Method and system for analyzing geometric characteristics of yield surface and plastic potential surface of material

Abstract

The invention discloses a material yield surface and plastic potential surface geometric feature analysis method and system, and relates to the technical field of material deformation feature analysis, and the method comprises the steps: 1, building a geometric model, converting the geometric model into a polar coordinate form at any height according to a yield function and a plastic potential function, and drawing a trajectory at the height on the polar coordinate; 2, rotating the track line into an isoclinal coordinate system to obtain an offset track line at the height; 3, judging that the height meets the required height, if not, adding 1 to the height, and skipping to the step 1; if yes, offset traces at all heights are recorded, and a yield surface and a plastic potential surface are drawn through a mesh function of Matlab. The characteristics of a yield function and a plastic potential function are understood more deeply, and an important reference basis is provided for selecting a constitutive relation model suitable for describing material deformation characteristics.

Description

Technical Field

[0001] The present invention relates to the technical field of material deformation characteristic analysis, and particularly relates to a method and a system for analyzing geometric characteristics of a material yield surface and a plastic potential surface. Background Art

[0002] In the fields of optical research, geotechnical engineering, and machining of mechanical parts, the stability of materials is crucial. For example, a slight deformation of optical materials can lead to very different optical experimental results; for example, after deformation of aircraft turbine blades under various environments, it can directly affect the safety of the aircraft. Therefore, it is of great significance to study the deformation characteristics of materials.

[0003] The difference between the classical plasticity theory and the simple plasticity model lies in the different yield functions and plastic potential functions. In order to select a plastic constitutive model that can objectively and accurately describe the behavior of materials, it is necessary to analyze the inherent characteristics of the yield function and the plastic potential function, and understand the advantages and disadvantages of the constitutive model in describing the deformation characteristics of materials. This often requires the help of the geometric expressions of the yield function and the plastic potential function. These two functions are generally expressed as complex functions of stress tensors, hardening parameters, and some field variables, and it is difficult to clearly understand their geometric structures.

[0004] For isotropic materials, the yield function and the plastic potential function are simplified to functions of principal stresses and hardening parameters. Based on the powerful symbolic operation and implicit function surface plotting capabilities of mathematical analysis software such as Mathematica, it is not very difficult to plot three-dimensional yield surfaces and plastic potential surfaces in the principal stress space. This intuitive geometric description enables researchers to no longer face boring mathematical expressions, but it is difficult to obtain detailed information on many geometric characteristics of the yield surface and the plastic potential surface only by observation. On the other hand, although the mathematical expressions of the yield function and the plastic potential function are very complex, in fact, the analytical method can better reveal the geometric forms and characteristics of the yield surface and the plastic potential surface.

[0005] Existing textbooks, theoretical monographs, and research literatures almost invariably give schematic geometric diagrams of the yield function and the plastic potential function in the principal stress space or the deviatoric plane, which are vague or even incorrect in local details. In addition, the geometric characteristics of the yield surface and the plastic potential surface are also directly given conclusions, leaving relatively little room for researchers to understand and deeply explore. Summary of the Invention

[0006] The purpose of the present invention is to provide a method and a system for analyzing geometric characteristics of a material yield surface and a plastic potential surface, which are used to accurately understand and comprehend the characteristics of the yield function and the plastic potential function, and facilitate the selection of a constitutive relation model that can objectively and accurately describe the deformation characteristics of materials during numerical calculation and simulation.

[0007] In a first aspect, a method for analyzing the geometric characteristics of the yield surface and plastic potential surface of a material provided by the present invention adopts the following technical solutions: A method for analyzing the geometric characteristics of the yield surface and plastic potential surface of a material, comprising: Step 1: Establish a geometric model, convert it into polar coordinates according to the yield function and plastic potential function at any height, and draw the trajectory line at this height on the polar coordinates; Step 2: Rotate the trajectory line into the isoclinic coordinate system to obtain the deviatoric trace line at this height; Step 3: Determine whether the height meets the required height. If not, add 1 to the height and jump to Step 1; if so, record the deviatoric trace lines at each height, and use the mesh function of Matlab to draw the yield surface and plastic potential surface.

[0008] A further technical solution lies in that the conversion into polar coordinates according to the yield function and plastic potential function at any height specifically includes: Obtain the stress state of any point P in the principal stress space; the stress state includes the deviatoric stress components of point P; Sweep the deviatoric stress components of point P by 360 degrees to obtain the yield curve and plastic potential curve; Obtain the polar coordinate forms of the yield function and plastic potential function according to the yield curve and plastic potential curve in the deviatoric plane passing through point P.

[0009] A further technical solution lies in that the stress state includes the deviatoric stress components of point P, which can be expressed as: ; wherein, are the deviatoric stress components of the stress state of a point in the principal stress space; N is the intersection point of the hydrostatic axis and the deviatoric plane.

[0010] A further technical solution lies in that the polar coordinate forms of the yield function and plastic potential function can be expressed as: ; wherein, is the polar coordinate form of the yield function, and A is a constant.

[0011] A further technical solution lies in that the rotation of the trajectory line into the isoclinic coordinate system specifically includes: Convert the polar coordinates into Cartesian coordinates; rotate clockwise by 45° around the X axis of the Cartesian coordinate system, and then rotate counterclockwise by around the Y axis to obtain the isoclinic coordinate system.

[0012] A further technical solution lies in that the conversion of the polar coordinates into Cartesian coordinates specifically includes: Establish the following relationship according to the deviatoric stress components of point P: ; ; According to the different sizes of the graph, add a function regarding the height change , and change the above formula to: ; ; Establish Cartesian coordinates in Matlab according to the above formula.

[0013] A further technical solution lies in that the isoclinic coordinate system can be represented by the following formula: ; ; ; where α = 45°, tan(β) = .

[0014] In a second aspect, a geometric feature analysis system for the yield surface and plastic potential surface of a material provided by the present invention adopts the following technical solution: A geometric feature analysis system for the yield surface and plastic potential surface of a material, comprising: Model establishment unit: Establish a geometric model, convert it into polar coordinate form according to the yield function and plastic potential function at any height, and draw the locus line at this height on the polar coordinates; Calculation unit: Rotate the locus line into the isoclinic coordinate system to obtain the deviatoric trace line at this height; Output unit: Judge whether the height meets the required height. If not, increase the height by 1 and jump to step 1; if so, record the deviatoric trace lines at each height, and use the mesh function of Matlab to draw the yield surface and plastic potential surface.

[0015] In summary, the present invention includes the following beneficial technical effects: Based on the geometric description of the stress state in the principal stress space and the general form of the Zienkiewicz-Pande yield function, this paper analyzes the mechanism of the polar coordinate method for describing the yield function and plastic potential function, and combines the Mohr-Coulomb and Drucker-Prager yield functions as well as the Menétrey-Willam and Gudehus-Arygris plastic potential functions to elaborate on the specific application of the polar coordinate description method, indicating that the polar coordinate description method has universal significance in the geometric feature analysis of the yield surface and plastic potential surface. With the help of this method, the characteristics of the yield function and plastic potential function can be understood more deeply, providing an important reference basis for selecting a constitutive relation model suitable for describing the deformation characteristics of materials. Description of the Drawings

[0016] Figure 1 is the flowchart of the method of the present invention; Figure 2 is a schematic diagram of the stress state of any point P in the principal stress space; Figure 3 is a schematic diagram of the Mohr-Coulomb yield envelope in the normal stress - shear stress plane; Figure 4 is a schematic diagram of the Mohr-Coulomb yield criterion in the deviatoric plane; Figure 5 is a schematic diagram of the Drucker-Prager yield criterion in the deviatoric plane; Figure 6 is a schematic diagram of the William-Warnke ellipse in the deviatoric plane; Figure 7 is a geometric interpretation diagram of the William-Warnke yield ellipse; Figure 8 is a schematic diagram of the Gudehus-Arygris plastic potential line in the deviatoric plane; Figure 9 is a schematic diagram of the geometric form of the Mohr-Coulomb yield function in the principal stress space; Figure 10 is a schematic diagram of the geometric form of the Drucker-Prager yield function in the principal stress space; Figure 11 and Figure 12 is a schematic diagram of the geometric form of the Gudehus-Arygris plastic potential function in the principal stress space. Detailed implementation manners

[0017] In order to clearly describe the technical solutions of the embodiments of the present invention, in the embodiments of the present invention, terms such as "first" and "second" are used to distinguish identical items or similar items with basically the same functions and effects. For example, the first threshold and the second threshold are only used to distinguish different thresholds, and do not limit their sequence. Those skilled in the art can understand that terms such as "first" and "second" do not limit the quantity and execution order, and "first", "second", etc. do not necessarily mean different.

[0018] It should be noted that in the present invention, words such as "exemplary" or "for example" are used to represent examples, illustrations or explanations. Any embodiment or design solution described as "exemplary" or "for example" in the present invention should not be construed as being more preferred or having more advantages than other embodiments or design solutions. Rather, the use of words such as "exemplary" or "for example" is intended to present related concepts in a specific manner.

[0019] For a certain material, the trajectory line is plotted in polar coordinates according to the yield function and the plastic potential function at a certain height, and then the trajectory line is rotated into the isoclinic coordinate system to obtain the deviatoric trace line at this height. Furthermore, the deviatoric trace lines at each height are plotted, and the yield surface and the plastic potential surface are plotted using the mesh function of Matlab.

[0020] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are some, but not all, of the embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0021] The following further describes the embodiments of the present invention in detail with reference to the accompanying drawings of the specification.

[0022] The embodiments of the present invention provide a method for analyzing the geometric characteristics of the yield surface and the plastic potential surface of a material, and the main process of the method is described as follows.

[0023] As Figure 1 shown: Step 1: Establish a geometric model, convert it into polar coordinate form according to the yield function and the plastic potential function at any height, and plot the trajectory line at this height on the polar coordinates.

[0024] Among them, converting it into polar coordinate form according to the yield function and the plastic potential function at any height specifically includes: obtaining the stress state of any point P in the principal stress space; the stress state includes the deviatoric stress components of point P; sweeping the deviatoric stress components of point P by 360 degrees to obtain the yield curve and the plastic potential curve; obtaining the polar coordinate forms of the yield function and the plastic potential function according to the yield curve and the plastic potential curve in the deviatoric plane passing through point P.

[0025] Specifically, the yield surface and the plastic potential surface are respectively composed of stress state points that satisfy the yield function and the plastic potential function. As Figure 2 shown, considering the stress state of any point P in the principal stress space, where ∆QRS is the deviatoric plane passing through point P, N is the intersection point of the hydrostatic axis and the deviatoric plane ∆QRS, and NQ, NR, and NS respectively represent the projections of the principal stress axes , , on the deviatoric plane ∆QRS , , . The unit vector along the hydrostatic axis passing through the origin, and the lengths of the components ON and NP of the vector OP along the hydrostatic axis and perpendicular to the hydrostatic axis are respectively represented as and : (1); (2); (3); Wherein, and are respectively the hydrostatic pressure component of the stress tensor and the second invariant of the deviatoric stress tensor, and are related to the hydrostatic pressure component and the deviatoric stress component of the stress state at point P in the principal stress space by and respectively.

[0026] The polar coordinate descriptions of the yield function and the plastic potential function are as follows: In the early 20th century, Mohr first truly proposed the Mohr-Coulomb yield criterion based on shear failure for geotechnical materials. It is relatively simple, and the two material parameters of cohesion and internal friction angle it contains have clear physical meanings and are easily obtained through experiments. It can reflect the characteristics of geotechnical materials such as dilatancy, the influence of hydrostatic pressure on yield, and different tensile and compressive strengths, so it has been widely used.

[0027] Zienkiewicz and Pande believe that using the Mohr-Coulomb yield criterion in engineering practice is relatively reliable. The disadvantage is that the non-smooth surface geometric characteristics of its six edges and one conical point lead to cumbersome calculations and slow convergence. Although Drucker and Prager introduced a conical yield surface to eliminate the six edges, the singular vertex still exists. Eliminating the vertex singularity helps to accelerate the numerical calculation convergence process, especially when the conditions of large friction angle and small cohesion (such as friction-type materials like sand) are present.

[0028] Zienkiewicz and Pande used a single smooth yield surface without edges and singular vertices to approximate the Mohr-Coulomb yield surface. The general form of the Zienkiewicz-Pande yield criterion is expressed as: (4); Wherein, , , , are material parameters, is the Lode angle, is a function that controls the shape of the yield curve in the deviatoric plane. If the Lode angle , , , is the ratio of tensile and compressive strengths. According to , , , different values of different forms of, the general form of the Zienkiewicz - Pande yield function almost covers the yield functions and plastic potential functions of all simple plastic models. From Equation (3), the general form of the Zienkiewicz - Pande yield criterion changes to: (5); The yield surface and the plastic potential surface are axisymmetric about the hydrostatic axis in the principal stress space. Their intersection lines with the deviatoric plane are the yield curve and the plastic potential line, which are also collectively called the deviatoric traces. In a certain deviatoric plane, is a constant. From Equation (5), the Zienkiewicz - Pande yield function is expressed as: (6); where, A is a determined constant. The yield curve and the plastic potential line are obtained by sweeping 360 degrees. Therefore, in Equation (6) actually represents the polar coordinate form of the Zienkiewicz - Pande yield function, which describes the geometric characteristics of the deviatoric traces, and the deviatoric traces corresponding to each deviatoric plane are geometrically similar. The geometric characteristics of any one deviatoric trace characterize the geometric characteristics of the entire yield surface and the plastic potential surface.

[0029] Next, combining the Mohr - Coulomb and Drucker - Prager yield functions often used in geotechnical materials and the Menétrey - Willam and Gudehus - Arygris plastic potential functions, their polar coordinate description forms are obtained and some of their important geometric characteristics are analyzed.

[0030] Examples of the application of the polar coordinate description method are listed as follows.

[0031] Polar coordinate description of the Mohr - Coulomb yield function: The Mohr - Coulomb yield criterion assumes that material yielding is controlled by the maximum shear stress, and the shear stress at yielding depends on the normal stress. According to the experimental results of the maximum and minimum principal stresses, a family of Mohr circles of the stress state at material yielding is plotted. The straight line tangent to each Mohr circle in it is the Mohr - Coulomb yield envelope. As Figure 3 shown, assuming tensile stress is positive and compressive stress is negative, the Mohr - Coulomb yield criterion is expressed as: (7); where, is the shear stress, is the normal stress, and They are the cohesive force of the material and the angle of internal friction respectively.

[0032] From Figure 3 the geometric relationships in (8); If and let there is: (9); Substituting Eqs. (8) and (9) into (7), the polar coordinate description form of the Mohr-Coulomb yield function is obtained: (10); Where: , ; Equation (10) is the straight-line equation in polar coordinates. Therefore, corresponding to the Lode angle , the Mohr-Coulomb yield criterion is represented as a straight line in the deviatoric plane. As Figure 4 shown, by symmetrically extending the straight line at , the Mohr-Coulomb yield criterion is an irregular hexagon in the deviatoric plane and an irregular open hexagonal pyramid in the principal stress space.

[0033] When and , they respectively correspond to the ratios of and of the tensile and compressive meridians (referred to as the flow stress ratio e in some engineering numerical analysis software): (11); The polar coordinate description of the Drucker-Prager yield function is as follows: The Drucker-Prager yield criterion is expressed as: (12); Where, and are material constants, is the first invariant of the stress tension, . From Eq. (3), the Drucker-Prager yield function in polar coordinates is expressed as: (13); From Eq. (13), in a certain deviatoric plane, is a constant. Therefore, as Figure 5 shown, The Drucker-Prager yield curve obtained by sweeping 360 degrees is a circle, which is represented as an open cone symmetric about the hydrostatic pressure axis in the principal stress space.

[0034] Polar coordinate description of the Menétrey-Willam plastic potential function: The Mohr-Coulomb model in software such as Abaqus usually uses the non-associated plastic potential function proposed by Menétrey and Willam: (14); where is the dilatancy angle of the plastic potential line in the meridian plane under high confining pressure; is the parameter controlling the shape of the plastic potential line in the meridian plane; and are defined as the equivalent compressive stress and the deviatoric stress respectively, , ; is defined as: ; where: ; The Menétrey-Willam plastic potential function expressed by the formula is very complex. It is a hyperbola in the meridian plane (p-q plane), but its geometric characteristics in the deviatoric plane are not intuitive. Through the polar coordinate description method, it can be relatively easily found that it is actually a Willam-Warnke piecewise ellipse in the deviatoric plane, as Figure 6 shown, and it meets the geometric characteristic requirements of plastic potential symmetry, smoothness and convexity.

[0035] The symmetry condition of the plastic potential line requires to be equal to and orthogonal to the William-Warnke ellipse at and ( ). Therefore, the axis of the local coordinate system of the ellipse is made to coincide with ( ), and the condition that Figure 7 is orthogonal to the ellipse is always satisfied when and . The axis of the local coordinate system of the ellipse is as (15); According to Figure 9 in the geometric description, there is: (16); The relationship between the local Cartesian coordinates ( ) and the polar coordinates with N as the origin ( ) is: (17); From equations (16) and (17), the elliptic equation (15) is transformed into: (18); Where: ; The Menétrey-Willam plastic potential function (14) is transformed into: ; So: ; Then, the Menétrey-Willam plastic potential function in polar coordinate form is expressed as: ; The above equation is exactly the same in form as the equation. Therefore, the Menétrey-Willam plastic potential function is a Willam-Warnke piecewise ellipse in the deviatoric plane. When ([[]]END]] ), , the Willam-Warnke ellipse degenerates into a circle; when ([[]]END]] ), from equation (16), , the Willam-Warnke ellipse almost degenerates into a triangle, and there are corner points on the compressive meridian of the Willam-Warnke ellipse. Therefore, to ensure the smoothness of the Menétrey-Willam plastic potential, it is required that .

[0036] Polar coordinate description of the Gudehus-Arygris plastic potential function: In engineering numerical analysis software, the plastic potential functions of some constitutive models use another simple form proposed by Gudehus and Arygris: Another simple form: (19); The deviatoric trace described by equation (19) in the deviatoric plane is as Figure 8 shown, which is a rounded-corner hexagon similar to the Willam-Warnke ellipse.

[0037] Since is a constant and the partial trace has the same shape within six 60-degree sectors, to ensure the convexity of the Gudehus-Arygris partial trace, it is only necessary to ensure that within . For this purpose,[[]] it is necessary to satisfy the discriminant condition for the convexity of the curve in polar coordinates: ; From equation (19), we obtain and , and the discriminant condition changes to: ; Because , and in addition , it can be proven that , and the problem is transformed into: (20); If , , the partial trace degenerates into a circle, and the convexity is always satisfied.

[0038] If , the Cauchy discriminant corresponding to inequality (20): ; Since , so , and thus, when ([[]] ), , regardless of taking any value, the convexity of the partial trace is always satisfied; when ([[]] ), , the two equal real roots of the quadratic equation corresponding to inequality (20) are: ; Within the semi-open and semi-closed interval , inequality (20) holds. Considering the symmetry property of the partial trace, when , the convexity of the partial trace is always satisfied; when ([[]] ), , the two real roots of the quadratic equation corresponding to inequality (20) must be on both sides of 1, and the convexity of the partial trace cannot always be satisfied.

[0039] Therefore, to ensure the convexity of the Gudehus-Arygris plastic potential line, , and because , so there is: (21); From Eqs. (11) and (21), , that is .

[0040] In short, if the Menétrey-Willam or Gudehus-Arygris non-associated plastic potential function is used, then in the deviatoric plane, the piecewise Willam-Warnke ellipse or the Gudehus-Arygris rounded-corner hexagon approximates the Mohr-Coulomb irregular hexagon, and in the meridian plane, a hyperbola is used instead of a straight line, eliminating the numerical difficulties caused by the non-uniqueness of the plastic flow direction at the six edges and one conical point of the Mohr-Coulomb yield surface. In addition, the use of the non-associated flow rule can eliminate excessive dilatancy. It should be noted that the associated flow rule ensures the uniqueness of the solution to the boundary value problem. If the non-associated flow rule is used, although the solution to the boundary value problem may still be unique, it cannot be proven at present.

[0041] Step 2: Rotate the said trajectory line into the isoclinic coordinate system to obtain the deviator trace line at this height.

[0042] Specifically: Convert polar coordinates to Cartesian coordinates, specifically including: Establish the following relational expressions based on the deviator stress components of point P: ; ; Since the sizes of the figures are different at different heights, the variable is added. The variable r is a function of the height h, then the above equation becomes: ; ; Establish Cartesian coordinates in Matlab according to the above equation.

[0043] The drawn trajectory line needs to be rotated to become the deviator trace line. To rotate the trajectory line in the Cartesian coordinate system to the isoclinic coordinate system, first, rotate it clockwise by 45° around the X-axis of the Cartesian coordinate system, and then rotate it counterclockwise around the Y-axis , to obtain the isoclinic coordinate system, that is, the transformation matrix is: ; ; ; where α = 45°, tan(β) = .

[0044] Step 3: Determine whether the height meets the required height. If not, increase the height by 1 and jump to Step 1; if so, record the deviatoric trace lines at each height, and use the mesh function of Matlab to plot the yield surface and plastic potential surface. The geometric forms of the Mohr-Coulomb yield function, Drucker-Prager yield function, and Gudehus-Arygris plastic potential function in the principal stress space are as Figures 9 to 12 shown.

[0045] The embodiment of the present application also discloses a geometric feature analysis system for the yield surface and plastic potential surface of materials, including: Model establishment unit: Establish a geometric model, convert it into polar coordinate form according to the yield function and plastic potential function at any height, and plot the trace line at this height on the polar coordinates; Calculation unit: Rotate the trace line into the isoclinic coordinate system to obtain the deviatoric trace line at this height; Output unit: Determine whether the height meets the required height. If not, increase the height by 1 and jump to Step 1; if so, record the deviatoric trace lines at each height, and use the mesh function of Matlab to plot the yield surface and plastic potential surface.

[0046] In a simple plastic model, the yield function and plastic potential function directly determine the ability of the constitutive relation model to describe the deformation characteristics of materials. Understanding the characteristics of the yield function and plastic potential function is of great significance for selecting a constitutive relation model that can objectively and accurately describe the deformation characteristics of materials during numerical calculation and simulation.

[0047] Based on the geometric description of the stress state in the principal stress space and the general form of the Zienkiewicz-Pande yield function, this paper analyzes the mechanism of using the polar coordinate method to describe the yield function and plastic potential function, and combines the Mohr-Coulomb and Drucker-Prager yield functions and the Menétrey-Willam and Gudehus-Arygris plastic potential functions to elaborate on the specific application of the polar coordinate description method, indicating that the polar coordinate description method has universal significance in the geometric feature analysis of the yield surface and plastic potential surface. With the help of this method, the characteristics of the yield function and plastic potential function can be understood more deeply, providing an important reference basis for selecting a constitutive relation model suitable for describing the deformation characteristics of materials.

[0048] As mentioned above, the above embodiments are only used to introduce the technical solutions of the present application in detail, but the descriptions of the above embodiments are only used to help understand the method and its core idea of the present invention, and should not be construed as a limitation of the present invention. Any changes or substitutions that can be easily thought of by those skilled in the art within the technical scope disclosed by the present invention should be covered within the protection scope of the present invention.

Claims

1. A method for analyzing geometric characteristics of material yield surface and plastic potential surface, characterized in that: include: Step 1: Establish a geometric model, convert the yield function and plastic potential function into polar coordinates at any height, and draw the trajectory line at that height on the polar coordinates; Step 2: Rotate the trajectory line into an isoclinic coordinate system to obtain a deflection trajectory line at the height; Step 3: Determine whether the height meets the required height. If not, add 1 to the height and jump to step 1; if so, record the deflection traces at each height and use Matlab's mesh function to draw the yield surface and plastic potential surface.

2. The method for geometrical characteristic analysis of material yield surface and plastic potential surface according to claim 1, characterized in that: The conversion into polar coordinate form according to the yield function and the plastic potential function at any height specifically includes: Obtaining the stress state of any point P in the principal stress space; the stress state includes the deviatoric stress component of point P; The deviatoric stress component of the point P is swept 360 degrees to obtain a yield curve and a plastic potential curve; In the deviatoric plane passing through point P, the polar coordinate forms of the yield function and the plastic potential function are obtained according to the yield curve and the plastic potential curve.

3. A geometric characteristic analysis method of material yield surface and plastic potential surface according to claim 2, characterized in that: The stress state includes the deviatoric stress component at point P, which can be expressed as: ; in, is the deviatoric stress component of the stress state at a point in the principal stress space; N is the intersection of the hydrostatic axis and the deviatoric plane.

4. The method for geometrical characteristic analysis of material yield surface and plastic potential surface according to claim 2, characterized in that: The polar coordinate form of the yield function and plastic potential function can be expressed as: ； in, is the polar coordinate form of the yield function, and A is a constant.

5. The method for geometrical characteristic analysis of material yield surface and plastic potential surface according to claim 1, characterized in that: The step of rotating the trajectory line into an isoclinic coordinate system specifically includes: Convert polar coordinates to Cartesian coordinates; rotate 45° clockwise around the X axis of the Cartesian coordinate system, then rotate counterclockwise around the Y axis , and we get an isoclinic coordinate system.

6. A method for analyzing geometric characteristics of material yield surface and plastic potential surface according to claim 5, characterized in that: The converting of polar coordinates into Cartesian coordinates specifically includes: The following relationship is established based on the deviatoric stress component at point P: ； ； Add a function for height change according to the size of the graph , the above formula becomes: ； ； According to the above formula, Cartesian coordinates are established in Matlab.

7. The method according to claim 6, characterized in that The isoclinic coordinate system can be expressed by the following formula: ； ； ； Where α=45°, tan(β)= .

8. A geometric characteristic analysis system for material yield surface and plastic potential surface, characterized in that: include: Model building unit: build a geometric model, convert it into polar coordinates at any height according to the yield function and plastic potential function, and draw the trajectory line at the height on the polar coordinates; Calculation unit: rotating the trajectory line into an isoclinic coordinate system to obtain a deviated trajectory line at the height; Output unit: Determine whether the height meets the required height. If not, add 1 to the height and jump to step 1; if yes, record the deflection traces at each height and use Matlab's mesh function to draw the yield surface and plastic potential surface.

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