GEOMETRICALLY TAILORED PYRAMIDAL LATTICE STRUCTURES WITH I-SHAPED STRUTS

BACKGROUND OF THE INVENTION

Pyramidal truss lattices may possess characteristics that make these lattices attractive as candidates for core materials of sandwich structures. The characteristics can include a low density, an ease of fabrication, tunable mechanical properties, and an ability to absorb kinetic energy in an event of an impact or blast. Mechanical properties of pyramidal truss lattices can be adjusted by altering geometrical parameters. The geometrical parameters can include strut slenderness, inclination angle, and strut spacing. Effects of the geometrical parameters on crush strength and energy absorption can be well understood. Despite attractive properties, such as high weight-specific shear strength, responses of pyramidal truss lattices can be highly bend-dominated and can limit out-of-plane strength and stiffness, particularly for low relative density lattices.

BRIEF SUMMARY

Geometrically tailored pyramidal lattice structures including inclined struts with I-shaped cross-sections can effectively support loads. For example, a truss system described herein can include a plurality of cells. Each of the cells can include a base plate. Additionally, each of the cells can include a plurality of struts. Each strut can include an I-shaped cross-section. Each strut can be attached at one end to the base plate. Each of the cells can also include a top plate. Each strut can be attached to the top plate at an opposite end. Additionally, each strut can extend at an inclined angle between the base plate and the top plate.

In another example, a unit cell of a truss system described herein can include a base plate. Additionally, the unit cell can include a plurality of struts. Each strut of the unit cell can include an I-shaped cross-section. Each strut can be attached at one end to the base plate. The unit cell can also include a top plate. Each strut can be attached to the top plate at an opposite end. Additionally, each strut can extend at an inclined angle between the base plate and the top plate.

In an example, a method described herein can include forming a base plate. The method can also include forming a plurality of struts. Each strut can include an I-shaped cross-section. Additionally, the method can include attaching one end of each strut to the base plate. The method can further include forming a top plate. The method can include attaching an opposite end of each strut to the top plate by extending an inclined angle of each strut between the base plate and the top plate.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a perspective view of a truss system including a number of unit cells according to some aspects of the present disclosure.

FIG. 2 is a head-on view along a y axis of a unit cell for a truss system according to some aspects of the present disclosure.

FIG. 3 is a head-on view of a cross-section A-A for a strut of a unit cell for a truss system according to some aspects of the present disclosure.

FIG. 4 is a flow chart of an example of a process for making a unit cell of a truss system according to some aspects of the present disclosure.

FIG. 5 is a flow chart of an example of a process that can be performed by a unit cell of a truss system according to some aspects of the present disclosure.

FIG. 6 is a diagram of five cross-sections of different shapes involved in a parametric study according to some aspects of the present disclosure.

FIG. 7 is a bar chart of normalized compressive modulus for five struts with cross-sections of different shapes involved in a parametric study according to some aspects of the present disclosure.

FIG. 8 is a bar chart of normalized peak strength for five struts with cross-sections of different shapes involved in a parametric study according to some aspects of the present disclosure.

FIG. 9 is a bar chart of normalized energy absorbed for five struts with cross-sections of different shapes involved in a parametric study according to some aspects of the present disclosure.

FIG. 10 is a graph with relative density contours for cross-sectional design parameters t and w for struts with I-shaped cross-sections involved in a parametric study according to some aspects of the present disclosure.

FIG. 11 is a bar chart of normalized compressive modulus for struts with I-shaped cross-sections of different parameters according to some aspects of the present disclosure.

FIG. 12 is a bar chart of normalized peak strength for struts with I-shaped cross-sections of different parameters according to some aspects of the present disclosure.

FIG. 13 is a bar chart of normalized energy absorbed for struts with I-shaped cross-sections of different parameters according to some aspects of the present disclosure.

DETAILED DESCRIPTION OF THE INVENTION

Certain aspects and examples of the present disclosure relate to geometrically tailored pyramidal lattice structures including inclined struts with I-shaped cross-sections. The I-shaped cross-sectional properties can be carefully tailored to obtain a more uniform stress distribution under various loading conditions in comparison to conventional prismatic, square, or circular cross-sections. The improved bending stiffness can enhance an out-of-plane compressive modulus, crush strength, or energy absorption of the pyramidal lattice structures. An enhancement of such properties can be verified experimentally using three-dimensional (3D) printed prototypes. Also, a finite element (FE) model can be developed to predict collapse response and can be validated through a comparison with experimental results. The FE model can assess a sensitivity of an out-of-plane mechanical response to variations in design parameters of the geometrically tailored pyramidal lattice structures. A parametric study can reveal that the geometrically tailored pyramidal lattice structures can outperform conventional square-strut pyramidal lattices of equal weight in terms of strength and energy absorption for a wide range of design parameters. The geometrically tailored lattice structures may be beneficial to applications in aerospace, defense, and automotive industries.

FIG. 1 is a perspective view of a truss system 100 including a number of unit cells according to some aspects of the present disclosure. For clarification, a unit cell 105 is highlighted in FIG. 1. The unit cell 105 can include a base plate 102, a top plate 104, and struts, such as struts 106A-C. The top plate 104 can have dimensions of width, w, and length, L. The width, w, can extend along a y axis and the length, L, can extend along an x axis. The base plate 102 can also have dimensions of width and length. In some examples, the width and length of the base plate 102 is equivalent to the width and length of the top plate 104. A height, h, can describe a vertical separation between the base plate 102 and the top plate 104. In some examples, the dimensions can all be equivalent, (e.g. w=L=h).

Each of the struts can have a first end attached to the base plate 102 and a second end attached to the top plate 104. Each of the struts can have an I-shaped cross-section and can extend at an inclined angle away from an xy plane. The inclined angle can be an obtuse, acute, or right angle. Although four struts (a fourth strut is hidden by strut 106B) are included in FIG. 1, the unit cell 105 can include any number of struts, including a single strut. In some examples, the base plate 102 and the struts can form a pyramid or a truncated pyramid. The struts can be angled toward one another so that the second end of each of the struts converge at a point on the top plate 104. In some examples, the struts can be angled towards each other so that imaginary lines extending from the second end of each of the struts converge at a point above the top plate 104.

In some examples, the pyramid can be an inverted pyramid or inverted truncated pyramid. For example, the top plate 104 and the struts can form the inverted pyramid or inverted truncated pyramid. Each of the struts can have an I-shaped cross-section and can extend at a declined angle away from an xy plane. The declined angle can be an obtuse, acute, or right angle.

The struts can be angled toward one another so that the first end of each of the struts converge at a point on the base plate 102. In some examples, the struts can be angled towards each other so that imaginary lines extending from the first end of each of the struts converge at a point below the base plate 102.

The truss system 100 can be formed by repeatedly copying and translating the unit cell 105. Translation can occur in both the x and y directions. Each unit cell can be in direct contact with neighboring unit cells so that no space separates adjacent unit cells. Once formed, the truss system 100 can have a length of a*L, where a is a number of adjacent unit cells in the truss system 100 along any x direction. The truss system 100 can also have a width b*w, where b is a number of adjacent unit cells in the truss system 100 along any y direction. A height, h, of the truss system 100 can be the same height as the height of the unit cell 105.

FIG. 2 is a head-on view along a y axis of a unit cell 105 for a truss system (e.g., truss system 100 from FIG. 1) according to some aspects of the present disclosure. The unit cell 105 can include a base plate 102, a top plate 104, and struts, such as struts 106A-C. Each of the struts can have a first end attached to the base plate 102 and a second end attached to the top plate 104. Each of the struts can extend at an inclined angle a away from an xy plane. The inclined angle a can be an obtuse, acute, or right angle.

Although four struts (a fourth strut is hidden by strut 106B) are included in FIG. 2, the unit cell 105 can include any number of struts, including a single strut. Each of the struts can have an I-shaped cross section. In some examples, the base plate 102 and the struts can form a pyramid or a truncated pyramid. The strutscan be angled toward one another so that the second end of each of the struts converge at a point on the top plate 104. In some examples, the struts can be angled towards each other so that imaginary lines extending from the second end of each of the struts converge at a point above the top plate 104. When the base plate 102 and the struts from a truncated pyramid (as in FIG. 2), a distance d along the top plate 104 can separate closest points of contact for the second end of strut 106A and the second end of strut 106C with the top plate 104. Each of the struts can have a length l and full cross sectional width S. FIG. 2 also shows a cross-section A-A associated with strut 106C.

FIG. 3 is a head-on view of a cross-section A-A for a strut of a unit cell (e.g., unit cell 105 of FIG. 1 or FIG. 2) for a truss system (e.g., truss system 100 of FIG. 1) according to some aspects of the present disclosure. The cross-section A-A can have a height and a full cross-sectional width. In some examples, the height and full cross-sectional width can each be equivalent with a value described as S. The cross-section A-A can be I-shaped and can include parallel bars 302A and 302B. The two parallel bars 302A and 302B can have a height, t, and a width equal to the full cross-sectional width. The cross-section A-A can also include a link 304 that extends perpendicularly between and connects the two parallel bars 302A and 302B. The link 304 can have a width, w, and a height, S−2t. The width, w, of the link 304 can be equal to, greater than, or smaller than the height, t, of each of the two parallel bars 302A and 302B.

The cross-section A-A can also include triangular portions 306A-D. Each of the triangular portions 306A-D can include a side that is affixed to the link 304. In some examples, the triangular portions 306A-D can be right triangles, such as right isosceles triangles with equivalent bases and heights. Each triangle can form a transition from the link 304 to either of the two parallel bars 302A and 302B. The bases and heights of the right isosceles triangles can be denoted by c as depicted in FIG. 3. A value of c can vary between 0 (e.g., the triangular portions 306A-D can be absent from the cross-section A-A) to a value of c=½(S−2t) or c=½(S−w), whichever is larger. In some examples, at least a portion of each of the triangular portions 306A-D can overlap with at least one other triangle of the triangular portions 306A-D.

FIG. 4 is a flow chart of an example of a process 2000 for making a unit cell of a truss system according to some aspects of the present disclosure. Once formed, multiple unit cells can be joined to form the truss system. The operations of the flowchart start at block 2010 and may be performed in the order indicated or in any suitable order. Other examples can involve more operations, fewer operations, different operations, or a different order of the operations shown in FIG. 4.

At block 2010, the process 2000 involves forming a base plate. The base plate can be formed of any suitable construction material including concrete, steel, stone, brick, 3D printed material such as graphene, carbon fiber, bioplastics, metals, etc. The base plate can have dimensions of width, w, length, L, and thickness, T (e.g., per unit cell) and/or width b*w, length a*L, and thickness, T (e.g., per the overall base plate).

At block 2020, the process 2000 involves forming a plurality of struts. The plurality of struts can include four or any other number of struts. Each of the struts can have an I-shaped cross-section. The I-shaped cross-section can have a height and a full cross-sectional width. In some examples, the height and full cross-sectional width can be equivalent with a value described as S. The I-shaped cross-section can include two parallel bars. The two parallel bars can have a height, t, and a width equal to the full cross-sectional width. The I-shaped cross-section can also include a link that extends perpendicularly between and connects the two parallel bars. The link can have a width, w, and a height, S−2t. The width, w, of the link can be equal to, greater than, or smaller than the height, t, of each of the two parallel bars.

The I-shaped cross-section can also include triangular portions. Each of the triangular portions can include a side that is affixed to the link. In some examples, the triangular portions can be right triangles, such as right isosceles triangles with equivalent bases and heights. The bases and heights of the right isosceles triangles can be denoted by c as depicted in FIG. 3. A value of c can vary between 0 (e.g., the triangular portions can be absent from the I-shaped cross-section) to a value of c=½(S−2t) or c=½(S−w), whichever is larger. In some examples, at least a portion of each of the triangular portions can overlap with at least one other triangle of the triangular portions.

At block 2030, the process 2000 involves attaching one end of each strut to the base plate. The one end of each strut can be attached to the base plate by any method including welding, fitting, bolting, pressing, additive manufacturing methods such as 3D printing, etc. Each strut can be attached to the base plate in a non-perpendicular manner. For example, the end of each strut can form an inclined angle with the base plate. The inclined angle can be an acute, obtuse, or right angle.

At block 2040, the process 2000 involves forming a top plate. The top plate can be formed of any suitable construction material including concrete, steel, stone, brick, 3D printed material such as graphene, carbon fiber, bioplastics, metals, etc. The top plate can have dimensions of width, w, length, L, and thickness, T (e.g., per unit cell) and/or width b*w, length a*L, and thickness, T (e.g., per the overall top plate). In some examples, the dimensions of the top plate can match the dimensions of the bottom plate. The top plate can be positioned at a height h above the bottom plate.

At block 2050, the process 2000 involves attaching an opposite end of each strut to the top plate. Such opposite end of each strut can be attached to the top plate by any method including welding, fitting, bolting, pressing, additive manufacturing methods such as 3D printing, etc. Each strut can be attached to the top plate in a non-perpendicular manner. For example, such opposite end of each strut can form an inclined angle with the top plate. The inclined angle can be an acute, obtuse, or right angle. In some examples, the base plate and the struts can form a pyramid or a truncated pyramid. The struts can be angled toward one another so that the opposite end of each of the struts converge at a point on the top plate. In some examples, the struts can be angled towards each other so that imaginary lines extending from the second end of each of the struts converge at a point above the top plate.

FIG. 5 is a flow chart of an example of a process 2500 that can be performed by a unit cell of a truss system according to some aspects of the present disclosure. The operations of the flowchart start at block 2510 and may be performed in the order indicated or in any suitable order. Other examples can involve more operations, fewer operations, different operations, or a different order of the operations shown in FIG. 5.

At block 2510, the process 2000 involves receiving a load through a top plate. The top plate can form a top portion of a unit cell of a truss system. The top plate can be formed of any suitable construction material including concrete, steel, stone, brick, 3D printed material such as graphene, carbon fiber, bioplastics, metals, etc. The top plate can have dimensions of width, w, length, L, and thickness, T (e.g., per unit cell) and/or width b*w, length a*L, and thickness, T (e.g., per an overall top plate of the truss system).

At block 2520, the process 2000 involves transferring the load from the top plate to a bottom plate through a plurality of struts. The plurality of struts can include four or any other number of struts. Each of the struts can have an I-shaped cross-section. I-shaped cross-sectional properties of each of the struts can be carefully tailored to obtain a more uniform stress distribution under the load in comparison to conventional prismatic, square, or circular cross-sections. The I-shaped cross-section can have a height and a full cross-sectional width. In some examples, the height and full cross-sectional width can be equivalent with a value described as S. The I-shaped cross-section can include two parallel bars. The two parallel bars can have a height, t, and a width equal to the full cross-sectional width. The I-shaped cross-section can also include a link that extends perpendicularly between and connects the two parallel bars.

Each strut can be attached to the base plate in a non-perpendicular manner. For example, the end of each strut can form an inclined angle with the base plate. The inclined angle can be an acute, obtuse, or right angle. Each strut can also be attached to the top plate in a non-perpendicular manner. For example, such opposite end of each strut can form an inclined angle with the top plate. The inclined angle can be an acute, obtuse, or right angle. In some examples, the base plate and the struts can form a pyramid or a truncated pyramid. The struts can be angled toward one another so that the opposite end of each of the struts converge at a point on the top plate. In some examples, the struts can be angled towards each other so that imaginary lines extending from the second end of each of the struts converge at a point above the top plate.

EXAMPLES

An enhancement of properties of struts with I-shaped cross-sections can be verified experimentally using three-dimensional (3D) printed prototypes. Geometrically tailored pyramidal lattice structures were fabricated using a Digital Light Processing (DLP) 3D printing technique. An ASIGA PRO2 desktop 3D printer with 75 μm XY resolution was utilized to print the structures using Plasgray™ photocurable resin supplied by ASIGA (Alexandria, Australia). The DLP 3D printer uses UV LEDs to polymerize the photocurable resin layer-by-layer. CAD models of the pyramidal lattice structures were prepared using SolidWorks software (version 2019) and were later converted into STL files for 3D printing. The STL files were transferred to the 3D printer through its support software (Asiga Composer™), and processing parameters were set as follows: slice thickness 0.05 mm, exposure time 100 s, heater temperature 30° C., and approach velocity 10 mm s⁻¹. Upon completion of printing sequence, the 3D printed samples were extracted from a build platform and rinsed with isopropyl alcohol to remove residual uncured resin from the structures. Specimens were then visually inspected for any defects and further post-cured under UV light for 180 s. To rule out an influence of print direction on mechanical performance, all structures were printed in a single orientation with a face sheet surface parallel to a print bed.

An FE model was applied to predict properties and performance of a variety of cross-sectional parameters. In an initial comparison, five cross-sections with different shapes were compared. FIG. 6 is a diagram of the five cross-sections 402-410 of different shapes involved in the initial comparison according to some aspects of the present disclosure. Cross-section 402 is a non-tailored square cross-section for comparison with conventional struts. Dimensions of the square cross-section were selected to equate a relative density of 10% for the square truss and for the other four cross-sections. Cross-section 404 is an I-shaped cross-section with a link width equivalent to a thickness of each parallel bar within the I shape. Cross-section 406 is an I-shaped cross-section with a link width larger than a thickness of each parallel bar within the I shape. Cross- section 408 is an I-shaped cross-section with triangular portions and a link width equivalent to a thickness of each parallel bar within the I shape. Cross-section 410 is an I-shaped cross-section with triangular portions and a link width larger than a thickness of each parallel bar within the I shape. In FIGS. 7-9, data labeled A is associated with cross-section 402, data labeled B associated with cross-section 404, data labeled C associated with cross-section 406, etc.

FIG. 7 is a bar chart 500 of normalized compressive modulus for five struts with cross-sections of different shapes involved in a parametric study according to some aspects of the present disclosure. Compressive modulus or Young's modulus can be a measure of elasticity, based on a ratio of stress acting on an object to strain produced. The bar chart 500 includes values predicted by the FE model alongside experimentally verified values. The FE model predicts an increase in the normalized compressive modulus over the square truss for each of the struts with I-shaped cross-sections 404-410 with a maximum predicted increase of 12% for a strut with cross-section 406. The experimentally verified values line up well with predicted values of the FE model, although the experimental values were all slightly higher than predicted.

FIG. 8 is a bar chart 600 of normalized peak strength for five struts with cross-sections of different shapes involved in a parametric study according to some aspects of the present disclosure. Peak strength can be a maximum value of shear stress for an object. The bar chart 600 includes values predicted by the FE model alongside experimentally verified values. The FE model predicts an increase in the normalized peak strength over the square truss for all but one of the struts with I-shaped cross-sections 404-410. The FE model predicts a slight decrease in the normalized peak strength for the strut with cross-section 404. For the other struts with I cross-section 406-410, an increase in normalized peak strength is predicted with a maximum predicted increase of 8% for struts with cross-section 408 and 410. The experimentally verified values line up well with predicted values of the FE model, although the experimental values were mostly slightly higher than predicted.

FIG. 9 is a bar chart 700 of normalized energy absorbed for five struts with cross-sections of different shapes involved in a parametric study according to some aspects of the present disclosure. The bar chart 700 includes values predicted by the FE model alongside experimentally verified values. Regarding energy absorption, the FE model predicted increases over the square truss for some struts with I-shaped cross-sections (cross-sections 404 and 408), but decreases in others (cross-sections 406 and 410). Experimentally, all of the struts with I-shaped cross-sections exhibited larger normalized energy absorption than the square shaped strut, with the strut with cross-section 408 exhibiting 168% of the energy absorbed by the square shaped strut. The experimentally verified values line up well with predicted values of the FE model, although the experimental values were slightly higher than predicted.

FIG. 10 is a graph 800 with relative density contours for cross-sectional dimensionless design parameters t and w for struts with I-shaped cross-sections involved in a parametric study according to some aspects of the present disclosure. Dimensionless quantities involved in the parametric study can be expressed in terms of dimensions illustrated in FIG. 2 and FIG. 3 as:

l = l/s
(1)

c = c/s
(2)

w = w/s
(3)

t = 2t/s
(4)

In the following equation (5), a strut distance at a top face of a unit cell is assumed to be equal to a strut thickness, d=S (see FIG. 2). Then, a relative density of the geometrically tailored pyramidal strut lattice can be expressed in terms of the non-dimensional parameters given in eq. (5):

\overline{ρ} = \frac{8 \sin α (\overline{t} + \overline{w} - \overline{t} \overline{w} + 2 {\overline{c}}^{2})}{{[2 (\sin α + 1) + \overline{l} \sin 2 α]}^{2}}

(5)

To form graph 800, some dimensionless design parameters were held constant. For example, dimensionless versions of a length of each truss, l, an inclined angle α, and a base/height of triangular portion, c, were held constant (l=7, c=0.1, α=75°). The graph 800 includes three contours, contour 802, contour 804, and contour 806. For all data points along contour 802, the relative density of the pyramidal lattice structure is 4%. For all data points along contour 804, the relative density of the pyramidal lattice structure is 7%. For all data points along contour 806, the relative density of the pyramidal lattice structure is 10%.

To demonstrate a dependence of truss performance on the cross-sectional dimensionless design parameters t and w, some data points on graph 800 are highlighted. One highlighted data point is on contour 802, three highlighted data points are on contour 804, and three more highlighted data points are on contour 806. The single highlighted data point 808 on contour 802 corresponds to a truss with 4% relative density and a link width that is nearly equivalent to a height of each of two parallel bars in the I shape (w≈t). Table 1 below summarizes characteristics of highlighted data points of graph 800.

TABLE 1

Characteristics of highlighted data

points from graph 800 of FIG. 8.

Highlighted Point
Contour
Relative Density
Description

808
802
4%

w ≈ t

810
804
7%

w < t

812
804
7%

w ≈ t

814
804
7%

w > t

816
806
10%

w < t

818
806
10%

w ≈ t

820
806
10%

w > t

FIG. 11 is a bar chart 900 of normalized compressive modulus for trusses with I-shaped cross-sections of different parameters according to some aspects of the present disclosure. The trusses have cross-sections consistent with the highlighted data points from graph 800 of FIG. 10. The compressive modulus values are normalized relative to square trusses with a same relative density as each of the trusses. Compressive modulus or Young's modulus can be a measure of elasticity, based on a ratio of stress acting on an object to strain produced. The bar chart 900 includes values predicted by an FE model. The FE model predicts an increase in the normalized compressive modulus over the square truss for each of the struts with I-shaped cross-sections associated with highlighted data points 808-820 with a maximum predicted increase of 14% for a truss with cross-section associated with data point 808.

FIG. 12 is a bar chart 1000 of normalized peak strength for struts with I-shaped cross-sections of different parameters according to some aspects of the present disclosure. The trusses have cross-sections consistent with the highlighted data points from graph 800 of FIG. 10. The peak strength values are normalized relative to square trusses with a same relative density as each of the trusses. Peak strength can be a maximum value of shear stress for an object. The bar chart 1000 includes values predicted by an FE model. The FE model predicts an increase in the normalized peak strength over the square truss for each of the struts with I-shaped cross-sections associated with data points 808-820 with a maximum predicted increase of 93% for a truss with cross-section associated with data point 812.

FIG. 13 is a bar chart 1100 of normalized energy absorbed for struts with I-shaped cross-sections of different parameters according to some aspects of the present disclosure. The trusses have cross-sections consistent with the highlighted data points from graph 800 of FIG. 10. The energy absorbed values are normalized relative to square trusses with a same relative density as each of the trusses. The bar chart 1100 includes values predicted by an FE model. The FE model predicts an increase in the normalized energy absorbed over the square truss for each of the struts with I-shaped cross-sections associated with highlighted data points 808-820 with a maximum predicted increase of 161% for a truss with cross-section associated with data point 810.

The foregoing description of certain examples, including illustrated examples, has been presented only for the purpose of illustration and description and is not intended to be exhaustive or to limit the disclosure to the precise forms disclosed. Numerous modifications, adaptations, and uses thereof will be apparent to those skilled in the art without departing from the scope of the disclosure.

GEOMETRICALLY TAILORED PYRAMIDAL LATTICE STRUCTURES WITH I-SHAPED STRUTS

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