Lattice structure elasticity modulus prediction method based on first principle

Abstract

The invention provides a lattice structure elastic modulus prediction method based on a first principle, which comprises the following steps: firstly, extracting RVE units, and enabling a lattice structure to be equivalent to periodic arrangement of the RVE units; secondly, performing numerical simulation on the RVE unit and calculating to obtain an elastic constant group; then, a Voigt-Russ-Hill average model is used for calculating an elastic modulus predicted value of the RVE unit; evaluating the accuracy of the predicted value of the elastic modulus through a test; and finally, determining an expected elastic modulus-wall thickness empirical function. According to the method, calculation is performed in combination with an energy-strain method in a finite element environment, so that the influence of the difference between the model and reality on a simulation result is reduced, and meanwhile, manual calculation of a complex equation can be avoided; calculation is carried out based on the elastic matrix, the numerical rule of elastic constant results in different lattice structures is effectively adapted, and the mechanical property of the structure is reflected more accurately.

Description

Technical Field

[0001] This invention belongs to the field of structural mechanics and numerical simulation, and relates to a method for predicting the elastic modulus of lattice structures based on first principles. Background Technology

[0002] With the aviation industry's dual pursuit of performance enhancement and economic efficiency, the demand for lightweight aircraft is increasing. Currently, at the forefront of design, some components are being lightweighted through structural design. However, weight reduction inevitably leads to a loss of rigidity in these components, so it is crucial to more accurately grasp the impact of structural changes on mechanical properties.

[0003] In recent years, lattice structures have gradually emerged with the development and application of additive manufacturing technology. Due to their advantages such as high specific strength and low filler ratio, they are widely used as lightweight feature structures. For complex structural components, since lattice structures often need to be generated according to shape, the irregular geometry increases the difficulty of load performance testing. The design side usually faces the problem of difficulty in obtaining effective data and forming design basis, so as not to accurately achieve the lightweight goal. If the lattice model is tested one-to-one, it will undoubtedly increase the cost burden. Therefore, it is particularly important to introduce finite element analysis technology and develop prediction methods for related performance. At present, there are few prediction methods for this problem at home and abroad, and they generally have the following shortcomings: (1) the calculation formula is complex and the manual analysis is difficult; (2) the methods for certain specific structures have low universality and are difficult to be used interchangeably. Finite element technology can effectively solve the cost pain point, but its calculation results will be affected by the similarity between simulation and real models. If finite element is combined with first principle calculation, thereby reducing the interference of real conditions, the accuracy of prediction data can be greatly improved. The advantage of first-principles calculations lies in their focus on the essence of things, utilizing only fundamental parameters that are relatively easy to obtain in a finite element environment. Combining these two approaches unlocks significant technological advantages and application prospects. Its engineering significance lies primarily in lattice design, where predicting the elastic modulus provides theoretical support for stress analysis and more effectively determines whether rigidity meets standards. However, considering the parameterization of characteristic structures, the increased computation time due to a large number of samples reduces process efficiency. To improve design speed and reduce workload, all predicted performance values ​​under parameterized characteristic structures are collected during the R&D phase, and constitutive equations for the lattice elastic modulus are generated through fitting. By establishing a theoretical workflow, designers only need to input the desired structural parameters to quickly obtain the target results within a short time. Summary of the Invention

[0004] This invention proposes a first-principles method for predicting the elastic modulus of lattice structures. It extracts representative volume elements (RVEs) of the lattice structure based on the homogenization method, establishing a mechanical model equivalent to different lattice structures formed by periodic arrangement of RVEs. By selecting elastic matrices corresponding to different crystal systems during density functional theory (DFT) calculations, the elastic modulus of the lattice is predicted using the Voigt-Reuss-Hill average model, and combined with finite element method (FEM) technology, all independent matrix elements required for calculation are quickly obtained in a simulation environment. After comparing the predicted data with experimental results, denser sample values ​​are collected, and all predicted values ​​are fitted using a regression algorithm.

[0005] The technical solution adopted in this invention is as follows:

[0006] A method for predicting the elastic modulus of a lattice structure based on first principles includes the following steps:

[0007] Step 1: Extract the RVE unit.

[0008] Since lattice structures are generally periodic, based on the homogenization method, Relative Value Vessels (RVEs) that reflect the basic characteristics of the lattice structure are extracted as equivalent unit cells. Thus, the lattice structure is equivalent to a periodic arrangement of RVEs. RVEs exhibit a certain field response relationship with the macroscopic structure; therefore, lattice structures of any geometric shape can be represented by RVEs.

[0009] Step 2: Perform numerical simulation on the RVE element and calculate the elastic constant array.

[0010] The three-dimensional digital model of the RVE element is meshed in a finite element environment. Numerical simulation of the RVE element based on the energy-strain method is then performed. According to the shear reciprocity theory, only minimal strain needs to be applied to the RVE element in six directions (X, Y, Z, XY, XZ, YZ) (based on the coordinate system of the finite element environment) during the numerical simulation. The first three are normal strains, and the latter three are shear strains. The CASTEP algorithm, to which the energy-strain method belongs, is based on density functional theory. The advantage of this algorithm is that it introduces the concept of pseudopotential during calculation to freeze internal electrons, which helps reduce the overall computational load. The elastic constant C is... ij The calculation process is as follows:

[0011] Within the range of linear elastic variation of the material, the relationship between stress and strain satisfies Hooke's Law, that is, the relationship between the two satisfies the relationship shown in formula (1):

[0012]

[0013] In the formula, σ represents stress, ε represents strain, subscript i represents the index number of the stress component, subscript j represents the index number of the strain component, and C ijRepresents a flexible constant array.

[0014] Strain energy is the change in the energy of the system relative to the ground state after a small strain is applied, as shown in Equation (2).

[0015]

[0016] Where U represents strain energy and V represents the equilibrium volume of the material.

[0017] Substituting equation (1) into equation (2), performing an arithmetic transformation on the stress part of equation (2), and taking a second-order approximation through Taylor expansion, we obtain equation (3):

[0018]

[0019] During the numerical simulation, the system automatically calculates the change in strain energy and obtains the elastic constant array by polynomial fitting using formula (3), which is then exported and denoted as C. 11 C 12 ...C 66 .

[0020] Since the stress component is a linear function of the strain component, Equation (1) can be transformed into a six-element matrix form for subsequent calculations, as shown in Equation (4).

[0021]

[0022] C ij The independent matrix elements are derived, and the elastic matrix that meets the characteristic conditions of the crystal system is selected according to the type and number of independent matrix elements.

[0023] Step 3: Calculate the predicted elastic modulus of the RVE element using the Voigt-Reuss-Hill average model.

[0024] Based on the determined elastic matrix of the crystal system, and using the equations in the Voigt method and the Reuss method respectively, C ij The predicted values ​​of the volume modulus and shear modulus of the RVE element under the stiffness matrix and compliance matrix are obtained by solving, and are denoted as the volume modulus B under the stiffness matrix. v Shear modulus G under stiffness matrix v Body modulus B under the flexibility matrix R Shear modulus G under the compliance matrix R .

[0025] B v With B R G v With G RThe average values ​​of the bulk modulus B and shear modulus G are obtained by taking the average values ​​respectively. The predicted value of the elastic modulus E of the RVE unit is calculated according to formula (5).

[0026]

[0027] The Voigt-Reuss-Hill average model combines the advantages of the Voigt and Reuss methods, making the results more reliable. It is important to note that the calculation formulas for estimating the elastic modulus using this method differ for elastic matrices of different crystal systems.

[0028] Step four: Evaluate the accuracy of the predicted elastic modulus value through experiments.

[0029] Periodic lattice structures of RVE units are manufactured using selective laser melting forming equipment. Since the elastic modulus of the lattice structure tends to stabilize as the size increases, the overall size of the lattice structure needs to be determined by comprehensively considering cost, experimental difficulty, and data reliability. The elastic modulus of the periodic lattice structure is measured using compression tests. Based on the homogenization principle, the measured elastic modulus of the periodic lattice structure can be equated to the measured elastic modulus of the RVE unit.

[0030] The predicted value of the elastic modulus of the lattice structure is compared with the measured value. If the error between the two does not exceed the set threshold, it means that the prediction value is accurate, that is, the prediction method in steps two to three is feasible. If the error exceeds the set threshold, the compression test is repeated or the meshing parameters of the three-dimensional digital model of the RVE unit are adjusted until the error is within the threshold range, where the threshold is 15%.

[0031] Because lattice structures generally have low fill rates, these porous media structures will exhibit offsets in multiple directions when subjected to loads. Therefore, during compression tests, extensometers are needed to assist in measuring the displacement generated in the direction of the applied force.

[0032] Step 5: Determine the desired elastic modulus-wall thickness empirical function.

[0033] The wall thickness parameterization model of the RVE element is performed, and the elastic modulus of the RVE element under different wall thicknesses is predicted using the methods in steps two and three.

[0034] The predicted elastic modulus is fitted to the wall thickness to obtain a fitted curve of wall thickness versus elastic modulus. A linear regression equation is then constructed using the least squares method. The deviation of the fitted curve is calculated, representing the degree of deviation between the linear regression equation and the actual situation. By optimizing the fitting parameters to bring the deviation close to zero, the optimal fitting parameters are obtained. The linear regression equation corresponding to the optimal fitting parameters is the desired elastic modulus-wall thickness empirical function. When applied to the design of aerospace lattice structures, the theoretical wall thickness value is input based on the empirical function to quickly obtain feedback results on the stiffness state of the lattice structure.

[0035] Furthermore, in step three, clamps are designed based on the lattice structure to mount the extensometer on the lattice structure. It is particularly important to note that the extensometer used in the experiment must have bidirectional measurement capabilities to avoid data acquisition failures. The clamps are rectangular plate-like structures with a protrusion at the center of one end along the vertical direction for mounting the extensometer. In use, two clamps are aligned, with the protrusions on opposite sides, clamping the lattice structure between them, ensuring the clamps are fully in contact with the surface of the lattice structure. The extensometer is then mounted on the protrusion. The contact surface between the clamps and the lattice structure must completely cover the lattice structure, and the clamp thickness must be ≤1mm to minimize its impact on the experimental process.

[0036] The beneficial effects of this invention are:

[0037] (1) By combining the energy-strain method in the finite element environment, the impact of the difference between the model and reality on the simulation results is reduced, and the complex equations can be avoided by manual calculation.

[0038] (2) Calculations based on the elastic matrix can effectively adapt to the numerical law of elastic constant results in different lattice structures by taking advantage of the differences in matrix morphology between different crystal systems, and more accurately reflect the mechanical properties of the structure.

[0039] (3) The Voigt-Reuss-Hill average model was introduced for mechanical property calculation, which combined the advantages of the Voigt method and the Reuss method, making the predicted value of the elastic modulus closer to the actual situation.

[0040] (4) By using linear regression to fit the predicted results of parameterized samples, the material cost required for the experiment is reduced and the process efficiency at the design end is improved.

[0041] (5) The working model of this method can not only be applied to the design of parts, but also to the pre-processing end of additive manufacturing technology, which can better guide the design of support structures. Attached Figure Description

[0042] Figure 1 A schematic diagram for extracting RVE units.

[0043] Figure 2 The diagram shows the clip, where (a) is the front view, (b) is the side view, (c) is the top view, and (d) is the axonometric view.

[0044] Figure 3 This is a comparison chart of the predicted and actual values ​​of the elastic modulus for the three structures.

[0045] Figure 4 This is a schematic diagram of P-structure linear regression.

[0046] Figure 5 This is a roadmap for prediction methods. Detailed Implementation

[0047] In this embodiment, the classic lattice structures, namely Gyroid structure, Primitive structure, and Diamond structure, are used as examples to further illustrate the method of the present invention. The three structures are referred to as G structure, P structure, and D structure, respectively.

[0048] A first-principles-based method for predicting the elastic modulus of lattice structures, the process of which is as follows: Figure 5 As shown, it includes the following steps:

[0049] (1) Based on the homogenization concept, RVE units that can reflect the basic characteristics of the lattice structure are extracted as the equivalent unit cell of the lattice structure, such as... Figure 1 As shown. The envelope dimensions of the RVE units in the three structures are all 10mm×10mm×10mm, and the wall thickness is 1mm.

[0050] (2) Mesh the three-dimensional digital models of G-structure, P-structure and D-structure in the RVE finite element environment. The parameters are uniformly set as follows: use tetrahedral mesh, surface / volume mesh length is 1mm, feature angle is 45° and growth factor is 1.5.

[0051] (3) Numerical simulation based on energy-strain method was performed on the RVE element, in which Ti-6Al-4V (TC4 titanium alloy) was uniformly selected as the material, and boundary conditions equivalent to micro-element were set.

[0052] (4) Calculate and derive the elastic constants of the G structure, P structure and D structure according to formula (3), and keep two decimal places. Convert the elastic constants of the three structures into six-element matrix form respectively to obtain formulas (7), (8) and (9), with the unit being GPa.

[0053]

[0054] According to the matrix forms in formulas (7), (8), and (9), the G and D structures each contain 9 independent matrix elements, while the P structure only has 3 independent matrix elements. Generally speaking, the higher the symmetry of the RVE, the fewer the number of independent matrix elements, and the simpler the calculation process. It can be seen that the P structure has a higher level of symmetry than the other two. Based on the type and number of independent matrix elements of the three structures, the elastic matrices of the G and D structures conform to the characteristics of an orthorhombic crystal system, while the elastic matrix of the P structure conforms to the characteristics of a cubic crystal system.

[0055] (5) The predicted values ​​of the bulk modulus and shear modulus of the G structure, P structure and D structure under the stiffness matrix and flexibility matrix were obtained by using the Voigt method and the Reuss method, respectively, and were recorded as follows, retaining three decimal places:

[0056] 1) G structure: B v =7.509 GPa; B R =7.526 GPa; G v = 3.605 GPa; G R =5.794 GPa.

[0057] 2) P structure: B v =B R = 6.987 GPa; G v =3.160 GPa; G R =2.644 GPa.

[0058] 3) D structure: B v =10.049 GPa; B R =11.195 GPa G v = 5.049 GPa; G R = 9.191 GPa.

[0059] It should be noted that, based on the three types of structural elastic matrix morphologies determined in (4), the predicted values ​​of bulk modulus and shear modulus of the P structure should be calculated using the cubic crystal system corresponding formula, i.e., formula (10), and the predicted values ​​of bulk modulus and shear modulus of the G and D structures should be calculated using the orthorhombic crystal system corresponding formula, i.e., formula (11).

[0060]

[0061] In the formula, Δ=C 13 (C 12 C 23 -C 13 C 22 )+C 23 (C 12 C 13 -C 23 C 11)+C 33 (C 11 C 22 -C 12 2 )

[0062] Calculate B under B structure, P structure, and D structure respectively. v With B R G v With G R The average values ​​are recorded below:

[0063] 1) Structure B: B = 7.518 GPa; G = 4.699 GPa.

[0064] 2) P structure: B = 6.987 GPa; G = 2.902 GPa.

[0065] 3) D structure: B = 10.622 GPa; G = 7.120 GPa.

[0066] (6) Substitute B and G of the three structures into formula (5) respectively to obtain the predicted elastic modulus values: Predicted elastic modulus value E of structure G G =11.66 GPa, predicted elastic modulus E of P structure P =7.65GPa, predicted elastic modulus E of structure D D =17.46 GPa.

[0067] (7) Use selective laser melting forming equipment to print lattice structure test pieces of G structure, P structure and D structure respectively. The overall size of the test pieces is 50mm×50mm×50mm.

[0068] The true values ​​of the elastic modulus of the three structures were obtained through compression tests. Comparison with the predicted values ​​showed that the trends in elastic modulus variation were consistent. Figure 3 As shown, the errors for all three structures are within 15%, indicating the feasibility of the prediction method. The predicted values ​​for structures G and P are slightly lower than the actual values, while the predicted value for structure D is the opposite. This is because the accuracy of drawing different structural features varies when using the same grid parameters, leading to fluctuations in the predicted values. In practical applications, prediction accuracy can be improved by fine-tuning the grid parameters.

[0069] During compression testing, an extensometer is needed to measure the displacement generated in the direction of the applied force. A clamp is designed based on the lattice structure to mount the extensometer on the lattice structure, such as... Figure 2As shown, the clamping plate is a rectangular plate structure, 70mm long and 50mm wide, capable of completely covering the lattice structure. It is made of 1mm thick plate, with a 20mm long and 20mm wide boss at the center of one end, used to mount the extensometer. Its forming process involves bending, laser cutting, and hand polishing. In use, the two clamping plates are aligned, with the bosses on opposite sides, clamping the lattice structure between them, ensuring the clamping plates are completely in contact with the surface of the lattice structure. The extensometer is then mounted on the boss.

[0070] (8) For the three structures, parameterized modeling of wall thickness is performed, with the wall thickness set to 0.5mm to 1mm. Steps (2) to (6) are executed to predict all samples. The obtained predicted elastic modulus values ​​are fitted with the wall thickness using linear regression and a linear regression equation is constructed. The deviation of the fitted curve is calculated. By optimizing the fitting parameters to make the deviation approach 0, the optimal fitting parameters are obtained. The corresponding linear regression equation is the empirical function of the elastic modulus of the lattice structure, such as... Figure 4 The fitted curve for the P structure is shown, exhibiting good goodness of fit and significance. The linear regression equations for the P, G, and D structures are as follows:

[0071] P-structure: y = 8.78x + 1.14

[0072] G structure: y = 9x + 0.81

[0073] Structure D: y = 11.28x + 0.89

[0074] Where x represents the wall thickness and y represents the elastic modulus.

[0075] This completes the prediction of the mechanical properties of the lattice structure. Under this workflow, for the same lattice structure, only the theoretical wall thickness value needs to be input to quickly obtain the feedback result of the stiffness state. Verification shows that the empirical function constructed using this method predicts a small deviation between the elastic modulus and the actual value, proving the effectiveness of this invention.

[0076] The above description is only a partial embodiment of the present invention and does not limit the implementation and protection scope of the present invention. Any improvements and modifications made by those skilled in the art based on the disclosure of the present invention without departing from the scope of the present invention should be within the protection scope of the present invention.

Claims

1. A method for predicting the elastic modulus of a lattice structure based on first-principles calculations, characterized in that, Includes the following steps: Step 1: Extract the RVE unit; Step 2: Perform numerical simulation on the RVE element and calculate the elastic constant array; Step 3: Calculate the predicted elastic modulus of the RVE element using the Voigt-Reuss-Hill average model; Step four: Evaluate the accuracy of the predicted elastic modulus value through experiments; Step 5: Determine the desired elastic modulus-wall thickness empirical function.

2. The method for predicting the elastic modulus of a lattice structure based on first principles, as described in claim 1, is characterized in that... Step 1 specifically involves: based on the homogenization method, extracting RVE units that reflect the basic characteristics of the lattice structure, then the lattice structure is equivalent to the periodic arrangement of RVE units.

3. The method for predicting the elastic modulus of a lattice structure based on first principles, as described in claim 1, is characterized in that... The second step is specifically as follows: The three-dimensional digital model of the RVE element is meshed in the finite element environment, and numerical simulation of the RVE element based on the energy-strain method is performed. Within the range of linear elastic variation, the relationship between stress and strain in a material satisfies Hooke's Law: In the formula, σ represents stress, ε represents strain, subscript i represents the index number of the stress component, subscript j represents the index number of the strain component, and C ij Represents a flexible constant array; The strain energy is shown in equation (2): Where U represents strain energy and V represents the equilibrium volume of the material; Substituting equation (1) into equation (2), performing an arithmetic transformation on the stress part of equation (2), and taking a second-order approximation through Taylor expansion, we obtain equation (3): During the numerical simulation, the system calculates the change in strain energy and obtains the elastic constant array by polynomial fitting using formula (3), which is then derived and denoted as C. 11 C 12 ...C 66 ; Since the stress components are linear functions of the strain components, equation (1) can be transformed into a six-element matrix form: C ij The independent matrix elements are derived, and the elastic matrix that meets the characteristic conditions of the crystal system is selected according to the type and number of independent matrix elements.

4. The method for predicting the elastic modulus of a lattice structure based on first principles, as described in claim 3, is characterized in that... According to the shear reciprocity theory, when performing numerical simulations of RVE elements based on the energy-strain method, minimum strains are applied to the six directions of the RVE elements: X, Y, Z, XY, XZ, and YZ. The first three are normal strains, and the latter three are shear strains.

5. A method for predicting the elastic modulus of a lattice structure based on first principles, as described in claim 1 or 3, characterized in that, Step three specifically involves: based on the determined elastic matrix of the crystal system, and using the equations in the Voigt method and Reuss method respectively, utilizing C... ij The predicted values ​​of the volume modulus and shear modulus of the RVE element under the stiffness matrix and compliance matrix are obtained by solving, and are denoted as the volume modulus B under the stiffness matrix. v Shear modulus G under stiffness matrix v Body modulus B under the flexibility matrix R Shear modulus G under the compliance matrix R ; B v With B R G v With G R The average values ​​of the bulk modulus B and the average value of the shear modulus G are obtained by taking the average values ​​respectively. The predicted value of the elastic modulus E of the RVE element is calculated according to formula (5).

6. The method for predicting the elastic modulus of a lattice structure based on first principles, as described in claim 1, is characterized in that... Step four specifically involves: manufacturing a periodic lattice structure for the RVE unit, measuring the elastic modulus of the periodic lattice structure using a compression test, comparing the predicted value of the elastic modulus of the lattice structure with the measured value, and if the error between the two does not exceed a set threshold, then the predicted value is accurate; if the error exceeds the set threshold, then the compression test is repeated or the meshing parameters of the three-dimensional digital model of the RVE unit are adjusted until the error is within the threshold range.

7. The method for predicting the elastic modulus of a lattice structure based on first principles, as described in claim 6, is characterized in that... During the compression test, an extensometer is used to assist in measuring the displacement generated in the direction of the applied force. The extensometer must have bidirectional measurement capabilities.

8. The method for predicting the elastic modulus of a lattice structure based on first principles, as described in claim 7, is characterized in that... An extensometer is installed on a lattice structure using clips. The clips are rectangular plates with a protrusion at one end in the vertical direction for mounting the extensometer. In use, two clips are aligned, with the protrusions on opposite sides, clamping the lattice structure between them and ensuring that the clips are fully in contact with the surface of the lattice structure. The extensometer is then mounted on the protrusion.

9. The method for predicting the elastic modulus of a lattice structure based on first principles, as described in claim 8, is characterized in that... The mating surface of the clip and the dot matrix structure must be able to completely cover the dot matrix structure, and the thickness of the clip must be ≤1mm.

10. The method for predicting the elastic modulus of a lattice structure based on first principles, as described in claim 1, is characterized in that... Step five specifically refers to: The wall thickness parameterization model of the RVE element is performed, and the elastic modulus of the RVE element under different wall thicknesses is predicted using the methods in steps two and three. The predicted value of elastic modulus is fitted to the wall thickness to obtain the fitting curve of wall thickness and elastic modulus. A linear regression equation is constructed using the least squares method. The deviation value of the fitting curve is calculated. The optimal fitting parameters are obtained by optimizing the fitting parameters to make the deviation approach 0. The linear regression equation corresponding to the optimal fitting parameters is the expected elastic modulus-wall thickness empirical function.

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