A micro-nano circular plate fracture analysis method considering coupling of surface effect and plastic zone

Abstract

The application discloses a micro-nano circular plate fracture analysis method considering coupling of surface effect and plastic zone, and firstly establishes a three-dimensional curved surface coordinate system suitable for a surface layer, and then further corrects a Dugdale cohesion force model according to a superposition principle after considering surface correlation of the plate, and then simplifies the Dugdale cohesion force model into a hyper-singular integral equation by using Hankel transformation, and finally obtains influences of the surface effect on the plastic zone size and crack surface displacement by numerically solving the hyper-singular integral differential equation.The method can accurately capture the nonlinear coupling relationship between the crack size, the surface shear modulus, the torsional displacement and the stress intensity factor under the micro-nano scale, effectively corrects calculation deviation generated by traditional classical theory when the surface effect is ignored, ensures the accuracy of the fracture analysis under different scales, and realizes more accurate and more comprehensive theoretical description of the elastic-plastic fracture behavior of the micro-scale material.

Description

Technical Field

[0001] This invention belongs to the field of fracture behavior analysis and evaluation technology of nanoscale or microscale elastomer structures, specifically a fracture analysis method for micro / nano discs that considers the coupling of surface effects and plastic regions. Background Technology

[0002] With the development of nanoscience and nanotechnology, the dimensions of components in nanoelectromechanical systems (NEMS), such as torsional resonators, rotor devices, and carbon nanotube torsional springs, are becoming increasingly miniaturized. At these micro- and nanoscales, the surface area to volume ratio of materials increases significantly, making the influence of surface free energy and surface stress on the mechanical properties of materials non-negligible. This differs significantly from classical elasticity theory at the macroscale, where classical theories often neglect surface effects. Furthermore, engineering components are frequently subjected to variable-amplitude cyclic loading during service, leading to plastic-induced fatigue cracks. A plastic yield zone typically exists near the crack tip. For macroscopic cracks, the classic Dugdale model (strip yield model) is widely used to estimate the size of the plastic zone and the crack tip opening displacement. However, existing technical solutions still have the following drawbacks: 1. Limitations of macroscopic fracture theory: Traditional macroscopic fracture mechanics (such as analysis based on stress intensity factors) fails when dealing with nanomaterials or microstructures. As structural dimensions shrink to the nanoscale, the effects of surface elasticity and surface stress become non-negligible, but existing models often fail to accurately incorporate these size effects.

[0003] 2. The problem of neglecting the plastic zone: Existing nanocrack analyses that consider surface effects are mostly based on linear elastic fracture mechanics. However, a plastic zone inevitably exists at the crack tip before fracture. Ignoring this elastoplastic behavior can lead to biased predictions of actual fracture toughness and critical load, especially when analyzing high-toughness materials.

[0004] 3. Lack of efficiency in analysis models: Existing analysis methods that simultaneously consider surface effects and plastic zones mostly rely on complex finite element methods (FEM), which involve large computational loads and make it difficult to obtain clear analytical relationships of size effects. Summary of the Invention

[0005] The purpose of this invention is to overcome the shortcomings of the prior art and provide a fracture analysis method for micro / nano circular plates that considers the coupling between surface effects and plastic zones. By establishing a three-dimensional curved surface coordinate system suitable for the surface layer, and after considering the surface correlation of the plate, the Dugdale cohesive model is further modified according to the superposition principle. Then, the Hankel transformation is used to simplify it into a hypersingular integral equation. Subsequently, the influence of surface effects on the size of the plastic zone and the crack surface displacement is obtained by numerically solving the hypersingular integral differential equation.

[0006] To achieve the above objectives, the present invention provides the following technical solutions.

[0007] A fracture analysis method for micro / nano circular plates considering the coupling of surface effects and plastic regions includes the following steps: Step S1: Establish the theoretical framework and geometric model: Based on the surface elasticity theory and the classical bulk elastic continuum model, establish a three-dimensional geometric model of an infinitely large isotropic elastic solid containing a circular plate-shaped crack. The surface elasticity theory model is applied to the crack surface to simulate surface effects. Step S2: Establish governing equations and boundary conditions: Establish the governing equations of the matrix material and the constitutive equations of the surface layer. Combine the applied torsional load and the yield stress in the plastic zone, introduce the surface equilibrium equation, and construct non-standard hybrid boundary conditions that couple surface effects. Step S3: Derive the hypersingular integral equation: Apply Hankel integral transform to transform the non-standard mixed boundary condition problem established in step S2 into a hypersingular integral equation with respect to the auxiliary function; Step S4: Numerical solution; Using the superposition principle, the original problem is decomposed into a first subproblem subjected only to external torsional load and a second subproblem subjected only to yield stress in the plastic zone. The hypersingular integral equations corresponding to the two subproblems are solved by numerical methods to obtain their respective stress intensity factors. Step S5: Determine the size of the plastic zone; Based on the Dagdale cohesive model, by eliminating the condition of stress singularity at the crack tip, the relationship between external load, material yield stress and plastic zone size is established, and then the stress intensity factor and plastic zone size of the two subproblems are solved. Step S6: Calculation and Evaluation; Based on the plastic zone size determined in step S5 and the auxiliary function obtained in step S3, the torsional displacement distribution of the crack surface and the stress distribution at the crack tip are calculated. The shielding or amplification effect of the surface effect on the size of the plastic zone is analyzed, thereby assessing the fracture risk of micro- and nano-components under specific loads.

[0008] Specifically, the three-dimensional geometric model of an infinitely large isotropic elastic solid containing a circular plate-shaped crack established in step S1 is assumed to have a crack radius of . A ring-shaped plastic zone exists around the crack tip, and the outer diameter of the ring-shaped plastic zone is... The range of the annular plastic region is , The width of the annular plastic region.

[0009] Specifically, in step S2, the governing equations of the matrix material and the constitutive equations of the surface layer are established, as follows: Step S21: Establish the governing equations for the matrix material; Based on the classical linear elastic constitutive equation, the equilibrium equation of the matrix material under torsional deformation is obtained: (1); In the above formula, For torsional displacement; Indicates to Find the partial derivative; The axial direction is perpendicular to the polar coordinate plane. Step S22: Establish the constitutive equation of the surface layer; Based on surface elasticity theory, the constitutive equation of the surface layer is obtained: (2); In the above formula, This represents the in-plane surface stress component. The residual surface tension under unconstrained zero-strain conditions; and These are the Lamé constants of the surface materials, respectively; These are in-plane strain components; The symbol for Kronecker; For volumetric strain; The in-plane displacement gradient; This represents the normal surface stress component; The normal displacement gradient; Based on the theory of surface elasticity, the relationship between surface stress and surface deformation is established for surface materials: (3); In the above formula, It is circumferential stress; For mechanical loads; This is the surface shear modulus.

[0010] Specifically, in step S2, the non-standard mixed boundary conditions with coupled surface effects are constructed as follows: To solve the boundary value problem caused by the equilibrium equation (1) constrained by formula (3), the following conditions must be added at infinity outside the surface: (4); In the above formula, For matrix stress components; Due to the actual crack zone The cracked surface is subjected to an applied torsional load. Function, in the plastic zone According to the Dagdale hypothesis, the stress is equal to the material's yield stress. Introducing the surface equilibrium equation: (5); In the above formula, This refers to the surface stress component; Constructing non-standard hybrid boundary conditions with coupled surface effects: (6); In the above formula, The matrix shear modulus; Apply an equivalent load to the nominal crack surface.

[0011] Specifically, the process of deriving the hypersingular integral equation using the Hankel integral transform in step S3 is as follows: Step S31: Construct the torsional displacement components using the Hankel integral transform. ; (7); In the above formula, , These are undetermined coefficients, determined by boundary conditions; is the base of the natural logarithm; It is a Bessel function of the first kind; For transforming variables; Since displacement and stress are continuous in the glued region of the upper and lower half-space of the model, the following continuity conditions apply in the glued region: (8); In the above formula, The displacement component is the surface displacement component of the crack. The stress component on the crack surface; On the crack surface, the displacement components are constructed based on the displacement-stress relationship given by formula (3) and formula (7). The stress components are obtained. : (9); Combining equations (8) and (9), we obtain the hypersingular integral equation for the auxiliary function: (10); In the above formula, For auxiliary functions; For integration variables; and To calculate variables, (11); In the above formula, For torsional load; For the Heaviside function.

[0012] Specifically, in step S4, the original problem is decomposed into two sub-problems. The original problem is the elastoplastic fracture problem of a micro / nano circular plate considering the coupling of surface effects and plastic regions. The two sub-problems after decomposition are: a circular elastic plate with cracks subjected to torsional loads on the crack surface. Fracture problems under action and cracked circular elastic plates subjected to stresses equal to the yield stress in the plastic yield region Fracture problem under action: For the first subproblem, the crack surface is subjected to torsional load. Its function, its hypersingular integral equation, is expressed as: (12); For the second subproblem, the plastic region is subjected to yield stress. Its function, its hypersingular integral equation, is expressed as: (13); Using the Chebyshev polynomial expansion method, the hypersingular integral equations corresponding to the two subproblems are transformed into a system of linear algebraic equations for solution, and the stress intensity factors under the two subproblems are calculated respectively. and ; First, we introduce the following dimensionless variables. (14); The boundary integral equations (12) and (13) of the two subproblems are then rewritten as follows: (15); (16); In the above formula, The width of the dimensionless plastic region; (17); To numerically solve the integral equation (15), a dimensionless auxiliary function is introduced. , (18); In the above formula, These are the unknown coefficients corresponding to the first subproblem; Indicates the order is Chebyshev polynomials of the first kind As the independent variable; Substituting equation (18) into equation (15) and simplifying, we get: (19); In the above formula, ; ; All of these are parameters to be calculated. It is a positive integer. It is a positive integer; Indicates the order is Chebyshev polynomials of the first kind; It is a Chebyshev polynomial of the second kind; (20); By integrating equation (19), we obtain information about the unknown coefficients. The system of linear algebraic equations: (twenty one); In the above formula, These are the initial coefficients corresponding to the first subproblem; These are the unknown coefficients corresponding to the first subproblem; , , , , , These are known constants determined by the integral formula; It is a positive integer; (twenty two); In the above formula, Indicates the order is Chebyshev polynomials of the first kind; Therefore, the stress intensity factor of the first subproblem is expressed as: (twenty three); Repeat the above solution process for the second subproblem to obtain the stress intensity factor in the plastic region: (twenty four); In the above formula, These are the unknown coefficients corresponding to the second subproblem.

[0013] Specifically, in step S5, the relationship between external load, material yield stress, and plastic zone size is established, as follows: Based on the Dagdale cohesive model, the elastoplastic region in front of the crack tip is equivalent to a hypothetical cohesive region. The body stress in front of the crack edge is obtained by integrating the hypersingular integral equation obtained in step S3. The expression: (25); In the above formula, For Bessel functions, ; Based on the properties of Bessel functions, formula (25) can be simplified to obtain: (26); From formula (26), we can obtain that when When, the second term on the right-hand side of formula (26) is integrable, when When the first term on the right side of equation (26) has a square root singularity, then the stress intensity factor of the coin-shaped crack surface under torsional load is... Represented as: (27); Combining equations (26) and (27), we get: (28); Similarly, the torsional displacement is obtained by integrating the boundary integral equations obtained in step S3. : (29); Simplifying formula (29) based on the properties of Bessel functions yields: (30); Performing an inverse Abelian transform on formula (30), the auxiliary function... Using torsional displacement The representation of "to proceed" includes: (31); Formulas (30) and (31) give the relationship between torsional displacement and auxiliary function. When there is an annular plastic yield zone around the coin-shaped crack, the singularity of the crack edge disappears, that is, the total stress intensity factor is satisfied. The requirement is that the dimensionless plastic region width at this time Compared with external load and yield stress ratio The relationship between them is represented as follows: (32); In the above formula, , For torsional load, The yield stress; According to the definition of formula (7), the torsional displacement of the coin-shaped Dugdale crack is obtained. : (33); When the first N terms of the series expression (18) are truncated: When N is a sufficiently large positive integer, it is possible to approximate the exact solution of the integral equation (15). Through numerical calculation, the stress intensity factor and plastic zone size of the two subproblems are obtained.

[0014] Compared with the prior art, the present invention has the following beneficial effects: 1. The method of this invention simultaneously couples surface elasticity and Dugdale plasticity during torsional fracture analysis, making the model closer to the actual fracture process of nanomaterials. It can accurately capture the nonlinear coupling relationship between crack size, surface shear modulus, torsional displacement, and stress intensity factor at the micro- and nano-scale, effectively correcting the calculation deviations caused by neglecting surface effects in traditional classical theories. This ensures the accuracy of fracture analysis at different scales and achieves a more accurate and comprehensive theoretical description of the elastoplastic fracture behavior of microscale materials.

[0015] 2. The method of this invention adopts Hankel transformation and numerical solution of hypersingular integral equations, which avoids complex and computationally time-consuming three-dimensional finite element modeling and improves analysis efficiency.

[0016] 3. The method of this invention can accurately quantify the influence of nanoscale size effect (surface effect) on the size of the plastic zone at the crack tip and the fracture toughness of the material. The calculation results can provide new ideas for the optimization design of multifunctional composite materials and coating technology in smart material applications (such as micro sensors and actuators), and provide direct data support for the toughening design of nanodevices.

[0017] 4. The method of this invention reveals the "shielding" mechanism of surface effect on the evolution of plastic zone, and can accurately assess the improvement of solid load-bearing capacity near defects. In the design of micro and nano structures such as nanoelectromechanical systems (NEMS), this method can be used to predict crack propagation and failure behavior, and can provide a reliable theoretical model and calculation tool for the fatigue resistance design and stability optimization of structures, which has important engineering application value. Attached Figure Description

[0018] To provide a more intuitive understanding of the technical implementation of this invention, the accompanying drawings involved in the embodiments of this invention are briefly described below. These drawings are used to assist in illustrating the implementation methods and are not intended to limit the invention. Those skilled in the art can make derivative designs based on the drawings without creative effort.

[0019] Figure 1 This is a flowchart of a fracture analysis method for micro / nano circular plates that considers the coupling of surface effects and plastic regions according to the present invention. Figure 2 The width of the plastic zone in the embodiment of the present invention without surface effect. and Relationship curve; Figure 3 The width of the plastic zone under different surface elastic parameters in the embodiments of the present invention. Follow The changing relationship curve; Figure 4 The width of the plastic zone in this embodiment of the invention With surface parameters The changing relationship curve. Detailed Implementation

[0020] To facilitate understanding and implementation of the present invention by those skilled in the art, the various steps of the method proposed in this invention are described in detail below. It should be understood that these embodiments are for illustrative purposes only and are not intended to limit the scope of the invention. Furthermore, it should be understood that after reading the teachings of this invention, those skilled in the art can make various modifications or alterations to the invention, and these equivalent forms also fall within the scope defined by the appended claims.

[0021] Example 1 like Figure 1 As shown, this embodiment discloses a fracture analysis method for micro / nano circular plates that considers the coupling of surface effects and plastic regions, including the following steps: Step S1: Establish the theoretical framework and geometric model: Based on the surface elasticity theory and the classical bulk elastic continuum model, establish a three-dimensional geometric model of an infinitely large isotropic elastic solid containing a circular plate-shaped crack. The surface elasticity theory model is applied to the crack surface to simulate surface effects. Step S2: Establish governing equations and boundary conditions: Establish the governing equations of the matrix material and the constitutive equations of the surface layer. Combine the applied torsional load and the yield stress in the plastic zone, introduce the surface equilibrium equation, and construct non-standard hybrid boundary conditions that couple surface effects. Step S3: Derive the hypersingular integral equation: Apply Hankel integral transform to transform the non-standard mixed boundary condition problem established in step S2 into a hypersingular integral equation with respect to the auxiliary function; Step S4: Numerical solution; Using the superposition principle, the original problem is decomposed into a first subproblem subjected only to external torsional load and a second subproblem subjected only to yield stress in the plastic zone. The hypersingular integral equations corresponding to the two subproblems are solved by numerical methods to obtain their respective stress intensity factors. Step S5: Determine the size of the plastic zone; Based on the Dagdale cohesive model, by eliminating the condition of stress singularity at the crack tip, the relationship between external load, material yield stress and plastic zone size is established, and then the stress intensity factor and plastic zone size of the two subproblems are solved. Step S6: Calculation and Evaluation; Based on the plastic zone size determined in step S5 and the auxiliary function obtained in step S3, the torsional displacement distribution of the crack surface and the stress distribution at the crack tip are calculated. The shielding or amplification effect of the surface effect on the size of the plastic zone is analyzed, thereby assessing the fracture risk of micro- and nano-components under specific loads.

[0022] Specifically, the three-dimensional geometric model of an infinitely large isotropic elastic solid containing a circular plate-shaped crack established in step S1 is assumed to have a crack radius of . A ring-shaped plastic zone exists around the crack tip, and the outer diameter of the ring-shaped plastic zone is... The range of the annular plastic region is , The width of the annular plastic region.

[0023] In this embodiment, the three-dimensional geometric model of an infinitely large isotropic elastic solid containing circular plate-shaped cracks is applicable to both the matrix material and the surface material. The matrix material refers to the main body of the model other than the surface material.

[0024] Specifically, in step S2, the governing equations of the matrix material and the constitutive equations of the surface layer are established, as follows: Step S21: Establish the governing equations for the matrix material; Based on the classical linear elastic constitutive equation, the equilibrium equation of the matrix material under torsional deformation is obtained: (1); In the above formula, For torsional displacement; Indicates to Find the partial derivative; The axial direction is perpendicular to the polar coordinate plane. Step S22: Establish the constitutive equation of the surface layer; Based on surface elasticity theory, the constitutive equation of the surface layer is obtained: (2); In the above formula, This represents the in-plane surface stress component. The residual surface tension under unconstrained zero-strain conditions; and These are the Lamé constants of the surface materials, respectively; These are in-plane strain components; The symbol for Kronecker; For volumetric strain; The in-plane displacement gradient; This represents the normal surface stress component; The normal displacement gradient; Based on the theory of surface elasticity, the relationship between surface stress and surface deformation is established for surface materials: (3); In the above formula, It is circumferential stress; For mechanical loads; This is the surface shear modulus.

[0025] In this embodiment, unless otherwise specified, in all parameter symbols, the superscript S represents the corresponding physical quantity defined on surface S, spatial index. Surface index The coordinates represent the surface coordinates. For repeated indices, a summation convention is used. The commas between subscripts indicate partial differentiation of the spatial coordinates corresponding to the comma.

[0026] Specifically, in step S2, the non-standard mixed boundary conditions with coupled surface effects are constructed as follows: To solve the boundary value problem caused by the equilibrium equation (1) constrained by formula (3), the following conditions must be added at infinity outside the surface: (4); In the above formula, For matrix stress components; Due to the actual crack zone The cracked surface is subjected to an applied torsional load. Function, in the plastic zone According to the Dagdale hypothesis, the stress is equal to the material's yield stress. Introducing the surface equilibrium equation: (5); In the above formula, This refers to the surface stress component; Constructing non-standard hybrid boundary conditions with coupled surface effects: (6); In the above formula, The matrix shear modulus; Apply an equivalent load to the nominal crack surface.

[0027] Specifically, the process of deriving the hypersingular integral equation using the Hankel integral transform in step S3 is as follows: Step S31: Construct the torsional displacement components using the Hankel integral transform. ; (7); In the above formula, , These are undetermined coefficients, determined by boundary conditions; is the base of the natural logarithm; It is a Bessel function of the first kind; For transforming variables; Since displacement and stress are continuous in the glued region of the upper and lower half-space of the model, the following continuity conditions apply in the glued region: (8); In the above formula, The displacement component is the surface displacement component of the crack. The stress component on the crack surface; On the crack surface, the displacement components are constructed based on the displacement-stress relationship given by formula (3) and formula (7). The stress components are obtained. : (9); Combining equations (8) and (9), we obtain the hypersingular integral equation for the auxiliary function: (10); In the above formula, For auxiliary functions; For integration variables; and To calculate variables, (11); In the above formula, For torsional load; For the Heaviside function.

[0028] Specifically, in step S4, the original problem is decomposed into two sub-problems. The original problem is the elastoplastic fracture problem of a micro / nano circular plate considering the coupling of surface effects and plastic regions. The two sub-problems after decomposition are: a circular elastic plate with cracks subjected to torsional loads on the crack surface. Fracture problems under action and cracked circular elastic plates subjected to stresses equal to the yield stress in the plastic yield region Fracture problem under action: For the first subproblem, the crack surface is subjected to torsional load. Its function, its hypersingular integral equation, is expressed as: (12); For the second subproblem, the plastic region is subjected to yield stress. Its function, its hypersingular integral equation, is expressed as: (13); Using the Chebyshev polynomial expansion method, the hypersingular integral equations corresponding to the two subproblems are transformed into a system of linear algebraic equations for solution, and the stress intensity factors under the two subproblems are calculated respectively. and ; First, we introduce the following dimensionless variables. (14); The boundary integral equations (12) and (13) of the two subproblems are then rewritten as follows: (15); (16); In the above formula, The width of the dimensionless plastic region; (17); To numerically solve the integral equation (15), a dimensionless auxiliary function is introduced. , (18); In the above formula, These are the unknown coefficients corresponding to the first subproblem; Indicates the order is Chebyshev polynomials of the first kind As the independent variable; Substituting equation (18) into equation (15) and simplifying, we get: (19); In the above formula, ; ; All of these are parameters to be calculated. It is a positive integer. It is a positive integer; Indicates the order is Chebyshev polynomials of the first kind; It is a Chebyshev polynomial of the second kind; (20); By integrating equation (19), we obtain information about the unknown coefficients. The system of linear algebraic equations: (twenty one); In the above formula, These are the initial coefficients corresponding to the first subproblem; These are the unknown coefficients corresponding to the first subproblem; , , , , , These are known constants determined by the integral formula; It is a positive integer; (twenty two); In the above formula, Indicates the order is Chebyshev polynomials of the first kind; Therefore, the stress intensity factor of the first subproblem is expressed as: (twenty three); Repeat the above solution process for the second subproblem to obtain the stress intensity factor in the plastic region: (twenty four); In the above formula, These are the unknown coefficients corresponding to the second subproblem.

[0029] Specifically, step S5 establishes the external load, material yield stress, and plastic zone dimensions. The relationship is as follows: Based on the Dagdale cohesive model, the elastoplastic region in front of the crack tip is equivalent to a hypothetical cohesive region. The body stress in front of the crack edge is obtained by integrating the hypersingular integral equation obtained in step S3. The expression: (25); In the above formula, For Bessel functions, ; Based on the properties of Bessel functions, formula (25) can be simplified to obtain: (26); From formula (26), we can obtain that when When, the second term on the right-hand side of formula (26) is integrable, when When the first term on the right side of equation (26) has a square root singularity, then the stress intensity factor of the coin-shaped crack surface under torsional load is... Represented as: (27); Combining equations (26) and (27), we get: (28); Similarly, the torsional displacement is obtained by integrating the boundary integral equations obtained in step S3. : (29); Simplifying formula (29) based on the properties of Bessel functions yields: (30); Performing an inverse Abelian transform on formula (30), the auxiliary function... Using torsional displacement The representation of "to proceed" includes: (31); Formulas (30) and (31) give the relationship between torsional displacement and auxiliary function. When there is an annular plastic yield zone around the coin-shaped crack, the singularity of the crack edge disappears, that is, the total stress intensity factor is satisfied. The requirement is that the dimensionless plastic region width at this time Compared with external load and yield stress ratio The relationship between them is represented as follows: (32); In the above formula, , For torsional load, The yield stress; According to the definition of formula (7), the torsional displacement of the coin-shaped Dugdale crack is obtained. : (33); When the first N terms of the series expression (18) are truncated: When N is a sufficiently large positive integer, it is possible to approximate the exact solution of the integral equation (15). Through numerical calculation, the stress intensity factor and plastic zone size of the two subproblems are obtained.

[0030] The technical effects of the method of the present invention will be further explained below using the fracture analysis of gallium nitride (GaN) nanostructures as an example.

[0031] I. Surface-effect-free fracture analysis of gallium nitride (GaN) nanostructures; First, we analyze the size of the plastic zone in an elastomer with a torsional coin-shaped crack in the absence of surface effects. We then set the matrix shear modulus. Surface Lame constant Set yield stress Define dimensionless surface parameters as material standard values. ,make According to formula (28), the stress intensity factor is... : (34); The calculation results of formula (34) are consistent with existing classical results, when yield stress exists in the plastic yield zone. At that time, by Obtain the plastic zone dimensions The following relationship must be satisfied: (35); Formula (35) is about The nonlinear equations, under normal circumstances, indicate that the plastic region size is relatively small, i.e. From formula (35), the approximate expression for the size of the plastic zone can be obtained as follows: (36); In the above formula, ; From formula (36), it can be seen that if the size of the plastic zone... If the accuracy of the obtained approximate value is greater than 94.4%, it can meet the engineering design requirements.

[0032] II. Fracture analysis of gallium nitride (GaN) nanostructures; (a) Comparison and verification of calculations considering surface effects with classical results; While keeping the material and load parameters unchanged in the above "surface-effect-free fracture analysis", dimensionless surface size parameters are introduced. (Determined by the surface shear modulus, matrix shear modulus and crack radius), and the dimensionless plastic zone width is obtained by solving formula (15) and formula (16) according to steps S3-S5.

[0033] To verify the correctness of the method of the present invention, firstly, take... (i.e., the surface effect disappears), then the numerical solution result of this invention should degenerate into the classic Dugdale coin-shaped crack result. For example... Figure 2 The figure shows the width of the plastic zone without surface effects. Follow The relationship curves are shown in Table 1 below, which lists the plastic zone widths under several load ratios. Comparing the data, the results show that... The maximum relative error between the results of this invention and the classical results does not exceed 1.14%, corresponding to an accuracy greater than 98.86%, which meets the requirements of fracture assessment calculation accuracy (usually not less than 95%) in engineering design.

[0034] Table 1 Under these conditions, the calculation results of this invention are compared with those of classical methods. ; Note: "Classical results" in the table are the calculated values ​​of the relationship shown in formula (35) / formula (36); "Results of this invention" are the values ​​obtained by solving formula (15) and formula (16) using Chebyshev polynomial truncation (N=60 truncation terms).

[0035] Further considering the surface effect, take ,like Figure 3 The figure shows different surface elastic parameters. Lower plastic zone width With load ratio The changing relationship curve. From Figure 3 It can be seen that when During (surface hardening), the width of the plastic zone decreases; when During surface softening, the width of the plastic zone increases. This indicates that the method of the present invention can effectively capture the influence of surface effects on the evolution of the plastic zone; (II) Comparison of calculation results after introducing surface effects with classical limit solutions; exist At this time, surface elasticity changes the equivalent boundary conditions of the crack surface, thus having a "shielding" or "amplifying" effect on the size of the plastic zone. For example... Figure 4 As shown, at different load ratios Below, the dimensionless plastic region width Depending on the dimensionless surface parameters The relationship between the changes shows a significant size dependence. Figure 4 The curve in the figure shows that, for a given load ratio The width of the plastic zone varies with surface parameters As the surface elasticity increases, the surface effect decreases, which inhibits the development of the plastic zone. Furthermore, the surface elasticity changes the equivalent boundary conditions of the crack surface, thus "shielding" or "amplifying" the size of the plastic zone.

[0036] To verify the correctness of the method of the present invention in considering surface effects, this example uses the classical limit solution for comparison: when the yield stress When the plastic region is sufficiently sized to approach zero (i.e., approximately the "no yield" limit), the method of this invention should degenerate into the existing linear elastic coin-shaped crack solution considering surface effects. For the same Calculated with load parameters, the displacement obtained by the present invention under the "no yield" limit has a maximum relative error of no more than 2.0% compared with the classical solution, and the accuracy is greater than 98.0%.

[0037] In addition, to visually demonstrate the quantitative impact of surface effects on the width of the plastic zone, Table 2 below shows the results under a fixed load ratio. Under the condition of 0.20, different dimensionless surface parameters Corresponding plastic zone width And the case without surface effect ( Compared to the rate of change of the plastic zone width, engineers can quickly estimate the impact of surface effects on the size of the plastic zone by referring to a table.

[0038] Table 2. Plastic zone width corresponding to different surface parameters under a fixed load ratio ; The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention in any other way. Any person skilled in the art may make changes or modifications to the above-disclosed technical content to create equivalent embodiments. However, any simple modifications, equivalent changes, and modifications made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention shall still fall within the protection scope of the present invention.

Claims

1. A fracture analysis method for micro / nano circular plates considering the coupling of surface effects and plastic regions, characterized in that, Includes the following steps: Step S1: Establish the theoretical framework and geometric model: Based on the surface elasticity theory and the classical bulk elastic continuum model, establish a three-dimensional geometric model of an infinitely large isotropic elastic solid containing a circular plate-shaped crack. The surface elasticity theory model is applied to the crack surface to simulate surface effects. Step S2: Establish governing equations and boundary conditions: Establish the governing equations of the matrix material and the constitutive equations of the surface layer. Combine the applied torsional load and the yield stress in the plastic zone, introduce the surface equilibrium equation, and construct non-standard hybrid boundary conditions that couple surface effects. Step S3: Derive the hypersingular integral equation: Apply Hankel integral transform to transform the non-standard mixed boundary condition problem established in step S2 into a hypersingular integral equation with respect to the auxiliary function; Step S4: Numerical solution; Using the superposition principle, the original problem is decomposed into a first subproblem subjected only to external torsional load and a second subproblem subjected only to yield stress in the plastic zone. The hypersingular integral equations corresponding to the two subproblems are solved by numerical methods to obtain their respective stress intensity factors. Step S5: Determine the size of the plastic zone; Based on the Dagdale cohesive model, by eliminating the condition of stress singularity at the crack tip, the relationship between external load, material yield stress and plastic zone size is established, and then the stress intensity factor and plastic zone size of the two subproblems are solved. Step S6: Calculation and Evaluation; Based on the plastic zone size determined in step S5 and the auxiliary function obtained in step S3, the torsional displacement distribution of the crack surface and the stress distribution at the crack tip are calculated. The shielding or amplification effect of the surface effect on the size of the plastic zone is analyzed, thereby assessing the fracture risk of micro- and nano-components under specific loads.

2. The fracture analysis method for micro / nano circular plates considering the coupling of surface effects and plastic regions according to claim 1, characterized in that, The three-dimensional geometric model of an infinitely large isotropic elastic solid containing a circular plate-shaped crack established in step S1 is given by the following formula: Let the crack radius of this model be... A ring-shaped plastic zone exists around the crack tip, and the outer diameter of the ring-shaped plastic zone is... The range of the annular plastic region is , The width of the annular plastic region.

3. The fracture analysis method for micro / nano circular plates considering the coupling of surface effects and plastic regions according to claim 2, characterized in that, In step S2, the governing equations of the matrix material and the constitutive equations of the surface layer are established, as follows: Step S21: Establish the governing equations for the matrix material; Based on the classical linear elastic constitutive equation, the equilibrium equation of the matrix material under torsional deformation is obtained: （1）； In the above formula, For torsional displacement; Indicates to Find the partial derivative; The axial direction is perpendicular to the polar coordinate plane. Step S22: Establish the constitutive equation of the surface layer; Based on surface elasticity theory, the constitutive equation of the surface layer is obtained: （2）； In the above formula, This represents the in-plane surface stress component. The residual surface tension under unconstrained zero-strain conditions; and These are the Lamé constants of the surface materials, respectively; These are in-plane strain components; The symbol for Kronecker; For volumetric strain; The in-plane displacement gradient; This represents the normal surface stress component; The normal displacement gradient; Based on the theory of surface elasticity, the relationship between surface stress and surface deformation is established for surface materials: （3）； In the above formula, It is circumferential stress; For mechanical loads; This is the surface shear modulus.

4. The fracture analysis method for micro / nano circular plates considering the coupling of surface effects and plastic regions according to claim 3, characterized in that, In step S2, non-standard mixed boundary conditions with coupled surface effects are constructed as follows: To solve the boundary value problem caused by the equilibrium equation (1) constrained by formula (3), the following conditions must be added at infinity outside the surface: （4）； In the above formula, For matrix stress components; Due to the actual crack zone The cracked surface is subjected to an applied torsional load. Function, in the plastic zone According to the Dagdale hypothesis, the stress is equal to the material's yield stress. Introducing the surface equilibrium equation: （5）； In the above formula, This refers to the surface stress component; Constructing non-standard hybrid boundary conditions with coupled surface effects: （6）； In the above formula, The matrix shear modulus; Apply an equivalent load to the nominal crack surface.

5. The fracture analysis method for micro / nano circular plates considering the coupling of surface effects and plastic regions according to claim 4, characterized in that, The process of deriving the hypersingular integral equation using the Hankel integral transform in step S3 is as follows: Step S31: Construct the torsional displacement components using the Hankel integral transform. ; （7）； In the above formula, , These are undetermined coefficients, determined by boundary conditions; is the base of the natural logarithm; It is a Bessel function of the first kind; For transforming variables; Since displacement and stress are continuous in the glued region of the upper and lower half-space of the model, the following continuity conditions apply in the glued region: （8）； In the above formula, The displacement component is the surface displacement component of the crack. The stress component on the crack surface; On the crack surface, the displacement components are constructed based on the displacement-stress relationship given by formula (3) and formula (7). The stress components are obtained. : （9）； Combining equations (8) and (9), we obtain the hypersingular integral equation for the auxiliary function: （10）； In the above formula, For auxiliary functions; For integration variables; and To calculate variables, （11）； In the above formula, For torsional load; For the Heaviside function.

6. The fracture analysis method for micro / nano circular plates considering the coupling of surface effects and plastic regions according to claim 5, characterized in that, In step S4, the original problem is decomposed into two sub-problems. The original problem is the elastoplastic fracture problem of a micro / nano circular plate considering the coupling of surface effects and plastic regions. The two sub-problems after decomposition are: a circular elastic plate with cracks subjected to torsional loads on the crack surface. Fracture problems under action and cracked circular elastic plates subjected to stresses equal to the yield stress in the plastic yield region Fracture problem under action: For the first subproblem, the crack surface is subjected to torsional load. Its function, its hypersingular integral equation, is expressed as: （12）； For the second subproblem, the plastic region is subjected to yield stress. Its function, its hypersingular integral equation, is expressed as: （13）； Using the Chebyshev polynomial expansion method, the hypersingular integral equations corresponding to the two subproblems are transformed into a system of linear algebraic equations for solution, and the stress intensity factors under the two subproblems are calculated respectively. and ; First, we introduce the following dimensionless variables. （14）； The boundary integral equations (12) and (13) of the two subproblems are then rewritten as follows: （15）； （16）； In the above formula, The width of the dimensionless plastic region; （17）； To numerically solve the integral equation (15), a dimensionless auxiliary function is introduced. , （18）； In the above formula, These are the unknown coefficients corresponding to the first subproblem; Indicates the order is Chebyshev polynomials of the first kind As the independent variable; Substituting equation (18) into equation (15) and simplifying, we get: （19）； In the above formula, , , All of these are parameters to be calculated. It is a positive integer. It is a positive integer; Indicates the order is Chebyshev polynomials of the first kind; It is a Chebyshev polynomial of the second kind; （20）； By integrating equation (19), we obtain information about the unknown coefficients. The system of linear algebraic equations: （21）； In the above formula, These are the initial coefficients corresponding to the first subproblem; These are the unknown coefficients corresponding to the first subproblem; , , , , , These are known constants determined by the integral formula; It is a positive integer; （22）； In the above formula, Indicates the order is Chebyshev polynomials of the first kind; Therefore, the stress intensity factor of the first subproblem is expressed as: （23）； Repeat the above solution process for the second subproblem to obtain the stress intensity factor in the plastic region: （24）； In the above formula, These are the unknown coefficients corresponding to the second subproblem.

7. The fracture analysis method for micro / nano circular plates considering the coupling of surface effects and plastic regions according to claim 6, characterized in that, In step S5, the relationship between external load, material yield stress, and plastic zone size is established. The process is as follows: Based on the Dagdale cohesive model, the elastoplastic region in front of the crack tip is equivalent to a hypothetical cohesive region. The body stress in front of the crack edge is obtained by integrating the hypersingular integral equation obtained in step S3. The expression: （25）； In the above formula, For Bessel functions, ; Based on the properties of Bessel functions, formula (25) can be simplified to obtain: （26）； From formula (26), we can obtain that when When, the second term on the right-hand side of formula (26) is integrable, when When the first term on the right side of equation (26) has a square root singularity, then the stress intensity factor of the coin-shaped crack surface under torsional load is... Represented as: （27）； Combining equations (26) and (27), we get: （28）； Similarly, the torsional displacement is obtained by integrating the boundary integral equations obtained in step S3. : （29）； Simplifying formula (29) based on the properties of Bessel functions yields: （30）； Performing an inverse Abelian transform on formula (30), the auxiliary function... Using torsional displacement The representation of "to proceed" includes: （31）； Formulas (30) and (31) give the relationship between torsional displacement and auxiliary function. When there is an annular plastic yield zone around the coin-shaped crack, the singularity of the crack edge disappears, that is, the total stress intensity factor is satisfied. The requirement is that the dimensionless plastic region width at this time Compared with external load and yield stress ratio The relationship between them is represented as follows: （32）； In the above formula, , For torsional load, The yield stress; According to the definition of formula (7), the torsional displacement of the coin-shaped Dugdale crack is obtained. : （33）； When the first N terms of the series expression (18) are truncated: When N is a sufficiently large positive integer, it is possible to approximate the exact solution of the integral equation (15). Through numerical calculation, the stress intensity factor and plastic zone size of the two subproblems are obtained.

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