Decoupling Interaction Integral Method for Macroscopic and Microscopic Fracture Strength Factors of Chiral Materials

Abstract

The present invention discloses an interaction integral method for decoupling the macro-micro fracture strength factors of chiral materials. First, according to the governing equations of chiral materials and the expressions of auxiliary fields, a J-integral expression characterized by macro-micro strength factors is obtained; second, the interaction part of the real field and the auxiliary field of chiral materials is extracted to obtain the line integral form of the interaction integral; third, by substituting the definition of the auxiliary field and the governing equations of the real field into the interaction integral, an interaction integral form considering chiral microstructures is obtained. Subsequently, the line integral along the material interface is derived to eliminate the influence of the chiral material interface integral. Finally, according to the J-integral expression, the relationship between the interaction integral and the macro-micro strength factors is obtained, and the macro-micro strength factors are separated by selecting the auxiliary strength factors. This method can achieve the decoupling of the macroscopic stress intensity factor and the microscopic couple stress intensity factor of chiral materials and accurately analyze the fracture behavior of materials.

Description

Technical Field

[0001] The present invention belongs to the technical field of fracture mechanics and relates to an interaction integral method that can decouple the intensity factors of macro - micro fracture modes of chiral materials. Background Art

[0002] Biological microstructures tend to adopt chiral shapes, which is even considered as a unified structural principle in biology. There are various complex chiral microstructures in nature, such as the Bouligand structure in the exoskeleton of crabs, the nanorod bundle microstructure in the teeth of limpets, and the triple - helix structure of collagen fibers in tendons. These chiral microstructures usually endow materials with excellent mechanical properties, including high strength, toughness, and impact resistance.

[0003] Chirality in materials usually manifests as the coupling of mechanical deformations, such as tension - torsion coupling or bending - torsion coupling. Due to the unique deformation characteristics of chiral microstructures, the chiral microstructures control the fracture and mechanical properties of materials. For example, chiral tissues can induce couple stresses, thereby significantly reducing the stress concentration near the crack tip. In addition, chiral tissues can hinder crack propagation, thus enhancing the fracture toughness of biological materials to a certain extent. Therefore, for such materials, establishing a correct fracture mechanics analysis model to deeply understand their potential deformation mechanisms is crucial for designing high - performance bionic composite materials.

[0004] In linear elastic fracture mechanics, the stress intensity factor and the energy release rate near the crack tip are important parameters for evaluating the fracture behavior of materials. Currently, the main methods for solving the stress intensity factor include the displacement method, the stress method, the J - integral, and the interaction integral. For the coupling between the macro - micro fracture modes of chiral materials, the interaction integral can well separate the intensity factors of different fracture modes.

[0005] However, there is currently a lack of means to decouple the macroscopic stress intensity factor and the microscopic couple stress intensity factor of chiral materials. Therefore, in order to study the fracture mechanics properties of such materials and guide the bionic optimization design of composite materials, it is of great engineering significance to establish an interaction integral that decouples the macro - micro fracture intensity factors of chiral materials. Summary of the Invention

[0006] In order to solve the current lack of means to decouple the macroscopic stress intensity factor and the microscopic couple stress intensity factor of chiral materials, the present invention takes into account the coupled deformation between the volumetric strain and the micro - rotation of chiral materials, and through strict J - integral theoretical derivation, obtains an interaction integral expression that decouples the macro - micro fracture intensity factors of chiral materials, and then proposes an interaction integral method for decoupling the macro - micro fracture intensity factors of chiral materials. This method can achieve the decoupling of the macroscopic stress intensity factor and the microscopic couple stress intensity factor of chiral materials and accurately analyze the fracture behavior of materials.

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

[0008] An interaction integral method for decoupling the macro-micro fracture strength factors of chiral materials, comprising the following steps:

[0009] Step 1: Considering the influence of the coupling of tension and torsion deformation, based on the governing equations of chiral materials and the expressions of the auxiliary fields, establish the J-integral theory of chiral materials. Through derivation and simplification, obtain the J-integral expression characterized by macro-micro strength factors;

[0010] Step 2: By extracting the interaction part of the real field and the auxiliary field of the chiral material, obtain the line integral form of the interaction integral of the chiral material;

[0011] Step 3: Through the divergence theorem, convert the line integral into a region integral, divide the line integral form of the interaction integral of the chiral material into two terms, and substitute the definition of the auxiliary field and the governing equation of the real field into the interaction integral of the chiral material for derivation to obtain the interaction integral form of the chiral material considering the chiral microstructure;

[0012] Step 4: Combining the structural characteristics of biological materials containing complex material interfaces, divide the integral region into different material parts, eliminate the influence of the chiral material interface integral by setting the characteristic of the intact bonding of the material interface, based on the curvilinear coordinate system, and using the chain rule to derive the line integral along the material interface;

[0013] Step 5: According to the J-integral expression characterized by macro-micro strength factors, obtain the relationship between the interaction integral and the macro-micro strength factors, and decouple the macro-micro strength factors by selecting the auxiliary strength factors.

[0014] Compared with the prior art, the present invention has the following advantages:

[0015] 1. For the first time, the J-integral expression characterized by macro-micro strength factors is completely derived, and an interaction integral method for separating the macro-micro strength factors of chiral materials is proposed according to the J-integral theory, expanding the application range of the interaction integral method.

[0016] 2. The present invention eliminates the influence of the material interface integral through strict theoretical derivation, making the interaction integral still applicable to biological materials containing complex material interfaces, and can realize the fracture mechanics analysis of biological materials with different structural characteristics, such as the nanorod bundle microstructure in limpet teeth.

[0017] 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 decouple the macro-micro fracture modes of chiral biomaterials, and can be developed into a commercial program to flexibly adapt to the changes of the required problems. Description of the Drawings

[0018] Figure 1 It is a flow chart for deriving the interaction integral of the decoupled macro-micro fracture strength factors of chiral materials;

[0019] Figure 2 It is a schematic diagram of the integration region bounded by a closed loop;

[0020] Figure 3 It is a schematic diagram of the integration domain A being divided into two domains A1 and A2 by the material interface;

[0021] Figure 4 It is a schematic diagram of a two-dimensional auxetic chiral elastic solid containing a planar crack under mixed-mode loading conditions;

[0022] Figure 5 It is a schematic diagram of the curvilinear coordinate system on the material interface;

[0023] Figure 6 It is a diagram of a chiral material square plate and a finite element mesh;

[0024] Figure 7 It is a diagram of the stress intensity factor and couple stress intensity factor at different chiral angles;

[0025] Figure 8 It is the error of the stress intensity factor and couple stress intensity factor decoupled from different integration regions. Detailed Embodiment

[0026] The technical solution of the present invention will be further described below in conjunction with the drawings, but it 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 shall be covered by the protection scope of the present invention.

[0027] The present invention provides an interaction integral method for decoupling the macro-micro fracture strength factors of chiral materials. The method takes into account the constitutive relationship between the volumetric strain and micro-rotation of chiral materials. First, according to the governing equations of chiral materials and the expressions of auxiliary fields, through derivation and simplification, a J-integral expression characterized by macro-micro strength factors is obtained. Secondly, by extracting the interaction part of the real field and the auxiliary field of chiral materials, the line integral form of the interaction integral is obtained. Thirdly, by substituting the definition of the auxiliary field and the governing equation of the real field into the interaction integral for derivation, the interaction integral form considering chiral microstructures is obtained. Subsequently, based on the curvilinear coordinate system and using the chain rule to derive the line integral along the material interface, the influence of the chiral material interface integral is eliminated. Finally, according to the J-integral expression characterized by macro-micro strength factors, the relationship between the interaction integral and the macro-micro strength factors is obtained, and the macro-micro strength factors are separated by selecting the auxiliary strength factors. As Figure 1 shown, the specific steps are as follows:

[0028] Step 1: Considering the influence of the coupling of tension and torsion, according to the governing equations of chiral materials and the expressions of auxiliary fields, establish the J-integral theory of chiral materials. Through derivation and simplification, obtain the J-integral expression characterized by macro-micro strength factors. The specific steps are as follows:

[0029] Step 1-1: Establish the governing equations of chiral materials:

[0030] Equilibrium equation:

[0031]

[0032] Geometric equation:

[0033]

[0034] Constitutive equation:

[0035]

[0036] The above equations adopt tensor component notation. The value ranges of the subscripts i, j, k, and l are 1 to 2 and follow the Einstein summation convention: the variables marked with the subscripts i, j, k, and 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 that 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 ε jiare strain components, m i3 are couple stress components, χ i3 are curvature components. e ij3 e 3ij and e 3lk are the Levi-Civita symbols. δ ij is the Kronecker symbol. A comma in a subscript denotes partial differentiation with respect to the corresponding coordinate component, i.e., u j are displacement components, φ3 is the micro-rotation, x i are coordinate components. are stress components without considering the chirality effect, are stress components considering the chirality effect. λ, μ, κ, and γ are elastic constants. A is a chirality material parameter characterizing the coupling of volumetric strain and micro-rotation. The relationships between Young's modulus E, shear modulus G, Poisson's ratio ν, and coupling number N are as follows:

[0037]

[0038] The expressions for each parameter are as follows:

[0039]

[0040] where ω represents the slenderness ratio of the thickness to the length within the ligament plane, E s ′ represents the plane elastic modulus. When E s ′ = E s it represents plane stress. 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 chirality angle, where β > 0 represents positive chirality and β < 0 represents negative chirality.

[0041] Steps One and Two: The auxiliary field stress field of the chiral material can be expressed as:

[0042]

[0043]

[0044] The auxiliary field displacement field of the chiral material can be expressed as:

[0045]

[0046]

[0047] The auxiliary field strain field of the chiral material can be expressed as:

[0048]

[0049] wherein and represent the auxiliary stress, and represent the auxiliary displacement. r, θ represent the polar coordinates of the crack tip, and represent the auxiliary strain and curvature respectively. and represent the auxiliary stress component and the couple stress component respectively, represents the auxiliary micro-rotation. A ijkl and B i3k3 are the material stiffnesses of the chiral elastomer. Where ξ = A / (λ + μ) is a dimensionless parameter, and its value range is The material parameter S can be expressed as:

[0050]

[0051] wherein in the case of plane stress, the Poisson's ratio v needs to be replaced by v / (1 + v). The macroscopic stress intensity factor K I and K II , the microscopic couple stress intensity factor K VI can be defined respectively as:

[0052]

[0053] wherein, r and θ represent the cylindrical coordinate components, lim represents the limit symbol, K I and K II represent the macroscopic mode I and mode II stress intensity factors respectively, K VI represents the microscopic mode VI stress intensity factor, σ 22 and σ 21 represent the stress components, m 23 represents the couple stress component.

[0054] For a two-dimensional non-uniform chiral elastomer, the J integral is defined as follows:

[0055]

[0056] where Γ0 represents an arbitrary integration path, taking a circular integration path, at this time Γ = rdθ. n i represents the unit outer normal vector of the integration path, as shown in Figure 2 . σ jk and σ ij represent the stress components, m j3 represents m i3 the couple stress component, u j,1 represents the displacement component, φ 3,1 represents the micro-rotation component, χj3 Denotes the curvature component, ε jk Denotes the strain component. δ i1 Is the Kronecker symbol. ∫ denotes the integral symbol.

[0057] Substitute the expression of the crack tip auxiliary field into the definition of the J integral. Through derivation and simplification, the J integral expression expressed by two macroscopic stress intensity factors and one microscopic couple stress intensity factor can be obtained:

[0058]

[0059] Step 2: By extracting the interaction part of the real field and the auxiliary field of the chiral material, the line integral form of the chiral material interaction integral is obtained. The specific steps are as follows:

[0060] Step 2-1: Convert the J integral (Equation (13)) in Step 1 into a closed-loop integral. As Figure 2 shown, assuming that there is no stress and couple stress on the crack surface, at this time the J integral can be expressed in the form of an integrand multiplied by a weight function:

[0061]

[0062] where, ω i Is the unit outer normal vector of the contour line , Γ1, Denotes the integration path, Γ B Denotes the closed loop composed of the integration path Γ1, , as Figure 2 shown. q is an arbitrary smooth weight function, whose value is 1 on Γ0 and 0 on Γ1.

[0063] Step 2-2: By superimposing the real field and the auxiliary field, obtain the J act+aux integral of the superimposed state:

[0064]

[0065] where, J act+aux Denotes the superimposed state of the real field and the auxiliary field, and the J act+aux integral can be divided into three parts J act+aux = J act + J aux + I, where J act Is the J integral form of the real field, J aux Is the J integral form of the auxiliary field, and Are the real field stress components without considering the chiral effect, and Are the real field stress components considering the chiral effect. and are the auxiliary field stress components, and are the auxiliary field couple stress components, is the auxiliary field strain component, is the auxiliary field curvature component, is the auxiliary field displacement component, is the auxiliary field micro-rotation. The comma in the subscript represents partial derivative with respect to the corresponding coordinate component, i.e., ∮ represents the integral symbol, and I represents the line integral form of the chiral material interaction integral. The line integral form of the chiral material interaction integral can be expressed as:

[0066]

[0067] Step 3: Through the divergence theorem, convert the line integral into a volume integral. Divide the line integral form of the chiral material interaction integral into two terms. By substituting the definitions of the auxiliary fields and the governing equations of the real fields into the chiral material interaction integral for derivation, obtain the chiral material interaction integral form considering the chiral microstructure. The specific steps are as follows:

[0068] Step 3-1: Through the divergence theorem, the line integral form of the chiral material interaction integral can be transformed into a volume integral form:

[0069]

[0070] Step 3-2: Considering that the auxiliary constitutive equation is consistent with the real field constitutive equation, the relationship between the auxiliary field and the real field can be expressed as According to the definition of the auxiliary field, there are the following relationships:

[0071]

[0072] where and represent the material constants at the crack tip.

[0073] Step 3-3: According to the definition of the auxiliary field and the governing equations of the real fields, the interaction integral can be simplified into a form without the derivative terms of the chiral material, obtaining the chiral material interaction integral form considering the chiral microstructure:

[0074]

[0075] Step 4. Combining the structural characteristics of the biological material containing the complex material interface, divide the integration region into different material parts. By setting the property of the intact adhesion of the material interface, based on the curvilinear coordinate system, and using the chain rule to derive the line integral along the material interface, the influence of the chiral material interface integral is eliminated. The specific steps are as follows:

[0076] Step 4-1. Consider a chiral material with an ideal bonding interface to discuss the influence of the material interface, as Figure 3 shown. An integration domain A is divided into two domains A1 and A2 by the material interface. Among them, A1 is surrounded by a closed integral , where Γ B1 , Γ interface , Γ B3 , all represent the integration paths, and A2 is surrounded by a closed integral , where represents the integration path in the opposite direction to Γ interface , and Γ B2 represents the integration path. The material properties of these two regions show diversity but remain continuous and differentiable, ensuring a smooth transition between different properties. At this time, the chiral material interaction integral considering the chiral microstructure in Step 3 can be decomposed into two loop integrals and and a boundary integral I interface :

[0077]

[0078] The boundary integral can be expressed as:

[0079]

[0080] where the superscript indicates that they are located in regions A1 and A2 respectively, and the material properties in regions A1 and A2 are taken respectively.

[0081] Step 4-2. The definition of the auxiliary field stipulates the continuity of the auxiliary stress, couple stress, displacement, micro-rotation and their derivatives at the interface. Therefore, the auxiliary field satisfies the following relationships:

[0082]

[0083] Step 4-3. Set the interface to be in an equilibrium state, which necessarily means that the resultant force and the resultant force acting on the interface are both zero:

[0084]

[0085] Step 4-4. Substitute the continuity condition of the auxiliary field and the interface equilibrium condition into the boundary integral to obtain:

[0086]

[0087] Step 4 and 5: As Figure 4 shown, define a curvilinear coordinate system (ξ1, ξ2), and a point on the curvilinear coordinates satisfies the relationship and In addition, the relationship between the Cartesian coordinate system and the curvilinear coordinate system is as follows:

[0088]

[0089] Step 4 and 5: On the premise that the interfaces are fully combined, the displacements, micro-rotations and their derivatives on the adjacent sides of the interfaces must be equal. At this time

[0090] Step 4 and 6: Substitute the expressions obtained in Step 4 and 5 and Equation (26) into the interface integral expression, and I interface = 0 can be obtained. This shows that the expression of the interaction integral obtained in Step 3 is still applicable to chiral materials with complex interfaces. Through Step 3 and Step 4, it is proved that the interaction integral proposed by the present invention for the fracture characteristics of chiral microstructures does not require the derivatives of material parameters, and is also applicable to the structural characteristics of complex material interfaces in biological materials.

[0091] Step 5: According to the J-integral expression characterized by macro-microscopic intensity factors, obtain the relationship between the interaction integral and the macro-microscopic intensity factors, and decouple the macro-microscopic intensity factors by selecting auxiliary intensity factors. The specific steps are as follows:

[0092] Step 5-1: According to the J-integral expression in formula (14) in Step 1, considering the superposition of the real field and the auxiliary field, the J-integral expression expressed by the auxiliary field intensity factor and the real field intensity factor can be obtained:

[0093]

[0094] where and respectively represent the auxiliary field macroscopic mode I and mode II stress intensity factors, represents the auxiliary field microscopic mode VI stress intensity factor.

[0095] Step 5-2: The interaction integral expressed by two macroscopic stress intensity factors and one microscopic couple stress intensity factor can be derived from the interaction between the real state and the auxiliary state:

[0096]

[0097] Step 5-3: By using the vector Take the values [1 0 0], [0 1 0] and [0 0 1] respectively to obtain the intensity factors expressed by the interaction integral:

[0098]

[0099] Among them, the material parameters S, μ, κ, γ need to be replaced by non-uniform chiral material parameters at the crack tip.

[0100] Step Four: Through solution, the macroscopic stress intensity factor and the microscopic couple stress intensity factor of the chiral material can be obtained.

[0101] To illustrate the applicability of the above technical solution of the present invention, first decouple the stress intensity factor and the couple stress intensity factor below. Then select different material functions to verify that the present invention is still applicable to chiral materials containing material interfaces.

[0102] Example 1: Decouple two macroscopic stress intensity factors and one couple stress intensity factor.

[0103] As Figure 5 shown, a square plate of homogeneous chiral material with a length of 2W = 2 and a central crack length of 2a = 1 is studied. Both the upper and lower edges of the chiral plate are subjected to stress σ0 and couple stress m0. At the same time, the corresponding finite element mesh of the square plate of homogeneous chiral material is given in Figure 5 . All numerical examples given in the results use normalized intensity factors. From the results of Figure 6 , it can be found that the present invention can easily decouple two macroscopic stress intensity factors and one microscopic couple stress intensity factor, proving that the method of the present invention is effective when decoupling the fracture intensity factors of chiral materials.

[0104] Example 2: The applicability of the interaction integral for chiral materials containing complex material interfaces.

[0105] As Figure 7 shown, a material function is selected to divide the chiral material plate in Figure 5 into a uniform material region and a non-uniform material region. At the same time, 10 integration regions (10, 25, 50, 100, 150, 200, 250, 300, 350 and 400) are selected, and the errors of the stress intensity factors and couple stress intensity factors decoupled in different integration regions are shown in Figure 8 . From Figure 8 , it can be found that the errors of all intensity factors are less than 0.2%. From this example, it can be proved that for chiral biomaterials containing complex interfaces, the interaction integral proposed by the present invention is not affected and is still applicable.

Claims

1. An interaction integral method for decoupling the macroscopic and microscopic fracture strength factors of chiral materials, characterized in that The method includes the following steps: Step 1: Considering the influence of the coupled tension-torsion deformation, establish the J-integral theory of chiral materials according to the control equations of chiral materials and the expressions of auxiliary fields. Through derivation and simplification, obtain the J-integral expression characterized by macro-microscopic strength factors; Step 2: By extracting the interaction part between the real field and the auxiliary field of chiral materials, obtain the line integral form of the interaction integral of chiral materials; Step 3: Through the divergence theorem, convert the line integral into a region integral. Divide the line integral form of the interaction integral of chiral materials into two terms. By substituting the definition of the auxiliary field and the control equation of the real field into the interaction integral of chiral materials for derivation, obtain the interaction integral form of chiral materials considering chiral microstructures. The specific steps are as follows: Step 3-1: Convert the line integral form of the interaction integral of chiral materials into a region integral form through the divergence theorem: Among them, is the real field stress component without considering the chiral effect, is the real field stress component considering the chiral effect, and are the auxiliary field stress components, and are the auxiliary field couple stress components, is the auxiliary field displacement component, is the auxiliary field micro-rotation. The comma in the subscript indicates partial derivative with respect to the corresponding coordinate component. I represents the line integral form of the chiral material interaction integral. m i3 is the couple stress component, χ j3 represents the curvature component, ε jk represents the strain component, u j,1 represents the displacement component, φ 3,1 represents the micro-rotation component, δ i1 is the Kronecker symbol, ∫ represents the integral symbol. The value range of each subscript i, j, k, l is 1 to 2. The variables marked with the subscripts i, j, k, l are the components of a vector or tensor; Step 32. Considering that the auxiliary constitutive equation is consistent with the true-field constitutive equation, the relationship between the auxiliary field and the true field is expressed as According to the definition of the auxiliary field, the following relationship holds: Among them, σ ij is the stress component, ε ij is the strain component, and respectively represent the auxiliary strain and curvature, and respectively represent the auxiliary stress component and the couple stress component, is the auxiliary field couple stress component, χ i3 is the curvature component, and represent the material constants at the crack tip; Step 3-3: Simplify the interaction integral according to the definition of the auxiliary field and the control equation of the real field into a form without derivative terms of chiral materials, and obtain the interaction integral form of chiral materials considering chiral microstructures: where A is a chiral material parameter characterizing the coupling of volumetric strain and micro-rotation, q is an arbitrary smooth weight function, A ijkl and B i3k3 are the material stiffnesses of the chiral elastomer, is the true field stress component without considering the chiral effect, e ij3 is the Levi-Civita symbol, is the auxiliary field displacement component, and the comma in the subscript indicates partial differentiation with respect to the corresponding coordinate component; Step 4: Combining the structural characteristics of biological materials with complex material interfaces, divide the integral region into different material parts. By setting the characteristic of good bonding at the material interface, based on the curvilinear coordinate system, and using the chain rule to derive the line integral along the material interface to eliminate the influence of the chiral material interface integral; Step 5: According to the J-integral expression characterized by macro-microscopic strength factors, obtain the relationship between the interaction integral and the macro-microscopic strength factors. By selecting the auxiliary strength factor, decouple the macro-microscopic strength factors.

2. The interactive integral method for decoupling the macroscopic and microscopic fracture strength factors of chiral materials according to claim 1, wherein The specific steps of Step 1 are as follows: Step 1-1: Establish the control equations of chiral materials: Equilibrium equation: σ ji,j = 0, m i3,i +e ij3 σ ij = 0, Geometric equation: ε ij = u j,i - e ij3 φ3, χ i3 = φ 3,i , Constitutive equation: m i3 =γχ i3 , Among them, σ ij is the stress component, ε kk , ε lk , ε ij and ε ji are the strain components, m i3 is the couple stress component, χ i3 is the curvature component, e ij3 , e 3ij and e 3lk are the Levi-Civita symbols, δ ij is the Kronecker symbol, the comma in the subscript represents partial differentiation with respect to the corresponding coordinate component, u j is the displacement component, φ3 is the micro-rotation, is the stress component without considering the chiral effect, is the stress component considering the chiral effect, λ, μ, κ and γ are elastic constants, A is the chiral material parameter characterizing the coupling of volumetric strain and micro-rotation, u j,1 represents the displacement component, φ 3,1 represents the micro-rotation component; Step 1-2: The auxiliary field stress field of chiral materials is expressed as: The auxiliary field displacement field of chiral materials is expressed as: The auxiliary field strain field of chiral materials is expressed as: Among them, and represent the auxiliary stress, and represent the auxiliary displacement, r, θ represent the polar coordinates of the crack tip, and represent the auxiliary strain and curvature respectively, and represent the auxiliary stress component and the couple stress component respectively, represents the auxiliary micro-rotation, A ijkl and B i3k3 are the material stiffnesses of the chiral elastic body, where ξ = A / (λ + μ) is a dimensionless parameter, and its value range is S is the material parameter, K I and K II are the macroscopic stress intensity factors, K VI is the microscopic couple stress intensity factor; For a two-dimensional inhomogeneous chiral elastic body, the J-integral is defined as follows: where Γ0 represents an arbitrary integration path, n i represents the unit outer normal vector of the integration path, σ jk and σ ij represent stress components, m j3 represents m i3 couple stress components, u j,1 represents displacement components, φ 3,1 represents the micro-rotation components, χ j3 represents the curvature components, ε jk represents strain components, δ i1 is the Kronecker symbol, and ∫ represents the integral symbol; Substitute the expression of the auxiliary field at the crack tip into the definition of the J-integral. Through derivation and simplification, obtain the J-integral expression expressed by two macroscopic stress intensity factors and one microscopic couple stress intensity factor:

3. The interactive integral method for decoupling the macro-micro fracture strength factors of 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 chiral materials are expressed as:

4. The interactive integral method for decoupling the macroscopic and microscopic fracture strength factors of 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 length, and E′ s represents the plane elastic modulus. When E′ s = E s , it represents plane stress. 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, and β represents the chiral angle, where β > 0 represents positive chirality and β < 0 represents negative chirality.

5. The interactive integral method for the macro-micro fracture strength factors of the decoupled chiral material according to claim 2, wherein The material parameter S is expressed as: G, ν, and N respectively represent the shear modulus, Poisson's ratio, and coupling number of chiral materials; Macroscopic stress intensity factor K I and K II , the microscopic couple stress intensity factor K VI are respectively defined as: Among them, lim represents the limit symbol, K I and K II represent the macroscopic mode I and mode II stress intensity factors respectively, K VI represents the microscopic mode VI stress intensity factor, σ 22 and σ 21 represent stress components, m 23 represents the couple stress component.

6. The interactive integral method for the macro-micro fracture strength factors of the decoupled chiral material according to claim 2, wherein The specific steps of Step 2 are as follows: Step 2-1: Convert the J-integral in Step 1 into a closed-loop integral. Assume that there is no stress and couple stress on the crack surface. At this time, the J-integral is expressed in the form of the integrand multiplied by a weight function: where ω i is the unit outer normal vector of the contour line , Γ1 denotes the integration path, and Γ B denotes the closed loop formed by the integration path Γ1 . q is an arbitrary smooth weight function that takes the value 1 on Γ0 and 0 on Γ1; Step 22: Obtain J in the superposition state by superposing the real field and the auxiliary field act+aux Integration: Among them, J act+aux represents the superposition state of the real field and the auxiliary field, and J act+aux The integral is divided into three parts, and J act+aux = J act + J aux + I, where J act is the J-integral form of the real field, and J aux is the J-integral form of the auxiliary field. and are the real field stress components without considering the chiral effect. and are the real field stress components considering the chiral effect. and are the auxiliary field stress components. and are the auxiliary field couple stress components. is the auxiliary field strain component. is the auxiliary field curvature component. is the auxiliary field displacement component. is the auxiliary field micro-rotation. The comma in the subscript indicates partial differentiation with respect to the corresponding coordinate component. represents the integral symbol, and I represents the line integral form of the chiral material interaction integral.

7. The interactive integral method for decoupling the macro-micro fracture strength factors of chiral materials according to claim 6, characterized in that The line integral form of the interaction integral of chiral materials is expressed as:

8. The interactive integral method for decoupling the macroscopic and microscopic fracture strength factors of chiral materials according to claim 1, characterized in that The specific steps of Step 5 are as follows: Step 5-1: According to the J-integral expression in Step 1, considering the superposition of the real field and the auxiliary field, obtain the J-integral expression expressed by the auxiliary field strength factor and the real field strength factor: Among them, and respectively represent the macroscopic mode-I and mode-II stress intensity factors of the auxiliary field, represents the microscopic mode-VI stress intensity factor of the auxiliary field; Step 52: Derive the interaction integral expressed by two macroscopic stress intensity factors and one microscopic couple stress intensity factor from the interaction between the real state and the auxiliary state. Step Five Three. By forming a vector consisting of three auxiliary intensity factors and respectively taking values of [100], [010], and [001], intensity factors expressed by interaction integrals are obtained: Step 54: Obtain the macroscopic stress intensity factor and the microscopic couple stress intensity factor of the chiral material by solving.

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