An orthotropic artificial structural material and its design method

Abstract

The present invention discloses an orthogonal anisotropic artificial structural material and a design method thereof. The artificial structural material is composed of units periodically arranged in space, and the units include two rectangular parallelepiped structures. The first structure is the basic structure of the unit, and the second structure is a transformation of the basic structure of the unit. The first structure of the unit is a rectangular parallelepiped structure composed of 24 rods connected by 8 nodes, and the second structure of the unit is a rectangular parallelepiped structure composed of 36 rods connected by 14 nodes. The artificial structural material has 6 independent elastic constants and has a large degree of freedom in material design. The elastic constant of the material can be specified relatively freely to realize the required artificial structural material, including mechanical metamaterials. The present invention has important value in the fields of mechanical metamaterials, propagation control of elastic waves, and vibration control of structures.

Description

Technical Field

[0001] The invention belongs to the technical field of material structure, and relates to an orthogonal anisotropic artificial structural material. The invention also relates to a design method of the artificial structural material. Background Art

[0002] Artificial structural materials refer to materials with artificially designed structures and special properties, including mechanical metamaterials. Mechanical metamaterials are strategic frontier technologies and emerging research hotspots that have attracted international attention. The propagation and control of elastic waves and the vibration of structures have great application value in aerospace, mechanical, civil engineering and other engineering fields, and are also long-lasting research hotspots. It can be seen from mechanical theory that the elastic constant of a material is a major factor affecting the propagation of elastic waves and the vibration of structures. Therefore, the propagation of elastic waves and the vibration of structures can be controlled by specifying the elastic constant of the material.

[0003] Orthotropic materials have three mutually orthogonal principal directions of materials, 1, 2, and 3, and three mutually orthogonal elastic symmetry planes, namely, the 1-2 plane, the 2-3 plane, and the 3-1 plane. In addition, orthotropic materials have 12 elastic constants, including 6 Poisson's ratios, 3 elastic moduli, and 3 shear moduli, but only 9 independent elastic constants, of which 3 Poisson's ratios can be represented by the other 3 Poisson's ratios and 3 elastic moduli. If the elastic constants of materials can be designed more freely, such as designing mechanical metamaterials with negative or zero Poisson's ratios, and designing artificial structural materials with specified elastic constants to regulate the propagation of elastic waves and the vibration of structures, it will be of great significance and value. However, it is very difficult to design orthotropic artificial structural materials with specified elastic constants, and no scholar has proposed a simple and effective universal design method.

[0004] The document "Ke Jiang. New unit model of elastic solid [J]. Shanxi Architecture, 2012, 38 (19): 58-59" proposed a truss unit model based on the generalized Hooke's law and the superposition principle to calculate orthogonal anisotropic elastic mechanics problems, that is, to calculate the stress, strain and displacement of each point in the solid under the action of load. Since the finite element model of the solid unit can already solve general orthogonal anisotropic elastic mechanics problems, it has not attracted the attention of scholars at home and abroad. At present, no scholar has proposed an artificial structural material and its design method based on the truss unit model. Summary of the invention

[0005] The purpose of the present invention is to provide an orthotropic artificial structural material and a design method thereof. The unit structure of the artificial structural material is simple, and it has a large degree of freedom in material design. The elastic constant of the material can be specified relatively freely to realize the required artificial structural material.

[0006] The present invention provides an orthotropic artificial structural material, and the technical scheme adopted is that the artificial structural material is composed of units arranged periodically in space; the units include two rectangular parallelepiped structures, the first structure is the basic structure of the unit, and the second structure is a transformation of the basic structure of the unit; the first structure of the unit is a rectangular parallelepiped structure composed of 24 rods connected by 8 nodes, including: 4 rods parallel to each main direction of the material, 3 main directions of the material, and a total of 12 parallel rods, in the rectangular parallelepiped structure Each face of the structure has 2 cross rods, and if there are 6 faces, there are a total of 12 cross rods, and the 8 vertices of the rectangular parallelepiped structure are used as 8 nodes connected to each rod; the second structure of the unit is different from the first structure of the unit in that the intersection of the two cross rods on each face of the rectangular parallelepiped structure is used as an intermediate node, so that the number of cross rods on each face becomes 4, and if there are 6 faces, there are a total of 6 intermediate nodes and 24 cross rods, and the others are the same, so that 36 rods are connected by 14 nodes to form a rectangular parallelepiped structure.

[0007] Any rod in the unit is allowed to be replaced by a spring or a variable-section rod in accordance with the principle of keeping the length and stiffness coefficient of the rod unchanged, wherein the stiffness coefficient of a rod refers to the axial force required to apply when the rod produces unit axial deformation, which is equal to the ratio of the axial stiffness of the rod to the length of the rod.

[0008] The design method of the artificial structure material comprises the following steps:

[0009] Step 1: The design target of the artificial structural material means specifying n independent elastic constants among 9 elastic constants, where 1≤n≤6;

[0010] According to the design goal of the artificial structure material, the design values ​​of the 9 elastic constants of the artificial structure material are determined. If the specified n independent elastic constants are known, (6-n) independent elastic constants can be freely selected. The other 3 elastic constants are obtained according to the following 3 conditions:

[0011] ,

[0012] ,

[0013] ;

[0014] in, E 1, E 2, E 3——Elastic modulus of the material in principal directions 1, 2, and 3, respectively,

[0015] v ij ——Poisson’s ratio, i.e. v ij = -ε i / ε j , is the normal stress acting alone in the j direction σ j When the other stress components are 0, the negative value of the ratio of the strain in the i direction to the strain in the j direction is

[0016] G 23 ,G 31 ,G 12 ——Shear moduli in 2-3, 3-1, and 1-2 planes respectively.

[0017] Step 2: Determine whether the nine elastic constants of the artificial structural material meet the allowable range of orthotropic elastic mechanics, and the allowable range is as follows:

[0018] E 1>0, E 2>0, E 3>0,

[0019] |v 12 |<( E 2 / E 1) 1 / 2 ,|v 23 |<( E 3 / E 2) 1 / 2 ,|v 13 |<( E 3 / E 1) 1 / 2 ,

[0020] G 12 >0,G 31 >0,G 23 >0,

[0021] ,

[0022] ;

[0023] If the nine elastic constants of the artificial structural material are all within the allowable range, proceed to the next step; otherwise, reselect the elastic constants and repeat the previous step and this step.

[0024] Step 3: Select the dimensions of the unit in three directions l 1. l 2. l 3. Then calculate the axial stiffness of each member in the unit, that is:

[0025] ,

[0026] ,

[0027] ,

[0028] ,

[0029] ,

[0030] ;

[0031] in, l 1. l 2. l 3——The dimensions of the unit in directions 1, 2, and 3, respectively.

[0032] K 1 - axial stiffness of a parallel rod parallel to the main direction 1 of the material,

[0033] K 2 – axial stiffness of a parallel rod parallel to the main direction 2 of the material,

[0034] K 3 – axial stiffness of a parallel rod parallel to the main direction 3 of the material,

[0035] K 4 – axial stiffness of a cross member parallel to the plane 1-2,

[0036] K 5 – axial stiffness of a cross member parallel to the plane 1-3,

[0037] K 6 – axial stiffness of a cross member parallel to the 2-3 plane;

[0038] For any rod in the unit, select the elastic modulus of the rod material, and then calculate the cross-sectional area of ​​the rod, that is, the cross-sectional area of ​​the rod = the axial stiffness of the rod / the elastic modulus of the rod material;

[0039] Check the cross-sectional area of ​​each rod obtained by calculation. If there is a negative cross-sectional area, adjust the size of the unit and then recalculate according to this step; if there is still a negative cross-sectional area, modify the elastic constant design value of the artificial structure material and redesign from step 1; if there is no negative cross-sectional area for each rod, select the cross-sectional shape of the rod and determine the cross-sectional size of the rod, so that the cross-sectional size of all rods in the unit can be obtained, wherein, if the cross-sectional area of ​​a rod is 0, it means that the rod is not included in the rod composition of the unit; then, from the two structures of the unit, choose to adopt the first structure or the second structure; finally, according to the size of the unit, judge whether there is a cross-sectional area of ​​each rod that is too large within the space range of the unit, resulting in crowded space and inability to connect the rods to form a unit. If this problem exists, increase the elastic modulus of the material of each rod and then re-determine the cross-sectional area and size of each rod;

[0040] The units are periodically arranged in space, and adjacent units share their adjacent nodes, thus forming the artificial structural material. If the number of rods in the artificial structural material is to be reduced, adjacent rods of adjacent units are combined into one rod by adding the cross-sectional areas.

[0041] If the accuracy of the design is to be evaluated and the actual values ​​of the elastic constants of the artificial structural materials are to be determined, experimental determination or finite element analysis is used, that is: take one of the units and perform calculations under uniaxial tension (or uniaxial compression) in the three main directions of the material, and then determine the three elastic moduli and three Poisson's ratios of the material according to the definitions of the elastic modulus and Poisson's ratio, and perform calculations under pure shear in the 1-2 plane, 2-3 plane, and 3-1 plane, and then determine the three shear moduli of the material according to the definition of the shear modulus.

[0042] The present invention has the following beneficial effects: the orthogonal anisotropic artificial structural material proposed in the present invention is composed of units periodically arranged in space, and the structure of the unit is very simple, which is convenient for manufacturing the artificial structural material; the artificial structural material of the present invention has 6 independent elastic constants, so when designing the material, it has a larger degree of design freedom, and the elastic constant of the material can be specified more freely to realize the required artificial structural material; and the material design method proposed in the present invention is very simple. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 This is a schematic diagram of an orthotropic artificial structural material of the present invention.

[0044] Figure 2This is a schematic diagram of a unit of an orthotropic artificial structural material of the present invention, (a) is a schematic diagram of a first structure of the unit, and (b) is a schematic diagram of a second structure of the unit; wherein 1, 2, 3, 4, 5, 6, 7, 8, a, b, c, d, e, and f represent nodes of the unit.

[0045] Figure 3 Schematic diagram of the calculation model of the unit of the artificial structure material of Example 1 under uniaxial tension in the main direction 2 of the material.

[0046] Figure 4 Schematic diagram of the calculation model of the solid unit corresponding to Example 1 under uniaxial tension in the main direction 2 of the material.

[0047] Figure 5 Schematic diagram of the calculation model of the unit cell of the artificial structural material of Example 1 under pure shear in the 2-3 plane.

[0048] Figure 6 It is a schematic diagram of the calculation model of the solid unit corresponding to Example 1 under pure shear in the 2-3 plane.

[0049] Figure 7 It is the deformation of the unit cell of the artificial structure material of Example 1 under uniaxial tension in the main direction 2 of the material.

[0050] Figure 8 is the deformation of the unit cell of the artificial structure material of Example 1 under pure shear in the 2-3 plane.

[0051] Fig. 9 It is the deformation of the unit cell of the artificial structure material of Example 2 under uniaxial tension in the main direction 2 of the material.

[0052] Fig.10 is the deformation of the unit cell of the artificial structure material of Example 2 under pure shear in the 2-3 plane.

[0053] Fig.11 It is the deformation of the unit cell of the artificial structure material of Example 3 under uniaxial tension in the main direction 2 of the material.

[0054] Fig.12 is the deformation of the unit cell of the artificial structure material of Example 3 under pure shear in the 2-3 plane.

[0055] Fig.13 It is the deformation of the unit cell of the artificial structure material of Example 4 under uniaxial tension in the main direction 2 of the material.

[0056] Fig.14 is the deformation of the unit cell of the artificial structure material of Example 4 under pure shear in the 2-3 plane.

[0057] Fig.15This is a unit schematic diagram of the artificial structural material of Example 2.

[0058] Fig.16 This is a unit schematic diagram of the artificial structure material of Example 4.

[0059] In the figure, each coordinate system is xyz, where x represents the main direction 1 of the material, y represents the main direction 2 of the material, z represents the main direction 3 of the material, the xy plane represents the 1-2 plane, the yz plane represents the 2-3 plane, and the zx plane represents the 3-1 plane. DETAILED DESCRIPTION

[0060] An orthotropic artificial structural material and a design method thereof of the present invention are described in detail below with reference to the accompanying drawings and embodiments.

[0061] The present invention provides an orthotropic artificial structural material, wherein the artificial structural material is composed of units arranged periodically in space, such as Figure 1 As shown; the unit includes two rectangular parallelepiped structures, the first structure is the basic structure of the unit, and the second structure is a transformation of the basic structure of the unit, such as Figure 2 shown.

[0062] The first structure of the unit is a rectangular parallelepiped structure composed of 24 rods connected by 8 nodes, such as Figure 2 As shown in (a), the unit includes 8 nodes, namely node 1, node 2, node 3, node 4, node 5, node 6, node 7, node 8, 4 rods parallel to the main direction 1 of the material, namely rod 14, rod 23, rod 67, rod 58, 4 rods parallel to the main direction 2 of the material, namely rod 12, rod 56, rod 87, rod 43, and rods parallel to the main direction 3 of the material. There are 4 rods, namely rod 15, rod 48, rod 37, and rod 26, totaling 12 parallel rods. There are 2 cross rods on each face of the rectangular structure. There are 6 faces, totaling 12 cross rods, namely rod 13, rod 24, rod 57, rod 68, rod 18, rod 54, rod 27, rod 63, rod 16, rod 25, rod 47, and rod 38.

[0063] The second structure of the unit, such as Figure 2As shown in (b), compared with the first structure of the unit, the difference is that the intersection of the two cross bars of each face of the rectangular parallelepiped structure is used as an intermediate node, so that the number of cross bars on each face becomes 4, and there are 6 faces, then a total of 6 intermediate nodes, namely node a, node b, node c, node d, node e, node f, and a total of 24 cross bars, namely rod 1a, rod 2a, rod 3a, rod 4a, rod 5b, rod 6b, rod 7b, rod 8b, rod 1c, rod 5c, rod 8c, rod 4c, rod 2d, rod 6d, rod 7d, rod 3d, rod 1e, rod 2e, rod 6e, rod 5e, rod 4f, rod 3f, rod 7f, rod 8f, and the others are the same. In this way, 36 rods are connected by 14 nodes to form a rectangular parallelepiped structure.

[0064] The dimensions of the unit in directions 1, 2, and 3 are:

[0065] l 1= l 14 = l 23 = l 67 = l 58 , l 2= l 12 = l 56 = l 87 = l 43 , l 3= l 15 = l 48 = l 37 = l 26 ,

[0066] in, l - length, e.g. l 1. l 2. l 3 are the dimensions of the unit in directions 1, 2, and 3 respectively, l 26 is the length of the rod 26 of the unit.

[0067] The rods in the unit are divided into 6 types according to the size of the axial stiffness, namely:

[0068] K 1= K 14 = K23 = K 67 = K 58 ， K 2= K 12 = K 56 = K 87 = K 43 ， K 3= K 15 = K 48 = K 37 = K 26 ，

[0069] K 4= K 13 = K 24 = K 57 = K 68 = K 1a = K 2a = K 3a = K 4a = K 5b = K 6b = K 7b = K 8b ，

[0070] K 5= K 18 = K 54 = K 27 = K 63 = K 1c = K 5c = K 8c = K 4c = K 2d = K 6d = K7d = K 3d ,

[0071] K 6= K 16 = K 25 = K 47 = K 38 = K 1e = K 2e = K 6e = K 5e = K 4f = K 3f = K 7f = K 8f .

[0072] in, K - axial stiffness of the member, e.g. K 14 is the axial stiffness of the rod 14, K 8f is the axial stiffness of rod 8f,

[0073] K 1 - axial stiffness of a parallel rod parallel to the main direction 1 of the material,

[0074] K 2 – axial stiffness of a parallel rod parallel to the main direction 2 of the material,

[0075] K 3 – axial stiffness of a parallel rod parallel to the main direction 3 of the material,

[0076] K 4 – axial stiffness of a cross member parallel to the plane 1-2,

[0077] K 5 – axial stiffness of a cross member parallel to the plane 1-3,

[0078] K 6 – Axial stiffness of a cross member parallel to the 2-3 plane.

[0079] Any rod in the unit can obviously be replaced by a spring or a variable-section rod in accordance with the principle of keeping the length and stiffness coefficient of the rod unchanged, wherein the stiffness coefficient of a rod refers to the axial force required to apply when the rod produces unit axial deformation, which is equal to the ratio of the axial stiffness of the rod to the length of the rod.

[0080] Example 1

[0081] The design goal of an orthotropic artificial structural material of this embodiment is to specify a Poisson's ratio among nine elastic constants: v 12 = -0.5.

[0082] Step 1: According to the design goal of the artificial structure material of this embodiment, determine the design values ​​of the nine elastic constants of the artificial structure material. v 12 = -0.5, and then freely select 5 independent elastic constants, v 23 =0.85, v 13 =0.75, E 1=480 N / mm 2 , E 2=400 N / mm 2 , E 3=500 N / mm 2 , the other three elastic constants are obtained according to the following three conditions:

[0083] =24.7423 N / mm 2 ,

[0084] =804.124 N / mm 2 ,

[0085] =824.742 N / mm 2 .

[0086] Step 2: E 1=480 N / mm 2 >0, E 2=400 N / mm 2 >0, E 3=500 N / mm 2 >0,

[0087] |v 12 |=0.5<( E 2 / E 1) 1 / 2=0.91,|v 23 |=0.85<( E 3 / E 2) 1 / 2 =1.12,|v 13 |=0.75<( E 3 / E 1) 1 / 2 =1.02,

[0088] G 12 =24.7423 N / mm 2 >0,G 31 =804.124 N / mm 2 >0,G 23 =824.742 N / mm 2 >0,

[0089] =0.194 >0,

[0090] ;

[0091] Therefore, the nine elastic constants of the artificial structure material of Example 1 are all within the allowable range of orthotropic elastic mechanics.

[0092] Step 3: Select the dimensions of the unit in three directions: l 1=10 mm, l 2 = 10 mm, l 3=10 mm, and then calculate the axial stiffness of each member in the unit, that is:

[0093] =5381.44 N,

[0094] =2474.23 N,

[0095] =4381.44 N,

[0096] =1749.54 N,

[0097] =56860.1 N,

[0098] =58318.1 N;

[0099] For simplicity, the elastic modulus of the material of all the bars in the unit is chosen to be the same value, i.e. E =100000 N / mm 2, then the cross-sectional area of ​​each member is:

[0100] A 1= K 1 / E =0.0538144 mm 2 , A 2= K 2 / E =0.0247423 mm 2 , A 3= K 3 / E =0.0438144 mm 2 ,

[0101] A 4= K 4 / E =0.0174954 mm 2 , A 5= K 5 / E =0.568601 mm 2 , A 6= K 6 / E =0.583181 mm 2 ;

[0102] in, A 1 – cross-sectional area of ​​a parallel rod parallel to the main direction 1 of the material,

[0103] A 2 – the cross-sectional area of ​​a parallel rod parallel to the main direction 2 of the material,

[0104] A 3 – cross-sectional area of ​​a parallel rod parallel to the main direction 3 of the material,

[0105] A 4——the cross-sectional area of ​​a cross bar parallel to the plane 1-2,

[0106] A 5——the cross-sectional area of ​​a cross member parallel to the plane 1-3,

[0107] A 6 – the cross-sectional area of ​​a cross bar parallel to the plane 2-3;

[0108] The cross-sectional area of ​​each rod is not a negative number, so there is no need to adjust the size of the unit, nor to modify the design value of the elastic constant of the artificial structural material; for simplicity, the cross-sectional shape of all rods is selected to be circular, and it is obvious that it is easy to determine the cross-sectional size of each rod; from the two structures of the unit, the first structure is selected; no drawing is required, and after a simple estimate, it is easy to determine that within the spatial range of a unit, the cross-sectional size of each rod is appropriate, and the rods can be connected to form a unit, and the units are periodically arranged in space, and adjacent units share their adjacent nodes, thus forming the artificial structural material.

[0109] In Example 1, if the elastic modulus of all rod materials is selected to be E1=480 N / mm 2 , recalculate the cross-sectional area and cross-sectional size of each rod, and it can be clearly determined that within the spatial range of a unit, the cross-sectional area of ​​each rod is too large, resulting in crowded space and the rods cannot be connected to form a unit. Therefore, after determining the cross-sectional shape and cross-sectional size of each rod and selecting the structure of the unit, it is necessary to determine whether the rods can be connected to form a unit within the spatial range of a unit. In fact, within the spatial range of a unit, as long as the elastic modulus of the material of each rod in the unit is selected to be large enough, the cross-sectional area and cross-sectional size of each rod will be small enough, and the rods can always be connected to form a unit.

[0110] Example 2

[0111] The design goal of an orthotropic artificial structural material of this embodiment is still: among the nine elastic constants, one Poisson's ratio is specified as: v 12 = -0.5.

[0112] Compared with Example 1, the design steps of Example 2 remain unchanged except that the structure of the unit is changed from the first structure to the second structure, that is, the design steps 1 and 2 of Example 2 are exactly the same as those of Example 1. In step 3, the size of the unit, the elastic modulus of the material of each rod, and the axial stiffness, cross-sectional area, cross-sectional shape and cross-sectional size of each rod are also exactly the same.

[0113] Example 3

[0114] The design goal of an orthotropic artificial structural material of this embodiment is to specify one Poisson's ratio and one elastic modulus among nine elastic constants: v 12 =0, E 3=1000 N / mm 2 .

[0115] Step 1: According to the design goal of the artificial structure material of this embodiment, determine the design values ​​of the nine elastic constants of the artificial structure material. v 12 =0, E 3=1000 N / mm 2 , and then freely select 4 independent elastic constants, v 23 =0.4, v 13 =0.3, E 1=800 N / mm 2 , E 2=1200 N / mm 2 , the other three elastic constants are obtained according to the following three conditions:

[0116] =156.522 N / mm 2 ,

[0117] =326.087 N / mm 2 ,

[0118] =652.174 N / mm 2 .

[0119] Step 2: E 1=800 N / mm 2 >0, E 2=1200 N / mm 2 >0, E 3=1000 N / mm 2 >0,

[0120] |v 12 |=0<( E 2 / E 1) 1 / 2 =1.22,|v 23 |=0.4<( E 3 / E 2) 1 / 2 =0.91,|v 13 |=0.3<( E 3 / E 1) 1 / 2 =1.12,

[0121] G 12 =156.522 N / mm 2 >0,G 31 =326.087 N / mm2 >0,G 23 =652.174 N / mm 2 >0,

[0122] =0.736 >0,

[0123] ;

[0124] Therefore, the nine elastic constants of the artificial structure material of Example 3 are all within the allowable range of orthotropic elastic mechanics.

[0125] Step 3: Select the dimensions of the unit in three directions: l 1=10 mm, l 2 = 11 mm, l 3=12 mm, and then calculate the axial stiffness of each member in the unit, that is:

[0126] K 1 = 17241.0 N, K 2 = 23269.3 N, K 3 = 3107.21 N,

[0127] K 4 = 14024.6 N, K 5 = 28481.9 N, K 6 = 53284.2 N;

[0128] For simplicity, the elastic modulus of the material of all the bars in the unit is selected to be E =50000 N / mm 2 , then the cross-sectional area of ​​each member is:

[0129] A 1= K 1 / E =0.344820 mm 2 , A 2= K 2 / E =0.465387 mm 2 , A 3= K 3 / E =0.0621443 mm 2 ,

[0130] A 4= K 4 / E =0.280493 mm 2 , A 5=K 5 / E =0.569639 mm 2 , A 6= K 6 / E =1.06568 mm 2 ;

[0131] The cross-sectional area of ​​each rod is not a negative number, so there is no need to adjust the size of the unit, nor to modify the design value of the elastic constant of the artificial structural material; for simplicity, the cross-sectional shape of all rods is selected to be a square. Obviously, it is easy to determine the cross-sectional size of each rod; from the two structures of the unit, the first structure is selected; no drawing is required, and after a simple estimate, it is easy to determine that within the spatial range of a unit, the cross-sectional size of each rod is appropriate, and the rods can be connected to form a unit, and the units are periodically arranged in space, and adjacent units share their adjacent nodes, thus forming the artificial structural material.

[0132] Example 4

[0133] The design goal of an orthotropic artificial structural material of this embodiment is to specify 3 Poisson's ratios and 3 elastic moduli among 9 elastic constants, namely: v 12 =0, v 23 =0.5, v 13 =0.5, E 1=100 N / mm 2 , E 2=100 N / mm 2 , E 3=100 N / mm 2 .

[0134] Step 1: According to the design goal of the artificial structure material of this embodiment, determine the design values ​​of the nine elastic constants of the artificial structure material. v 12 = 0, v 23 =0.5, v 13 =0.5, E 1=100 N / mm 2 , E 2=100 N / mm 2 , E 3=100 N / mm 2 , the other three elastic constants are obtained according to the following three conditions:

[0135] = 50 N / mm 2 ,

[0136] = 100 N / mm 2 ,

[0137] = 100 N / mm 2 .

[0138] Step 2: E 1=100 N / mm 2 >0, E 2=100 N / mm 2 >0, E 3=100 N / mm 2 >0,

[0139] |v 12 |=0<( E 2 / E 1) 1 / 2 =1,|v 23 |=0.5<( E 3 / E 2) 1 / 2 =1,|v 13 |=0.5<( E 3 / E 1) 1 / 2 =1,

[0140] G 12 = 50 N / mm 2 >0,G 31 = 100 N / mm 2 >0,G 23 = 100 N / mm 2 >0,

[0141] =0.5 >0,

[0142] ;

[0143] Therefore, the nine elastic constants of the artificial structure material of Example 4 are all within the allowable range of orthotropic elastic mechanics.

[0144] Step 3: Select the dimensions of the unit in three directions: l 1=4 mm, l 2 = 4 mm, l 3=4 mm, and then calculate the axial stiffness of each member in the unit, that is:

[0145] K1=0 N, K 2=0 N, K 3=0 N,

[0146] K 4 = 565.685 N, K 5=1131.37 N, K 6 = 1131.37 N;

[0147] For simplicity, the elastic modulus of the material of all the bars in the unit is selected to be E =10000 N / mm 2 , then the cross-sectional area of ​​each member is calculated as:

[0148] A 1= K 1 / E =0 mm 2 , A 2= K 2 / E =0 mm 2 , A 3= K 3 / E =0 mm 2 ,

[0149] A 4= K 4 / E =0.0565685 mm 2 , A 5= K 5 / E =0.113137 mm 2 , A 6= K 6 / E =0.113137 mm 2 ;

[0150] The cross-sectional area of ​​each rod is not a negative number, among which the cross-sectional areas of the 12 parallel rods parallel to the main directions 1, 2, and 3 of the material are all 0, indicating that the unit does not include parallel rods but only consists of cross rods; then the cross-sectional shapes of all rods are selected to be circular, and obviously, it is easy to determine the cross-sectional size of each rod; from the two structures of the unit, the second structure is selected; without drawing, it is easy to judge that within the spatial range of a unit, the cross-sectional size of each rod is appropriate after a simple estimation, and each rod can be connected to form a unit, and the unit is periodically arranged in space, and adjacent units share their adjacent nodes. In order to reduce the number of rods in the artificial structural material, adjacent rods of adjacent units are merged into one rod by adding the cross-sectional areas, thus constituting the artificial structural material.

[0151] If the accuracy of the design is to be evaluated and the actual values ​​of the elastic constants of the artificial structural materials are to be determined, experimental determination or finite element analysis is used, that is: take one of the units and perform calculations under uniaxial tension (or uniaxial compression) in the three main directions of the material, and then the three elastic moduli and three Poisson's ratios of the material can be determined based on the definitions of the elastic modulus and Poisson's ratio. The calculations are performed under pure shear in the 1-2 plane, 2-3 plane, and 3-1 plane, and then the three shear moduli of the material can be determined based on the definition of the shear modulus.

[0152] According to the designed artificial structural material, various methods such as 3D printing are used for production and manufacturing. Although various connection methods can be used between the rods of the unit, the actual connection method is generally neither an ideal hinge nor an ideal rigid connection, but is between a hinge and a rigid connection. However, in order to facilitate finite element analysis, it can be simplified according to specific circumstances. If the actual connection method is close to a hinge, it is calculated according to the hinge, otherwise, it is calculated according to the rigid connection.

[0153] An example is given below. A unit in each embodiment is taken and calculated under uniaxial tension in the main direction 2 of the material and pure shear in the 2-3 plane, and compared with a corresponding solid unit. The elastic constant of the solid unit is the design value of the elastic constant of the artificial structure material given in step 1 of the embodiment. The sizes, loads and boundary conditions of the two units are also exactly the same, and finite element software is used for calculation. When finite element analysis is performed on each embodiment, the connection mode between the rods of the unit is hinged for embodiment 1 and embodiment 3, and rigidly connected for embodiment 2 and embodiment 4.

[0154] When using the traditional entity unit model for calculation based on finite element software, attention should be paid to the definition of Poisson's ratio used by the software. In various textbooks on elasticity mechanics or composite material mechanics at home and abroad, orthotropic materials have 6 Poisson's ratios, and Poisson's ratio has two completely opposite definitions; this invention adopts one of the definitions, and the corresponding generalized Hooke's law is as follows:

[0155] ;

[0156] The three Poisson's ratios can be expressed by the other three Poisson's ratios and three elastic moduli, namely: 21 =v 12 E 1 / E 2, v 31 =v 13 E 1 / E 3, v 32 =v 23 E 2 / E 3.

[0157] For Example 1, the calculation model of the two units under uniaxial tension in the main direction 2 of the material is as follows: Figure 3 , Figure 4 As shown, the calculation model of the two units under pure shear in the 2-3 plane is as follows Figure 5 , Figure 6 As shown; the loads acting on the two units are determined by the following method: Under uniaxial tension in the main direction 2 of the material, any positive stress is taken σ 2=40 N / mm 2 , then in finite element analysis, the concentrated force F applied to each node is σ 2×( l 1× l 3)÷4=40×(10×10)÷4=1000 N. Under pure shear in the 2-3 plane, any shear stress τ is 23 =τ 32 =80 N / mm 2 , then in finite element analysis, the concentrated force F1=τ applied to each node 23 ×( l 1× l 3)÷4=80×(10×10)÷4=2000 N, F2=τ 32 ×( l 1× l 2)÷4=80×(10×10)÷4=2000 N. For Example 2, under uniaxial tension in the main direction 2 of the material, the normal stress is σ 2=40 N / mm 2, and in pure shearing in the 2-3 plane, take the shear stress τ 23 =τ 32 =80 N / mm 2 , according to the same method as in Example 1, determine the loads acting on the two units and the calculation models of the two units; for Example 3, under uniaxial tension in the main direction 2 of the material, take the normal stress σ 2=120 N / mm 2 , and in pure shearing in the 2-3 plane, take the shear stress τ 23 =τ 32 =88 N / mm 2 , according to the same method as in Example 1, determine the loads acting on the two units and the calculation models of the two units; for Example 4, under uniaxial tension in the main direction 2 of the material, take the normal stress σ 2=10 N / mm 2 , and in pure shearing in the 2-3 plane, take the shear stress τ 23 =τ 32 =10 N / mm 2 , according to the same method as in Example 1, determine the loads acting on the two units and the calculation models of the two units.

[0158] By using finite element software for calculation, it can be seen that the displacements of the corresponding nodes of the units and corresponding entity units in Examples 1 and 3 are exactly the same, while the displacements of the corresponding nodes of the units and corresponding entity units in Examples 2 and 4 are close, and the strains of the units and corresponding entity units in Examples 1 to 4 under the uniaxial tension in the main direction 2 of the material and the pure shear in the 2-3 plane are calculated, as shown in Table 1, wherein the strains of the units in each main direction of the material can be obtained according to the strains of the bars parallel to the main direction of the material, or calculated according to the displacements of the nodes of the units, and the shear strains can be calculated according to the node displacements of the units. As can be seen from Table 1, the strains of the units in Examples 1 and 3 are exactly the same as the strains of the corresponding entity units; while the strains of the units in Examples 2 and 4 are close to the strains of the corresponding entity units.

[0159] Table 1. Summary of strains of the units and corresponding solid units of Examples 1 to 4 under uniaxial tension in the main direction 2 of the material and pure shear in the 2-3 plane

[0160]

[0161] The design values ​​and actual values ​​of the elastic constants of the artificial structural materials of Examples 1 to 4 are shown in Table 2, wherein the design values ​​in Table 2 are the design values ​​of the elastic constants of the artificial structural materials given in step 1 of each embodiment, and the actual values ​​are calculated based on the definition of the elastic constants. Taking Example 1 as an example, the actual value of the elastic constant is: v 12 = -ε1 / ε2= -0.05 / 0.1= -0.5, v 32 = -ε3 / ε2=0.068 / 0.1=0.68, E2= σ 2 / ε2=40 / 0.1=400 N / mm 2 , G 23 =|τ 23 / γ 23 |=80 / 0.0970=824.742 N / mm 2 .

[0162] It can be seen from Table 2 that if the connection mode between the rods of the unit is calculated according to the hinge connection, and the structure of the unit is the first structure, the actual value of the elastic constant of the artificial structural material is exactly the same as the design value, for example, Example 1 and Example 3; In addition, for Example 1, a finite element comparative analysis was performed, that is, except that the connection mode between the rods of the unit is changed to rigid connection calculation, the other remains unchanged. It can be known through calculation that: under the uniaxial tension of the material in the main direction 2, the strains in the directions 1, 2, and 3 of the unit are ε1= 0.0495, ε2= 0.0994, ε3=-0.0674, respectively, and the shear strain under the pure shear action in the 2-3 plane is γ 23 = -0.0969, which is very close to the strain of the corresponding solid unit in Table 1. Then, the actual value of the elastic constant is calculated based on the definition of the elastic constant, that is: v 12 = -0.498, v 32 = 0.678, E2 = 402 N / mm 2 , G 23 = 825.6 N / mm 2 , are very close to the design value of the elastic constant of the artificial structural material of Example 1 in Table 2. According to Example 2 and Example 4, it can be seen that if the connection between the rods of the unit is rigid, no matter the structure of the unit adopts the first structure or the second structure, the actual value of the elastic constant of the artificial structural material deviates from the design value, but the design accuracy is good. Generally speaking, the closer the actual connection between the rods of the unit is to the hinge, the higher the design accuracy.

[0163] The deformation of the unit in Examples 1 to 4 under uniaxial tension in the main direction 2 of the material and pure shear in the 2-3 plane is as follows: Figure 7 , Figure 8 , Fig. 9 , Fig.10 , Fig.11 , Fig.12 , Fig.13 , Fig.14 As shown in Table 1, it is consistent with the design expectations. Figure 7 , Fig. 9 , Fig.11 , Fig.13 As shown in Table 2, the units of Example 1 and Example 2 have a negative Poisson's ratio v 12 , while the unit of Example 3 has zero Poisson's ratio v 12 , the Poisson's ratio of the unit of Example 4 v 12 It is basically equal to 0, so the artificial structure materials in Examples 1 to 4 are all mechanical metamaterials. Fig.15 is a schematic diagram of a unit of the artificial structure material of Example 2, Fig.16 This is a schematic diagram of a unit of the artificial structure material of Example 4, from which one can intuitively see the unit structures of the two metamaterials proposed in Example 2 and Example 4, respectively, which can obviously be easily produced.

[0164] Table 2. Summary of design values ​​and actual values ​​of elastic constants of the artificial structural materials of Examples 1 to 4

[0165]

[0166] In summary, the orthotropic artificial structural material and the design method thereof of the present invention can design the elastic constants of the material relatively freely, and can design orthotropic artificial structural materials with specified elastic constants, of course including mechanical metamaterials with negative Poisson's ratio or zero Poisson's ratio.

[0167] The contents not described in detail in the present invention belong to the technology known to the professional and technical personnel in the field. Obviously, the described embodiments are only exemplary and are not intended to limit the scope of protection of the present invention. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention should be included in the scope of protection of the present invention.

Claims

1. An orthotropic artificial structural material, characterized in that: The artificial structure material is composed of units arranged periodically in space; the units include two rectangular parallelepiped structures, the first structure is the basic structure of the unit, and the second structure is a transformation of the basic structure of the unit; The first structure of the unit is a rectangular parallelepiped structure composed of 24 rods connected by 8 nodes, including: 4 rods parallel to each main direction of the material, 3 main directions of the material, a total of 12 parallel rods, 2 cross rods on each face of the rectangular parallelepiped structure, 6 faces, a total of 12 cross rods, and 8 vertices of the rectangular parallelepiped structure as 8 nodes connected to each rod; the second structure of the unit, compared with the first structure of the unit, is different in that the intersection of the two cross rods on each face of the rectangular parallelepiped structure is used as an intermediate node, so that the number of cross rods on each face becomes 4, there are 6 faces, a total of 6 intermediate nodes and 24 cross rods, and the others are the same, so that 36 rods are connected by 14 nodes to form a rectangular parallelepiped structure; Any rod in the unit is allowed to be replaced by a spring or a rod with a variable cross-section according to the principle of keeping the length and stiffness coefficient of the rod unchanged, wherein the stiffness coefficient of a rod refers to the axial force required to apply when the rod produces a unit axial deformation, which is equal to the ratio of the axial stiffness of the rod to the length of the rod; The design method of the artificial structure material comprises the following steps: Step 1: The design target of the artificial structural material means specifying n independent elastic constants among 9 elastic constants, where 1≤n≤6; According to the design goal of the artificial structure material, the design values ​​of the 9 elastic constants of the artificial structure material are determined. If the specified n independent elastic constants are known, (6-n) independent elastic constants can be freely selected. The other 3 elastic constants are obtained according to the following 3 conditions: Among them, E1, E2, and E3 are the elastic moduli of the material in the main directions 1, 2, and 3 respectively. v ij is Poisson’s ratio, specifically v ij =-ε i / ε j , is the normal stress σ acting alone in the j direction j When the other stress components are 0, the negative value of the ratio of the strain in the i direction to the strain in the j direction is G 23 ,G 31 ,G 12 are the shear moduli in the 2-3, 3-1, and 1-2 planes respectively; Step 2: Determine whether the nine elastic constants of the artificial structural material meet the allowable range of orthotropic elastic mechanics, and the allowable range is as follows: E1>0, E2>0, E3>0, <h2 style=";text-align:left;direction:ltr">|v<h2 style=";text-align:left;direction:ltr"> 12 <h2 style=";text-align:left;direction:ltr"> |<(E2 / E1)<h2 style=";text-align:left;direction:ltr"> 1 / 2 <h2 style=";text-align:left;direction:ltr"> ,|v<h2 style=";text-align:left;direction:ltr"> 23 <h2 style=";text-align:left;direction:ltr"> |<(E3 / E2)<h2 style=";text-align:left;direction:ltr"> 1 / 2 <h2 style=";text-align:left;direction:ltr"> ,|v<h2 style=";text-align:left;direction:ltr"> 13 <h2 style=";text-align:left;direction:ltr"> |<(E3 / E1)<h2 style=";text-align:left;direction:ltr"> 1 / 2 <h2 style=";text-align:left;direction:ltr"> , G 12 >0,G 31 >0,G 23 >0, If the 9 elastic constants of the artificial structure material are all within the allowable range, proceed to the next step; otherwise, reselect the elastic constant and repeat the previous step and this step; Step 3: Select the dimensions l1, l2, and l3 of the unit in three directions, and then calculate the axial stiffness of each rod in the unit, specifically: Among them, l1, l2, l3 are the dimensions of the unit in the 1, 2, and 3 directions respectively. K1 is the axial stiffness of a parallel rod parallel to the main direction 1 of the material, K2 is the axial stiffness of a parallel rod parallel to the main direction 2 of the material, K3 is the axial stiffness of a parallel rod parallel to the main direction 3 of the material, K4 is the axial stiffness of a cross bar parallel to the 1-2 plane, K5 is the axial stiffness of a cross bar parallel to the 1-3 plane, K6 is the axial stiffness of a cross bar parallel to the 2-3 plane; For any rod in the unit, the elastic modulus of the rod material is selected, and then the cross-sectional area of ​​the rod is calculated, specifically: the cross-sectional area of ​​the rod = the axial stiffness of the rod / the elastic modulus of the rod material; Check the cross-sectional area of ​​each rod obtained by calculation. If there is a negative cross-sectional area, adjust the size of the unit and then recalculate according to this step; if there is still a negative cross-sectional area, modify the elastic constant design value of the artificial structure material and redesign from step 1; if there is no negative cross-sectional area for each rod, select the cross-sectional shape of the rod and determine the cross-sectional size of the rod, so that the cross-sectional size of all rods in the unit can be obtained, wherein, if the cross-sectional area of ​​a rod is 0, it means that the rod is not included in the rod composition of the unit; then, from the two structures of the unit, choose to adopt the first structure or the second structure; finally, according to the size of the unit, judge whether there is a cross-sectional area of ​​each rod that is too large within the space range of the unit, resulting in crowded space and inability to connect the rods to form a unit. If this problem exists, increase the elastic modulus of the material of each rod and then re-determine the cross-sectional area and size of each rod; The units are periodically arranged in space, and adjacent units share their adjacent nodes, thus forming the artificial structural material. If the number of rods in the artificial structural material is to be reduced, adjacent rods of adjacent units are combined into one rod by adding the cross-sectional areas.

2. The orthotropic artificial structural material and the design method thereof according to claim 1, characterized in that: If the accuracy of the design is to be evaluated and the actual values ​​of the elastic constants of the artificial structural materials are to be determined, experimental determination or finite element analysis is used, specifically: take one of the units, perform calculations under uniaxial tension or uniaxial compression in the three main directions of the material, and then determine the three elastic moduli and three Poisson's ratios of the material according to the definitions of the elastic modulus and Poisson's ratio, perform calculations under pure shear in the 1-2 plane, 2-3 plane, and 3-1 plane, and then determine the three shear moduli of the material according to the definition of the shear modulus.

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