Method and device for calculating bearing capacity envelope line of apron board foundation in marine clay

Abstract

The invention provides a method and a device for calculating a bearing capacity envelope line of an apron board foundation in marine clay. According to the method, Cosseerat continuum regularization is realized by utilizing a UEL interface and FORTRAN programming of ABAQUS, a rotational degree of freedom and an internal length parameter are introduced, and a Drucker-Prager constitutive model is established. Modeling is carried out in the ABAQUS, an inp file is modified after grid division, and a user-defined unit and Cosseerat parameters are embedded. And applying a load by adopting a side friction method, extracting the ultimate bending moment and the bearing capacity after solving, and drawing an envelope line. According to the method, the mesh generation problem is solved through the Cosseerat theory, the mesh dependence is avoided, and the calculation accuracy is improved in combination with the Drucker-Prager model.

Description

Technical Field

[0001] The present application belongs to the field of bearing capacity calculation of skirt foundations, and in particular to a method and device for calculating the bearing capacity envelope of skirt foundations in marine clay. Background Art

[0002] The bearing capacity of the skirt foundation is affected by many factors, such as soil type, physical and mechanical properties of the soil, geometric dimensions of the foundation (such as depth-to-diameter ratio), load action mode, and changes in the surrounding environment of the foundation. Therefore, accurate numerical simulation and analysis of the bearing capacity of the skirt foundation is an effective means to solve these complex influencing factors.

[0003] Saturated marine clay is a special type of soil. Its complex mechanical behaviors such as nonlinearity and strain localization make it difficult for traditional numerical calculation methods to accurately simulate. Although the Moore-Coulomb criterion can effectively describe the strength of the soil, it has a strong grid dependence when considering the strain localization phenomenon, resulting in unstable and inaccurate calculation results. In order to solve this problem, researchers have tried to introduce the Cosserat continuum theory. The Cosserat continuum theory introduces the microstructural effects of the material. By introducing rotational degrees of freedom and length scale parameters, it can effectively describe and simulate the non-local behavior of the soil, especially in strain localization, plastic failure and foundation bearing capacity analysis. It shows good results.

[0004] However, the existing analysis methods of skirt foundation bearing capacity based on Cosserat continuum theory still have certain technical bottlenecks. Although some studies have adopted Cosserat continuum theory to analyze skirt foundation, most of them are mainly focused on simple two-dimensional or three-dimensional models and are mostly limited to static analysis. How to conduct in-depth research on numerical solution capabilities, refinement of theoretical models, and how to achieve efficient calculations under complex working conditions have become issues that need to be solved urgently. Summary of the invention

[0005] The purpose of the present application is to overcome the defects in the above-mentioned prior art and to provide a method and device for calculating the bearing capacity envelope of a skirt foundation in marine clay.

[0006] The present application provides a method for calculating the bearing capacity envelope of a skirt foundation in marine clay, including: Through the UEL subroutine interface of ABAQUS, the regularization mechanism of Cosserat continuum theory is implemented by FORTRAN code programming. The regularization mechanism includes the introduction of rotational degrees of freedom and internal length parameters, and the establishment of the Drucker-Prager constitutive model matching the Moore-Coulomb criterion to simulate the elastic-plastic behavior of saturated marine clay. A two-dimensional plane strain model of a skirt foundation with different depth-to-diameter ratios is established in the ABAQUS, an inp file is exported after meshing, and the inp file is modified to embed a custom unit, Cosserat material parameters, and virtual unit connection settings; The side rubbing method is used to apply rotation load and horizontal displacement load to the two-dimensional plane strain model in the modified inp file. The ultimate bending moment and horizontal bearing capacity are extracted after solving by ABAQUS, and the bearing capacity envelope is drawn.

[0007] Optionally, the regularization mechanism is implemented according to the strain-displacement relationship and equilibrium equation of a two-dimensional Cosserat continuum, expressed as: ; ; Among them, ε is the Cosserat strain tensor, L is the differential operator matrix, u is the displacement vector, σ is the Cosserat stress tensor, and f is the body force vector; The differential operator matrix L is defined as: ; in, represents the positive strain in the x direction, represents the positive strain in the y direction; For ε, defined as the sum of linear elastic strain and plastic strain, the isotropic elastic modulus matrix is satisfy: ; Among them, λ=2Gν / (1-2ν), G is the shear modulus, ν is the Poisson's ratio, is the Cosserat shear modulus.

[0008] Optionally, establishing a Drucker-Prager constitutive model matching the Mohr-Coulomb criterion includes: establishing a Drucker-Prager constitutive model matching the Mohr-Coulomb criterion according to an equivalent relationship between the Drucker-Prager criterion and the Mohr-Coulomb criterion under two-dimensional plane strain conditions.

[0009] Optionally, it also includes: According to the custom field variable output requirements, a cloud map containing Cosserat rotational degrees of freedom, plastic strain localization area and stress vector is generated in the ABAQUS result file.

[0010] Optionally, a two-dimensional plane strain model of the skirt foundation with different depth-to-diameter ratios is established in the ABAQUS, including: According to the dimensional requirements of depth-to-diameter ratio H / D=0.2, 0.5, and 1.0, a two-dimensional plane strain model with a soil domain size of 10B×3B is established; The mesh of the area around the skirt foundation is encrypted, and the mesh element type is an eight-node plane strain quadrilateral element. A fixed constraint is imposed on the bottom boundary and a horizontal displacement constraint is imposed on the lateral boundary.

[0011] Optionally, applying a rotational load and a horizontal displacement load to the modified inp file by a side rubbing method, including: Apply a rotation load with an angle increment of Δθ=0.001rad until the failure state θmax=0.1rad is reached; Apply horizontal displacement loads in stages at the limit rotation angle, with a displacement increment of Δu=0.01B, until the calculation fails to converge.

[0012] Optionally, the ultimate bending moment and horizontal bearing capacity are extracted after solving by ABAQUS, and the bearing capacity envelope is drawn, including: According to the ODB result file of ABAQUS, the integral value of the base reaction under each load step is extracted, the horizontal bearing capacity Hult and bending moment Mult are calculated, and the VHM three-dimensional envelope is generated by data fitting.

[0013] Optionally, the value range of the internal length parameter is 0.1m≤lc≤1.0m, and lc=2d50 is satisfied according to the average diameter d50 of soil particles. The present application also provides a device for calculating the bearing capacity envelope of a skirt foundation in marine clay, comprising: The regularization modeling module is configured to implement the regularization mechanism of Cosserat continuum theory using FORTRAN code through the UEL subroutine interface of ABAQUS, including: a rotational degree of freedom embedding unit, which is used to add the microscopic rotation component around the z axis to the node degree of freedom; an internal length parameter loading unit, which stores and associates the mapping relationship between the internal length parameter and the average diameter of the soil particles; a differential operator matrix generator, which generates a 7×3-dimensional differential operator matrix according to the formula; The DP-MC constitutive module is configured to establish a Drucker-Prager constitutive model that matches the Moore-Coulomb criterion, including: a parameter converter that converts the input cohesion c and friction angle into DP-MC coefficients; a yield surface calculation unit that updates the yield surface in real time; The model building module is configured to generate and preprocess finite element model data, including: a depth-to-diameter ratio parameterized modeling unit, which receives the depth-to-diameter ratio and automatically adjusts the model geometry; an intelligent mesh generator, which performs local encryption on the area around the skirt foundation; an inp file editor, which writes Cosserat material parameters into the inp file; The calculation execution module is configured to drive the ABAQUS solver to perform bearing capacity analysis, including: a side-rubbing loading controller that applies loads in stages according to the rotation angle increment and displacement increment; a regularized convergence monitor that triggers mesh adaptive reconstruction when the node rotation wz>0.1rad; The visualization output module is configured to generate engineering analysis results, including: an envelope plotter that fits the extracted Hult-Mult data into a three-dimensional envelope surface; a strain localization cloud map generator that displays the κxz, κyz curvature distribution in the form of a heat map.

[0014] Optionally, the regularized modeling module uses FORTRAN code to implement the regularization mechanism of Cosserat continuum theory, expressed as: ; ; Among them, ε is the Cosserat strain tensor, L is the differential operator matrix, u is the displacement vector, σ is the Cosserat stress tensor, and f is the body force vector; The differential operator matrix L is defined as: ; in, represents the positive strain in the x direction, represents the positive strain in the y direction; For ε, defined as the sum of linear elastic strain and plastic strain, the isotropic elastic modulus matrix is satisfy: ; Among them, λ=2Gν / (1-2ν), G is the shear modulus, ν is the Poisson's ratio, is the Cosserat shear modulus.

[0015] The beneficial effects of this application are: The present application provides a method for calculating the bearing capacity envelope of a skirt foundation in marine clay, comprising: implementing a regularization mechanism of Cosserat continuum theory by FORTRAN code programming through the UEL subroutine interface of ABAQUS, wherein the regularization mechanism includes introducing rotational degrees of freedom and internal length parameters, and establishing a Drucker-Prager constitutive model matching the Moore-Coulomb criterion to simulate the elastic-plastic behavior of saturated marine clay; establishing a two-dimensional plane strain model of a skirt foundation containing different depth-to-diameter ratios in the ABAQUS, exporting an inp file after meshing, and modifying the inp file to embed a custom unit, Cosserat material parameters, and virtual unit connection settings; applying a rotational load and a horizontal displacement load to the two-dimensional plane strain model in the modified inp file by the side rubbing method, extracting the ultimate bending moment and horizontal bearing capacity after solving by ABAQUS, and drawing a bearing capacity envelope. This application solves the problems of local strain and numerical instability caused by grid division in traditional numerical methods by introducing the regularization mechanism of Cosserat continuum theory; utilizes the non-local effect of Cosserat continuum theory to effectively avoid the common grid dependence problem in traditional finite element methods; combines the Drucker-Prager constitutive model and Cosserat continuum theory to accurately describe the elastic-plastic behavior of saturated marine clay, thereby improving the accuracy of bearing capacity calculation. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Figure 1 It is a schematic diagram of the calculation process of the bearing capacity envelope of the skirt foundation in marine clay in this application; Figure 2 It is a schematic diagram of the mesh division and boundary conditions of the skirt foundation in this application; Figure 3 is a schematic diagram of a deformed grid in this application; Figure 4 It is a schematic diagram comparing the envelope of the bearing capacity of the skirt board foundation in this application; Figure 5 It is a schematic diagram of the deviation of different internal length parameters lc in this application. DETAILED DESCRIPTION

[0017] The exemplary embodiments of the present disclosure will be described in more detail below with reference to the accompanying drawings. Although the exemplary embodiments of the present disclosure are shown in the accompanying drawings, it should be understood that the present disclosure can be implemented in various forms and should not be limited by the embodiments set forth herein. On the contrary, the embodiments are provided in order to enable a more thorough understanding of the present disclosure and to fully convey the scope of the present disclosure to those skilled in the art.

[0018] Please refer to Figure 1As shown, the present application provides a method for calculating the bearing capacity envelope of a skirt foundation in marine clay, comprising: S101. Using the UEL subroutine interface of ABAQUS, a regularization mechanism of Cosserat continuum theory is implemented by FORTRAN code programming. The regularization mechanism includes introducing rotational degrees of freedom and internal length parameters, and establishing a Drucker-Prager constitutive model matching the Moore-Coulomb criterion to simulate the elastic-plastic behavior of saturated marine clay. Constructing regularization mechanism and elastoplastic constitutive model: For the secondary development UEL subroutine interface provided by ABAQUS, FORTRAN code is used for programming. The regularization mechanism of Cosserat continuum theory is introduced into the matrix to be solved, and the constitutive model is programmed as the Drucker-Prager criterion matching the Moore-Coulomb criterion to simulate the elastoplastic behavior of saturated marine soil.

[0019] Develop visualization modules to customize variable output.

[0020] Furthermore, by introducing the regularization mechanism of rotational freedom and internal length parameter lc, the strain-displacement relationship and equilibrium equation of the two-dimensional Cosserat continuum are: ; ; Among them, ε is the Cosserat strain tensor, L is the differential operator matrix, u is the displacement vector, σ is the Cosserat stress tensor, and f is the body force vector; The differential operator matrix L is defined as: ; in, represents the positive strain in the x direction, represents the positive strain in the y direction; For ε, a linear elastic strain is assumed and plastic strain The sum of the isotropic elastic modulus matrix is for: ; Among them, λ=2Gν / (1-2ν), G is the shear modulus, ν is the Poisson's ratio, is the Cosserat shear modulus.

[0021] is the Lame constant, and its relationship with the shear modulus and Poisson's ratio satisfies the equation: ; The shear strain in a two-dimensional Cosserat continuum can be expressed as: ; in, is the asymmetric Cosserat shear strain component, is the displacement gradient component, is the microscopic rotational freedom around the z-axis.

[0022] Reasonable values ​​of the coefficients and in the Drucker-Prager yield criterion (DP criterion) under two-dimensional plane strain conditions can make the DP yield criterion and the Mohr-Coulomb yield criterion (MC criterion) have a high consistency when calculating rock and soil strength related problems, that is, the DP criterion matching the MC criterion (DP-MC criterion) is obtained.

[0023] Under two-dimensional plane strain conditions, the yield surface equation of DP-MC is expressed as follows: ; Where: ; ; ; ; ; Where q is the deviatoric stress, is the hydrostatic pressure, is the friction angle related parameter, is the cohesion-related parameter, is the soil friction angle, and c is the cohesion.

[0024] S102, establishing a two-dimensional plane strain model of a skirt foundation with different depth-to-diameter ratios in the ABAQUS, exporting an inp file after meshing, and modifying the inp file to embed a custom unit, Cosserat material parameters, and virtual unit connection settings; The analytical model of the skirt foundation is established in Abaqus software, including different depth-to-diameter ratios to simulate the influence of size. After dividing the mesh and establishing the relevant unit set and node set, the inp file containing various parameters of the model that can be used for calculation is exported.

[0025] Modify the inp file, add custom units and material parameter settings, and add virtual units to achieve connection with the developed solver part.

[0026] Take the rigid skirt foundation located in a homogeneous marine saturated clay as an example: An eight-node plane strain quadrilateral element was used in the calculation. To avoid the influence of boundary effects, the size of the soil domain was set to 10BX3B (B=3m).

[0027] In this numerical simulation, the depth-to-diameter ratios are 0.2, 0.5, and 1.0 to simulate the influence of foundation size effect on bearing capacity.

[0028] Horizontal constraints are imposed on the vertical boundaries of the soil domain, fixed constraints are imposed on the bottom boundary, and the top boundary is free.

[0029] like Figure 2 As shown in the figure, in order to improve the calculation accuracy, the mesh within a certain range near the anchor plate is refined. The basic mesh division and boundary conditions of the skirt plate with a depth-to-diameter ratio of 1.0.

[0030] For undrained conditions, the same material parameters are used and the soil Poisson's ratio , friction angle and expansion angle , cohesion , directly using the undrained shear strength of the soil, and the elastic modulus is taken as , and it is assumed that the skirt base is rigid, that is, it is considered that it does not deform during the drawing process.

[0031] The interface between the foundation and the soil is rough and binds the foundation to the soil.

[0032] S103, applying rotational load and horizontal displacement load to the two-dimensional plane strain model in the modified inp file by using the side rubbing method, extracting the ultimate bending moment and horizontal bearing capacity after solving by ABAQUS, and drawing the bearing capacity envelope.

[0033] Use the side-wiping method and the displacement loading method to set different rotation angles and displacements.

[0034] Submit the inp file and the programmed FOR program in the job module of the ABAQUS platform for calculation. After the calculation is completed, open the odB result file, extract the calculation results of the skirt foundation bearing capacity, organize the data, and draw the envelope line.

[0035] like Figure 3 As shown in the figure, the deformed mesh schematic diagram obtained greatly improves the mesh distortion, improves the mesh quality and overcomes the mesh dependence compared with the classical finite element method.

[0036] like Figure 4 As shown, the bearing capacity envelopes of skirt foundations with different depth-to-diameter ratios are arranged. This application ensures the accuracy of the bearing capacity envelope while taking advantage of the Cosserat continuum.

[0037] like Figure 5As shown in the figure, the influence of different values ​​of the regularization mechanism of the internal length parameter on the bearing capacity envelope is analyzed, and the deviation comparison of the internal length parameter lc is obtained, which reveals the change law of different lc values ​​on the envelope shape and the ultimate bearing capacity.

[0038] The present application also provides a device for calculating the bearing capacity envelope of a skirt foundation in marine clay, comprising: The regularization modeling module is configured to implement the regularization mechanism of Cosserat continuum theory using FORTRAN code through the UEL subroutine interface of ABAQUS, including: a rotational degree of freedom embedding unit, which is used to add the microscopic rotation component around the z axis to the node degree of freedom; an internal length parameter loading unit, which stores and associates the mapping relationship between the internal length parameter and the average diameter of the soil particles; a differential operator matrix generator, which generates a 7×3-dimensional differential operator matrix according to the formula; The DP-MC constitutive module is configured to establish a Drucker-Prager constitutive model that matches the Moore-Coulomb criterion, including: a parameter converter that converts the input cohesion c and friction angle into DP-MC coefficients; a yield surface calculation unit that updates the yield surface in real time; The model building module is configured to generate and preprocess finite element model data, including: a depth-to-diameter ratio parameterized modeling unit, which receives the depth-to-diameter ratio and automatically adjusts the model geometry; an intelligent mesh generator, which performs local encryption on the area around the skirt foundation; an inp file editor, which writes Cosserat material parameters into the inp file; The calculation execution module is configured to drive the ABAQUS solver to perform bearing capacity analysis, including: a side-rubbing loading controller that applies loads in stages according to the rotation angle increment and displacement increment; a regularized convergence monitor that triggers mesh adaptive reconstruction when the node rotation wz>0.1rad; The visualization output module is configured to generate engineering analysis results, including: an envelope plotter that fits the extracted Hult-Mult data into a three-dimensional envelope surface; a strain localization cloud map generator that displays the κxz, κyz curvature distribution in the form of a heat map.

[0039] Furthermore, the regularized modeling module uses FORTRAN code to implement the regularization mechanism of Cosserat continuum theory, which is expressed as: ; ; Among them, ε is the Cosserat strain tensor, L is the differential operator matrix, u is the displacement vector, σ is the Cosserat stress tensor, and f is the body force vector; The differential operator matrix L is defined as: ; in, represents the positive strain in the x direction, represents the positive strain in the y direction; For ε, defined as the sum of linear elastic strain and plastic strain, the isotropic elastic modulus matrix is satisfy: ; Among them, λ=2Gν / (1-2ν), G is the shear modulus, ν is the Poisson's ratio, is the Cosserat shear modulus.

[0040] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims

1. A method for calculating the bearing capacity envelope of a skirt foundation in marine clay, characterized in that: include: Through the UEL subroutine interface of ABAQUS, the regularization mechanism of Cosserat continuum theory is implemented by FORTRAN code programming. The regularization mechanism includes the introduction of rotational degrees of freedom and internal length parameters, and the establishment of the Drucker-Prager constitutive model matching the Moore-Coulomb criterion to simulate the elastic-plastic behavior of saturated marine clay. A two-dimensional plane strain model of a skirt foundation with different depth-to-diameter ratios is established in the ABAQUS, an inp file is exported after meshing, and the inp file is modified to embed a custom unit, Cosserat material parameters, and virtual unit connection settings; The side rubbing method is used to apply rotation load and horizontal displacement load to the two-dimensional plane strain model in the modified inp file. The ultimate bending moment and horizontal bearing capacity are extracted after solving by ABAQUS, and the bearing capacity envelope is drawn.

2. The method for calculating the bearing capacity envelope of a skirt foundation in marine clay according to claim 1, characterized in that: The regularization mechanism is implemented based on the strain-displacement relationship and equilibrium equation of the two-dimensional Cosserat continuum, which is expressed as: ； ； Among them, ε is the Cosserat strain tensor, L is the differential operator matrix, u is the displacement vector, σ is the Cosserat stress tensor, and f is the body force vector; The differential operator matrix L is defined as: ； in, represents the positive strain in the x direction, represents the positive strain in the y direction; For ε, defined as the sum of linear elastic strain and plastic strain, the isotropic elastic modulus matrix is satisfy: ； Among them, λ=2Gν / (1-2ν), G is the shear modulus, ν is the Poisson's ratio, is the Cosserat shear modulus.

3. The method for calculating the bearing capacity envelope of a skirt foundation in marine clay according to claim 1, characterized in that: A Drucker-Prager constitutive model matching the Mohr-Coulomb criterion is established, including: establishing a Drucker-Prager constitutive model matching the Mohr-Coulomb criterion according to the equivalent relationship between the Drucker-Prager criterion and the Mohr-Coulomb criterion under two-dimensional plane strain conditions.

4. The method for calculating the bearing capacity envelope of a skirt foundation in marine clay according to claim 1, characterized in that: Also includes: According to the custom field variable output requirements, a cloud map containing Cosserat rotational degrees of freedom, plastic strain localization area and stress vector is generated in the ABAQUS result file.

5. The method for calculating the bearing capacity envelope of a skirt foundation in marine clay according to claim 1, characterized in that: A two-dimensional plane strain model of the skirt foundation with different depth-to-diameter ratios is established in the ABAQUS, including: According to the dimensional requirements of depth-to-diameter ratio H / D=0.2, 0.5, and 1.0, a two-dimensional plane strain model with a soil domain size of 10B×3B is established; The mesh of the area around the skirt foundation is encrypted, and the mesh element type is an eight-node plane strain quadrilateral element. A fixed constraint is imposed on the bottom boundary and a horizontal displacement constraint is imposed on the lateral boundary.

6. The method for calculating the bearing capacity envelope of a skirt foundation in marine clay according to claim 1, characterized in that: The side rubbing method is used to apply rotation load and horizontal displacement load to the two-dimensional plane strain model in the modified inp file, including: Apply a rotation load with an angle increment of Δθ=0.001rad until the failure state θmax=0.1rad is reached; Apply horizontal displacement loads in stages at the limit rotation angle, with a displacement increment of Δu=0.01B, until the calculation fails to converge.

7. The method for calculating the bearing capacity envelope of a skirt foundation in marine clay according to claim 1, characterized in that: After solving with ABAQUS, the ultimate bending moment and horizontal bearing capacity are extracted, and the bearing capacity envelope is drawn, including: According to the ODB result file of ABAQUS, the integral value of the base reaction under each load step is extracted, the horizontal bearing capacity Hult and bending moment Mult are calculated, and the VHM three-dimensional envelope is generated by data fitting.

8. The method for calculating the bearing capacity envelope of a skirt foundation in marine clay according to claim 2, characterized in that: The value range of the internal length parameter is 0.1m≤lc≤1.0m, and according to the average diameter d50 of soil particles, lc=2d50 is satisfied.

9. A device for calculating the bearing capacity envelope of a skirt foundation in marine clay, characterized in that: include: The regularization modeling module is configured to implement the regularization mechanism of Cosserat continuum theory using FORTRAN code through the UEL subroutine interface of ABAQUS, including: a rotational degree of freedom embedding unit, which is used to add the microscopic rotation component around the z axis to the node degree of freedom; an internal length parameter loading unit, which stores and associates the mapping relationship between the internal length parameter and the average diameter of the soil particles; a differential operator matrix generator, which generates a 7×3-dimensional differential operator matrix according to the formula; The DP-MC constitutive module is configured to establish a Drucker-Prager constitutive model that matches the Moore-Coulomb criterion, including: a parameter converter that converts the input cohesion c and friction angle into DP-MC coefficients; a yield surface calculation unit that updates the yield surface in real time; The model building module is configured to generate and preprocess finite element model data, including: a depth-to-diameter ratio parameterized modeling unit, which receives the depth-to-diameter ratio and automatically adjusts the model geometry; an intelligent mesh generator, which performs local encryption on the area around the skirt foundation; an inp file editor, which writes Cosserat material parameters into the inp file; The calculation execution module is configured to drive the ABAQUS solver to perform bearing capacity analysis, including: a side-rubbing loading controller that applies loads in stages according to the rotation angle increment and displacement increment; a regularized convergence monitor that triggers mesh adaptive reconstruction when the node rotation wz>0.1rad; The visualization output module is configured to generate engineering analysis results, including: an envelope plotter that fits the extracted Hult-Mult data into a three-dimensional envelope surface; a strain localization cloud map generator that displays the κxz, κyz curvature distribution in the form of a heat map.

10. The device for calculating the bearing capacity envelope of a skirt foundation in marine clay according to claim 9, characterized in that: The regularized modeling module uses FORTRAN code to implement the regularization mechanism of Cosserat continuum theory, which is expressed as: ； ； Among them, ε is the Cosserat strain tensor, L is the differential operator matrix, u is the displacement vector, σ is the Cosserat stress tensor, and f is the body force vector; The differential operator matrix L is defined as: ； in, represents the positive strain in the x direction, represents the positive strain in the y direction; For ε, defined as the sum of linear elastic strain and plastic strain, the isotropic elastic modulus matrix is satisfy: ； Among them, λ=2Gν / (1-2ν), G is the shear modulus, ν is the Poisson's ratio, is the Cosserat shear modulus.

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