**MATERIALS WITH STRUCTURES EXHIBITING ZERO POISSON'S RATIO**

JPH10128767 | PRODUCTION OF FOAMED SYNTHETIC RESIN HEAT INSULATING MATERIAL |

JPH06226890 | MANUFACTURE OF FIREPROOF PANEL |

JP5514470 | Structures and molded products |

NAGHAVI ZADEH SEYED MOHAMMAD (GB)

*;*

**B32B3/12***;*

**B31D3/00**

**B31D3/02**US7678440B1 | 2010-03-16 | |||

US8652602B1 | 2014-02-18 |

DATABASE WPI Week 201515, Derwent World Patents Index; AN 2015-12877W, XP002796686

Claims: 1. A material for exhibiting Zero Poisson's Ratio (ZPR), the material having a tessellated cellular structure, each cell of the cellular structure comprising: a substantially fish shaped configuration. 2. The material of claim 1 wherein the fish shaped configuration comprises a head portion (convex part) and a tail portion (concave part). 3. The material of claim 1 or 2 wherein the cells of the cellular structure are tessellated in x-direction and in y-direction. 4. The material of any preceding claim wherein the fish shaped configuration may be alternatively described by any of the following analogous terms alone or in combination: glass cell, saddle-shaped honeycombs, vase cell, N-shaped rib honeycomb; mixed honeycomb, bidirectional hybrid honeycomb, re-entrant mixed honeycomb, wavy honeycomb, corrugated rib honeycomb, modified chequered honeycomb, zig-zag honeycomb, convex-concave honeycomb, curved rib honeycomb, hexagonal re-entrant coupled honeycomb, modified auxhex honeycomb, hybrid honeycomb; odd function rib honeycomb, spline rib honeycomb, strain isolated honeycomb, cosine rib honeycomb, trigonometric rib honeycomb, sine rib honeycomb, polynomial curved rib honeycomb, Fourier series rib honeycomb, eight sided polyhedron as a combination of three trapezoids. 5. The material of claim 2 wherein a mutual base that would interface between the respective head and tail portions is omitted, resulting in at least an eight sided shape with a line of symmetry through the respective head and tail portions. 6. The material of claim 2 wherein the geometry of the head portion and tail portion in a fish shaped configuration is defined by at least six parameters corresponding to eight nodes. 7. The material of claim 6 wherein the parameters are base B, inclined ribs horizontal component b, inclined ribs vertical component h, height H, a connector length and uniform thickness T. 8. The material of claim 7 wherein an inclined rib connected to the base of the head portion is equal in orientation and length to an inclined rib connected to the base of the tail portion. 9. The material of any preceding claims 2 or 6 to 8 wherein the tail portion is functionally a re-entrant cell portion and the head portion is functionally a hexagonal cell portion with negative and positive Poisson's ratio, respectively. 10. The material of any preceding claim wherein the fish shaped cell configuration of the tessellated cellular structure are arranged in columns, positioned next to each other. 11. The material of claim 10 wherein the fish shaped cells of a first column are connected to fish shaped cells of a second column by connectors. 12. The material of claim 10 or 11 wherein the fish shaped cells in any column are connected directly to each other, such that a head portion of a first fish shaped cell forms a base of a tail portion of a second fish shaped cell. 13. The material of any one of claims 10 to 12 wherein the columns are arranged in opposite directions such that each head portion of a fish shaped cell will be connected to an adjacent tail portion of a fish shaped cell of an adjacent column. 14. The material of any one of claims 10 to 12 wherein the columns are arranged in the same direction such that one fish shaped cell is offset and higher with respect to a next horizontally adjacent fish shaped cell of an adjacent column. 15. The material of claim 13 wherein at least one edge of the tessellated cellular structure is modified by removal or addition of parts of the fish shaped configuration. 16. The material of claim 10 wherein at least two vertical connectors connect adjacent cells in a column. 17. The material of any preceding claim wherein the fish shaped configuration is comprised from a concave polygon with an even number of walls and a convex polygon with the same even number of walls, or section parts of the concave and convex polygon shapes. 18. The material of claim 17 wherein an innermost wall of the concave polygon is parallel with an outermost wall of the convex polygon. 19. The material of claim 17 or 18 wherein an inclined wall of the concave polygon is common and continuous with an inclined wall of the convex polygon. 20. The material of claim 17, 18 or 19 further including connector elements extending laterally outwardly from corners of the convex polygon and/or concave polygon, for connection to a laterally adjacent cell. 21. The material of any preceding claim 1 to 16 wherein the fish shaped configuration is comprised of a continuous curve from a convex head portion to a concave tail portion and wherein the head portion is wider than the tail portion. 22. The material of claim 21 where the continuous curves on two sides of the cell are an odd function with respect to points (±B, H/2). 23. A morphing skin incorporating the material of any preceding claim. |

Technical Field

The present invention relates to a material having a novel cellular structure design exhibiting Zero Poisson's Ratio (ZPR). Such a material, or 'metamaterial', can have particular application in the field of aerospace and morphing structures, e.g. as a skin to be installed on an airframe structures. The applications may extend from aerospace structures and wind energy to biomedical engineering for tailoring scaffold and implant properties and even go further into electronics as a substrate for stretchable displays and circuits.

Background to the invention

The term 'metamaterial' generally refers to a group of materials designed in micro/nano scale to exhibit exotic behaviour in the macroscale. These engineered material properties can be applied to different physical disciplines such as electromagnetics, optics, acoustics, and structural mechanics. The smallest fundamental representative volume element or unit cell of the metamaterial is designed with a specific geometry to be repeated in two or three dimensions to create larger-scale structures with the desired exotic property. Recent advances in additive manufacturing have enabled the fabrication of structures with arbitrarily complex nano/micro-architecture, which has attracted increasing attention to the field of metamaterials.

Mechanical metamaterials have a wide range of applications in aerospace, biomechanics, and renewable energy structures. Such materials are important because they provide an exotic range of values for mechanical properties, e.g. elasticity modulus, bulk modulus, Poisson's ratio and coefficient of thermal expansion. Each property offers flexibility for the designer to tune mechanical properties according to requirements. Particularly, Negative Poisson's Ratio (NPR) and Zero Poisson's Ratio (ZPR), negative and zero coefficient of thermal expansion, negative stiffness, and negative bulk modulus are exotic mechanical properties for which metamaterials can be developed.

Poisson's ratio is defined as the ratio of lateral contraction to the longitudinal extension of a material subject to tensile loading, and the ratio of lateral expansion to longitudinal contraction in the case of compressive loading. Materials with a Positive Poisson's Ratio (PPR), i.e. that become thinner in a first direction as they extend/stretch in a second direction, are common in nature (typically with a value in the vicinity of 0.3 and potentially up to 0.5), while ZPR and NPR behaviour can be achieved through metamaterial designs. NPR metamaterials, referred to as 'auxetics', become thicker perpendicular to the applied force. Based on its deformation mechanism, NPR metamaterials may be classified in three main categories including re-entrant, chiral and rotating rigid structures.

Similar to hexagonal honeycombs, the deformation of re-entrant structures is mainly due to the local bending of ribs at hinges (hinging mechanism) where rib angle increases under extension, leading to NPR. Chiral geometry exploits both bending and rotation of the elements where ribs are attached tangentially to cylinders. Rotating rigid structures refer to solid elements connected mutually by tiny joints on vertices.

An important difference between NPR/PPR and ZPR metamaterials is the out of plane deformation. Specifically, a rectangular panel with NPR and PPR will experience synclastic and anticlastic curvatures under bending moment, while the ZPR panel experiences no curvature, i.e. it is unclastic.

ZPR metamaterials have a wide range of applications in many disciplines. For example: in tissue engineering where ZPR scaffolds may have similar function as natural tissue and provide better wound healing and tissue regrowth; in amorphous materials like viscous thin films, where ZPR materials can be tightly rolled to make soft, ultrathin coatings without thickness modification requirements or squeezing force increment; and in the textile industry with ZPR weaving patterns of fabrics. These applications will require ZPR behaviour in different scales and dimensions. In a planar context, ZPR may exist in either one or two directions. Three dimensional ZPR materials are also possible.

Various ZPR structures have been proposed in the prior art, although these have tended not to find a useful commercial application. However, complex geometry and manufacturing are the main issues in development and commercialization of mechanical metamaterials and the advent and development of additive manufacturing techniques in recent years has solved an important part of this issue, paving the way for ZPR metamaterials commercialization.

UK industries are concerned with renewable energy and biomechanics through superior structural integrity and smart self-adapting capabilities. Hence, the results of research associated with the invention can be used in the realization of advanced wind turbine blades with superior performance, tidal turbine with a reliable infrastructure and increased resilience as well as medical implants and drug delivery systems with a great conformity and efficiency. For example, design of earthquake resilient buildings and transport infrastructures through hierarchical metamaterials capable of self-adapting in extreme vibration to maximise the vibration damping and crashworthiness.

Summary of the invention

The present invention was devised in the context of developing a metamaterial with zero Poisson's ratio (ZPR) in at least two orthogonal directions for integration in morphing structures. The invention seeks to provide such a material or at least provide a material with useful properties to offer as an alternative to available prior art.

In a broad aspect, the invention is defined according to claim 1. The material of the invention has a structure exhibiting a substantially zero Poisson's ratio, embodied by a recurring fish-shaped cell which serves as the fundamental representative volume element or unit cell. A sheet of material comprised of a dense array of fish-cells provides a Poisson's ratio result closest to zero. For example, the value of Poisson's ratios obtained for a 41x41 tessellation are v _{y } = -7.43xl0 ^{5 } and v _{yx } = 4.77xl0 ^{5 } which are small enough to be negligible. Hence, the fish cell metamaterial of the invention has an effective and substantially ZPR behaviour in two orthogonal directions when there are sufficient tessellations. In other words, the overall material does not stretch in either direction since deformations at the microscale are neutralized, therefore no lateral expansion or contraction occurs at the macroscale. Notably, it is preferable to select odd numbers of tessellations to maintain symmetry along the y-direction.

The invention can be defined by the fish-shaped cell or unit. However, the fish shape may be alternatively described by any one or combination of the following analogous terms: glass cell, saddle-shaped honeycombs, vase cell, N-shaped rib honeycomb; or geometric terms: mixed honeycomb, bidirectional hybrid honeycomb, re-entrant mixed honeycomb, wavy honeycomb, corrugated rib honeycomb, modified chequered honeycomb, zig-zag honeycomb, convex-concave honeycomb, curved rib honeycomb, hexagonal re-entrant coupled honeycomb, modified auxhex honeycomb, hybrid honeycomb; or mathematical terms: odd function rib honeycomb, spline rib honeycomb, strain isolated honeycomb, cosine rib honeycomb, trigonometric rib honeycomb, sine rib honeycomb, polynomial curved rib honeycomb, Fourier series rib honeycomb. An exemplary embodiment of fish cell shape is generally an eight sided polyhedron as a combination of three trapezoids.

Brief description of drawings

Figure 1 illustrates a general view of the parameters/components of a fish cell unit according to the invention;

Figure 2 illustrates a view of a first arrangement of fish cells where the respective fish assemble in opposite directions, connected together; Figure 3 illustrates a view of an array of fish cells according to Figure 2, configured to be stressed/stretched perpendicular to the fish cell direction;

Figure 4 illustrates a view of an array of fish cells according to Figure 2, configured to be stressed/stretched along the fish cell direction;

Figure 5 illustrates a view of a second arrangement of fish cells where the cells are side by side and assemble in the same direction;

Figure 6 illustrates a view of an array of fish cells according to Figure 5;

Figure 7 illustrates a graphical representation of displacement to force applied experimentally to the material of the invention;

Figures 8A and 8B illustrate modified edge structures;

Figure 9 illustrates a first variation of cell structure;

Figure 10 illustrates a tessellated cell structure of the first variation;

Figure 11 illustrates a second variation of cell structure;

Figure 12 illustrates a third variation of cell structure;

Figure 13 illustrates a tessellated cell structure of the second variation;

Figure 14 illustrates a fourth variation of cell structure;

Figure 15 illustrates a tessellated cell structure of the fourth variation;

Figure 16 illustrates a fifth variation of cell structure;

Figure 17 illustrates a tessellated cell structure of the fifth variation;

Detailed description of the invention

Advantages of the invention will become apparent from the following detailed description, taken in conjunction with the accompanying drawings that illustrate exemplary embodiments of the invention. However, the scope of the invention is not intended to be limited to the precise details of the embodiments, with variations and equivalent constructions apparent to a skilled person deemed also to be included. Furthermore, terms for components used herein should be given a broad interpretation that also encompasses equivalent functions and features. Descriptive terms should also be given the broadest possible interpretation; e.g. the term "comprising" as used in this specification means "consisting at least in part of" such that interpreting each statement in this specification that includes the term "comprising", features other than that or those prefaced by the term may also be present. Related terms such as "comprise" and "comprises" are to be interpreted in the same manner. The description herein may also referto spatial directions such as 'top', 'bottom', 'horizontal', 'vertical', 'front' and 'rear'. These are terms relative to the context of the invention used for ease of explanation. It should be clear that these are not ultimately limiting if a construction has otherwise the same function. For example, a vertical orientation may become a horizontal orientation if it is turned wholly or partially to a new orientation.

The present description refers to embodiments with particular combinations of features, however, it is envisaged that further combinations and cross-combinations of compatible features between embodiments will be possible.

An architecture for a metamaterial unit cell according to the invention is shown in Figure 1. Particularly, the unit has the appearance of a fish with a head/body portion and a tail portion. The structure is a hybrid configuration having effectively juxtaposing re-entrant (the tail) and hexagonal (the head) cells. The mutual base that would interface between the two shapes is omitted resulting in an eight sided shape with a vertical (as pictured) line of symmetry. It can be noted that the horizontal base of the tail portion is an equivalent width to the uppermost edge of the head portion (i.e. length of 7 - 8 = 2 - 1) and the inclined ribs between nodes 7 -^ 5 and 3 -^ 2 respectively are preferably equal, i.e. same angle and length. The head portion has a greater width than the tail portion since it is a convex polygon shape with inclined ribs compared to a concave polygon shape with inclined ribs.

The geometry of the fish cell can be defined in general by ten parameters, however considering geometric constraints, this reduces to six parameters corresponding to eight nodes numbered 1-8 according to Figure 1. These parameters are base B, inclined ribs horizontal component b, inclined ribs vertical component h, height H, a connector length C and uniform thickness T (not shown). It will be apparent hereinbelow that connector lengths laterally across the tessellated cellular structure enable flexing of adjacent ribs to occur in opposite directions.

The geometry may be considered as a modified assembly of re-entrant and hexagonal cells with negative and positive Poisson's ratio, respectively. Therefore, by using a proper and consistent geometric ratio between re-entrant and hexagonal shapes, the fish geometry will have ZPR under loading in the x-direction, i.e. v _{xy } = 0 as shown in Figure 3.

In a first application of the shape devised according to the invention the fish cells are positioned next to each other (in columns or rows depending on the orientation) in opposite directions (see Figure 2) so each hexagonal cell will be connected to an adjacent re-entrant cell. As a result, the two fish assembled in opposite directions cancel each other's lateral displacement (along x-direction) when subjected to tension or compression along the y-direction, i.e. v _{yx } = 0 as shown in Figure 4. The tessellation is presented as mxn where m is the number of cells in x-direction and n defines number of cells along y- direction.

According to Figures 3 and 4, specific area abgh in the substantive central region of the metamaterial is considered for the study of ZPR behaviour. In this regard, lines ah and 6y should experience negligible length change and rotation when loaded in the x-direction (Figure 3), and similarly lines ab and gh should experience negligible length change when stretched in the y-direction (Figure 4).

It is noteworthy that, based on the parameters defined, the vertical and horizontal components of the inclined ribs should be in an acceptable range to satisfy the geometric conformability. These ranges of parameters are: h

0 < - < 0.5

H

The main property obtained by employing the exemplified geometry is a unit cell of a metamaterial structure having zero Poisson's ratio in two directions.

Creation of a large metamaterial structure composed of a fish cell unit requires the cells to be arranged together and connected in an order which satisfies the ZPR requirements. In this regard, the hexagonal head portion of each cell should be attached to a re-entrant tail portion of the other cell (convex to concave attachment). Considering this connection constraint, there will be two sorts of tessellations, e.g. as already described by Figure 2 and that of Figure 5.

Referring to Figures 2, 3 and 4, two adjacent cells/columns juxtapose in an opposite 'swimming' direction where hexagonal and re-entrant section connect to each other laterally. For an opposite sided tessellation of this type, the total height will be H _{tes } = n x H _{ceU } and the length of the tessellation will be L _{tes } = m x (2 B + 2 C) .

With reference to Figures 5 and 6, two adjacent cells/columns retain the same swimming direction, however, one cell is offset and shifts higher with respect to the next horizontally adjacent cell to comply with a convex to concave connection requirement. In the tessellation of Figure 6, the outer (at least upper and lower) edges of the array will not be as smooth as the opposite sided orientation of Figures 3 and 4 since jagged indentations exist due to the shift of cells in even columns. For the same direction tessellation, the total height will be H _{tes } = (n + 0.5) x H _{ceU } and the length of the tessellation will be the same as opposite sided tessellation equal with L _{tes } = m x (2 B + 2 C).

Regarding the structural behaviour of a fish cell metamaterial, this is best understood by analysis of Poisson's ratio. Poisson's ratio is defined as the ratio of strains in the lateral direction to actuation direction as presented in Eq. 1 and Eq. 2 below. Considering rectangle abgh of Figure 3, e _{c } is defined as the ratio of length change in line ab to its initial length and s _{y } (Figure 4) is defined as the ratio of length change in line 6y to its initial length.

The elastic modulus is calculated according to Eq. 3 and Eq. 4 below where P indicates the total reaction forces along loading direction, W is the panel thickness, l _{x } and l _{y } are the lengths of the panel in x and y directions, respectively.

Structural behaviour of the metamaterial is dependent on the length ratio of the unit cell to the structure domain. So, it is important to achieve a tessellation with a right number of cells to represent a homogenized behaviour. Properties of a homogenous metamaterial structure are expected to be independent of unit cell size. For this purpose, four models with tessellation numbers of 5x5, 11x11, 21x21 and 41x41 were investigated. The results show good convergence and homogeneity. The reason for selecting odd numbers of tessellation is maintaining symmetry along the y-direction.

To conclude parametric analysis results, it was observed that a 41x41 tessellation (i.e. the greatest number of fish cells in an array) is a good representative of a homogenous metamaterial for further studies. It was also shown there was a convergence of results for the trend of elastic modulus in two directions, where an elastic modulus is normalized with respect to the original constructing material elastic modulus. The elasticity modulus of the metamaterial is significantly smaller than the original constructing material, i.e. is an order of 10 ^{7 } for T/C = 1/20, however the ratio is dependent on the thickness of the members. This very low stiffness ratio is useful for biomedical applications such as implants, where the tuning of a high compliance structure is required in interaction with soft tissues.

By way of summary, the value of Poisson's ratios obtained for 41x41 tessellation are v _{y } = - 7.43xl0 ^{5 } and v _{yx } = 4.77xl0 ^{5 } which are small enough to be negligible. Hence, the proposed fish cell metamaterial has ZPR behaviour in two orthogonal directions.

During testing the best manufacturing technique was additive manufacturing where fine tolerance can be achieved, however, alternative techniques may be employed as will be known or become known to a skilled person. In general, a high density of cells for the material area is preferable. The material can be manufactured at any practical scale, limited only by production techniques.

Figure 7 shows a force-displacement diagram of various metamaterial samples according to the invention in x- and y-direction. For the specific case of experiment where constructing material is Nylon PA2200 and certain dimensions for Fish Cell is employed, the failure load and displacement of experimental metamaterial samples in x-direction and y- direction are 659.3 N, 628.9 N as well as 23.0 mm and 41.7 mm, respectively. The ultimate displacement capacity of the metamaterial in the y-direction is approximately 1.8 times larger than x-direction. Therefore, for applications where large deformations are required, installation of metamaterial with loading along y-direction may be considered. Moreover, the variation of structural stiffness is smaller in the y-direction and approximately maintains stiffness value up to failure, while metamaterial shows softening behaviour along the x-direction.

Application of the invention will now be described with reference to morphing technologies. Specifically for a vehicle, morphing refers to a set of technologies employed to increase performance by manipulating geometry of the structure and setting it at the most optimum shape and configuration.

Morphing technology has diverse applications in many sectors including aerospace, automotive, biomechanics, renewable energy systems, and structures. Research on the performance of flying vehicles shows that employing morphing structures allows continuous shape changing and multi-point adaptation that provides much higher efficiency for developing the next generation of aircraft. However, the main challenge for the development of morphing structures is a suitable morphing skin for providing a smooth aerodynamic surface, transferring the distributed aerodynamic loads to the main structural elements and enabling large shape changes through large strain capacity in the actuation direction. Hence, low in-plane stiffness and high out-of-plane stiffness are needed in the design of morphing skins to enable large shape changes with less actuation energy and to transfer aerodynamic loads efficiently.

The skins for morphing aircraft are usually made of elastomeric covers supported by a core with different geometries and deformation mechanisms. For example, a flexible skin made of a corrugated core and elastomer coatings has the main advantage of anisotropic behaviour, i.e. stiffness along the corrugation direction while flexibility in the transverse direction.

Corrugated skins have also other remarkable characteristics such as high strength to density ratio, good energy absorption, and easy fabrication. However, the intrinsic weakness of a corrugated core is thickness reduction under deformation, resulting in a significant decrease of the moment of inertia and bending stiffness. A proposed solution may be using cellular structures like honeycombs for the core of morphing skin but, although satisfying thickness requirements, such structures have PPR which leads to dependency of longitudinal strains and transverse stresses; in other words actuation in morphing direction causes stress in the orthogonal direction. This stress coupling will result in stiffness augmentation and synclastic or anticlastic curvatures under actuation, which is not desirable.

Substituting ZPR metamaterials according to the invention in place of conventional PPR honeycombs can solve the PPR issue. Metamaterials with ZPR behavior could be very beneficial for morphing application because of preventing stiffness augmentation and unclastic curvature which provides better geometric conformability. Particularly, using a fish cell metamaterial according to the invention in a morphing skin should solve stiffness augmentation issues as the stress coupling will no longer exist.

Although theoretically a homogenized fish cell metamaterial exhibits ZPR behaviour in two orthogonal directions, in practice perfect homogeneity might not be always be achieved due to manufacturing limits where the ratio of the unit cell to whole panel size is not small. Therefore, it is important to have a criterion for evaluating integration such as edge deformations and debonding stress. It will be apparent that it is also important to ensure the ratio of the unit cell to whole panel size is small, i.e. a high density of cells for a given material surface area.

The edge of a material surface may also be modified in order to take into account integration problems such as stiffness augmentation, edge smoothness fluctuating trends, shear-extension coupling effects and dependency of edge smoothness to odd and even tessellations that may arise in the absence of perfect homogeneity. Further topology and shape optimizations may be performed on the edges to improve these issues.

Particularly, a proposed enhancement for the geometries of edges+ _{y } is shown in Figure 8A where fish cells are removed alternately. In addition, a further/alternative proposed enhancement for the geometries of edges+ consists of an additional half of fish cells for each edge, as shown in Figure 8B. Enhancements were found to have a positive effect. An important achievement due to the edge enhancement might be preventing stiffness augmentation, as presented in Tables 1 and 2 below, where stiffness remains approximately constant for both cases of free and constrained side edges. This means that the fish cell metamaterial has excellent integration capabilities in addition to its ZPR performance in the x- and y-direction.

Table 1- Stiffness ratios for free and constrained panel with enhanced edge in the x-direction

Table 2- Stiffness ratios for free and constrained panel with enhanced edge in the y-direction

The foregoing concepts for geometric edge enhancements are proposed numerically to alleviate stiffness augmentation as the main issue in the development of morphing structures.

Variations on the fish cell structure of the invention are possible. Several derivatives are outlined with reference to Figures 9 to 17.

Referring to Figures 9 and 10, two vertical connectors can be added to the cell. There will be no significant effect on ZPR characteristics so long as there are a sufficient number of tessellations.

Particularly, the inventive concept behind the fish cell geometry is mainly related to the topology of the inclined ribs and their connection to other cells. To achieve a suitable deformation mechanism leading to ZPR, the inclined ribs geometry should be an odd function with respect to the points (B, H/2) and (-B, H/2) for right and left ribs, respectively. This means that several geometries can theoretically be included in a fish cell design so long as this constraint is satisfied. In this case, any set of concave and convex polygons with an even number of edges can replace hexagonal and re-entrant cells while maintaining symmetry along the y-axis. A fish cell modification with octagonal cells is shown in Figure 11 (and Figure 13 tessellation). It should be noted that fish cells consisting of even numbered polygons with more than six edges, which are multiples of 4, require four connectors instead of two on each side. This can be compared to a decagonal cell configuration as shown by Figure 12.

In addition to polygons, the odd functionality of inclined ribs geometry can theoretically be achieved with continuous curves. In the case illustrated by Figures 14 and 15, two joined half circles can also satisfy this condition and provide ZPR characteristics in two directions.

In general, the odd function for the inclined ribs can be drawn by a parametric curve such as B-spline which passes through the point (±B, H/2) as shown in Figure 16 and generates tessellation as shown in Figure 17. The change of function for inclined ribs does not affect the range of parameters which exist naturally for a fish cell design but will directly affect the elastic properties. Based on the number of control points considered to define the B- spline, an optimum curve to achieve certain properties can be obtained. This way, the B- spline based fish cell can be very useful from an optimization point of view.

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