Patent Publication Number: US-7910193-B2

Title: Three-dimensional auxetic structures and applications thereof

Description:
FIELD OF THE INVENTION 
     This invention relates generally to negative Poisson&#39;s ratio (NPR) or auxetic structures and, in particular, to three-dimensional auxetic structures and applications thereof. 
     BACKGROUND OF THE INVENTION 
     Poisson&#39;s ratio (ν), named after Simeon Poisson, is the ratio of the relative contraction strain, or transverse strain (normal to the applied load), divided by the relative extension strain, or axial strain (in the direction of the applied load). Some materials, called auxetic materials, have a negative Poisson&#39;s ratio (NPR). If such materials are stretched (or compressed) in one direction, they become thicker (or thinner) in perpendicular directions. 
     The vast majority of auxetic structures are polymer foams. U.S. Pat. No. 4,668,557, for example, discloses an open cell foam structure that has a negative Poisson&#39;s ratio. The structure can be created by triaxially compressing a conventional open-cell foam material and heating the compressed structure beyond the softening point to produce a permanent deformation in the structure of the material. The structure thus produced has cells whose ribs protrude into the cell resulting in unique properties for materials of this type. 
     Auxetic and NPR structures have been used in a variety of applications. According to U.S. Pat. No. 7,160,621, an automotive energy absorber comprises a plurality of auxetic structures wherein the auxetic structures are of size greater than about 1 mm. The article also comprises at least one cell boundary that is structurally coupled to the auxetic structures. The cell boundary is configured to resist a deformation of the auxetic structures. 
     NPR structures can react differently under applied loads.  FIG. 1  illustrates a reactive shrinking mechanism, obtained through a topology optimization process. The unique property of this structure is that it will shrink in two directions if compressed in one direction.  FIG. 1  illustrates that when the structure is under a compressive load on the top of the structure, more material is gathered together under the load so that the structure becomes stiffer and stronger in the local area to resist against the load. 
     SUMMARY OF THE INVENTION 
     This invention is directed to negative Poisson&#39;s ratio (NPR) or auxetic structures and, in particular, to three-dimensional auxetic structures and applications thereof. One such structure comprises a pyramid-shaped unit cell having four base points A, B, C, and D defining the corners of a square lying in a horizontal plane. Four stuffers of equal length extend from a respective one of the base points to a point E spaced apart from the plane. Four tendons of equal length, but less than that of the stuffers, extend from a respective one of the base points to a point F between point E and the plane. 
     The stuffers and tendons have a rectangular, round, or other cross sections. For example, the stuffers may have a rectangular cross section with each side being less than 10 millimeters, and the tendons may have a rectangular cross section with each side being less than 10 millimeters. As one specific but non-limiting example, the stuffers may be 5 mm×3 mm, and the tendons may be 5 mm×2 mm. 
     According to one preferred embodiment, the angle formed between opposing stuffers from points A and C or B and D is on the order of 60 degrees, and the angle formed between opposing tendons from points A and C or B and D is on the order of 130 degrees, though other angles may be used. 
     In three-dimensional configurations, a plurality of unit cells are arranged as tiles in the same horizontal plane with the base points of each cell connected to the base points of adjoining cells, thereby forming a horizontal layer. A plurality of horizontal layers are stacked with each point E of cells in one horizontal layer being connected to a respective one of the points F of cells in an adjacent layer. In certain applications, the structure may further including a pair of parallel plates made sandwiching a plurality of horizontal layers of unit cells. The plates may be made of any suitable rigid materials, including metals, ceramics and plastics. The structure may further include an enclosure housing a plurality of horizontal layers of unit cells, thereby forming a mattress. 
     The stuffers and the tendons may be of equal or unequal length, and may have equal or unequal cross sections. The tiles may be arranged in parallel or diagonal patterns, and different layers may include unit cells with different dimensions or compositions, resulting in a functionally-graded design. 
     The stuffers may be made of metals, ceramics, plastics, or other compressive materials, and the tendons may be made of metals, plastics, fibers, fiber ropes, or other tensile materials. In one preferred embodiment, the stuffers and tendons are made of steel, with the cross-sectional area of the tendons being less than the cross-sectional area of the stuffers. pair of parallel plates sandwiching a plurality of horizontal layers of unit cells. 
     A pair of parallel plates or panels may be used to sandwich a plurality of horizontal layers of unit cells. Such plates or panels may be composed of metals such as aluminum, fabrics, fiber-reinforced polymer composites or other materials or layers. For example, the structure may further include an enclosure housing a plurality of horizontal layers of unit cells, thereby forming a mattress. 
     The geometry, dimensions or composition of the tendons or stuffers may be varied to achieve different effective material properties along different directions, to achieve a different effective Young&#39;s modulus along different directions, or to achieve different effective Poisson&#39;s ratios along different directions. The structures may achieve different material densities in different layers. 
    
    
     
       BRIEF DESCRIPTION OF TIE DRAWINGS 
         FIG. 1  illustrates a reactive shrinking mechanism, obtained through a topology optimization process; 
         FIG. 2  illustrates a particular negative Poisson ratio (NPR) structure. 
         FIG. 3A  illustrates the material of  FIG. 2  with θ 1 =60° and θ 2 =120°; 
         FIG. 3B  illustrates the material of  FIG. 2  with θ 1 =30° and θ 2 =60°; 
         FIG. 4  illustrates how an NPR structure can be used in load-bearing application; 
         FIG. 5  illustrates a three-dimensional version of the NPR structure; 
         FIG. 6A  is an example parallel-arranged 3D NPR structure; 
         FIG. 6B  is an example diagonally-arranged 3D NPR structure; 
         FIGS. 7A and 7B  illustrate a three-dimensional NPR structure having two negative (effective) Poisson&#39;s ratios in a horizontal plane; 
         FIGS. 8A and 8B  illustrate a three-dimensional NPR structure having one negative (effective) Poisson&#39;s ratio and one positive (effective) Poisson&#39;s ratio; and 
         FIGS. 9A and 9B  illustrate a three-dimensional NPR structure having a functionally-graded arrangement in the vertical direction, in which each layer of the structure a different effective Young&#39;s modulus and Poisson&#39;s ratio. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Having discussed basic two-dimensional shrinking and shearing structures in  FIG. 1 , the reader&#39;s attention is now directed to  FIG. 2  which illustrates a negative Poisson&#39;s ratio (NPR) structure having the unique property that it will shrink along all directions when compressed in one direction. A nonlinear finite element method has been developed with a multi-step linearized analysis method to predict nonlinear behavior of this material. Effective material properties, such as Young&#39;s modulus, Poisson&#39;s ratio, material density, and load-bearing efficiency can then be calculated with consideration of the geometric nonlinear effect for any large load amplitudes. 
       FIG. 3  shows two example designs that were evaluated.  FIG. 3A  illustrates a material design with θ 1 =60° and θ 2 =120°, while  FIG. 3B  illustrates another design with θ 1 =30° and θ 2 =60°.  FIG. 3  also illustrates the predicted deformation shapes and effective material properties of the two designs, in which, v denotes the effective Poisson&#39;s ratio and E is the effective Young&#39;s modulus. In  FIGS. 3A  and B, dashed lines represent the undeformed shape, and solid lines represent the deformed shape. Comparing  FIGS. 3A  and B, it is seen that the deformation shapes of the two designs are very different under the same loading condition. The effective Poisson&#39;s ratio changed from ν=−0.96 to ν=−7.4 from design #1 to design #2, while the effective Young&#39;s modulus changed from E=1.4e3 MPa to E=2.7e3 MPa. This suggests that the second design is better suited to problems that require a large absolute value of NPR and a higher Young&#39;s modulus. 
       FIG. 4  illustrates how the NPR structure of  FIG. 1A  can be used in a typical application, wherein localized pressure is applied to an NPR structure. The original structure configuration is shown in dashed lines, and solid lines illustrate the deformed structure obtained from the simulation. As shown in the Figure, the surrounding material is concentrated into the local area due to the negative Poisson&#39;s ratio effect as the force is applied. Therefore the material becomes stiffer and stronger in the local area. 
       FIG. 5  shows how the shrinking mechanism can be extended to a three-dimensional auxetic structure. The structure is based upon a pyramid-shaped unit cell having four base points A, B, C, and D defining the corners of a square lying in a horizontal plane  502 . Four stuffers  510 ,  512 ,  514 ,  516  of equal length extend from a respective one of the base points to a point E spaced apart from plane  502 . Four tendons  520 ,  522 ,  524 ,  526  of equal length, but less than that of the stuffers, extend from a respective one of the base points to a point F between point E and the plane  502 . While this and other structures disclosed herein depict points E and F positioned downwardly from the horizontal plane, it will be appreciated that the structure and those in  FIGS. 1 ,  2 - 4  and  7  may be flipped over and produce the same effect. 
     The stuffers and tendons may be made of any suitable rigid materials, including metals, ceramics and plastics. In one embodiment, the stuffers and tendons are made of steel, with the cross-sectional area of the tendons being less than the cross-sectional area of the stuffers. For example, the stuffers may have a rectangular cross section with each side being less than 10 millimeters, and the tendons may have a rectangular cross section with each side being less than 10 millimeters. As one specific but non-limiting example, the stuffers may be 5 mm×3 mm, and the tendons may be 5 mm×2 mm. 
     According to one preferred embodiment, the angle formed between opposing stuffers from points A and C or B and D is on the order of 60 degrees, and the angle formed between opposing tendons from points A and C or B and D is on the order of 130 degrees, though other angles may be used as described in further detail below 
     In the three-dimensional embodiment, a plurality of unit cells are arranged as tiles in the same horizontal plane with the base points of each cell connected to the base points of adjoining cells, thereby forming a horizontal layer. A plurality of horizontal layers are stacked with each point E of cells in one horizontal layer being connected to a respective one of the points F of cells in an adjacent layer. In some applications, the structure may further including a pair of parallel plates made sandwiching a plurality of horizontal layers of unit cells. The plates may be made of any suitable rigid materials, including metals, ceramics and plastics. 
     The example of  FIG. 4  shows that an NPR structure can improve its performance by redistributing its materials and morphine its shape in a load-bearing event without utilizing extra energy supply. Using the new design possibilities for three-dimensional designs, more advanced load-bearing structures can be designed and tailored to a wide range of applications. For example, the configuration of  FIG. 5  may be used in applications such as the construction of mattresses. In such applications, the upper and lower “plates” would be replaced with flexible padding or fabric. As with other embodiments, the space around the unit cells may be filled with a material such as foam. 
     According to the invention, different three-dimensional NPR structures can be formed with the same unit cell but different arrangements of the unit cells.  FIG. 6A  is an example of a parallel-arranged 3D NPR structure, whereas  FIG. 6B  is an example of a diagonally-arranged 3D NPR structure. Arranging 147 unit cells (7 by 7 in each layer) in a parallel pattern, as one example of many, results in a NPR structure with a dimension of 200 mm×200 mm×60.9 mm. Arranging the same number of unit cells in a diagonal pattern results in a different NPR structure with a dimension of 141.4 mm×141.4 mm×60.9 mm and different material properties. The following table compares material properties of the above two designs for this typical example: 
     
       
         
           
               
               
               
               
               
             
               
                   
               
               
                   
                 Young&#39;s 
                 Poisson 
                 Material 
                 Material 
               
               
                 NPR Structure 
                 Modulus (MPa) 
                 Ratio 
                 Density (%) 
                 Efficiency 
               
               
                   
               
             
            
               
                 Parallel pattern 
                 2.1e2 
                 −0.76 
                 14.4 
                 14.6 
               
               
                 Diagonal pattern 
                 6.5e2 
                 −0.66 
                 21.9 
                 29.7 
               
               
                   
               
            
           
         
       
     
     By adjusting geometry, the dimensions (i.e., cross-section and/or length), and/or the composition of the tendons and/or stuffers, three-dimensional NPR structures may be designed with different Poisson&#39;s ratios in different directions. Such structures may have two negative Poisson&#39;s ratios; one negative Poisson&#39;s ratio and one positive Poisson&#39;s ratio; or two positive Poisson&#39;s ratios.  FIGS. 7A and 7B  illustrate a three-dimensional NPR structure that has two negative (effective) Poisson&#39;s ratios (−2.5 in the example) in the horizontal orientation.  FIGS. 8A and 8B  illustrate the three-dimensional NPR structure that has one negative (effective) Poisson&#39;s ratio (−8.3 in the example) and one positive (effective) Poisson&#39;s ratio (1.8 in the example) in the horizontal plan. 
     Three-dimensional structures according to the invention may also exhibit a functionally-graded arrangement, in which each layer of the NPR structure has a different effective Young&#39;s modulus and Poisson&#39;s ratio.  FIGS. 9A and 9B  show such a structure. This embodiment of the invention may be applied to various applications, including self-locking fastener mechanisms.