An analytical method for solving the crack tip field of auxetic chiral materials

Abstract

The present invention discloses an analytical method for solving the crack tip field of auxetic chiral materials. First, the chiral lattice material is regarded as a chiral continuum to obtain the governing equations of the auxetic chiral material. Secondly, the compatibility equation expressed by stress and couple stress is obtained in the cylindrical coordinate system. Thirdly, based on the singular analysis, a system of equations characterizing the singular terms of stress and couple stress is obtained. Subsequently, by derivation, the singular terms of stress and couple stress, and the zero-order terms of displacement and micro-rotation expressed by stress intensity factors and angular functions in the cylindrical coordinate system are obtained, and the first-order terms of stress, couple stress, displacement and micro-rotation are obtained. Finally, the asymptotic expansion expressions of stress, couple stress, displacement and micro-rotation in the Cartesian coordinate system are derived. This method takes into account the coupled deformation between the volumetric strain and micro-rotation of the auxetic chiral material. Based on the chiral elasticity theory, the displacement field and stress field near the crack tip of the auxetic chiral material under tensile-shear loading are obtained through strict theoretical derivation.

Description

Technical Field

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

[0002] Auxetic materials or microstructures with negative Poisson's ratio have been found in many biological materials, such as the brick - mud structure in shells, the porous microstructure in bones, and the nanorod bundle structure in limpet teeth. Inspired by these biological materials, 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 materials. Chirality refers to the geometric property that an object cannot coincide with its mirror image. Chirality in materials usually manifests as the coupling of mechanical deformations, such as tension - torsion coupling or bending - torsion coupling. In continuum mechanics, chiral lattices are usually homogenized into 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 deeply explore the solution of the key fracture parameters of auxetic chiral materials containing cracks.

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

[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 calculate the key fracture parameters of auxetic chiral materials comprehensively 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 fracture mechanics research 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 auxetic chiral materials, for a two - dimensional auxetic chiral elastic body containing a plane crack under tensile - shear loading, the present invention gives the analytical form of the asymptotic solution of the crack tip field, and further provides an analytical method for solving the crack tip field of auxetic chiral materials. This method takes into account the coupled deformation between the volumetric strain and the micro - rotation of the auxetic chiral material, and based on the chiral elasticity theory, obtains the displacement field and stress field near the crack tip of the auxetic chiral material under tensile - shear loading through strict theoretical derivation.

[0007] The object of the present invention is achieved by the following technical solutions:

[0008] An analytical method for solving the crack tip field of auxetic chiral materials, comprising the following steps:

[0009] Step 1: Considering the influence of the coupling of volumetric strain and micro-rotation, regarding the chiral lattice material as a continuum, and obtaining the control equations of the auxetic chiral material;

[0010] Step 2: Transforming the control equations of the auxetic chiral material in the Cartesian coordinate system obtained in Step 1 into the control equations of the auxetic chiral material in the cylindrical coordinate system. According to the boundary conditions of the auxetic chiral material under tensile-shear loading, a compatibility equation expressed by stress and couple stress is obtained through strict derivation;

[0011] Step 3: Substituting the Williams' asymptotic expansions of the force stress and couple stress at the crack tip into the stress and couple stress equilibrium equations in the cylindrical coordinate system obtained in Step 1 and the compatibility equation characterized by stress obtained in Step 2, and conducting derivation. Based on singular analysis, a system of equations characterizing the singular terms of stress and couple stress is obtained, and Cramer's rule is used to solve the singularity of stress and couple stress;

[0012] Step 4: By examining the plane strain case, setting no traction and no couple stress on the crack surface, the singular terms of stress and couple stress expressed by stress intensity factors and angular functions in the cylindrical coordinate system are obtained through derivation. Then, they are substituted into the equilibrium equations and constitutive equations of the chiral elastic body and derived to obtain the zero-order terms of displacement and micro-rotation in the cylindrical coordinate system;

[0013] Step 5: Using the derivation method in Step 4, the first-order terms of stress, couple stress, displacement, and micro-rotation in the cylindrical coordinate system are obtained through strict derivation and calculation;

[0014] Step 6: Using the coordinate transformation principle, the zero-order terms and first-order terms of stress, couple stress, displacement, and micro-rotation in the cylindrical coordinate system in Steps 4 and 5 are transformed into the corresponding cases 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 auxetic chiral materials 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 auxetic chiral material is expressed in the 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 the finite element method, extended finite element method, interaction integral method, etc. to solve the relevant physical quantities and fracture parameters of two-dimensional planar auxetic chiral materials under the coupling action of multiple physical fields such as force, electricity, and magnetism, and accurately analyze the fracture behavior of the materials. It can be developed into a commercial program to flexibly adapt to the changes of the required problems. Description of the Drawings

[0019] Figure 1 It is a flow chart for deriving the basic analytical form of the crack tip field of auxetic chiral materials;

[0020] Figure 2 It is a schematic diagram of a two-dimensional auxetic chiral elastic body with coupling between volumetric strain and micro-rotation;

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

[0022] Figure 4 It is a schematic diagram of a two-dimensional auxetic chiral elastic solid containing a planar crack under mixed-mode loading conditions;

[0023] Figure 5 It is a distribution diagram of angular stress in the opening mode (mode I);

[0024] Figure 6 It is a distribution diagram of angular stress in the sliding mode (mode II). Detailed Embodiments

[0025] The technical solutions of the present invention will be further described below in conjunction with the accompanying drawings, but are not limited thereto. Any modification or equivalent replacement of the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention shall be covered by the protection scope of the present invention.

[0026] The present invention provides an analytical method for solving the crack tip field of auxetic chiral materials. Considering the constitutive relationship between the volumetric strain and the micro-rotation of auxetic chiral materials, first, taking into account the influence of the coupling of volumetric strain and micro-rotation, regarding the chiral lattice material as a chiral continuum, the governing equations of the auxetic chiral material are obtained; second, through strict derivation in the cylindrical coordinate system, the compatibility equations expressed by stress and couple stress are obtained; third, according to the Williams' asymptotic expansion of stress and couple stress at the crack tip, a system of equations characterizing the singular terms of stress and couple stress is obtained based on singular analysis; subsequently, by examining the case of plane strain, setting the crack surface without traction and without couple stress, the singular terms of stress and couple stress, the zero-order terms of displacement and micro-rotation expressed by stress intensity factors and angular functions are obtained through derivation, and the first-order terms of stress, couple stress, displacement and micro-rotation are also obtained; finally, using the coordinate transformation principle, the asymptotic expansion expressions of stress, couple stress, displacement and micro-rotation in the Cartesian coordinate system widely used in general engineering practice are derived. As Figure 1 shown, the specific steps are as follows:

[0027] Step 1: Considering the influence of the coupling of volumetric strain and micro-rotation, regarding the chiral lattice material as a continuum, the governing equations of the auxetic chiral material are obtained. The specific steps are as follows:

[0028] As Figure 2 shown, considering an auxetic chiral elastic body, the equilibrium equations under the action of no body force, charge and current are:

[0029]

[0030] The geometric equations are:

[0031]

[0032] The constitutive equations of the auxetic chiral material are:

[0033]

[0034] The above equations adopt the tensor component notation. The value ranges of each subscript i, j, k, l are 1 to 2 and abide by the Einstein summation convention: the variables marked by the subscripts i, j, k, l are components of a vector or tensor. When a subscript appears only once, it is a "free index" and must traverse all the values of this subscript; in the same term, if a subscript appears in pairs, it is called a "dummy index" indicating summation over its value range. As long as the free indices on both 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 are strain components, m i3 are couple stress components, χ i3 and χ k3 are curvature components. A ijkl and B i3k3 are the material stiffnesses of the auxetic chiral elastomer. e ij3 、e 3ij and e 3lk are the Levi-Civita symbols. δ ij is the Kronecker symbol. A comma indicates partial differentiation with respect to the corresponding coordinate component, i.e., u j is the displacement component, φ3 is the micro-rotation, x i are the coordinate components. λ, μ, κ and γ are elastic constants, and A is the chiral material parameter characterizing the coupling of the volumetric strain and the micro-rotation. Taking a representative auxetic chiral material as an example, as Figure 2 shown. The expressions of the various parameters are as follows:

[0035]

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

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

[0038]

[0039] Step 2: Convert the governing equations of the auxetic chiral material in the rectangular coordinate system obtained in Step 1 into the governing equations of the auxetic chiral material in the cylindrical coordinate system. According to the boundary conditions of the auxetic chiral material under tensile-shear loading, a compatibility equation expressed in terms of stress and couple stress is obtained through strict derivation. The specific steps are as follows:

[0040] Step 2-1: As Figure 4 shown, the plane crack in the two-dimensional defective chiral elastic solid is studied. Under the condition of plane strain, the non-zero terms of the stress and couple stress in the 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 equations describing the auxetic chiral material can be expressed in the form of cylindrical coordinates, where:

[0041] The equilibrium equation can be expressed as:

[0042]

[0043]

[0044] where r, θ represent the polar coordinates of the crack tip, σ rr , σ rθ , σ θr , σ θθ represent the stress components in the cylindrical coordinate system, m zr and m zθ represent the couple - stress components 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 represent the stress components in the cylindrical coordinate system, χ zr and χ zθ represent the curvature components in the cylindrical coordinate system, u r , u θ represent the displacement components in the cylindrical coordinate system, and the comma in the subscript indicates partial differentiation with respect to the corresponding coordinate component, i.e., φ z represents the micro - rotation component in the cylindrical coordinate system, and the comma in the subscript indicates partial differentiation with respect to the corresponding coordinate component, i.e.,

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

[0050]

[0051] where the comma in the subscript represents taking the partial derivative with respect to the corresponding coordinate component, i.e.,

[0052] The constitutive equation can be expressed as:

[0053]

[0054] m rz =γχ rz , m θz =γχ θz 。 (12)

[0055] Step Two: Substitute the constitutive equation into the compatibility equation and, through strict derivation, obtain the compatibility equation expressed in terms of stress and couple stress:

[0056]

[0057]

[0058] m zr,θ -m zθ -rm zθ,r = 0, (15)

[0059] where S 11 、S 12 、S 13 、S 33 、S 34 represent material constants, and the material parameters S 11 、S 12 、S 13 、S 33 、S 34 are expressed as:

[0060]

[0061] Step Three: Substitute the Williams’ asymptotic expansions of the force stress and couple stress at the crack tip into the stress and couple stress equilibrium equations in cylindrical coordinates obtained in Step One and the compatibility equation characterized by stress obtained in Step Two, and conduct derivation. Based on singular analysis, obtain a system of equations characterizing the singular terms of stress and couple stress, and use Cramer's rule to solve for 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 Williams’ asymptotic expansion:

[0063]

[0064] where the subscripts η, ζ take r, θ respectively, and p represents the singular parameter, and represent the zero - order stress and couple - stress components in the cylindrical coordinate system respectively, and represent the first - order stress and couple - stress components in the cylindrical coordinate system respectively, and represent the second - order stress and couple - stress components in the cylindrical coordinate system respectively;

[0065] Step 32. Substitute the zero - order terms of the stress and couple - stress near the crack tip into the equilibrium equations of the stress and couple - stress in the cylindrical coordinate system obtained in Step 21 and the compatibility equation of the stress characterization obtained in Step 22. Through derivation and simplification, a system of equations characterizing the singular terms of the stress and couple - stress can be obtained:

[0066]

[0067]

[0068]

[0069]

[0070]

[0071]

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

[0073] Step 34. According to Cramer's rule, for the system of equations characterizing the singular terms of the stress and couple - stress to have a solution, the determinant of the coefficient matrix needs to be non - zero. Through matrix operations, p = - 1 / 2 can be obtained.

[0074] Step 4. By examining the case of plane strain, assuming no traction and no couple - stress on the crack surface, the singular terms of the stress and couple - stress expressed by the stress intensity factor and angular functions in the cylindrical coordinate system are obtained through derivation. Then, substituting them into the equilibrium equation and constitutive equation of the chiral elastic body and conducting derivation, the zero - order terms of the displacement and micro - rotation in the cylindrical coordinate system are obtained. The specific steps are as follows:

[0075] Step 41. In the case of plane strain, assuming no traction and no couple - stress on the crack surface, the boundary conditions of the crack surface can be obtained at this time:

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

[0077] Step Four Two: The type-I, type-II stress intensity factors and type-VI couple stress intensity factors can be defined as follows:

[0078]

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

[0080] Step Four Three: By solving the system of equations characterizing the stress and couple stress singular terms in Step Three and substituting the type-I, type-II stress intensity factors and type-VI couple stress intensity factors, the stress and couple stress singular terms expressed by the stress intensity factors and angular functions in the cylindrical coordinate system can be represented as:

[0081]

[0082]

[0083]

[0084]

[0085]

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

[0087]

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

[0089] Step Four Four: Substitute the obtained stress singular terms into the equilibrium equation and constitutive equation of the chiral elastic body and conduct the derivation to obtain the zero-order terms of the displacement and micro-rotation in the cylindrical coordinate system:

[0090]

[0091]

[0092]

[0093] Among them, 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 Five: Using the derivation method in Step Four, the first-order terms of stress, couple stress, displacement, and micro-rotation in the cylindrical coordinate system can be obtained through strict derivation and calculation. The specific steps are as follows:

[0095] Substitute the first-order terms of the force stress and the couple stress near the crack tip into the equilibrium equations of stress and couple stress in the cylindrical coordinate system obtained in Step Twenty-One and the compatibility equation characterized by stress obtained in Step Twenty-Two. Repeating the theoretical derivation in Step Four, the first-order terms of stress, couple stress, displacement, and micro-rotation in the cylindrical coordinate system can be obtained:

[0096]

[0097]

[0098]

[0099]

[0100]

[0101] In the formula, σ I and σ II are the T-stress terms related to Mode I and Mode II loadings respectively, m II is the constant couple stress term related to Mode II loading, represents the first-order stress component in the cylindrical coordinate system, and represent the first-order couple stress components 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 the higher-order terms.

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

[0103] Step Six-One: According to the conversion formula between the cylindrical coordinate system and the Cartesian coordinate system, that is:

[0104]

[0105]

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

[0107] Using the coordinate transformation principle, the zero-order and first-order terms of stress, couple stress, displacement, and micro-rotation in the cylindrical coordinates in Steps 4 and 5 are transformed into the corresponding cases in the Cartesian coordinate system.

[0108] Step 6 II. Through calculation, the asymptotic expansions of stress and couple stress in the Cartesian coordinate system at this time can be expressed as:

[0109]

[0110]

[0111] The asymptotic expansions of displacement and micro-rotation in the Cartesian coordinate system 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 auxetic chiral material is obtained, which has very important engineering significance for the fracture mechanics research of the auxetic chiral material.

[0117] To illustrate the influence of auxeticity on the stress angle distribution at the crack tip, the singular term distribution diagrams of the opening mode (Mode I) stress and the sliding mode (Mode II) stress are plotted, as shown in Figure 5 and 6 . When plotting the singular term distribution diagram of the opening mode (Mode I) stress, K I = 1, K II = 0, r = 1 / 2π. When plotting the singular term distribution diagram of the sliding mode (Mode 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 includes the following steps: Step 1: Considering the influence of the coupling of volumetric strain and micro-rotation, regarding the chiral lattice material as a continuum, and obtaining the governing equations of the auxetic chiral material; Step 2: Transforming the governing equations of the auxetic chiral material in the Cartesian coordinate system obtained in Step 1 into the governing equations of the auxetic chiral material in the cylindrical coordinate form. According to the boundary conditions of the auxetic chiral material under tensile-shear loads, a compatibility equation expressed by stress and couple stress is obtained through strict derivation; Step 3: Substituting the William’s asymptotic expansion of the stress and couple stress at the crack tip into the equilibrium equations of the stress and couple stress in the cylindrical coordinate system obtained in Step 2 and the compatibility equation characterized by stress obtained in Step 2, and conducting derivation. Based on the singular analysis, a system of equations characterizing the singular terms of stress and couple stress is obtained, and Cramer's rule is used to solve for the singularities of stress and couple stress; Step 4: By examining the plane strain case, setting no traction and no couple stress on the crack surface, the singular terms of stress and couple stress expressed by stress intensity factors and angular functions in the cylindrical coordinate system are obtained through derivation. Then, substituting them into the equilibrium equation and constitutive equation of the chiral elastic body and conducting derivation to obtain the zero-order terms of displacement and micro-rotation in the cylindrical coordinate system; Step 5: Using the derivation method in Step 4, the first-order terms of stress, couple stress, displacement, and micro-rotation in the cylindrical coordinate system are obtained through strict derivation and calculation; Step 6: Using the coordinate transformation principle, the zero-order terms and first-order terms of stress, couple stress, displacement, and micro-rotation in the cylindrical coordinate system in Steps 4 and 5 are transformed into the corresponding cases 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 1 are as follows: Consider an auxetic chiral elastic body, and the equilibrium equation under the action of no body force, charge, and current is: The geometric equation is: The constitutive equation of the auxetic chiral material is: wherein, is the stress component, , , and are the strain components, is the couple stress component, and are the curvature components, and are the material stiffnesses of the auxetic chiral elastomer, , and are the Levi-Civita symbols, is the Kronecker symbol, and a comma indicates partial differentiation with respect to the corresponding coordinate component, is the displacement component, is the micro-rotation, is the coordinate component, , , and are the elastic constants, and A is the chiral material parameter characterizing the coupling between the volumetric strain and the 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 of the auxetic chiral material , the shear modulus , the Poisson's ratio and the coupling number N 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 said , , , , the expression of A is as follows: Among them, represents the slenderness ratio of the thickness to the length within the ligament plane, represents the planar elastic modulus. When it represents plane stress, and when it represents plane strain, where and are the Young's modulus and Poisson's ratio of the underlying chiral lattice material respectively, represents the distance between the centers of the circles, represents the chiral angle, represents positive chirality, 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 2-1. Under the condition of plane strain, the non-zero terms of stress and couple stress in cylindrical coordinates are , , , , and . The non-zero terms of displacement and micro-rotation are , and . At this time, the governing equations describing the auxetic chiral material are expressed in the form of cylindrical coordinates, where: The equilibrium equation is expressed as: Among them, represents the polar coordinates of the crack tip, , , , represent the stress components in the cylindrical coordinate system, and represent the couple stress components in the cylindrical coordinate system; The geometric equation is expressed as: Among them, , , and represent stress components in the cylindrical coordinate system, and represent curvature components in the cylindrical coordinate system, , represent displacement components in the cylindrical coordinate system, and the comma in the subscript indicates partial differentiation with respect to the corresponding coordinate component, represents the micro-rotation component in the cylindrical coordinate system; The compatibility equation is obtained by differentiating the geometric equation: The constitutive equation is expressed as: Step 22: Substitute the constitutive equation into the compatibility equation, and a compatibility equation expressed by stress and couple stress is obtained through strict derivation: Among them, , , , , represent material constants.

6. The analytical method for solving the crack tip field of auxetic chiral materials according to claim 5, characterized in that The said , , , , are expressed as:

7. The analytical method for solving the crack tip field of the auxetic chiral material according to claim 5, characterized in that The specific steps of Step 3 are as follows: Step 31: The stress and couple stress near the crack tip can be expressed by William’s asymptotic expansion: where the subscript takes respectively , p represents the singular parameter, and represent the zero-order stress and couple stress components in the cylindrical coordinate system respectively, and represent the first-order stress and couple stress components in the cylindrical coordinate system respectively, and represent the second-order stress and couple stress components in the cylindrical coordinate system respectively; Step 3.2: Substitute the zero-order terms of the stress and couple stress near the crack tip , into the equilibrium equations of stress and couple stress in cylindrical coordinates obtained in Step 2.1 and the compatibility equation of stress representation obtained in Step 2.

2. Through derivation and simplification, a system of equations characterizing the singular terms of stress and couple stress is obtained: Among them , , , represent the zero-order stress components in the cylindrical coordinate system. The comma in the subscript indicates partial differentiation with respect to the corresponding coordinate component, and represent the zero-order couple stress components in the cylindrical coordinate system; Steps three and four: From Cramer's rule, we know that for the system of equations representing the singular terms of stress and couple stress to have a solution, the determinant of the coefficient matrix needs to be non-zero. Through matrix operations, we obtain .

8. The analytical method for solving the crack tip field of auxetic chiral materials according to claim 1, wherein The specific steps of Step 4 are as follows: Step 41: In the case of plane strain, setting no traction and no couple stress on the crack surface, the boundary conditions of the crack surface are obtained at this time: Step 42: The stress intensity factors of Mode I and Mode II and the couple stress intensity factor of Mode VI can be defined respectively as: Among them, represents the limit symbol, and represent the macroscopic mode I and mode II stress intensity factors respectively, represents the microscopic mode VI stress intensity factor, represents the couple stress component in the cylindrical coordinate system; Step 43: By solving the system of equations characterizing the singular terms of stress and couple stress in Step 3 and substituting the stress intensity factors of Mode I and Mode II and the couple stress intensity factor of Mode VI, the singular terms of stress and couple stress expressed by stress intensity factors and angular functions in the cylindrical coordinate system are expressed as: Among them, is a dimensionless parameter, and its value range is , and the material parameter S is expressed as: Step 44: Substitute the obtained stress singular terms into the equilibrium equation and constitutive equation of the chiral elastic body and conduct derivation to obtain the zero-order terms of displacement and micro-rotation in the cylindrical coordinate system: Among them, , represent the zero-order displacement components 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 5 are as follows: The force stress near the crack tip and couple stress The first-order terms are substituted into the equilibrium equations of stress and couple stress in cylindrical coordinates obtained in Step 21 and the compatibility equation of stress characterization obtained in Step 22, and the theoretical derivation of Step 4 is repeated to obtain the first-order terms of stress, couple stress, displacement, and micro-rotation in cylindrical coordinates: In the formula, and are the T-stress terms related to type-I and type-II loadings respectively, is the constant couple stress term related to type-II loading, , , , represent the first-order stress components in the cylindrical coordinate system, and represent the first-order couple stress components in the cylindrical coordinate system, , represent the first-order displacement components in the cylindrical coordinate system, represents the first-order micro-rotation component in the cylindrical coordinate system, represents the high-order terms.

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 6 are as follows: Step 6.

1. According to the conversion formula between cylindrical coordinates and Cartesian coordinates, i.e.: Among them, , , , represent stress components in the Cartesian coordinate system, and represent couple stress components in the cylindrical coordinate system, and represent couple stress components in the Cartesian coordinate system, and 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 cases in the Cartesian coordinate system; Step 6.

2. Through calculation, the stress and couple stress in the Cartesian coordinate system are asymptotically expanded as: The asymptotic expansion of displacement and micro-rotation in the Cartesian coordinate system is expressed as: Among them, , represent the crack tip displacement components in the Cartesian coordinate system.

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