Patent Application: US-201113077611-A

Abstract:
a shear layer for a shear band that is used in a tire is provided that has multiple cells or units having an auxetic configuration and that are constructed from aluminum or titanium alloys . the cells may have an angle of − 10 °.

Description:
brief review of linear elastic shear deformation of honeycombs and plastic collapse unit cell geometries with conventional and npr hexagonal honeycombs are shown in fig1 . the critical geometric parameters include the cell angle ( θ ), the vertical cell length ( h ), the inclined cell length ( l ), and the wall thickness ( t ). cmt has been validated with experimental and numerical work , and describes the honeycombs elastic behavior well including configurations with negative cell angles [ 2 - 6 ]. in plane effective shear modulus from cmt is given by [ 3 ]: g 12 * = e s ⁡ ( t l ) 3 ⁢ ( h l + sin ⁢ ⁢ θ ) ( h l ) 2 ⁢ ( 1 + 2 ⁢ h l ) ⁢ cos ⁢ ⁢ θ ( 1 ) where es is the young &# 39 ; s modulus of the cell wall material . not many reports on analytical and numerical models to provide a yield point of honeycombs as a function of materials &# 39 ; strength and nonlinear deformation of honeycombs are available in the literature likely due to the complexity needed to handle geometric and material nonlinearity . nonlinear constitutive relations of in - plane tensile and shear behaviors of regular honeycombs were developed using the elastic bending theory of beams in large deflection [ 19 ]. cmt assumes that honeycombs start collapsing plastically when the bending moment in the cell walls reaches the fully plastic moment and provides a yield point of honeycombs as a function of materials &# 39 ; strength over a material &# 39 ; s linear elastic range . maximum in - plane effective strains at which the honeycomb meso - structures can tolerate deformation without local cell wall failure when subjected to in - plane shear loading are given by [ 3 ]: ( γ pl * ) 12 = 1 4 ⁢ σ ys g 12 * ⁢ ( t l ) 2 ⁢ 1 h l ⁢ cos ⁢ ⁢ θ ( 2 ) where σ ys is the yield strength of the cell wall material . it should be noted that the analytical expressions for the effective properties and maximum effective strains are restricted to be used in the linear elastic range . the relative density is often used for the cellular materials design . the relative density of hexagonal honeycombs is given by [ 3 ], ρ * ρ s = t l ⁢ ( h l + 2 ) 2 ⁢ ⁢ cos ⁢ ⁢ θ ( h l + sin ⁢ ⁢ θ ) ( 3 ) for a shear compliant structural design of honeycomb meso - structures , the layer height , h , of a honeycomb sample is chosen to be 0 . 5 inch ( 12 . 7 mm ) in the x2 direction as shown in fig1 . this aligns with other design considerations of the structure that are outside the scope of this paper . further , it allows for a refined design space in which to explore the influences of the defined honeycomb parameters on the effective mesa - structural properties . for a given honeycomb of the layer height , h , and cell angle , θ , the cell height h is defined as h = h 2 ⁢ n · ( 1 + sin ⁢ ⁢ θ α ) ( 4 ) where n is the number of unit cells in the vertical direction ( the x2 direction in fig1 ) and α = h / l . in the present study , n is restricted to be 1 for convenience . for negative cell angles , there is a geometric constraint such as there are additional constrains for the honeycomb design . for example , the simple beam theory is valid for t / l or t / h & lt ; ¼ [ 3 ]. related to the manufacturing limitation , a minimum cell wall thickness should be set : in this study , the minimum cell wall thickness of 7075 - t6 is set to be 0 . 1 mm . cell angles also have a limitation to avoid collision with adjacent cell walls : we set the range of the cell angle as − 75 ≦ θ ≦ 75 °. to avoid elastic nonlinear deformation such as buckling , the relative density should be high enough . for example , thompson et al . used the minimum criteria of the relative density of 0 . 07 , which we use in this study [ 21 ]. the goal of the shear compliant honeycomb design are to maximize both the effective shear yield strength , ( t * pl ) 12 , and the effective shear yield strain , ( υ pl ) 12 , of hexagonal honeycomb meso - structures . the detailed formulation for the shear compliant honeycomb design is shown in fig2 . available dimensions of h and l satisfying both equations ( 3 ) and ( 4 ) are shown in fig3 as a function of α and θ . for α = ½ , 1 , 2 , and 4 ,: the corresponding unit cell geometries are shown in table 1 as a function of cell angles for α = 1 / 2 , 1 , 2 , and 4 . as base materials of compliant honeycomb meso - structures , an aluminum alloy , al - 7075 - t6 and a titanium alloy , ti - 6al - 4v are selected because they have higher yield strains than conventional metals . mechanical properties of al - 7075 - t6 and ti - 6al - 4v are shown in table 2 . these two materials are selected for prototyping and experimental validation purposes . the physical validation of this work will be presented in other work . in this study , honeycomb meso - structures are designed to have an elastomer &# 39 ; s shear modulus of 10mpa . the maximum effective shear strains , ( υ pl *) 12 of honeycombs are shown in fig3 . at negative cell angles , ( υ pl *) 12 shows a high value ; about 0 . 2 and 0 . 17 for α = 2 and 4 , respectively , at θ =− 75 ° as shown in fig3 ( a ). the corresponding cell wall thicknesses of honeycombs for a g 12 * of 10 mpa are shown in fig4 ( b ) as a function of θ and α . the cell wall thickness over a positive θ has a lower value than that of a negative θ due to the closely - packed cells for positive θ . t increases with a negative θ , but starts decreasing at θ =− 60 ° and − 45 ° for α = 2 and 4 , respectively . the angles are the transition points changing from expanding cell shapes to the contraction ones . effective poisson &# 39 ; s ratios as a function of e and a are shown in fig5 . in general , hexagonal honeycombs are not isotropic except for θ = 30 ° with α = 1 and − 30 ° with α = 2 ; v 12 * and v 21 * of the honeycomb for θ = 30 ° are 1 and v 12 * and v 21 * of the honeycombs for θ =− 30 ° are − 1 . both v 12 * and v 21 * have negative values over the negative cell angles . v 12 * shows a singular value near θ = 0 ° as shown in fig5 ( a ). however , v 21 * shows a continuous value over the whole range of θ . fig6 shows relative density ( ρ */ ρ s ) of honeycombs as a function of θ and α . generally , a honeycomb having a negative θ has a higher relative density than that of positive θ due to the higher t for a negative θ to meet the required g 12 *. a higher α (= h / l ) shows a higher relative density over a positive e as shown in fig6 . however , a lower α shows a higher relative density over negative range of θ . with 7075 - t6 ( ρ s = 2800 kg / m3 ) as a base material , honeycombs structures can be designed to have a density from about 140 to 1260 kg / m3 . fig7 shows t / l and t / l as a function of θ and α to make honeycombs have a g 12 * of 10 mpa . at a higher α , l has a lower value , resulting in a higher t / l as shown in fig7 ( a ). on the other hand , h has a higher value at a higher α , resulting in a lower t / h as shown in fig7 ( b ). effective plastic shear strengths of 7075 - t6 honeycombs as a function of density ( ρ *) and a are shown in fig8 . the corresponding cell wall thicknesses and cell angles have already been shown in fig3 ( b ). a higher plastic shear strength is expected with a higher α with a higher npr honeycombs , e . g ., the maximum values of ( t pl *) 12 of npr honeycombs for α = 2 and 4 are about 1 . 6 and 1 . 8 mpa , respectively , when the honeycombs are designed to be a g 12 * of 10 mpa . the higher npr honeycombs have a high cell wall thickness which causes a high density . the effective shear plastic strains can also be shown as a function of ρ * ( fig9 ). a higher a with npr has a higher ( υ pl *) 12 , e . g ., the maximum values of ( υ pl *) 12 are about 0 . 16 and 0 . 18 for α = 2 and 4 , respectively , when the honeycombs are designed to be a g 12 * of 10 mpa . a npr honeycomb with a high t show a high ( υ pl *) 12 , resulting in a high ρ * value ; e . g ., 0 . 25 mm . when honeycombs are designed to have a g 12 * of 10 mpa , effective shear strength and effective maximum shear strain are plotted for various a ( fig1 ). at a higher α , the dual property — high shear strength and high effective shear strain are obtained . the re - entrant geometry with npr shows the high dual target property . the higher cell wall thickness at a lower cell angle has cell walls avoid the elastic buckling . therefore , npr structures appear to be good to avoid the elastic buckling . the npr structures &# 39 ; high cell wall thickness also appears to tend to avoid fracture . auxetic honeycombs having negative poisson &# 39 ; s ratio show lower effective shear modulus and higher maximum effective shear strain than the regular counterparts , which means that the auxetic honeycombs are candidate geometries for a shear flexure design . as can be seen , the present invention provides a method for designing a suitable substitute for elastomeric materials such as rubber or polyurethane that are used in shear layer of a tire . hence , a tire with lower mass , hysteresis and rolling resistance can be designed and manufactured quickly and effectively . looking at fig1 , a the 100 having a rotational axis , x , in accordance with the design disclosed by u . s . pat . no . 7 , 201 , 194 is shown having a reinforced annular band 110 that has a shear layer with an auxetic honeycomb design that is made from an aluminum alloy . on either side of the shear layer 110 are found inextensible membranes 130 , 140 . the reinforced annular band 110 is connected to the mounting band 160 by flexible spokes 150 . the mounting band 160 in turn connects the tire to a hub or wheel 10 . also , a tread 106 is connected to the outward radial portion of the annular band 110 . when the tire is loaded with a force or load , l , the tread and annular band will flatten ( not shown ) to create a pressure distribution in the contact patch of the tire that is similar to that found in a pneumatic tire that lacks such a shear layer or annular band . while these embodiments are shown using spokes as disclosed by u . s . pat . no . 7 , 201 , 194 , it is equally contemplated that the honeycomb structures could be used in embodiments with solid sidewalls such as is disclosed by u . s . pat . no . 6 , 769 , 465 . for a shear compliant structural design of honeycomb meso - structures , the layer height , h , of a honeycomb sample is chosen to be 0 . 5 inch ( 12 . 7 mm ) in the x2 direction as shown in fig1 . this aligns with other design considerations of the structure that are outside the scope of this paper . further , it allows for a refined design space in which to explore the influences of the defined honeycomb parameters on the effective meso - structural properties . it should be understood that the present invention includes various other modifications that can be made to the exemplary embodiments described herein that come within the scope of the appended claims and their equivalents . these and other embodiments of the present invention are with the spirit and scope of the claims which follow . ju , j . d . summers , j . ziegert , and g . fadel , 2009 , design of honeycomb meta - materials for high shear flexure , in proceedings of the asme international design engineering technical conferences , detc2009 - 87730 , san diego , calif . abd el - sayed , f . k ., jones , r ., and burgess , i . w . 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