A method for testing the elastic modulus of 3D printed metal components

Abstract

This invention discloses a method for testing the elastic modulus of 3D-printed metal components. The method uses the root mean square error (RMSE) of the load-displacement curve obtained through simulation analysis and the load-displacement curve obtained experimentally as the objective function. A quadratic polynomial is then used as the response surface function. Undetermined coefficients are determined through regression analysis, and the polynomial is simplified through variance analysis, followed by accuracy verification. Finally, an optimization algorithm is applied to obtain the optimal parameter combination that minimizes the objective function, thus determining the elastic modulus of the material. This method solves the problem of measuring the elastic modulus of 3D-printed metal components that do not conform to the assumptions of uniformity, continuity, and isotropy.

Description

Technical Field

[0001] This invention relates to a method for testing the elastic modulus of 3D printed metal components. Background Technology

[0002] Currently, testing methods for the mechanical properties of homogeneous, continuous, and isotropic metallic materials are becoming increasingly mature. Various related standards and methods have become the norms for testing the elastic modulus of materials, such as the national standards for tensile, compressive, bending, and hardness tests. At present, the main methods for testing the elastic modulus of materials are tensile and compressive tests: the metal to be tested is processed into a standard specimen, subjected to tensile or compressive tests, and the stress-strain curve is obtained, thereby yielding the elastic modulus of the material.

[0003] Experimental studies show that 3D-printed metal rods and components exhibit a porous surface layer and a dense interior layer. Metal specimens prepared using 3D printing methods no longer satisfy the assumptions of homogeneity, continuity, and isotropy with orthopedic implants. Specimens of different sizes prepared from the same size rod exhibit different elastic moduli. Tensile and compressive specimens of the same size not only have different yield stresses but also different elastic moduli. The elastic modulus of a component is related not only to its size but also to its structure. Therefore, traditional methods for determining the elastic modulus of metallic materials through specimen analysis are no longer suitable for determining the elastic modulus of 3D-printed metal components.

[0004] With the implementation of precision medicine, 3D-printed prostheses are gradually entering clinical practice. However, considering the wide variety of implantable prostheses, their significant variations in structure, shape, and size, and the fact that 3D-printed metal components have a porous surface and a dense interior, violating the assumptions of uniformity, continuity, and isotropy of metallic materials, their elastic modulus depends on the component's structure and dimensions. Therefore, it is impossible to obtain the stress-strain curve of 3D-printed metal materials and determine the elastic modulus of the implantable prosthesis using standard mechanical testing methods.

[0005] Based on the above, a new testing method needs to be developed to determine the elastic modulus of 3D printed metal components for the design and manufacture of orthopedic implants. Summary of the Invention

[0006] To address the problems in the prior art, this invention provides a method for testing the elastic modulus of 3D printed metal components, solving the problem that the elastic modulus of 3D printed metal components cannot be obtained using traditional methods for testing the elastic modulus of metal materials.

[0007] This invention is achieved through the following technical solution:

[0008] A method for testing the elastic modulus of 3D printed metal components includes the following steps:

[0009] 3D printing technology was used to prepare components for orthopedic implants. The components were then embedded and tested according to the experimental standards for orthopedic implants to obtain the load-displacement curves of the components. The testing machine used was a hydraulic or electronic universal testing machine.

[0010] A solid model of the orthopedic implant was established, imported into finite element software, meshed, and boundary conditions were applied, which were the same as those in the experiment. The initial values ​​of the material parameters of the model were set to the elastic modulus of the same material after forging or casting. The load-displacement curve of the component was obtained by calculation. The calculated load-displacement curve and the experimental load-displacement curve were compared. By increasing or decreasing the value of the elastic modulus, the error between the calculated load-displacement curve and the experimental load-displacement curve was minimized, and the elastic modulus of the 3D printed metal component was initially obtained.

[0011] The calculation results are optimized by using an optimization design method, which reduces the number of calculations and improves accuracy.

[0012] The comparison of the calculated and experimental load-displacement curves, by increasing or decreasing the value of the elastic modulus, minimizes the error between the calculated and experimental load-displacement curves. The elastic modulus of the 3D printed metal component is obtained by: using the root mean square error between the load-displacement curve obtained from the material parameter combination simulation analysis and the experimental load-displacement curve as the objective function, and using a quadratic polynomial as the response surface function. The undetermined coefficients are determined through regression analysis, and the polynomial is simplified through variance analysis, and the accuracy is verified.

[0013] The optimization of the calculation results by using optimization design methods refers to applying the response surface methodology to obtain the optimal scheme with the minimum objective function.

[0014] The elastic modulus and Poisson's ratio of the 3D printed metal component are selected as independent variables, and the standard error of the finite element analysis results and experimental results is used as the objective function. Its expression is as follows:

[0015]

[0016] In equation (1), E is the elastic modulus; μ is Poisson's ratio; Xtn(E, μ) is the elastic modulus. n Support reactions in finite element analysis under displacement; ε tn The experimental load is denoted by N, where N is the total number of selected measurement points, and i is the number of material parameter combination schemes.

[0017] The beneficial effects of this invention are:

[0018] Response surface methodology (RSM) is a multidisciplinary parameter optimization method suitable for solving problems related to nonlinear data processing. It offers advantages such as short design cycles, low cost, and simple computation. Combining mechanical experimental results with the RSM method allows for the acquisition of precise elastic moduli for 3D-printed artificial prostheses, laying the foundation for optimized design and widespread clinical application of artificial prostheses.

[0019] The advantages of the method of the present invention are that it can detect the elastic modulus of 3D printed metal prosthesis components one by one, reduce the error caused by predicting the elastic modulus of prosthesis components using the material mechanics test results of small specimens, and has the advantages of not damaging the original specimens, reducing experimental costs, being simple in method and reliable in results. Attached Figure Description

[0020] Figure 1 Flowchart for identifying material parameters of 3D-printed titanium alloy prosthesis femoral stem;

[0021] Figure 2 Specimen positioning diagram; 1 Femoral head offset; 2 Loading point; 3 Neck axis; 4 Loading axis; 5 Embedding surface; 6 Femoral stem axis; T Farthest end of femoral stem; C Center of femoral head; D Embedding height; K, L Points at specified distances from T, used to define the femoral stem axis; α Angle between loading axis (4) and femoral stem axis (6) on the frontal plane (CKL); β Angle between loading axis (4) and femoral stem axis (6) on the lateral plane perpendicular to the frontal plane (CKL);

[0022] Figure 3 Hip joint model and finite element mesh generation; (A) Assembled 3D model; (B) Finite element mesh of 3D model;

[0023] Figure 4 Residual analysis: A. Normal probability distribution of residuals; B. Relationship between residuals and predicted values;

[0024] Figure 5 The interaction between elastic modulus and Poisson's ratio on the objective function; (A) response surface; (B) contour lines;

[0025] Figure 6 A comparison chart of simulation results and experimental results. Detailed Implementation

[0026] The specific embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0027] Example

[0028] The flowchart of the method of this invention is as follows Figure 1 As shown. This method can also be extended to the determination of parameters for other non-standard materials.

[0029] 1. Mechanics Experiment

[0030] 1.1 Sample Preparation

[0031] The experimental subject was a 3D-printed titanium alloy femoral prosthesis stem (provided by Jiasite Huajian Medical Devices (Tianjin) Co., Ltd.). According to the test principles and requirements of the *Chinese Pharmaceutical Industry Standard* (YY / T 0809.4-2018 / ISO 7206-4:2010), the distance CT from the center of the femoral head to the end of the femoral stem was measured to be 135 mm. The coronal and sagittal angles were α = 10° and β = 9°, respectively, with an angle error guaranteed within ±1°. The sample head was clamped onto the fixation device, and the sample angle was adjusted. The lower end of the femoral stem was embedded with filling resin to a height of 80 mm. Figure 2 .

[0032] 1.2 Experimental Methods

[0033] The embedded specimen was placed on the SUNS890-20P electro-hydraulic servo dynamic and static universal testing machine (Shenzhen Sansi Zongheng Technology Co., Ltd.), and the loading line of the testing machine was made to pass through the center point of the femoral head of the specimen (as defined by ISO7206-1). A compressive load was applied at a compression speed of 2 mm / min and a test termination force of 2300 N.

[0034] 2. Finite element simulation analysis

[0035] 2.1 Model Establishment

[0036] A 135mm hip joint femoral prosthesis stem model was established. The ball head pressure plate and femoral head were modeled using tetrahedral elements with a mesh size of 0.002m, while the femoral stem was modeled using tetrahedral elements with a mesh size of 0.001m. The elastic modulus of the pressure plate was 206GPa and the Poisson's ratio was 0.3. The elastic modulus and Poisson's ratio of the prosthesis were customized using response surface methodology, as shown in the figure.

[0037] 2.2 Boundary Conditions

[0038] The ball-end platen and the femoral head are set to frictional contact with a coefficient of friction of 0.1, and the prosthesis stem and the femoral head are set to be bonded. The area below the prosthesis stem D is set to full displacement constraint, and a downward displacement is applied to the upper surface of the ball-end platen.

[0039] 3. Response Surface Methodology

[0040] 3.1 Central Composite Design Central composite design (CCD) is a commonly used design method in response surface methodology (RSM), often used to fit second-order models. The elastic modulus and Poisson's ratio of the 3D-printed titanium alloy prosthetic stem are selected as independent variables, and the standard errors of the finite element analysis results and experimental results are used as the objective function. Its expression is:

[0041]

[0042] In equation (1), E is the elastic modulus; μ is Poisson's ratio; Xtn(E, μ) is the elastic modulus. n Support reactions in finite element analysis under displacement; ε tn The experimental load is denoted by N, where N is the total number of selected measurement points, and i is the number of material parameter combination schemes.

[0043] 3.2 Central Composite Design and Results Response surface methodology was designed based on the Central Composite Design principle. A two-factor, five-level experimental design was performed using Design Expert software. A and B represent the codes for elastic modulus and Poisson's ratio, respectively, and -1.414, -1, 0, 1, and 1.414 represent the five levels of the two factors. The codes and level values ​​for each design variable are shown in Table 1. The experimental schemes and objective function values ​​are shown in Table 2.

[0044] Table 1. Design factors and levels of the central composite design

[0045]

[0046] Table 2. Centralized design experiment and results

[0047]

[0048]

[0049] 4. Results Analysis

[0050] 4.1 Analysis of Variance Results: A quadratic regression was performed on the 13 sets of data in Table 2 to obtain a response surface model with the target outcome as the objective function. The quadratic multiple regression equation is as follows:

[0051] f(E,μ)=8605.2-151.9*A-1972.8*B+14*A*B+0.68*A 2 -754.6*B 2 (2)

[0052] The variance of the fitted quadratic multiple regression equation was analyzed, and the results are shown in Table 3.

[0053] Table 3. Results of ANOVA and Significance Test for Regression Equations

[0054]

[0055] As shown in the table, the regression terms A and A 2 P < 0.05 indicates a significant effect for AB, B, and B. 2Since P > 0.05, the effect is not significant. After removing insignificant influencing factors, the quadratic multiple regression equation can be optimized as follows:

[0056] f(E,μ)=8605.2-151.9*A+0.68*A^2 (3)

[0057] As shown in Table 3, within the experimental design range, the overall model has a P-value of <0.01, indicating that the response surface model is highly significant, has a high degree of fit, and the approximate model is effective.

[0058] Table 4 presents the error statistics of the multiple regression model, including the correlation coefficient R. 2 With the corrected correlation coefficient The correlation coefficient is close to 1, and the corrected correlation coefficient is... Correlation coefficient with prediction A difference of less than 0.2 indicates that the model's predicted values ​​are very close to the experimental data and are reliable.

[0059] Table 4. Error Statistical Analysis Results

[0060]

[0061] 4.2 Residual Analysis: The normal probability distribution of the residuals of the objective function is as follows: Figure 4 As shown in Figure A, the residuals at each point are roughly distributed around a straight line, indicating that the residuals of the model follow a normal distribution and have a high degree of fit. Figure 4 B is the distribution plot of residuals and predicted values, which shows that the residuals change with the predicted values ​​of the response surface model. The points are discretely distributed without a clear pattern, indicating that the residual distribution is reasonable and has randomness, and the experimental design is reasonable.

[0062] 4.3 Response Surface Analysis: Interaction analysis uses contour plots and response surface plots to analyze the significance of the interactions between independent variables on the target response. Based on the response surface regression model, the response surface and contour plots of the interaction between the elastic modulus and Poisson's ratio on the objective function are obtained as follows: Figure 5 As shown.

[0063] Depend on Figure 5 As shown in A, compared to the Poisson's ratio in direction B, the elastic modulus curve in direction A is steeper, indicating that the elastic modulus has a greater impact on the objective function than the Poisson's ratio. Figure 5 As shown in B, the contour density of the elastic modulus in direction A is significantly greater than that in direction B, indicating that the elastic modulus has a large impact on the objective function.

[0064] 4.4 Optimization Result Analysis To obtain the optimal parameter combination of elastic modulus and Poisson's ratio, based on the established multiple regression equation (3) and the comprehensive response surface analysis results, with the objective function minimization as the optimization objective, an optimization model was constructed according to the range of each factor:

[0065]

[0066] The optimal combination of response surfaces is shown in Table 5.

[0067] Table 5. Results of Prosthetic Stem Parameter Identification

[0068]

[0069] Figure 6 A schematic diagram comparing the simulation results of 13 sets of experiments centered on the optimal parameter combination and the simulation results of the optimal parameter combination from the optimization analysis with the compression test results of the 3D-printed titanium alloy prosthetic stem. Data processing software showed a correlation coefficient of 0.99 between the optimal factor combination and the compression test results, indicating a high degree of fit. This demonstrates that the finite element calculation results of the optimized factor combination and the compression test results have good consistency.

[0070] The proposed method for measuring the elastic modulus of 3D-printed components is significant for determining the elastic modulus of 3D-printed metal components of different sizes and structures. Accurately obtaining the elastic modulus of implanted prostheses and selecting artificial prostheses with elastic modulus values ​​close to those of real bone are crucial for reducing stress shielding and improving surgical outcomes. Therefore, obtaining accurate elastic modulus values ​​for prostheses is essential. Currently, the main methods for measuring elastic modulus include static tensile testing, dynamic resonance testing, beam bending testing, and ultrasonic testing. Due to the irregular shapes and diverse models of prostheses, traditional compression tests can only obtain load-displacement curves, which cannot be converted into stress-strain curves to determine the elastic modulus. There is no unified standard for measuring such complex pre-formed structures. In recent years, research on inverse biological tissue materials has become very popular, typically combining experimental-finite element simulation with optimization algorithms. Hua Xin et al. proposed an inverse method combining experiment, simulation analysis, and optimization algorithms to identify the material parameters of the dummy head and knee skin in a finite element model. Chen Jiqing et al. established a finite element model of the human chest ribs and obtained the rib material parameters through inverse analysis of animal tissue experiments. The results showed that the simulation results obtained by inversely determining the material parameters had good biofidelity. The advantage of this method is that it can detect the elastic modulus of the 3D-printed prosthesis one by one, reducing the error of predicting the elastic modulus of the prosthesis using the material mechanics test results of small specimens. It has the advantages of not destroying the original specimen, reducing experimental costs, and being simple and reliable. By using response surface methodology, the mechanical experimental results and simulation analysis results are combined to transform the parameter inverse problem into an optimization problem, and the elastic modulus of the 3D-printed hip joint femoral prosthesis stem is inversely determined. First, the load-displacement curve of the prosthesis is obtained through mechanical compression experiments. Then, a finite element model of the prosthesis is established, experimental boundary conditions are applied, an objective function is constructed, and the central composite design method is applied to randomly combine the elastic modulus and Poisson's ratio parameters to obtain the load-displacement calculation results after different parameter responses. Finally, the response surface methodology was applied to find the optimal solution for the minimum objective function, and the results were effectively verified using experimental data.

[0071] To determine the material performance parameters of the femoral stem for 3D-printed titanium alloy prostheses, this invention proposes a method for inversely calculating the elastic modulus of 3D-printed metal materials, combining experimental, computational, and optimization methods. This transforms the parameter inverse calculation problem into an optimization problem. Response surface methodology is applied for continuous optimization until a result satisfying the given conditions is obtained. Given fixed boundary conditions, the number of simulations depends on the number of material factor combinations and the capability of the optimization algorithm. This invention employs the Central Composite Design method for two-parameter, five-level finite element simulations, which provides a comprehensive range of material factor values ​​and a smaller number of combinations, thus reducing the number of simulations and saving costs. The elastic modulus of the prosthesis stem measured using this method is 109.07 GPa, with an error rate of only 0.8% compared to results reported in the literature, indicating high accuracy. This method also provides a new approach for determining other material parameters and has significant application and reference value.

Claims

1. A method for testing the elastic modulus of 3D printed metal components, characterized in that, Includes the following steps: 3D printing technology was used to prepare components for orthopedic implants. The components were then embedded and tested according to the experimental standards for orthopedic implants to obtain the load-displacement curves of the components. The testing machine used was an electronic or hydraulic universal testing machine. A solid model of the orthopedic implant was established, imported into finite element software, meshed, and boundary conditions were applied, identical to those in the experiment. The initial material parameters of the model were set to the elastic modulus of the same material after forging or casting. The load-displacement curve of the component was calculated. The calculated and experimental load-displacement curves were compared. By increasing or decreasing the value of the elastic modulus, the error between the calculated and experimental load-displacement curves was minimized, thus preliminarily obtaining the elastic modulus of the 3D-printed metal component. The elastic modulus of the 3D-printed metal component is defined as follows: using the root mean square error between the load-displacement curve obtained from the material parameter combination simulation analysis and the experimental load-displacement curve as the objective function, and using a quadratic polynomial as the response surface function, the undetermined coefficients were determined through regression analysis, and the polynomial was simplified through variance analysis, with accuracy verification performed. The calculation results are optimized by using an optimization design method, which reduces the number of calculations and improves accuracy.

2. The method for testing the elastic modulus of 3D printed metal components according to claim 1, characterized in that, The optimization of the calculation results using the optimization design method refers to applying the response surface methodology to obtain the optimal scheme with the minimum objective function.

3. The method for testing the elastic modulus of 3D printed metal components according to claim 1, characterized in that, The elastic modulus and Poisson's ratio of the 3D printed metal components are selected as independent variables, and the standard errors of the finite element analysis results and experimental results are used as the objective function, the expression of which is: （1）； In equation (1), E is the elastic modulus; μ is Poisson's ratio; (E, μ) is Support reactions under displacement in finite element analysis; The experimental load is denoted by N, where N is the total number of selected measurement points, and i is the number of material parameter combination schemes.

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