Analytic method for solving crack tip field of auxetic chiral material

Abstract

The invention discloses an analytical method for solving a crack tip field of an auxetic chiral material, which comprises the following steps of: firstly, taking a chiral lattice material as a chiral continuum to obtain a control equation of the auxetic chiral material; secondly, obtaining a compatibility equation expressed by stress and even stress under a cylindrical coordinate system; thirdly, obtaining an equation set representing stress and even stress singular terms on the basis of singular analysis; then, singular terms of stress and even stress expressed by a stress intensity factor and an angular function and zero-order terms of displacement and micro-rotation under cylindrical coordinates are obtained through derivation, and first-order terms of the stress, the even stress, the displacement and the micro-rotation are obtained; and finally, deriving to obtain a progressive expansion expression of the stress, even stress, displacement and micro-rotation of the Cartesian coordinate system. According to the method, coupling deformation between strain and micro-rotation of an auxetic chiral material is considered, and a displacement field and a stress field of the auxetic chiral material near a crack tip under the action of a tensile-shear load are obtained through strict theoretical derivation based on a chiral elasticity theory.

Description

Technical Field

[0001] The invention belongs to the technical field of fracture mechanics and relates to a method for solving a crack tip field of a tensile chiral material. Background Art

[0002] Auxetic materials or microstructures with negative Poisson's ratio have been found in many biomaterials, such as brick-mud structures in shells, porous microstructures in bones, and nanorod bundle structures in limpet teeth. Inspired by these biomaterials, engineering materials with auxetic effects are prepared through microstructure design, such as foam materials, microporous polymers, chiral lattices, etc. Due to their unusual behavior, these materials usually exhibit excellent mechanical properties such as high strength, high stiffness, and high toughness.

[0003] Chiral lattices are a special type of auxetic material. Chirality refers to the geometric property that an object cannot coincide with its mirror image. Chirality in materials usually manifests itself as coupling of mechanical deformations, such as tension-torsion coupling or bend-torsion coupling. In continuum mechanics, chiral lattices are usually homogenized as continuum media. However, due to the unique deformation mechanism of auxetic chiral materials, as well as their typical brittle and quasi-brittle characteristics, the materials are very sensitive to the presence of defects such as initial cracks and holes. Therefore, it is necessary to conduct in-depth research on the solution of the key fracture parameters of auxetic chiral materials with cracks.

[0004] In linear elastic fracture mechanics, the stress intensity factor near the crack tip is a key fracture parameter. The interaction integral method has been proven to be an effective means to extract the key fracture mechanics parameters at the crack tip. However, applying the interaction integral method to the fracture analysis of auxetic chiral materials requires obtaining the basic analytical form of the crack tip field of the auxetic chiral material.

[0005] However, there is currently a lack of proof and derivation of the basic analytical form of the crack tip field of auxetic chiral materials. Therefore, in order to fully and comprehensively calculate the key fracture parameters of auxetic chiral materials and accurately analyze the fracture behavior of auxetic chiral materials, it is necessary to give the basic analytical form of the crack tip field of auxetic chiral materials, which has very important engineering significance for the study of fracture mechanics of auxetic chiral materials. Summary of the invention

[0006] In order to solve the current lack of proof and derivation of the basic analytical form of the crack tip field of a dextrinsic chiral material crack, the present invention provides an analytical form of the asymptotic solution of the crack tip field for a two-dimensional dextrinsic chiral elastic body with a plane crack under a tensile-shear load, and further provides an analytical method for solving the crack tip field of a dextrinsic chiral material. The method takes into account the coupled deformation between the body strain and micro-rotation of the dextrinsic chiral material, and based on the chiral elasticity theory, obtains the displacement field and stress field near the crack tip of the dextrinsic chiral material under a tensile-shear load through rigorous theoretical derivation.

[0007] The objective of the present invention is achieved through the following technical solutions:

[0008] An analytical method for solving the crack tip field of a auxetic chiral material comprises the following steps:

[0009] Step 1: Considering the effects of volume strain and micro-rotation coupling, the chiral lattice material is regarded as a continuum, and the governing equation of the auxetic chiral material is obtained;

[0010] Step 2: convert the control equation of the chiral auxetic material in the rectangular coordinate system obtained in step 1 into the control equation of the chiral auxetic material in cylindrical coordinate form, and obtain the compatibility equation expressed in stress and couple stress through rigorous derivation according to the boundary conditions of the auxetic chiral material under the tensile-shear load;

[0011] Step 3: By bringing the William's asymptotic expansion of the force stress and couple stress at the crack tip into the equilibrium equation of stress and couple stress in cylindrical coordinates obtained in step 2.1 and the compatibility equation of stress characterization obtained in step 2.2, and deriving, a set of equations representing the singular terms of stress and couple stress is obtained on the basis of singular analysis, and Cramer's rule is used to solve the singularities of stress and couple stress;

[0012] Step 4: By examining the plane strain situation, it is assumed that the crack surface has no traction and no couple stress, and the singular terms of stress and couple stress expressed by stress intensity factor and angle function in cylindrical coordinates are obtained by deduction. Then, they are brought into the equilibrium equation and constitutive equation of the chiral elastic body and deduced to obtain the zero-order terms of displacement and micro-rotation in cylindrical coordinates;

[0013] Step 5: Using the derivation method in step 4, obtain the first-order terms of stress, couple stress, displacement and micro-rotation in cylindrical coordinates through rigorous derivation calculation;

[0014] Step 6: Using the principle of coordinate transformation, convert the zero-order and first-order terms of stress, couple stress, displacement and micro-rotation in cylindrical coordinates in steps 4 and 5 into the corresponding conditions in the Cartesian coordinate system.

[0015] Compared with the prior art, the present invention has the following advantages:

[0016] 1. For the first time, the basic analytical form of the crack tip field of a tensile chiral material is completely derived, and a specific analytical method for solving the field variables near the crack tip is given.

[0017] 2. The crack tip field of the tensile chiral material is expressed as a basic form corresponding to the Cartesian coordinate system in general engineering practice, which is convenient for direct fracture mechanics analysis.

[0018] 3. The method of the present invention has good applicability and can be combined with existing numerical calculation methods such as finite element, extended finite element, interaction integral method, etc. to realize the solution of relevant physical quantities and fracture parameters of two-dimensional planar expansion chiral materials under the coupling of multiple physical fields such as force, electricity, and magnetism, and accurately analyze the fracture behavior of materials. It can be developed into a commercial program to flexibly adapt to changes in required problems. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] Figure 1 A flowchart for deriving the basic analytical form of the crack tip field in auxetic chiral materials;

[0020] Figure 2 Schematic diagram of a two-dimensional auxetic chiral elastomer with coupling between volume strain and micro-rotation;

[0021] Figure 3 is a schematic diagram of the chiral lattice structure;

[0022] Figure 4 Schematic diagram of a two-dimensional auxetic chiral elastic solid with plane cracks under mixed-mode loading conditions;

[0023] Figure 5 is the angular stress distribution diagram of the open type (type I);

[0024] Figure 6 This is the angular stress distribution diagram of the sliding type (Type II). DETAILED DESCRIPTION

[0025] The technical solution of the present invention is further described below in conjunction with the accompanying drawings, but is not limited thereto. Any modification or equivalent replacement of the technical solution of the present invention without departing from the spirit and scope of the technical solution of the present invention should be included in the protection scope of the present invention.

[0026] The present invention provides an analytical method for solving the crack tip field of an axial chiral material. The method takes into account the constitutive relationship of the coupling between the volume strain and micro-rotation of the axial chiral material. First, the influence of the coupling of volume strain and micro-rotation is considered, and the chiral lattice material is regarded as a chiral continuum to obtain the control equation of the axial chiral material; secondly, the compatibility equation expressed by stress and couple stress is obtained by rigorous deduction in a cylindrical coordinate system; thirdly, according to the William's asymptotic expansion of the stress and couple stress at the crack tip, a set of equations characterizing the singular terms of stress and couple stress is obtained on the basis of singular analysis; then, by examining the plane strain situation, it is assumed that the crack surface has no traction and no couple stress, and the singular terms of stress and couple stress expressed by stress intensity factor and angular function in cylindrical coordinates, the zero-order terms of displacement and micro-rotation are obtained by deduction, and the first-order terms of stress, couple stress, displacement and micro-rotation are obtained at the same time; finally, the coordinate transformation principle is used to derive the asymptotic expansion expressions of stress, couple stress, displacement and micro-rotation in a Cartesian coordinate system widely used in general engineering practice. Figure 1 As shown, the specific steps include:

[0027] Step 1: Considering the effects of volume strain and micro-rotation coupling, the chiral lattice material is regarded as a continuum, and the governing equation of the auxetic chiral material is obtained. The specific steps are as follows:

[0028] like Figure 2 As shown, consider a dextrinsic chiral elastic body, where the equilibrium equation without body forces, charges and currents is:

[0029]

[0030] The geometric equation is:

[0031]

[0032] The constitutive equation of the auxetic chiral material is:

[0033]

[0034] The above formula uses tensor component notation, and the subscripts i, j, k, and l range from 1 to 2 and follow the Einstein summation convention: variables marked with symbolic subscripts i, j, k, and l are components of a vector or tensor. When a subscript appears only once, it is called a "free index" and all values ​​of the subscript must be traversed; in the same term, if a subscript appears in pairs, it is called a "dummy index" and the sum is traversed over its value range. As long as the free indices on the left and right sides of the equation can correspond, the specific letters used do not change the meaning of the equation. Among them, σ ij is the stress component, ε kk , ε lk , ε ij and εji is the strain component, m i3 is the couple stress component, χ i3 and χ k3 is the curvature component. ijkl and B i3k3 is the material stiffness of the auxetic chiral elastic body. ij3 、e 3ij and e 3lk is the Levitra symbol. ij is the Kronecker symbol. The comma indicates the partial derivative with respect to the corresponding coordinate component, that is, u j is the displacement component, φ3 is the micro-rotation, x i are coordinate components. λ, μ, κ and γ are elastic constants, and A is a chiral material parameter that characterizes the coupling between volume strain and microrotation. Taking a representative auxetic chiral material as an example, Figure 2 The expressions of each parameter are as follows:

[0035]

[0036] Where ω represents the slenderness ratio of the ligament's in-plane thickness to its length, E s ′ represents the plane elastic modulus, when E s ′=E s When , it indicates plane stress, when When , it represents plane strain, where E s and v s are Young’s modulus and Poisson’s ratio of the underlying chiral lattice material, respectively. s represents the distance between the centers of the circles. β represents the chiral angle, such as Figure 3 As shown, β>0 indicates positive chirality, and β<0 indicates negative chirality.

[0037] In practical implementation, the auxetic chiral material is usually regarded as a chiral continuum, and its Young's modulus E, shear modulus G, Poisson's ratio ν and coupling number N can be expressed as:

[0038]

[0039] Step 2: The control equation of the chiral auxetic material in the rectangular coordinate system obtained in step 1 is converted into the control equation of the chiral auxetic material in cylindrical coordinate form. According to the boundary conditions of the auxetic chiral material under the tensile-shear load, the compatibility equation expressed in stress and couple stress is obtained through rigorous derivation. The specific steps are as follows:

[0040] Step 21: Figure 4 As shown in Figure 2, a plane crack in a two-dimensional chiral elastic solid is studied. Under plane strain conditions, the non-zero terms of stress and couple stress in cylindrical coordinates are σ rr , σrθ , σ θr , σ θθ 、m zr and m zθ , the non-zero terms of displacement and micro-rotation are u r (r,θ),u θ (r,θ) and φ z (r,θ). At this time, the governing equation describing the auxetic chiral material can be expressed in cylindrical coordinates, where:

[0041] The equilibrium equation can be expressed as:

[0042]

[0043]

[0044] Among them, r,θ represents the polar coordinates of the crack tip, σ rr , σ rθ , σ θr , σ θθ represents the stress component in the cylindrical coordinate system, m zr and m zθ represents the couple stress component in the cylindrical coordinate system.

[0045] The geometric equation can be expressed as:

[0046]

[0047]

[0048] Where r represents the polar coordinate component of the crack tip, ε rr , ε rθ , ε θr and ε θθ represents the stress component in the cylindrical coordinate system, χ zr and χ zθ represents the curvature component in the cylindrical coordinate system, u r 、u θ represents the displacement component in the cylindrical coordinate system, and the comma in the subscript represents the partial derivative with respect to the corresponding coordinate component, that is, φ z represents the micro-rotation component in the cylindrical coordinate system, and the comma in the subscript represents the partial derivative with respect to the corresponding coordinate component, that is,

[0049] The consistent equation can be obtained by differentiating the geometric equation:

[0050]

[0051] The comma in the subscript indicates that the partial derivative is taken with respect to the corresponding coordinate component, that is,

[0052] The constitutive equation can be expressed as:

[0053]

[0054] m rz =γχ rz , m θz =γχ θz (12)

[0055] Step 2: Substitute the constitutive equation into the compatibility equation and obtain the compatibility equation expressed in stress and couple stress through rigorous derivation:

[0056]

[0057]

[0058] m zr,θ -m zθ -rm zθ,r =0, (15)

[0059] Among them, S 11 , S 12 , S 13 , S 33 , S 34 Represents material constant, material parameter S 11 , S 12 , S 13 , S 33 , S 34 It is expressed as:

[0060]

[0061] Step 3: By bringing the William's asymptotic expansion of the force stress and couple stress at the crack tip into the equilibrium equation of stress and couple stress in cylindrical coordinates obtained in step 2.1 and the compatibility equation of stress characterization obtained in step 2.2, and deriving, a set of equations representing the singular terms of stress and couple stress is obtained on the basis of singular analysis, and Cramer's rule is used to solve the singularities of stress and couple stress. The specific steps are as follows:

[0062] Step 3.1. The stress and couple stress near the crack tip can be expressed by William's asymptotic expansion:

[0063]

[0064] Where, the subscripts η and ζ are r and θ respectively, and p represents the singular parameters. and denote the zero-order stress and couple stress components in the cylindrical coordinate system, respectively. and denote the first-order stress and couple stress components in the cylindrical coordinate system, respectively. and denote the second-order stress and couple stress components in the cylindrical coordinate system respectively;

[0065] Step 3.2: Substitute the zero-order terms of stress and couple stress near the crack tip into Substitute the equilibrium equations of stress and couple stress in cylindrical coordinates obtained in step 2.1 and the compatibility equations of stress characterization obtained in step 2.2 into the equation group representing the singular terms of stress and couple stress through derivation and simplification:

[0066]

[0067]

[0068]

[0069]

[0070]

[0071]

[0072] in represents the zero-order stress component in the cylindrical coordinate system. The comma in the subscript indicates the partial derivative with respect to the corresponding coordinate component, that is, and represents the zero-order couple stress component in the cylindrical coordinate system. The comma in the subscript indicates the partial derivative with respect to the corresponding coordinate component, that is,

[0073] Step 3 and 4: From Cramer's rule, we know that in order to have a solution to the system of equations that characterize the singular terms of stress and even stress, the determinant of the coefficient matrix must not be zero. Through matrix operations, we can get p = -1 / 2.

[0074] Step 4: By examining the plane strain situation, it is assumed that the crack surface has no traction and no couple stress, and the singular terms of stress and couple stress expressed by stress intensity factor and angle function in cylindrical coordinates are obtained by deduction. Then, they are brought into the equilibrium equation and constitutive equation of the chiral elastic body and deduced to obtain the zero-order terms of displacement and micro-rotation in cylindrical coordinates. The specific steps are as follows:

[0075] Step 41: In the case of plane strain, set the crack surface to have no traction and no couple stress. At this time, the boundary conditions of the crack surface can be obtained:

[0076] σ θθ (θ=±π)=0, σ θr (θ=±π)=0, m θz (θ=±π)=0. (20)

[0077] Step 4.2. Type I, Type II stress intensity factors and Type VI couple stress intensity factors can be defined as:

[0078]

[0079] Among them, lim represents the limit symbol, K I and K II denote the macroscopic type I and type II stress intensity factors, K VI represents the microscopic type VI stress intensity factor, m θz represents the couple stress component in the cylindrical coordinate system.

[0080] Step 4.3. By solving the equations representing the singular terms of stress and couple stress in step 3 and substituting the type I, type II stress intensity factors and type VI couple stress intensity factors, the singular terms of stress and couple stress expressed by stress intensity factors and angle functions in cylindrical coordinates can be expressed as:

[0081]

[0082]

[0083]

[0084]

[0085]

[0086] Among them, ξ=A / (λ+μ) is a dimensionless parameter with a value range of The material parameter S can be expressed as:

[0087]

[0088] Here, Poisson's ratio v needs to be replaced by v / (1+v) in the case of plane stress.

[0089] Step 4. Substitute the obtained stress singular terms into the equilibrium equation and constitutive equation of the chiral elastic body and derive them to obtain the zero-order terms of displacement and micro-rotation in cylindrical coordinates:

[0090]

[0091]

[0092]

[0093] in, represents the zero-order displacement component in the cylindrical coordinate system, represents the zero-order micro-rotation component in the cylindrical coordinate system.

[0094] Step 5: Using the derivation method in step 4, the first-order terms of stress, couple stress, displacement and micro-rotation in cylindrical coordinates can be obtained through rigorous derivation and calculation. The specific steps are as follows:

[0095] The stress near the crack tip and couple stress The first-order term of is brought into the equilibrium equation of stress and couple stress in cylindrical coordinates obtained in step 2.1 and the compatibility equation of stress characterization obtained in step 2.2. Repeating the theoretical derivation of step 4 can obtain the first-order terms of stress, couple stress, displacement and micro-rotation in cylindrical coordinates:

[0096]

[0097]

[0098]

[0099]

[0100]

[0101] In the formula, σ I and σ II are the T stress terms related to type I and type II loading, respectively, and m II is the constant stress term associated with type II loading, represents the first-order stress component in the cylindrical coordinate system, and represents the first-order couple stress component in the cylindrical coordinate system, represents the first-order displacement component in the cylindrical coordinate system, represents the first-order micro-rotation component in the cylindrical coordinate system, O(r 3 / 2 ) represents a higher-order term.

[0102] Step 6: Using the coordinate transformation principle, transform the zero-order and first-order terms of stress, couple stress, displacement and micro-rotation in the cylindrical coordinates in steps 4 and 5 into the corresponding conditions of the Cartesian coordinate system widely used in general engineering practice. The specific steps are as follows:

[0103] Step 61: According to the conversion formula between cylindrical coordinates and Cartesian coordinate system, that is:

[0104]

[0105]

[0106] Among them, σ 11 , σ 12 , σ 21 , σ 22 represents the stress component in the Cartesian coordinate system, m zr and m zθ represents the couple stress component in the cylindrical coordinate system, m 13 and m 23 represents the couple stress component in the Cartesian coordinate system, and sinθ and cosθ represent trigonometric functions.

[0107] Using the principle of coordinate transformation, the zero-order and first-order terms of stress, couple stress, displacement and micro-rotation in cylindrical coordinates in steps four and five are converted into the corresponding conditions in the Cartesian coordinate system.

[0108] Step 62: Through calculation, the asymptotic expansion of stress and couple stress in the Cartesian coordinate system can be expressed as:

[0109]

[0110]

[0111] The asymptotic expansion of displacement and micro-rotation in Cartesian coordinates can be expressed as:

[0112]

[0113]

[0114]

[0115] Among them, u1 and u2 represent the crack tip displacement components in the Cartesian coordinate system.

[0116] Through the above six steps, the basic analytical form of the crack tip field of the axial chiral material is obtained, which has very important engineering significance for the fracture mechanics research of the axial chiral material.

[0117] In order to illustrate the effect of tensile expansion on the stress angle distribution at the crack tip, the distribution diagrams of the singular terms of the opening type (type I) stress and the sliding type (type II) stress are plotted, as shown in Figure 2. Figure 5 and 6 When plotting the singular term distribution diagram of the opening (I-type) stress, K I =1,K II =0, r = 1 / 2π. When drawing the distribution diagram of the singular term of the slip-type (Type II) stress, K I =1,KII =0, r = 1 / 2π. At the same time, three cases of β = -30°, 0°, and 30° were selected for comparison. From the angular stress distribution, it can be found that the tensile expansion of the microstructure can destroy the symmetry of type I and type II stress distribution.

Claims

1. An analytical method for solving the crack tip field of auxetic chiral materials, characterized in that The method comprises the following steps: Step 1: Considering the effects of volume strain and micro-rotation coupling, the chiral lattice material is regarded as a continuum, and the governing equation of the auxetic chiral material is obtained; Step 2: convert the control equation of the chiral auxetic material in the rectangular coordinate system obtained in step 1 into the control equation of the chiral auxetic material in cylindrical coordinate form, and obtain the compatibility equation expressed in stress and couple stress through rigorous derivation according to the boundary conditions of the auxetic chiral material under the tensile-shear load; Step 3: By bringing the William's asymptotic expansion of the force stress and couple stress at the crack tip into the equilibrium equation of stress and couple stress in cylindrical coordinates obtained in step 2.1 and the compatibility equation of stress characterization obtained in step 2.2, and deriving, a set of equations representing the singular terms of stress and couple stress is obtained on the basis of singular analysis, and Cramer's rule is used to solve the singularities of stress and couple stress; Step 4: By examining the plane strain situation, it is assumed that the crack surface has no traction and no couple stress, and the singular terms of stress and couple stress expressed by stress intensity factor and angle function in cylindrical coordinates are obtained by deduction. Then, they are brought into the equilibrium equation and constitutive equation of the chiral elastic body and deduced to obtain the zero-order terms of displacement and micro-rotation in cylindrical coordinates; Step 5: Using the derivation method in step 4, obtain the first-order terms of stress, couple stress, displacement and micro-rotation in cylindrical coordinates through rigorous derivation calculation; Step 6: Using the principle of coordinate transformation, convert the zero-order and first-order terms of stress, couple stress, displacement and micro-rotation in cylindrical coordinates in steps 4 and 5 into the corresponding conditions in the Cartesian coordinate system.

2. The analytical method for solving the crack tip field of auxetic chiral materials according to claim 1, characterized in that The specific steps of step one are as follows: Consider a auxetic chiral elastic body, where the equilibrium equation without body forces, charges and currents is: s ji,j =0, m i3,i +e ij3 s ij =0. The geometric equation is: ε ij =u j,i -e ij3 φ3, x i3 =φ 3,i 。 The constitutive equation of the auxetic chiral material is: s ij =A ijkl e kl =le kk d ij +(μ+k)e ij +with ji +Aδ ij e 3lk e lk +Ae 3ij e kk , m i3 =B i3k3 x k3 =gx i3 , Among them, σ ij is the stress component, ε kk , ε lk , ε ij and ε ji is the strain component, m i3 is the couple stress component, χ i3 and χ k3 is the curvature component, A ijkl and B i3k3 is the material stiffness of the auxetic chiral elastic body, e ij3 、e 3ij and e 3lk is the Levitra symbol, δ ij is the Kronecker symbol. The comma indicates the partial derivative with respect to the corresponding coordinate component. j is the displacement component, φ3 is the micro-rotation, x i are coordinate components, λ, μ, κ and γ are elastic constants, and A is the chiral material parameter that characterizes the coupling between volume strain and micro-rotation.

3. The analytical method for solving the crack tip field of auxetic chiral materials according to claim 2, characterized in that The Young's modulus E, shear modulus G, Poisson's ratio ν and coupling number N of the auxetic chiral material are expressed as:

4. The analytical method for solving the crack tip field of auxetic chiral materials according to claim 2 or 3, characterized in that The expressions of λ, μ, κ, γ, and A are as follows: Where ω represents the slenderness ratio of the ligament’s in-plane thickness to its length, E s ′ represents the plane elastic modulus, when E s ′=E s When , it indicates plane stress, when When , it represents plane strain, where E s and v s are Young’s modulus and Poisson’s ratio of the underlying chiral lattice material, a s represents the distance between the centers of the circles, β represents the chirality angle, β>0 represents positive chirality, and β<0 represents negative chirality.

5. The analytical method for solving the crack tip field of auxetic chiral materials according to claim 2, characterized in that The specific steps of step 2 are as follows: Step 21. Under the condition of plane strain, the non-zero terms of stress and couple stress in cylindrical coordinates are σ rr , σ rθ , σ θr , σ θθ 、m zr and m zθ , the non-zero terms of displacement and micro-rotation are u r (r,θ),u θ (r,θ) and φ z (r,θ), the governing equations describing the auxetic chiral material are now expressed in cylindrical coordinates, where: The equilibrium equation is expressed as: Where r,θ represents the polar coordinates of the crack tip, σ rr , σ rθ , σ θr , σ θθ represents the stress component in the cylindrical coordinate system, m zr and m zθ represents the couple stress component in the cylindrical coordinate system; The geometric equation is expressed as: e rr =u r,r ,he rθ =u θ,r -f z , Among them, ε rr , ε rθ , ε θr and ε θθ represents the stress component in the cylindrical coordinate system, χ zr and χ zθ represents the curvature component in the cylindrical coordinate system, u r 、u θ represents the displacement component in the cylindrical coordinate system. The comma in the subscript indicates the partial derivative with respect to the corresponding coordinate component. z represents the micro-rotation component in the cylindrical coordinate system; The consistent equations are obtained by differentiating the geometric equations: x zr,θ -(rx zθ ) ,r =0. The constitutive equation is expressed as: s rr =(λ+2μ+κ)ε rr +le θθ +A(e rθ -e θr ), s θθ =(λ+2μ+κ)ε θθ +le rr +A(e rθ -e θr ), s rθ =(μ+κ)ε rθ +with θr +A(e rr +e θθ ), s θr =(μ+κ)ε θr +with rθ -A(e rr +e θθ ), m rz =gx rz ,m θz =gx θz 。 Step 2: Substitute the constitutive equation into the compatibility equation and obtain the compatibility equation expressed in stress and couple stress through rigorous derivation: m zr,θ -m zθ -rm zθ,r =0, Among them, S 11 , S 12 , S 13 , S 33 , S 34 Represents material constants.

6. The analytical method for solving the crack tip field of auxetic chiral materials according to claim 5, characterized in that The S 11 , S 12 , S 13 , S 33 , S 34 It is expressed as:

7. The analytical method for solving the crack tip field of auxetic chiral materials according to claim 5, characterized in that The specific steps of step three are as follows: Step 3.

1. The stress and couple stress near the crack tip can be expressed by William's asymptotic expansion: Where, the subscripts η and ζ are r and θ respectively, and p represents the singular parameters. and denote the zero-order stress and couple stress components in the cylindrical coordinate system, respectively. and denote the first-order stress and couple stress components in the cylindrical coordinate system, respectively. and denote the second-order stress and couple stress components in the cylindrical coordinate system respectively; Step 3.2: Substitute the zero-order terms of stress and couple stress near the crack tip into Substitute the equilibrium equations of stress and couple stress in cylindrical coordinates obtained in step 2.1 and the compatibility equations of stress characterization obtained in step 2.2 into the equation group representing the singular terms of stress and couple stress through derivation and simplification: in represents the zero-order stress component in the cylindrical coordinate system. The comma in the subscript indicates the partial derivative with respect to the corresponding coordinate component. and represents the zero-order couple stress component in the cylindrical coordinate system; Step 3 and 4: From Cramer's rule, we know that in order to have a solution to the system of equations that characterize the singular terms of stress and even stress, the determinant of the coefficient matrix must be non-zero, and through matrix operations we can get p = -1 / 2.

8. The analytical method for solving the crack tip field of auxetic chiral materials according to claim 1, characterized in that The specific steps of step 4 are as follows: Step 41: In the case of plane strain, set the crack surface to have no traction and no couple stress. At this time, the boundary conditions of the crack surface are obtained: s θθ (θ=±π)=0,σ θr (θ=±π)=0,m θz (θ=±π)=0. Step 4.

2. Type I, Type II stress intensity factors and Type VI couple stress intensity factors can be defined as: Among them, lim represents the limit symbol, K I and K II are macroscopic type I and type II stress intensity factors, K VI represents the microscopic type VI stress intensity factor, m θz represents the couple stress component in the cylindrical coordinate system; Step 4.

3. By solving the equations representing the singular terms of stress and couple stress in step 3, and substituting the type I, type II stress intensity factors and type VI couple stress intensity factors, the singular terms of stress and couple stress expressed by stress intensity factors and angle functions in cylindrical coordinates are expressed as follows: Among them, ξ=A / (λ+μ) is a dimensionless parameter with a value range of The material parameter S is expressed as: Step 4. Substitute the obtained stress singular terms into the equilibrium equation and constitutive equation of the chiral elastic body and derive them to obtain the zero-order terms of displacement and micro-rotation in cylindrical coordinates: in, represents the zero-order displacement component in the cylindrical coordinate system, represents the zero-order micro-rotation component in the cylindrical coordinate system.

9. The analytical method for solving the crack tip field of auxetic chiral materials according to claim 8, characterized in that The specific steps of step five are as follows: The stress near the crack tip and couple stress The first-order term of is introduced into the equilibrium equation of stress and couple stress in cylindrical coordinates obtained in step 2.1 and the compatibility equation of stress characterization obtained in step 2.

2. Repeat the theoretical derivation of step 4 to obtain the first-order terms of stress, couple stress, displacement and micro-rotation in cylindrical coordinates: In the formula, σ I and σ II are the T stress terms related to type I and type II loading, respectively, and m II is the constant stress term associated with type II loading, represents the first-order stress component in the cylindrical coordinate system, and represents the first-order couple stress component in the cylindrical coordinate system, represents the first-order displacement component in the cylindrical coordinate system, represents the first-order micro-rotation component in the cylindrical coordinate system, O(r 3 / 2 ) represents a higher-order term.

10. The analytical method for solving the crack tip field of auxetic chiral materials according to claim 9, characterized in that The specific steps of step six are as follows: Step 61: According to the conversion formula between cylindrical coordinates and Cartesian coordinate system, that is: Among them, σ 11 , σ 12 , σ 21 , σ 22 represents the stress component in the Cartesian coordinate system, m zr and m zθ represents the couple stress component in the cylindrical coordinate system, m 13 and m 23 represents the couple stress component in the Cartesian coordinate system, sinθ and cosθ represent trigonometric functions; Using the principle of coordinate transformation, transform the zero-order and first-order terms of stress, couple stress, displacement and micro-rotation in cylindrical coordinates in steps 4 and 5 into the corresponding conditions in Cartesian coordinate system; Step 62: Through calculation, the asymptotic expansion of stress and couple stress in the Cartesian coordinate system is expressed as: The asymptotic expansion of displacement and micro-rotation in Cartesian coordinates is expressed as: Among them, u1 and u2 represent the crack tip displacement components in the Cartesian coordinate system.

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