Method for quickly obtaining material shear modulus

Abstract

The application relates to a method for quickly obtaining the shear modulus of a material, which comprises the following steps: obtaining a virtual field method basic equation, wherein the virtual field method basic equation contains material stiffness coefficients; expressing the material stiffness coefficients as a basic coefficient expression; reasonably constructing a virtual deformation field under the condition that a real strain is known, controlling the value of the basic coefficients, simplifying the problem, so that the basic coefficient expression of the virtual field method basic equation is equal to a unit matrix, and obtaining a stiffness coefficient expression; simulating a real strain field by using a finite element method, and substituting the simulated real strain field and a virtual displacement field expression into the stiffness coefficient expression to obtain an equation group; solving the equation group, substituting the solution into the virtual displacement field expression, obtaining a virtual displacement field, actually measuring a real strain field, and obtaining the shear modulus according to the virtual displacement field, the real strain field and the virtual field method basic equation. The shear modulus of the material can be accurately and quickly obtained, and the defects of the traditional method, such as complexity, tediousness and high cost in obtaining various material parameters by multiple tests, are avoided.

Description

Technical Field

[0001] This invention relates to a method for testing material properties, particularly suitable for the rapid and accurate measurement of the shear modulus of materials. Background Technology

[0002] Shear modulus is an important parameter characterizing the shear mechanical properties of materials. Unlike elastic modulus, which can be measured through uniaxial tension, shear modulus measurement is more complex. Currently, depending on the material type, common methods for measuring shear modulus include Iosipescu, Arcan disk, and thin-walled tube tests. Among these, the commonly used thin-walled tube test is only applicable to metallic materials, and its specimen preparation process is complex, difficult, and costly. The Iosipescu test requires additional equipment and has high requirements for fixtures. Therefore, a rapid and accurate experimental method for measuring the shear modulus of materials is currently lacking. Summary of the Invention

[0003] The technical problem solved by this application is to overcome the shortcomings of the prior art and provide a theoretical and experimental method that can quickly and accurately measure the shear modulus of a material. This method can accurately obtain the shear modulus of the material and avoids the disadvantages of traditional methods, which require multiple experiments to obtain various material parameters, which are complex, cumbersome, and costly. It saves experimental time and costs and improves the efficiency of experimental testing and analysis.

[0004] The technical solution provided in this application is as follows:

[0005] A method for rapidly obtaining the shear modulus of a material includes:

[0006] S1: For the test specimen, the principle of virtual work combined with the material constitutive equation yields the basic equation of the virtual field method (Equation 3). The basic equation of the virtual field method includes the four stiffness coefficients Q of orthotropic materials. xx Q xy Q yy and Q s ;

[0007] S2: Stiffness coefficient Q in the basic equations of the virtual field method xx Q xy Q yy and Q s The coefficients are expressed as the basic coefficient expression C. xx C xy C yy C ss This makes the basic coefficient expression of the basic equation of the virtual field method equal to the identity matrix, thus obtaining the stiffness coefficient expression (Formula 10).

[0008] S3: Using a polynomial function as the basic form of the virtual displacement field, considering the test format and boundary conditions of the test specimen, the virtual displacement field expression (Formula 11) is obtained. Each virtual displacement field expression contains two unknown coefficients (a... ij and b ij );

[0009] S4: Using the finite element method, obtain the simulated strain field under ideal experimental conditions, and use the simulated strain field as the real strain field ε of the test specimen. x ε y and ε s ;

[0010] S5: Obtain the true strain field ε from S4 x ε y and ε s Substituting the virtual displacement field expression obtained from S3 into the stiffness coefficient expression of S2 (Formula 10), we obtain the system of equations.

[0011] S6: Solve the system of equations to obtain a combination of multiple solutions for the unknown coefficients of the system of equations; select any one of the multiple solutions and combine it with the virtual displacement field expression (Formula 11) to obtain a set of virtual displacement fields for the test piece;

[0012] S7: Create a random speckle pattern on the surface of the test piece, and obtain the full-field strain ε on the surface of the test piece using optical measurement methods. x ε y and ε s ;

[0013] S8: Combining full-field strain ε x ε y and ε s The virtual displacement field of the test piece obtained from S6 is used to calculate the shear modulus Q of the material by inversion based on the basic equation of the virtual field method (Formula 3). s .

[0014] In step S1, the basic equation of the virtual field method is:

[0015]

[0016] Where S and e represent the surface area and thickness of the test piece, respectively; (i = x, y, or s) represents the virtual strain field, where ε represents virtual shear strain; i (i = x, y, or s) represents the true strain field of the test specimen; T i (i = x or y) represents the unit distributed force on the load boundary of the test piece; (i = x or y) represents the virtual displacement field, and the derivative of the virtual displacement field yields the virtual strain field; Qxx Q xy Q yy and Q s is the stiffness coefficient of the test specimen.

[0017] In step S2, the basic coefficient expression C xx C xy C yy C ss They are respectively:

[0018]

[0019] In step S2, the expression for the stiffness coefficient is:

[0020]

[0021]

[0022] in, Equal to the identity matrix; (i = x or y) (i = x or y) (i = x or y) and (i = x or y) represent the material stiffness coefficient Q, respectively. xx Q xy Q yy and Q s The virtual displacement field contained in the expression.

[0023] In step S3, the test form of the test piece is a three-point bend test.

[0024] In step S3, the expression for the virtual displacement field is:

[0025] Where L and h represent the span and height of the three-point bend specimen, respectively; a ij and b ij For the unknown coefficients of the virtual displacement field.

[0026] Step S4 includes: presetting a set of material parameters Q based on the constitutive relation of orthotropic materials under plane stress. xx Q xy Q yy and Q s The numerical value is obtained by establishing a finite element model based on the actual dimensions of the test specimen, using preset material parameters as input, and obtaining the full-field strain of the test specimen under its experimental conditions using the finite element method. The full-field strain is then taken as the true strain field ε of the test specimen. x ε y and ε s .

[0027] In step S6, the system of equations is solved to obtain combinations of multiple solutions to the system of equations, including:

[0028] Choose any four unknown coefficients from the unknown coefficients in the system of equations as unknowns to be solved, and set the other unknown coefficients to 0, so as to obtain the values ​​of these four unknown coefficients. Repeat this process to iterate through all the unknown coefficients and obtain multiple combinations of solutions to the system of equations.

[0029] In step S6, after obtaining the virtual displacement field that meets the screening conditions for the basic coefficient values, experiments of the same experimental form can continue to be used.

[0030] The test specimen is a specimen of any shape with uniform thickness in a plane.

[0031] In summary, this application includes at least the following beneficial technical effects:

[0032] The present invention describes a theoretical and experimental method for rapidly and effectively measuring the shear modulus of a material. Based on the principle of virtual work and combined with the material constitutive equation, it considers the boundary conditions and external loads in the experiment. It utilizes advanced optical measurement methods to obtain the full-field deformation of the material surface, accurately obtaining the shear modulus of the material. It does not require ensuring the pure shear stress state of all or part of the specimen, avoiding the disadvantages of traditional methods that require multiple experiments to obtain various material parameters, which are complex, cumbersome, and costly. This saves experimental time and costs, and improves the efficiency of experimental testing and analysis. Attached Figure Description

[0033] Figure 1 It is a specimen of arbitrary shape subjected to external loads;

[0034] Figure 2 This is a schematic diagram of a three-point bend test. The distance between the two support points of the three-point bend is L, the height of the specimen is h, and the thickness of the specimen is e. Usually, e is much smaller than L and h, so the specimen is under plane stress. Detailed Implementation

[0035] The term “exemplary” as used herein means “serving as an example, embodiment, or illustration.” Any embodiment illustrated herein as “exemplary” is not necessarily to be construed as superior to or better than other embodiments. Although various aspects of embodiments are shown in the accompanying drawings, the drawings are not necessarily drawn to scale unless specifically indicated otherwise.

[0036] This application discloses a method for rapidly obtaining the shear modulus of a material, such as... Figure 1 and Figure 2 As shown, it includes the following steps:

[0037] like Figure 1As shown, for a specimen of arbitrary shape and uniform thickness in a plane, consider its application under external loads. According to the principle of virtual work, the virtual work of internal forces equals the virtual work of external forces, which leads to the following:

[0038]

[0039] Where V, S, and e represent the volume, surface area, and thickness of the specimen, respectively. σ i (i = x, y, or s) represents the stress field, S f and T i (i = x or y) represent the load boundary and the unit distributed force on the load boundary of the specimen, respectively, S u and (i = x or y) represent the displacement boundary and the displacement on the displacement boundary, respectively. (i = x or y) represents the virtual displacement field. (i = x, y or s) represents the virtual strain field. (This represents virtual shear strain).

[0040] The in-plane constitutive relation of orthotropic materials can be expressed as:

[0041]

[0042] Where ε i (i = x, y, s) represents the actual strain field of the material (ε). s σ represents shear strain. s (representing shear stress), Q xx Q xy Q yy and Q s Let be the stiffness coefficient of the material. Combining equations (1) and (2), we can obtain the equation:

[0043]

[0044] It can be seen that equation (3) contains four stiffness coefficients of orthotropic materials. Since the real strain field is obtained through experimental measurement, if four independent virtual deformation fields (virtual displacement fields) can be found... (i = x, y) or virtual strain field (i=x,y,s)) can be used to establish a system of four equations to simultaneously solve for these unknown material parameters. In other words, to determine Q xx Q xy Q yy and Q s First, we need to find four independent virtual deformation fields.

[0045] To find a suitable virtual deformation field, four fundamental coefficients C are first introduced into equation (3).xx C xy C yy C ss Thus, we obtained

[0046]

[0047] The four basic coefficients are respectively

[0048]

[0049] Given the actual strain, by constructing a virtual deformation field, the values ​​of these fundamental coefficients can be controlled. Then, a virtual displacement field that meets the selection criteria for the fundamental coefficient values ​​can be found, thereby simplifying the problem and improving efficiency. For example, Q can be obtained by considering the values ​​of the fundamental coefficients in equation (6). xx :

[0050]

[0051] Substituting equation (6) into equation (4), we can obtain...

[0052]

[0053] Similarly, the other three stiffness coefficients Q xy Q yy Q s It can also be obtained by taking different values ​​of the basic coefficients. Therefore, by selecting appropriate virtual deformation field to form the equations in equation (8), the four stiffness coefficients Q of the orthogonal anisotropic material under plane stress state can be solved. xx Q xy Q yy and Q s .

[0054]

[0055] in, (i = x or y) (i = x or y) (i = x or y) (i = x or y) represent the material stiffness coefficient Q, respectively. xx Q xy Q yy Q s The virtual displacement field contained in the expression.

[0056] Among them, matrix It consists of fundamental coefficients comprising four virtual strain fields:

[0057]

[0058] If the four virtual displacement fields can satisfy the conditions in equation (10), then the fundamental coefficient matrix This will become an identity matrix, thus Q can be directly obtained from equation (8). xx Q xy Q yy and Q s The expression.

[0059]

[0060] Virtual displacement field Differentiating (i = x, y) yields the virtual strain field. (i = x, y, s).

[0061] To more accurately construct the virtual displacement field that satisfies the condition of equation (10) (i=x,y), we can first use the finite element method to obtain the simulated strain field under the ideal experimental condition, and use it as the real strain field ε in equation (9). x ε y and ε s Then, for the virtual displacement field... (i = x, y) Assume a specific functional form, here we use a polynomial function as the basic form of the virtual displacement field. Thus, the virtual displacement field can be expressed as:

[0062]

[0063] Among them, such as Figure 2 As shown, the concentrated load F is applied at point A, and points B and C are the support points on the bottom surface of the test specimen (i.e., the three-point bend specimen). The horizontal direction of the bottom surface of the test specimen is taken as the X-axis, and the left and right symmetrical center lines of the specimen are taken as the Y-axis. x and y represent the X-axis coordinates and Y-axis coordinates, respectively. L and h represent the span (distance between support points B and C) and height of the three-point bend specimen, respectively. The higher the power of the polynomial m, n, p, q in equation (11), the more unknown coefficients there are to be determined, and the more complex the displacement distribution form that the virtual field can represent. Generally, the experimental conditions and calculation costs should be considered comprehensively.

[0064] It can be seen that by combining the real strain field obtained from the finite element simulation and the virtual strain field expressed by equation (11) with equations (9) and (10), the virtual displacement field coefficient a can be obtained. ij and b ij , will a ij and b ij Substituting into equation (11), we obtain four independent virtual displacement fields.

[0065] When using the method described in this invention, the full-field strain of the specimen surface needs to be obtained by a full-field deformation measurement method (such as digital image correlation method, moiré interferometry, grid method, etc.). Then, the stiffness coefficient of the material to be tested is calculated by using the four independent virtual displacement fields obtained by the aforementioned method and Equations (8) and (9). Then, the shear modulus G of the material can be obtained by using the relationship between the material's elastic modulus, Poisson's ratio, shear modulus and stiffness coefficient in Equation (12).

[0066]

[0067] As can be seen from equation (12), the shear modulus G of the material is related to the stiffness coefficient Q. s They are equal. Therefore, when the stiffness coefficient Q of the material is obtained using the method described in this paper... s Then, the shear modulus of the material is obtained.

[0068] It should be noted that in the actual application of the method described in this paper, it is not necessary to redetermine the virtual displacement field that satisfies the condition of equation (10) in each experiment. Once an independent virtual displacement field is obtained under a certain experimental condition, it can be used in future experiments of the same type. Even if the real strain field and virtual displacement field no longer satisfy equation (10) due to different specific conditions such as load and specimen size in the experiment, the accurate shear modulus of the material under test can still be obtained. Considering the simplicity and universality of the experiment, the three-point bend test commonly used in mechanics of materials experiments is used as an example to illustrate the specific implementation method.

[0069] Step 1, according to equation (11), assume the specific form of the virtual displacement field, such as Figure 2 As shown, considering factors such as the displacement distribution and computational cost of the three-point bend experiment, and taking m = n = p = q = 3, then equation (11) becomes

[0070]

[0071] Step 2, combined with the appendix Figure 2 Consider the boundary conditions in the three-point bend experiment: the displacement boundary condition is that the displacement in the y-direction at the two support points is 0, i.e., u By =0, u Cy =0. The load boundary condition is that point A is subjected to a downward concentrated load F, i.e., T. Ay =-F. Additionally, in the three-point bend experiment, the symmetry condition of the displacement fields in the x and y directions about the y-axis should also be considered, i.e., u x (x,y)=-u x (-x,y) and u y (x,y)=u y (-x,y). Based on the above conditions, the virtual displacement field expression in equation (12) is simplified to:

[0072]

[0073] Step 3, differentiate equation (13) to obtain the virtual strain expression (14):

[0074]

[0075] Step 4: Based on the constitutive relation of orthotropic materials under plane stress in equation (2), and combined with experience, arbitrarily assume a set of material parameters Q. xx Q xy Q yy and Q s The specific value;

[0076] Step 5: Establish a finite element model based on the actual dimensions of the test specimen. Use the material parameters assumed in Step 4 as input, and use the finite element method to obtain the full-field strain of the specimen under the three-point bending test state. Use this strain as the true strain field ε in Equation (9). x ε y and ε s ;

[0077] Step 6: Substitute the virtual strain field and the real strain field obtained in Steps 3 and 5 into Equations (9) and (10) to obtain the corresponding four virtual displacement fields. A system of four equations (i = x, y) containing an unknown coefficient a. ij and b ij Since the number of unknowns in the system of equations is greater than the number of equations, it is an underdetermined system of equations.

[0078] Steps 7 and 8 need to be implemented using a computer program:

[0079] Step 7: To solve the underdetermined system of equations, arbitrarily select four coefficients from the unknown coefficients as the unknowns to be solved, and set the other coefficients to 0, thereby obtaining these four coefficients. Repeat this process to iterate through all the unknown coefficients 'a'. ij and b ij This leads to combinations of multiple solutions for each system of equations;

[0080] Step 8: Since each virtual displacement field has multiple combinations of unknown coefficients, but not all combinations can provide accurate and effective material parameters (e.g., multiple solutions exist or the results obtained are not accurate enough), it is necessary to set screening conditions in the calculation program, namely, a unique solution condition and a relative error condition. The unique solution condition is achieved by making the rank of the coefficient matrix in the system of equations for solving the unknown coefficients equal to the rank of the augmented matrix. The relative error condition is achieved by setting the accuracy of the material parameters. To ensure accuracy, it is recommended to set the relative error condition to less than 0.1%.

[0081] By following steps 7 and 8, the calculation program can obtain multiple sets of four independent virtual displacement fields, any one of which satisfies the condition in equation (10), and equation (15) gives any one of them;

[0082]

[0083] It should be noted that once the virtual displacement field that meets the screening criteria is obtained, it can be used in future experiments of the same type. Even if the real strain field and virtual displacement field no longer satisfy equation (10) due to different specific conditions such as load and specimen size in the experiment, accurate material parameters can still be obtained.

[0084] Step 9: Create a random speckle pattern on the surface of the three-point bend specimen to be tested;

[0085] Step 10: Measure the full-field strain ε on the surface of the three-point bending specimen using digital image correlation method. x ε y and ε s ;

[0086] Step 11: Combining the true strain field of the specimen measured in Step 10 with any set of virtual displacement fields obtained in Steps 1 to 8, such as Equation (15), the shear modulus Q of the material can be calculated by inversion according to Equation (3). s .

[0087] The contents not described in detail in this application specification are common knowledge to those skilled in the art.

[0088] The present application has been described in detail above with reference to specific embodiments and exemplary examples; however, these descriptions should not be construed as limiting the present application. Those skilled in the art will understand that various equivalent substitutions, modifications, or improvements can be made to the technical solutions and implementation methods of the present application without departing from the spirit and scope of the present application, and all such modifications and improvements fall within the scope of the present application. The scope of protection of the present application is determined by the appended claims.

Claims

1. A method for rapidly obtaining the shear modulus of a material, characterized in that: include S1: For the test specimen, the principle of virtual work combined with the material constitutive equation yields the basic equation of the virtual field method. The basic equation of the virtual field method includes the four stiffness coefficients of orthotropic materials. and ; S2: Stiffness coefficients in the fundamental equations of the virtual field method and The coefficients are expressed as the basic coefficient expressions. This makes the basic coefficient expression of the basic equation of the virtual field method equal to the identity matrix, thus obtaining the stiffness coefficient expression; S3: Using polynomial functions as the basic form of virtual displacement fields, considering the test form and boundary conditions of the test piece, the virtual displacement field expression is obtained. Each virtual displacement field expression contains unknown coefficients. S4: Using the finite element method, obtain the simulated strain field under ideal experimental conditions, and use the simulated strain field as the real strain field of the test specimen. , and ; S5: The actual strain field obtained in S4 , and Substituting the virtual displacement field expression obtained from S3 into the stiffness coefficient expression of S2, we obtain the system of equations. S6: Solve the system of equations to obtain a combination of multiple solutions for the unknown coefficients of the system of equations; select any one of the multiple solutions and combine it with the virtual displacement field expression to obtain a set of virtual displacement fields for the test piece; S7: Create a random speckle pattern on the surface of the test piece and obtain the full-field strain of the test piece surface using optical measurement methods. , and ; S8: Combining full-field strain , and The virtual displacement field of the test piece obtained from S6 is used to calculate the shear modulus of the material by inversion based on the basic equations of the virtual field method. ; In step S3, the expression for the virtual displacement field is: in, and These represent the span and height of the three-point bend specimen, respectively. and These are the unknown coefficients of the virtual displacement field; In step S6, the system of equations is solved to obtain combinations of multiple solutions to the system of equations, including: Choose any four unknown coefficients from the unknown coefficients in the system of equations as unknowns to be solved, and set the other unknown coefficients to 0, so as to obtain the values ​​of these four unknown coefficients. Repeat this process to iterate through all the unknown coefficients and obtain multiple combinations of solutions to the system of equations.

2. The method for rapidly obtaining the shear modulus of a material according to claim 1, characterized in that: In step S1, the basic equation of the virtual field method is: Where S and e represent the surface area and thickness of the test piece, respectively; Represents a virtual strain field. , Indicates virtual shear strain; This represents the actual strain field of the test specimen. ; This represents the unit distributed force on the load boundary of the test piece. ; Represents a virtual displacement field. The virtual strain field is obtained by differentiating the virtual displacement field; and is the stiffness coefficient of the test specimen.

3. The method for rapidly obtaining the shear modulus of a material according to claim 2, characterized in that: In step S2, the basic coefficient expression They are respectively: 。 4. The method for rapidly obtaining the shear modulus of a material according to claim 2, characterized in that: In step S2, the expression for the stiffness coefficient is: in, Equal to the identity matrix; , , and These represent the material stiffness coefficients. , , and The virtual displacement field contained in the expression. .

5. The method for rapidly obtaining the shear modulus of a material according to claim 1, characterized in that: In step S3, the test form of the test piece is a three-point bend test.

6. The method for rapidly obtaining the shear modulus of a material according to claim 1, characterized in that: Step S4 includes: Based on the constitutive relation of orthotropic materials under plane stress, a set of material parameters is pre-defined. and The value; A finite element model is established based on the actual dimensions of the test specimen. Preset material parameters are used as inputs, and the full-field strain of the test specimen under its experimental conditions is obtained using the finite element method. The full-field strain is then taken as the true strain field of the test specimen. , and .

7. The method for rapidly obtaining the shear modulus of a material according to claim 1, characterized in that: In step S6, after obtaining the virtual displacement field that meets the screening conditions for the basic coefficient values, experiments of the same experimental form can continue to be used.

8. The method for rapidly obtaining the shear modulus of a material according to claim 1, characterized in that: The test specimen is a specimen of any shape with uniform thickness in a plane.

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