TRUSS-LATTICE MATERIALS CONSISTING OF MIXED POLYGONS

Abstract

Disclosed are lattice materials including a truss structure obtained by arranging struts to form at least two triangles on each side of a polygon (i.e., a triangle, a quadrilateral, or a hexagon), which are used in many engineering fields, especially in materials engineering. The elastic modulus values per unit weight of many of the lattice materials produced from the truss structure are higher than those of the most existing cellular solids with stochastic cell distributions and many existing lattice materials. Using the truss structure disclosed herein, it is possible to produce two-dimensional (planar) lattice materials as well as spherical or tubular lattice structures in the form of a first cylindrical tube or a second cylindrical tube.

Claims

1. A truss structure which forms the basis of lattice materials obtained by tessellating unit cells in the plane and ensures that the elastic modulus values per unit weight of the formed lattice materials are greater than those of many existing cellular solids with stochastic distribution and many existing lattice materials, the truss structure comprising: at least one truss structure's inner element in a triangular, quadrilateral, or hexagonal geometric form, comprising at least three inner element edges and formed by combining the inner element edges; at least twelve outer struts placed on the inner element edges of the truss structure's inner element; placed on the inner element edge joined to form at least two triangles, and change the mechanical properties of the truss structure as the number of triangles increases; the truss structure's inner element obtained by joining equal-length inner element edges to form a triangle; the truss structure's inner element obtained by joining equal-length inner element edges to form a quadrilateral; or the truss structure's inner element obtained by joining equal-length inner element edges to form a hexagonal; and the outer struts added on each inner element edge of the truss structure's inner element to form an equilateral triangle.

2. (canceled)

3. (canceled)

4. The truss structure according to claim 1, wherein the outer struts are added on each inner element edge of the truss structure's inner element to form two equal-sized triangles.

5. The truss structure according to claim 1, wherein the truss structure's inner element provides the truss structure named Hierarchical-2 Triangles by adding outer struts on each inner element edge to form two triangles of equal size.

6. The truss structure according to claim 1, wherein the outer struts are added on each inner element edge of the truss structure's inner element to form three equal-sized triangles.

7. The truss structure according to claim 1, wherein the truss structure's inner element provides the truss structure named Hierarchical-3 Triangles by adding outer struts on each inner element edge to form three triangles of equal size.

8. (canceled)

9. (canceled)

10. The truss structure according to claim 1, wherein the outer struts are added on each inner element edge of the truss structure's inner element to form two equal-sized triangles.

11. The truss structure according to claim 1, wherein the truss structure's inner element provides the truss structure named Mixed Quadrilateral-2 Triangles by adding outer struts on each inner element edge to form two triangles of equal size.

12. The truss structure according to claim 1, wherein the outer struts are added on each inner element edge of the truss structure's inner element to form three equal-sized triangles.

13. The truss structure according to claim 1, wherein the truss structure's inner element provides the truss structure named Mixed Quadrilateral-3 Triangles by adding outer struts on each inner element edge to form three triangles of equal size.

14. (canceled)

15. (canceled)

16. The truss structure according to claim 1, wherein the outer struts are added on each inner element edge of the truss structure's inner element to form two equal-sized triangles.

17. The truss structure according to claim 1, wherein the truss structure's inner element provides the truss structure named Mixed Hexagonal-2 Triangles by adding outer struts on each inner element edge to form two triangles of equal size.

18. The truss structure according to claim 1, wherein the outer struts are added on each inner element edge of the truss structure's inner element to form three equal-sized triangles.

19. The truss structure according to claim 1, wherein the truss structure's inner element provides the truss structure named Mixed Hexagonal-3 Triangles by adding outer struts on each inner element edge to form three triangles of equal size.

20. The truss structure according to claim 1, comprising a unit cell comprising six truss structures when the truss structure's inner element has a triangular geometric form, and one truss structure when the truss structure's inner element has a quadrilateral or hexagonal geometric form.

21. The truss structure according to claim 1, comprising a two-dimensional lattice created by joining of the unit cells comprising the truss structure at the apex of the triangles formed by the outer struts.

22. The truss structure according to claim 1, comprising auxiliary struts placed in the truss structure's inner element and improving the mechanical performance of the truss structure's inner element to which they are attached.

23. The truss structure according to claim 1, characterized by comprising a lattice structure in the form of a first cylindrical tube obtained by folding a two-dimensional lattice created by joining unit cells comprising the truss structure at the apex of the triangles formed by the outer struts around any axis on the plane.

24. The truss structure according to claim 1, comprising a lattice structure in the form of a second cylindrical tube obtained by folding a two-dimensional lattice created by joining unit cells comprising the truss structure at the apex of the triangles formed by the outer struts around any axis on the plane.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0023] The truss structure and lattice materials realized to achieve the objectives of the present invention are depicted in the following figures:

[0024] FIG. 1. The Kagome lattice material according to the prior art (S. Hyun, S. Torquato, 2002. Optimal and manufacturable two-dimensional Kagome-like cellular solids. Journal of Materials Research 17 (1), 137-144).

[0025] FIG. 2. A unit cell of the Kagome lattice material according to the prior art (S. Hyun, S. Torquato, 2002. Optimal and manufacturable two-dimensional Kagome-like cellular solids. Journal of Materials Research 17 (1), 137-144).

[0026] FIG. 3. The lattice material referred to as Kagome with concentric triangles in the known status of the art (W. E. D. Nelissen, C. Ayas, C. Tekoglu, 2019. 2D lattice material architectures for actuation. Journal of the Mechanics and Physics of Solids 124, 83-101).

[0027] FIG. 4. A unit cell of the Kagome with concentric triangles lattice material encountered in the known status of the art (W. E. D. Nelissen, C. Ayas, C. Tekoglu, 2019. 2D lattice material architectures for actuation. Journal of the Mechanics and Physics of Solids 124, 83-101).

[0028] FIG. 5. A view of the Hierarchical-2 Triangles truss structure, which consists of a triangular inner element and outer struts forming two triangles on each edge of the inner element.

[0029] FIG. 6. A view of the Hierarchical-2 Triangles unit cell, which was formed by joining together six Hierarchical-2 Triangles truss structures.

[0030] FIG. 7. A view of a part of the Hierarchical-2 Triangles lattice material formed by combining Hierarchical-2 Triangles unit cells.

[0031] FIG. 8. A view of the Hierarchical-3 Triangles truss structure, which consists of a triangular inner element and outer struts forming three triangles on each edge of the inner element.

[0032] FIG. 9. A view of the Hierarchical-3 Triangles unit cell, which was formed by joining together six Hierarchical-3 Triangles truss structures.

[0033] FIG. 10. A view of a part of the Hierarchical-3 Triangles lattice material formed by combining Hierarchical-3 Triangles unit cells.

[0034] FIG. 11. A view of the Mixed Quadrilateral-2 Triangles truss structure, which consists of a quadrilateral inner element and outer struts forming two triangles on each edge of the inner element.

[0035] FIG. 12. A view of the Mixed Quadrilateral-2 Triangles unit cell, which consists of a single Mixed Quadrilateral-2 Triangles truss structure.

[0036] FIG. 13. A view of a part of the Mixed Quadrilateral-2 Triangles lattice material formed by combining Mixed Quadrilateral-2 Triangles unit cells.

[0037] FIG. 14. A view of the Mixed Quadrilateral-3 Triangles truss structure, which consists of a quadrilateral inner element and outer struts forming three triangles on each edge of the inner element.

[0038] FIG. 15. A view of the Mixed Quadrilateral-3 Triangles unit cell, which consists of a single Mixed Quadrilateral-3 Triangles truss structure.

[0039] FIG. 16. A view of a part of the Mixed Quadrilateral-3 Triangles lattice material formed by combining Mixed Quadrilateral-3 Triangles unit cells.

[0040] FIG. 17. A view of the Mixed Hexagonal-2 Triangles truss structure, which consists of a hexagonal inner element and outer struts forming two triangles on each edge of the inner element.

[0041] FIG. 18. A view of the Mixed Hexagonal-2 Triangles unit cell, which consists of a single Mixed Hexagonal-2 Triangles truss structure.

[0042] FIG. 19. A view of a part of the Mixed Hexagonal-2 Triangles lattice material formed by combining Mixed Hexagonal-2 Triangles unit cells.

[0043] FIG. 20. A view of the Mixed Hexagonal-3 Triangles truss structure, which consists of a hexagonal inner element and outer struts forming three triangles on each edge of the inner element.

[0044] FIG. 21. A view of the Mixed Hexagonal-3 Triangles unit cell, which consists of a single Mixed Hexagonal-3 Triangles truss structure.

[0045] FIG. 22. A view of a part of the Mixed Hexagonal-3 Triangles lattice material formed by combining Mixed Hexagonal-3 Triangles unit cells.

[0046] FIG. 23. A view of a part of Hierarchical-3-Y1 Triangles lattice material created by adding six auxiliary struts to each hollow hexagonal region of the Hierarchical-3 Triangles lattice material, which is composed of Hierarchical-3 Triangles unit cells.

[0047] FIG. 24. A view of a part of Hierarchical-3-Y2 Triangles lattice material created by adding six auxiliary struts to each hollow hexagonal region and one auxiliary strut to each hollow quadrilateral region surrounded by the outer struts (so that the auxiliary strut would be perpendicular to the inner element edges of the truss structure) of the Hierarchical-3 Triangles lattice material, which is composed of Hierarchical-3 Triangles unit cells.

[0048] FIG. 25. A view of a part of Hierarchical-3-Y3 Triangles lattice material created by adding six auxiliary struts to each hollow hexagonal region and one auxiliary strut to each hollow quadrilateral region surrounded by the outer struts (so that the auxiliary strut would be parallel to the inner element edges of the truss structure) of the Hierarchical-3 Triangles lattice material, which is composed of Hierarchical-3 Triangles unit cells.

[0049] FIG. 26. A view of a part of Hierarchical-3-Y4 Triangles lattice material created by adding twenty-four auxiliary struts to each hollow hexagonal region and one auxiliary strut to each hollow quadrilateral region surrounded by the outer struts (so that the auxiliary strut would be perpendicular to the inner element edges of the truss structure) of the Hierarchical-3 Triangles lattice material, which is composed of Hierarchical-3 Triangles unit cells.

[0050] FIG. 27. A view of a part of Hierarchical-3-Y5 Triangles lattice material created by adding twelve auxiliary struts to each hollow hexagonal region and one auxiliary strut to each hollow quadrilateral region surrounded by the outer struts (so that the auxiliary strut would be perpendicular to the inner element edges of the truss structure) of the Hierarchical-3 Triangles lattice material, which is composed of Hierarchical-3 Triangles unit cells.

[0051] FIG. 28. A different view of a part of the Hierarchical-2 Triangles lattice material formed by combining Hierarchical-2 Triangles unit cells.

[0052] FIG. 29. A view of a circular cylindrical tube obtained by folding a piece of the Hierarchical-2 Triangles lattice material formed by combining Hierarchical-2 Triangles unit cells around the x-axis.

[0053] FIG. 30. A view of a circular cylindrical tube obtained by folding a piece of the Hierarchical-2 Triangles lattice material formed by combining Hierarchical-2 Triangles unit cells around the y-axis.

[0054] FIG. 31. A different view of a part of the Hierarchical-3 Triangles lattice material formed by combining Hierarchical-3 Triangles unit cells.

[0055] FIG. 32. A view of a circular cylindrical tube obtained by folding a piece of the Hierarchical-3 Triangles lattice material formed by combining Hierarchical-3 Triangles unit cells around the x-axis.

[0056] FIG. 33. A view of a circular cylindrical tube obtained by folding a piece of the Hierarchical-3 Triangles lattice material formed by combining Hierarchical-3 Triangles unit cells around the y-axis.

[0057] FIG. 34. A different view of a part of the Mixed Hexagonal-2 Triangles lattice material formed by combining Mixed Hexagonal-2 Triangles unit cells.

[0058] FIG. 35. A view of a circular cylindrical tube obtained by folding a piece of the Mixed Hexagonal-2 Triangles lattice material formed by combining Mixed Hexagonal-2 Triangles unit cells around the x-axis.

[0059] FIG. 36. A view of a circular cylindrical tube obtained by folding a piece of the Mixed Hexagonal-2 Triangles lattice material formed by combining Mixed Hexagonal-2 Triangles unit cells around the y-axis.

[0060] FIG. 37. A different view of a part of the Mixed Hexagonal-3 Triangles lattice material formed by combining Mixed Hexagonal-3 Triangles unit cells.

[0061] FIG. 38. A view of a circular cylindrical tube obtained by folding a piece of the Mixed Hexagonal-3 Triangles lattice material formed by combining Mixed Hexagonal-3 Triangles unit cells around the x-axis.

[0062] FIG. 39. A view of a circular cylindrical tube obtained by folding a piece of the Mixed Hexagonal-3 Triangles lattice material formed by combining Mixed Hexagonal-3 Triangles unit cells around the y-axis.

[0063] FIG. 40. A different view of a part of Hierarchical-3-Y1 Triangles lattice material created by adding six auxiliary struts to each hollow hexagonal region of the Hierarchical-3 Triangles lattice material, which is composed of Hierarchical-3 Triangles unit cells.

[0064] FIG. 41. A view of a circular cylindrical tube obtained by folding a piece of Hierarchical-3-Y1 Triangles lattice material around the x-axis. Hierarchical-3-Y1 Triangles lattice material is created by adding six auxiliary struts to each hollow hexagonal region of the Hierarchical-3 Triangles lattice material, which is composed of Hierarchical-3 Triangles unit cells.

[0065] FIG. 42. A view of a circular cylindrical tube obtained by folding a piece of Hierarchical-3-Y1 Triangles lattice material around the y-axis. Hierarchical-3-Y1 Triangles lattice material is created by adding six auxiliary struts to each hollow hexagonal region of the Hierarchical-3 Triangles lattice material, which is composed of Hierarchical-3 Triangles unit cells.

[0066] FIG. 43. A different view of a part of Hierarchical-3-Y2 Triangles lattice material created by adding six auxiliary struts to each hollow hexagonal region and one auxiliary strut to each hollow quadrilateral region surrounded by the outer struts (so that the auxiliary strut would be perpendicular to the inner element edges of the truss structure) of the Hierarchical-3 Triangles lattice material, which is composed of Hierarchical-3 Triangles unit cells.

[0067] FIG. 44. A view of a circular cylindrical tube obtained by folding a piece of Hierarchical-3-Y2 Triangles lattice material around the x-axis. Hierarchical-3-Y2 Triangles lattice material created by adding six auxiliary struts to each hollow hexagonal region and one auxiliary strut to each hollow quadrilateral region surrounded by the outer struts (so that the auxiliary strut would be perpendicular to the inner element edges of the truss structure) of the Hierarchical-3 Triangles lattice material, which is composed of Hierarchical-3 Triangles unit cells.

[0068] FIG. 45. A view of a circular cylindrical tube obtained by folding a piece of Hierarchical-3-Y2 Triangles lattice material around the y-axis. Hierarchical-3-Y2 Triangles lattice material created by adding six auxiliary struts to each hollow hexagonal region and one auxiliary strut to each hollow quadrilateral region surrounded by the outer struts (so that the auxiliary strut would be perpendicular to the inner element edges of the truss structure) of the Hierarchical-3 Triangles lattice material, which is composed of Hierarchical-3 Triangles unit cells.

[0069] The components given in the figures are enumerated individually, and the meanings of the numbers are provided below. [0070] 1. Truss structure [0071] 2. Truss structure's inner element [0072] 2.1. Inner element edge [0073] 3. Outer strut [0074] 4. Unit cell [0075] 5. Two-dimensional (2D) lattice [0076] 6. Auxiliary strut [0077] 7. First cylindrical tube (obtained by folding the two-dimensional (2D) lattice around the x-axis) [0078] 8. Second cylindrical tube (obtained by folding the two-dimensional (2D) lattice around the y-axis)

[0079] In its most basic form, a truss structure (1), which forms the basis of lattice materials obtained by tessellating unit cells in the plane and ensures that the elastic modulus values per unit weight of the formed lattice materials are greater than those of many existing cellular solids with stochastic distribution and many existing lattice materials, consists of; [0080] at least one truss structure's inner element (2) in a triangular, quadrilateral, or hexagonal geometric form, comprising at least three inner element edges (2.1) and formed by combining the inner element edges (2.1), [0081] at least twelve outer struts (3) placed on the inner element edges (2.1) of the truss structure's inner element (2), [0082] placed on the inner element edge (2.1) joined to form at least two triangles, and [0083] change the mechanical properties of the truss structure (1) as the number of triangles increases.

[0084] The unit cell (4) consists of six truss structures (1) if the truss structure's inner element (2) is in a triangular geometric form and of one truss structure (1) if the truss structure's inner element (2) is in a quadrilateral or hexagonal geometric form.

[0085] A two-dimensional (2D) lattice (5) of the desired size can be obtained by combining the required number of unit cells (4).

[0086] The truss structure (1) may optionally contain auxiliary struts (6) that can be placed in different regions, in different numbers, and in different directions within the truss structure's inner element (2), increasing the resistance of the truss structure (1) to deformation.

[0087] The two-dimensional (2D) lattice (5) obtained by joining the unit cells (4) to each other can be folded around an axis in the plane to obtain lattice structures in the form of a first cylindrical tube (7) or a second cylindrical tube (8).

[0088] The truss structure (1) at issue in this application is utilized in numerous engineering fields, especially in materials engineering. The truss structure (1) at issue in this application consists of the truss structure's inner element (2) and the outer struts (3). The truss structure (1) at issue in this application is created by placing outer struts (3) on the edges of the inner element of the truss structure, which has the geometric shape of a triangle, quadrilateral, or hexagon, to form at least two triangles on each inner element edge (2.1). The truss structures (1) combine to create the unit cell (4). The unit cell (4) is composed of six truss structures (1) if the truss structure's inner element (2) has the geometric shape of a triangle, and one truss structure (1) if the truss structure's inner element (2) has the geometric shape of a quadrilateral or hexagon. Two-dimensional (2D) lattice materials (5) are obtained by tessellating the unit cells (4) in the plane. In other words, the truss structures (1) compose the unit cells (4), and the unit cells (4) form the lattice materials.

[0089] The truss structure (1) ensures that the elastic modulus values per unit weight of the lattice materials formed from it are greater than those of existing cellular solids with stochastic distribution and the vast majority of existing lattice materials. Depending on the geometric form of the truss structure's inner element (2) and the number of triangles placed on the inner element edges (2.1), it is possible to produce lattice materials with in-plane isotropic mechanical properties and high shear elastic and modulus values. Depending on the geometric form of the truss structure's inner element (2) and the number of triangles placed on the inner element edges (2.1), it is possible to produce lattice materials with a negative Poisson's ratio (NPO). The truss structure (1) at issue in this application may optionally include auxiliary struts (6). The auxiliary struts (6) increase the resistance of the truss structure (1) against deformation. The lattice structure (5) formed from the truss structure (1) can be folded around an axis in the plane to obtain lattice structures in the form of a first cylindrical tube (7) or a second cylindrical tube (8). Spherical lattice structures can be created by joining truss structures (1).

[0090] In one embodiment of the invention at issue in this application, the truss structure's inner element (2) can be in the geometric form of a triangle, a quadrilateral, or a hexagon. The truss structure's inner element (2) includes the inner element edge (2.1). The truss structure's inner element (2) is formed by combining the inner element edges (2.1).

[0091] In one embodiment of the invention at issue in this application, the outer struts (3) are placed on the inner element edges (2.1) of the truss structure's inner element (2). The outer struts (3) are assembled to form at least two triangles on each inner element edge (2.1). As the number of triangles formed by the outer struts (3) increases, the mechanical properties of the lattice material (5) produced from the truss structure (1) change.

[0092] The unit cell (4) in one embodiment of the invention at issue in this application is formed by combining truss structures (1). The unit cell (4) consists of six truss structures (1) if the truss structure's inner element (2) has the geometric shape of a triangle, and one truss structure (1) if the truss structure's inner element (2) has the geometric shape of a quadrilateral or hexagon.

[0093] In one embodiment of the invention at issue in this application, the two-dimensional (2D) lattice (5) is obtained by combining unit cells (4). By combining truss structures (1), unit cells (4) are produced, and by combining unit cells (4), a two-dimensional (2D) lattice (5) is generated.

[0094] In one embodiment of the invention at issue in this application, auxiliary struts (6) may optionally be placed inside the truss structure (1). The auxiliary struts (6) enhance the resistance to deformation of the truss structure (1) in which they are placed.

[0095] In one embodiment of the invention at issue in this application, lattice structures in the form of the first cylindrical tube (7) or the second cylindrical tube (8) may be obtained by folding the two-dimensional (2D) lattice (5), which is obtained by joining the unit cells (4), around an axis in the plane. Truss structures (1) can be combined to create spherical lattice structures.

[0096] In this embodiment of the invention at issue in this application, the inner element (2) of the truss structure (1) has a triangular shape (FIG. 5 and FIG. 8). As shown in FIG. 5, the inner element (2) of the truss structure (1) is obtained by combining the inner element edges (2.1) of equal length to form a triangle. On each inner element edge (2.1) of the inner element (2) of the truss structure (1), outer struts (3) are assembled into equilateral triangles.

[0097] In this embodiment of the invention at issue in this application, the inner element (2) of the truss structure (1) is formed by joining three equal-length inner element edges (2.1), as shown in FIG. 5. On each inner element edge (2.1) of the inner element (2) of the truss structure (1), outer struts (3) are added to create two equal-sized triangles. By adding outer struts (3) to each inner element edge (2.1) of the inner element (2) of the truss structure (1) to form two triangles of equal size, the Hierarchical-2 Triangles truss structure is obtained.

[0098] In this embodiment of the invention at issue in this application, as shown in FIG. 6, the unit cell (4) named Hierarchical-2 Triangles is obtained by adding six truss structures (1) named Hierarchical-2 Triangles at the apex of the triangles formed by the outer struts (3).

[0099] In this embodiment of the invention at issue in this application, as shown in FIG. 7, a two-dimensional (2D) lattice (5) named Hierarchical-2 Triangles is obtained by combining unit cells (4) called Hierarchical-2 Triangles at the apex of the triangles formed by the outer struts (3).

[0100] In this embodiment of the invention at issue in this application, the inner element (2) of the truss structure (1) is formed by joining three equal-length inner element edges (2.1), as shown in FIG. 8. On each inner element edge (2.1) of the inner element (2) of the truss structure (1), outer struts (3) are added to create three equal-sized triangles. By adding outer struts (3) to each inner element edge (2.1) of the inner element (2) of the truss structure (1) to form three triangles of equal size, the Hierarchical-3 Triangles truss structure is obtained.

[0101] In this embodiment of the invention at issue in this application, as shown in FIG. 9, the unit cell (4) named Hierarchical-3 Triangles is obtained by adding six truss structures (1) called Hierarchical-3 Triangles at the apex of the triangles formed by the outer struts (3).

[0102] In this embodiment of the invention at issue in this application, as shown in FIG. 10, a two-dimensional (2D) lattice (5) named Hierarchical-3 Triangles is obtained by combining unit cells (4) called Hierarchical-3 Triangles at the apex of the triangles formed by the outer struts (3).

[0103] The two-dimensional (2D) lattice materials (5) composed of truss structures (1) named Hierarchical-2 Triangles (FIG. 5) and Hierarchical-3 Triangles (FIG. 8) have different mechanical properties. In addition, the two-dimensional (2D) lattice materials (5) composed of the truss structures (1) named Hierarchical-2 Triangles (FIG. 5) and Hierarchical-3 Triangles (FIG. 8) have transverse isotropic mechanical properties. That is, the in-plane mechanical properties of these materials are isotropic (direction independent). Similar to the Hierarchical-2 Triangles and Hierarchical-3 Triangles truss structures (1), a truss structure named Hierarchical-S Triangles can be created by increasing S, or the number of triangles formed by the outer struts (3) on the inner element edges (2.1), where S is an integer greater than or equal to 2. Two-dimensional (2D) lattice materials (5) composed of Hierarchical-S Triangles truss structures (1) have transverse isotropic mechanical properties.

[0104] In this embodiment of the invention, the Hierarchical-S Triangles lattice materials (where S is the number of triangles formed by outer struts (3)) are analyzed in the range S=2 to S=12 by increasing the number of triangles formed by the outer struts (3) on the inner element edges (2.1). The analyses revealed that the lattice material (5) named Hierarchical-3 Triangles exhibits the most stretching-dominated behavior with an exponential n value of 1.05, while the lattice material (5) named Hierarchical-2 Triangles exhibits the most bending-dominated behavior with an exponential n value of 2.91 (see Equation 2). Variations in deformation behavior influence mechanical properties. The effects of relative density on the elastic modulus, shear modulus, and Poisson's ratio were studied for lattice materials (5) consisting of Hierarchical-S Triangles truss structures (1) with S ranging from 2 to 12. The elastic (E) and shear (G) modulus values of the lattice materials (5) are normalized by dividing the elastic (E.sub.s) and shear (G.sub.s) modulus values of the solid from which the lattice is fabricated by the product of the lattice's relative density value (*). As determined by this investigation, the Hierarchical-3 Triangles lattice material (5) has the highest elastic and shear modulus values among the Hierarchical-S Triangles lattice materials (5).

[0105] For all S (2S12) and relative density values (0<*0.2) examined in this embodiment of the invention, the Poisson's ratio of the Hierarchical-S Triangle lattice materials (5) increases as the relative density increases. Poisson's ratios are negative in all relative density values studied for S6. For S>7, the Poisson ratio, which is negative at low relative density values, reaches positive values as the relative density value increases.

[0106] In one embodiment of the invention, when determining the mechanical properties of the lattice materials (5) formed by the truss structure (1) Hierarchical-S Triangles, it is assumed that all the struts forming the truss structure (1) are made from the same solid (i.e., they have the same mechanical properties) and have the same geometric properties. To modify the mechanical properties of the lattice material to meet different design specifications, it is also possible to produce lattices with struts that have different material and/or geometric properties.

[0107] In this embodiment of the invention, the truss structure's inner element (2) has a quadrilateral shape (FIG. 11 and FIG. 14). As shown in FIG. 11, the truss structure's inner element (2) is obtained by joining the equal-length inner element edges (2.1) into a quadrilateral. Outer struts (3) are added to each inner element edge (2.1) of the truss structure's inner element (2) to form equilateral triangles.

[0108] In this embodiment of the invention, as shown in FIG. 11, the truss structure's inner element (2) is obtained by joining the equal-length inner element edges (2.1) into a quadrilateral. Outer struts (3) are placed on each inner element edge (2.1) of the truss structure's inner element (2) to form two equilateral triangles. By placing outer struts (3) to form two equilateral triangles on each inner element edge (2.1) of the truss structure's inner element (2), the truss structure (1) named Mixed Quadrilateral-2 Triangles is obtained.

[0109] In this embodiment of the invention, as depicted in FIG. 12, a truss structure (1) named Mixed Quadrilateral-2 Triangles corresponds to a unit cell (4) also named Mixed Quadrilateral-2 Triangles.

[0110] In this embodiment of the invention, as depicted in FIG. 13, a two-dimensional (2D) lattice (5) named Mixed Quadrilateral-2 Triangles is obtained by joining the unit cells (4) named Mixed Quadrilateral-2 Triangles at the apex of the triangles formed by the outer struts (3).

[0111] In this embodiment of the invention, as shown in FIG. 14, the truss structure's inner element (2) is obtained by joining the equal-length inner element edges (2.1) into a quadrilateral. Outer struts (3) are placed on each inner element edge (2.1) of the truss structure's inner element (2) to form three equilateral triangles. By placing outer struts (3) to form three equilateral triangles on each inner element edge (2.1) of the truss structure's inner element (2), the truss structure (1) named Mixed Quadrilateral-3 Triangles is obtained.

[0112] In this embodiment of the invention, as depicted in FIG. 15, a truss structure (1) named Mixed Quadrilateral-3 Triangles corresponds to a unit cell (4) also named Mixed Quadrilateral-3 Triangles.

[0113] In this embodiment of the invention, as depicted in FIG. 16, a two-dimensional (2D) lattice (5) named Mixed Quadrilateral-3 Triangles is obtained by joining the unit cells (4) named Mixed Quadrilateral-3 Triangles at the apex of the triangles formed by the outer struts (3).

[0114] The two-dimensional (2D) lattice materials (5) composed of truss structures (1) named Mixed Quadrilateral-2 Triangles (FIG. 11) and Mixed Quadrilateral-3 Triangles (FIG. 14) have different mechanical properties. Similar to the Mixed Quadrilateral-2 Triangles and Mixed Quadrilateral-3 Triangles truss structures (1), a truss structure named Mixed Quadrilateral-S Triangles can be created by increasing S, or the number of triangles formed by the outer struts (3) on the inner element edges (2.1), where S is an integer greater than or equal to 2.

[0115] In one embodiment of the invention, when considering the lattice materials (5) formed by the truss structure (1) Mixed Quadrilateral-S Triangles, it is assumed that all the struts forming the truss structure (1) are made from the same solid (i.e., they have the same mechanical properties) and have the same geometric properties. To modify the mechanical properties of the lattice material to meet different design specifications, it is also possible to produce lattices with struts that have different material and/or geometric properties.

[0116] In this embodiment of the invention, the truss structure's inner element (2) has a hexagonal shape (FIG. 17 and FIG. 20). As shown in FIG. 17, the truss structure's inner element (2) is obtained by joining the equal-length inner element edges (2.1) into a hexagon. Outer struts (3) are added to each inner element edge (2.1) of the truss structure's inner element (2) to form equilateral triangles.

[0117] In this embodiment of the invention, as shown in FIG. 17, the truss structure's inner element (2) is obtained by joining the equal-length inner element edges (2.1) into a hexagon. Outer struts (3) are placed on each inner element edge (2.1) of the truss structure's inner element (2) to form two equilateral triangles. By placing outer struts (3) to form two equilateral triangles on each inner element edge (2.1) of the truss structure's inner element (2), the truss structure (1) named Mixed Hexagonal-2 Triangles is obtained.

[0118] In this embodiment of the invention, as depicted in FIG. 18, a truss structure (1) named Mixed Hexagonal-2 Triangles corresponds to a unit cell (4) also named Mixed Hexagonal-2 Triangles.

[0119] In this embodiment of the invention, as depicted in FIG. 19, a two-dimensional (2D) lattice (5) named Mixed Hexagonal-2 Triangles is obtained by joining the unit cells (4) named Mixed Hexagonal-2 Triangles at the apex of the triangles formed by the outer struts (3).

[0120] In this embodiment of the invention, as shown in FIG. 20, the truss structure's inner element (2) is obtained by joining the equal-length inner element edges (2.1) into a hexagon. Outer struts (3) are placed on each inner element edge (2.1) of the truss structure's inner element (2) to form three equilateral triangles. By placing outer struts (3) to form three equilateral triangles on each inner element edge (2.1) of the truss structure's inner element (2), the truss structure (1) named Mixed Hexagonal-3 Triangles is obtained.

[0121] In this embodiment of the invention, as depicted in FIG. 21, a truss structure (1) named Mixed Hexagonal-3 Triangles corresponds to a unit cell (4) also named Mixed Hexagonal-3 Triangles.

[0122] In this embodiment of the invention, as depicted in FIG. 22, a two-dimensional (2D) lattice (5) named Mixed Hexagonal-3 Triangles is obtained by joining the unit cells (4) named Mixed Hexagonal-3 Triangles at the apex of the triangles formed by the outer struts (3).

[0123] The two-dimensional (2D) lattice materials (5) composed of truss structures (1) named Mixed Hexagonal-2 Triangles (FIG. 17) and Mixed Hexagonal-3 Triangles (FIG. 20) have different mechanical properties. In addition, the two-dimensional (2D) lattice materials (5) composed of the truss structures (1) named Mixed Hexagonal-2 Triangles (FIG. 17) and Mixed Hexagonal-3 Triangles (FIG. 20) have transverse isotropic mechanical properties. That is, the in-plane mechanical properties of these materials are isotropic (direction independent). Similar to the Mixed Hexagonal-2 Triangles and Mixed Hexagonal-3 Triangles truss structures (1), a truss structure named Mixed Hexagonal-S Triangles can be created by increasing S, or the number of triangles formed by the outer struts (3) on the inner element edges (2.1), where S is an integer greater than or equal to 2.

[0124] In one embodiment of the invention, when determining the mechanical properties of the lattice materials (5) formed by the truss structure (1) Mixed Hexagonal-S Triangles, it is assumed that all the struts forming the truss structure (1) are made from the same solid (i.e., they have the same mechanical properties) and have the same geometric properties. To modify the mechanical properties of the lattice material to meet different design specifications, it is also possible to produce lattices with struts that have different material and/or geometric properties.

[0125] In one embodiment of the invention, auxiliary struts (6) are added to the truss structure's inner element (2) to increase the resistance of the truss structure (1) against deformation. Consequently, the mechanical properties of the lattice material (5) consisting of the truss structure (1) can be improved. In this embodiment of the invention, the truss structure (1) formed by adding auxiliary struts (6) to the truss structure's inner element (2) is named Hierarchical-3-YX Triangles (X={1, 2, 3, 4, 5}; X depends on the number and direction of auxiliary struts (6) and the region where the auxiliary struts (6) are placed). For a lattice material (5) reinforced by auxiliary struts (6) to have in-plane isotropic mechanical properties, the addition of auxiliary struts (6) must not disturb the six-fold rotational symmetry of the unit cell (4).

[0126] In this embodiment of the invention, as shown in FIG. 23, a two-dimensional (2D) lattice (5) named Hierarchical-3-Y1 Triangles is obtained by adding six auxiliary struts (6) to each hollow hexagonal region in the two-dimensional (2D) lattice (5) named Hierarchical-3 Triangles

[0127] In this embodiment of the invention, as shown in FIG. 24, a two-dimensional (2D) lattice (5) named Hierarchical-3-Y2 Triangles is obtained by adding six auxiliary struts (6) to each hollow hexagonal region and one auxiliary strut (6) to each hollow quadrilateral region surrounded by the outer struts (3) (so that the auxiliary strut would be perpendicular to the inner element edges (2.1) of the truss structure (1)) of the two-dimensional (2D) lattice (5) named Hierarchical-3 Triangles.

[0128] In this embodiment of the invention, as shown in FIG. 25, a two-dimensional (2D) lattice (5) named Hierarchical-3-Y3 Triangles is obtained by adding six auxiliary struts (6) to each hollow hexagonal region and one auxiliary strut (6) to each hollow quadrilateral region surrounded by the outer struts (3) (so that the auxiliary strut would be parallel to the inner element edges (2.1) of the truss structure (1)) of the two-dimensional (2D) lattice (5) named Hierarchical-3 Triangles.

[0129] In this embodiment of the invention, as shown in FIG. 26, a two-dimensional (2D) lattice (5) named Hierarchical-3-Y4 Triangles is obtained by adding twenty-four auxiliary struts (6) to each hollow hexagonal region and one auxiliary strut (6) to each hollow quadrilateral region surrounded by the outer struts (3) (so that the auxiliary strut would be perpendicular to the inner element edges (2.1) of the truss structure (1)) of the two-dimensional (2D) lattice (5) named Hierarchical-3 Triangles.

[0130] In this embodiment of the invention, as shown in FIG. 27, a two-dimensional (2D) lattice (5) named Hierarchical-3-Y5 Triangles is obtained by adding twelve auxiliary struts (6) to each hollow hexagonal region and one auxiliary strut (6) to each hollow quadrilateral region surrounded by the outer struts (3) (so that the auxiliary strut would be perpendicular to the inner element edges (2.1) of the truss structure (1)) of the two-dimensional (2D) lattice (5) named Hierarchical-3 Triangles.

[0131] All of the five different two-dimensional (2D) lattice materials (5) named Hierarchical-3-Y1 Triangles, Hierarchical-3-Y2 Triangles, Hierarchical-3-Y3 Triangles, Hierarchical-3-Y4 Triangles and Hierarchical-3-Y5 Triangles obtained by adding auxiliary struts (6) to the two-dimensional (2D) lattice material (5) named Hierarchical-3 Triangles have in-plane isotropic mechanical properties. Poisson's ratio is positive for the two-dimensional (2D) lattice materials (5) named Hierarchical-3-Y3 Triangles, Hierarchical-3-Y4 Triangles, and Hierarchical-3-5 Triangles, whereas it is negative for the two-dimensional (2D) lattice materials (5) named Hierarchical-3-Y1 Triangles and Hierarchical-3-2 Triangles.

[0132] In one embodiment of the invention, while the elastic and shear modulus values of both lattice materials (5) named Hierarchical-3-Y1 Triangles and Hierarchical-3-Y2 Triangles are larger than that of the lattice material named Hierarchical-3 Triangles (5), the lattice material (5) named Hierarchical-3-Y2 Triangles has the largest elastic modulus.

[0133] In one embodiment of the invention, spherical lattice structures can be created by joining the truss structures (1). In another embodiment of the invention, lattice materials (5), which are produced from the truss structure (1) and have a broad range of mechanical properties, can be folded along any axis in the plane to obtain lattice structures in the form of cylindrical tubes.

[0134] In this embodiment of the invention, a lattice structure in the form of a first cylindrical tube (7) is obtained by folding the two-dimensional (2D) lattice (5), named Hierarchical-2 Triangles and shown in FIG. 28, around the x-axis (FIG. 29).

[0135] In this embodiment of the invention, a lattice structure in the form of a second cylindrical tube (8) is obtained by folding the two-dimensional (2D) lattice (5), named Hierarchical-2 Triangles and shown in FIG. 28, around the y-axis (FIG. 30).

[0136] In this embodiment of the invention, a lattice structure in the form of a first cylindrical tube (7) is obtained by folding the two-dimensional (2D) lattice (5), named Hierarchical-3 Triangles and shown in FIG. 31, around the x-axis (FIG. 32).

[0137] In this embodiment of the invention, a lattice structure in the form of a second cylindrical tube (8) is obtained by folding the two-dimensional (2D) lattice (5), named Hierarchical-3 Triangles and shown in FIG. 31, around the y-axis (FIG. 33).

[0138] In this embodiment of the invention, a lattice structure in the form of a first cylindrical tube (7) is obtained by folding the two-dimensional (2D) lattice (5), named Mixed Hexagonal-2 Triangles and shown in FIG. 34, around the x-axis (FIG. 35).

[0139] In this embodiment of the invention, a lattice structure in the form of a second cylindrical tube (8) is obtained by folding the two-dimensional (2D) lattice (5), named Mixed Hexagonal-2 Triangles and shown in FIG. 34, around the y-axis (FIG. 36).

[0140] In this embodiment of the invention, a lattice structure in the form of a first cylindrical tube (7) is obtained by folding the two-dimensional (2D) lattice (5), named Mixed Hexagonal-3 Triangles and shown in FIG. 37, around the x-axis (FIG. 38).

[0141] In this embodiment of the invention, a lattice structure in the form of a second cylindrical tube (8) is obtained by folding the two-dimensional (2D) lattice (5), named Mixed Hexagonal-3 Triangles and shown in FIG. 37, around the y-axis (FIG. 39).

[0142] In this embodiment of the invention, a lattice structure in the form of a first cylindrical tube (7) is obtained by folding the two-dimensional (2D) lattice (5), named Hierarchical-3-Y1 Triangles and shown in FIG. 40, around the x-axis (FIG. 41).

[0143] In this embodiment of the invention, a lattice structure in the form of a second cylindrical tube (8) is obtained by folding the two-dimensional (2D) lattice (5), named Hierarchical-3-Y1 Triangles and shown in FIG. 40, around the y-axis (FIG. 42).

[0144] In this embodiment of the invention, a lattice structure in the form of a first cylindrical tube (7) is obtained by folding the two-dimensional (2D) lattice (5), named Hierarchical-3-Y2 Triangles and shown in FIG. 43, around the x-axis (FIG. 44).

[0145] In this embodiment of the invention, a lattice structure in the form of a second cylindrical tube (8) is obtained by folding the two-dimensional (2D) lattice (5), named Hierarchical-3-Y2 Triangles and shown in FIG. 43, around the y-axis (FIG. 45).