and ^"Cahn- Hilliard' corrugations is essentially the same as for a flat plate. However this ignores the extreme sensitivity of this geometry under buckling to imperfections.

The elastic buckling of cylindrical shells is very sensitive to imperfections, for example deviations from a perfect circular cross-section. Therefore even though the buckling forces of perfect shells can be very high, this level of strength is never observed in practice. The strength of imperfect cylinders under axial loads with imperfection amplitudes on the order of the wall thickness can be several times lower than for the case of perfect cylinders. This represents a severe example of sensitivity, since in real applications, cylindrical shells are likely to be thin-walled, so that imperfections of this magnitude are unavoidable.

One advantage of corrugated shells is that the effective thickness (from an elastic point of view; not the actual volume of material per unit projected area) is greatly increased. Therefore, even if the product Vsl? remains largely unchanged, we expect structures made from corrugated walls with isotropic elastic behaviour to be significantly less sensitive to imperfections than the corresponding non-corrugated structures.

It seems plausible to suppose, that the corrugated structures can be made with imperfections less than the new effective elastic thickness, and therefore the strength of the packaging under axial loads increased by a factor of more than two. Since axial strength varies as f², then in this scenario replacing a smooth wall with

a thinner corrugated wall could lead to a 30% saving in material. Conclusions

The most interesting pattern studied is based on the Cahn-Hilliard theory of incomplete polymer phase separation, which gives good results for bending stiffness compared to most other patterns, and has a similar value for the quantity VSZ? (where S is the effective stretching stiffness and B the bending stiffness of the corrugated plate) to that of a flat plate. The quantity Vsl? is the key parameter setting the linear buckling force of thin-walled hollow cylinders under axial compressive loads. The enhanced and isotropic bending stiffness can be an advantage on its own if packaging contains large flat regions, such as the front and back of a liquid detergent bottle. However, the similarity of the quantity Vsz? to the uncorrugated plate suggests that there will be no advantage to isotropic corrugations in the case of axial

compressive loading of cylinders. This however neglects the sensitivity of this loading case to imperfections. It is difficult to be quantitative without further experiments, but we suggest tentatively that the diminished imperfection sensitivity of the corrugated panels could lead to materials savings of up to 30%. The Cahn-Hilliard corrugation pattern investigated has other useful properties: it gives statistically isotropic behaviour for bending and stretching stiffness, and it can be applied to any surface, whether curved or flat or of complex topology. When applied to such a surface, the pattern does not need to be locally scaled, nor do ^"seams' need to be introduced between incompatible regions of the pattern: instead, the pattern is statistically the same everywhere over the surface. Furthermore, this process of applying the pattern to an arbitrary surface can be iterated, using a similar pattern at a smaller length scale, to create hierarchical corrugations.

In order to show that the present invention can be applied to more complex geometries, figure 9 shows a 3D printed object made of ABS in the form of a liquid detergent container. Figure 10 is the same container with the surface altered according to the method described herein to apply the random undulations. The nature of the patterning means that the application of the undulations is not affected and no seams exist despite the irregular shape of the object.

Claims

A resilient body according to claim 1 , wherein the surface has statistically isotropic material properties.

A resilient body according to claim 1 or claim 2, wherein the notional surface is curved.

A resilient body according to claim 3, wherein the undulating surface forms a three-dimensional shape.

A resilient body according to claim 4, wherein the body is in the form of a container, preferably for a consumer product.

A resilient body according to claim 4, wherein the body is in the form of a mould.

A resilient body according to any one of the preceding claims, wherein the body is made up of at least 50wt% of the undulating surface, more preferably at least 75 wt%, most preferably at least 85% wt%.

A resilient body according to any one of the preceding claims, wherein the radius of curvature at any point on the undulating surface is not less than 0.1 times the mean magnitude of the amplitudes of undulation, preferably is not less than 0.2 times, more preferably not less than 0.5 times.

A resilient body according to any one of the preceding claims, wherein no part of the undulating surface exhibits a slope relative to the notional surface of greater than 60 degrees, preferably greater than 50 degrees.

A resilient body according to any one of the preceding claims, wherein the mean magnitude of the undulations is from 1 mm to 10mm.

A resilient body according to any one of the preceding claims wherein the undulating surface has a material thickness of from 0.1 to 3mm, more preferably less than 2mm, most preferably less than 1 mm.

A resilient body according to any one of the preceding claims, wherein the ratio of the mean amplitude of the undulations to the thickness of the surface is greater than 2:1 , preferably greater than 5:1 , more preferably greater than 10:1.

A resilient body according to any one of the preceding claims, wherein the magnitude of the amplitude of the undulations for a plurality of positive and negative undulations extending across the undulating surface is essentially the same across the surface.

A resilient body according to any one of the preceding claims, wherein the distance, parallel to the notional surface, between the "peak" of a positive undulation and its nearest neighbouring "peak" of a positive undulation is substantially the same for all such "peaks" on the undulating surface.

A resilient body according to any one of the preceding claims, wherein the distance, parallel to the notional surface, between the "peak" of a positive undulation and its nearest "trough" of a negative undulation is substantially the same for all such "peaks" on the undulating surface.

A resilient body according to any one of the preceding claims, wherein the mean distance, parallel to the notional surface, between the "peaks" of the positive undulations are essentially equal to the mean of the distance between the "troughs" of the negative undulations.

17. A resilient body according to any one of the preceding claims, wherein the ratio of (1 ) the mean distance, parallel to the notional surface, between the "peak" of a positive undulation and its nearest "trough" of a negative undulation, to (2) the mean amplitude of the undulations, is from 10:1 to 1 :3, preferably 5:1 to 1 :1 , more preferably from 3:1 to 1.5:1.

A resilient body according to any one of the preceding claims, wherein the profile of the undulating surface between a "peak" and its neighbouring "trough" is S-shaped.

A resilient body according to any one of the preceding claims, wherein the undulating surface comprises at least 10 positive undulations and at least 10 negative undulations.

A resilient body according to any one of the preceding claims, wherein the ratio of (1 ) the minimum lateral dimension of the surface, to (2) the mean distance, parallel to the notional surface, between the "peak" of a positive undulation and its nearest "trough" of a negative undulation, is greater than 5:1 , preferably greater than 10:1 , more preferably greater than 20:1.

A resilient body according to claim 1 , wherein the notional two-dimensional surface is itself an undulating surface as defined in any one of the preceding claims, and the plurality of positive and negative undulations normal to the undulating surface are smaller in amplitude than the undulations on the undulating surface and may be as defined in any one of claims 1 to 18.

A resilient body according to claim 21 , wherein the ratio of the magnitude of (1 ) the amplitude of the undulations on the notional two-dimensional surface, to (2) the amplitude of the positive and negative undulations normal to the undulating surface is greater than 3:1.

A process for the generation of a resilient body according to any one of the preceding claims, the process involving (1 ) generating a mathematical three- dimensional scalar field, (2) mathematically placing the notional two- dimensional surface into the three-dimensional scalar field, (3) recording the value of the scalar field over the two-dimensional surface, (4) creating a mathematical representation the undulating surface by applying undulations normal to the two-dimensional surface in proportion to the value of the scalar field at that point on the surface, (5) generating the body from the representation so obtained in step (4).

A process according to claim 23, wherein the scalar field is generated by mathematical analogy with Cahn-Hilliard theory of incomplete polymer ph separation.