A Method for Evaluating the Performance of Silicone Rubber Based on Hyperelastic and Viscoplastic Constitutive Models

Abstract

This invention belongs to the field of polymer material aging performance evaluation technology, specifically involving a method for evaluating the performance of silicone rubber based on hyperelastic and viscoplastic constitutive models. The method involves conducting uniaxial tensile tests on silicone rubber samples within a target temperature range to obtain engineering stress-strain curves, and fitting the experimental data to determine the constitutive model. Subsequently, the macroscopic shear modulus corresponding to each temperature point is fitted, and the shear modulus is defined as an interpolation function with temperature as the independent variable. A three-dimensional geometric model is created, coupling hyperelastic material nodes and viscoplastic material sub-nodes. Simulation calculations are performed using parametric scanning combined with transient studies to extract the equivalent viscoplastic strain of the sample after unloading and calculate the compression permanent deformation rate. Based on conventional uniaxial tensile experimental data, a hyperelastic constitutive model with high fitting accuracy is determined, and a viscoplastic model is fitted, achieving accurate simulation of the entire process of silicone rubber compression-holding-unloading.

Description

Technical Field

[0001] This invention belongs to the field of polymer material aging performance evaluation technology, specifically involving a method for evaluating the performance of silicone rubber based on a hyperelastic and viscoplastic constitutive model. This method is used to simulate the viscoplastic response of silicone rubber during the compression-holding-unloading process and to indirectly evaluate the compression permanent deformation behavior through stress-strain curves. Background Technology

[0002] Silicone rubber, as an important polymeric elastic material, is widely used in critical engineering components such as sealing, vibration damping, and insulation due to its excellent high and low temperature resistance, weather resistance, and insulation properties. In these applications, silicone rubber components are often under compression for extended periods, and their compression set behavior is crucial for assessing the material's long-term service performance and the risk of seal failure. Traditional assessment methods rely entirely on standard compression aging tests (such as GB / T7759.1), which are inherently flawed due to their lengthy processes, high costs, and inability to observe the evolution of the material's internal mechanical state, making them unsuitable for rapid research and development needs.

[0003] With the development of computer simulation technology, the finite element method (FEM) has become an important supplementary tool for numerically simulating and analyzing the mechanical behavior of rubber materials. Various finite element software programs offer diverse analytical methods for describing the time-dependent irreversible deformation of materials. These models are implemented to varying degrees in commercial software, enabling the simulation of complex mechanical behaviors.

[0004] However, existing simulation schemes mostly use hyperelastic constitutive models to simulate instantaneous elastic responses, which cannot describe time-dependent irreversible deformation and therefore cannot be used to evaluate compression permanent deformation behavior. Although some studies have mentioned viscoplastic models, there is a lack of methods to couple hyperelastic and viscoplastic constitutive models in a general multiphysics field and systematically apply them to the evaluation of silicone rubber compression behavior.

[0005] Therefore, there is an urgent need for a systematic approach that can start from experimental data, couple key constitutive models, and achieve efficient and trend-based assessment of the irreversible deformation behavior of silicone rubber after compression, providing a reference for material selection and design optimization. Summary of the Invention

[0006] To address the aforementioned technical problems, this invention provides a method for evaluating the performance of silicone rubber based on hyperelastic and viscoplastic constitutive models. This method overcomes the limitations of existing simulation techniques in assessing the compression set behavior of silicone rubber, providing a standardized and reproducible finite element simulation evaluation method. Based on conventional uniaxial tensile test data, this method can determine a hyperelastic constitutive model and its parameters with high fitting accuracy. By coupling the hyperelastic and viscoplastic models, it achieves accurate simulation of the entire compression-holding-unloading process of silicone rubber. By extracting and analyzing the stress-strain curves obtained from the simulation, the residual strain after unloading is observed, and the influence trend of material parameters (such as macroscopic shear modulus) on the residual strain is studied. This indirectly assesses the strength trend of the compression set behavior of silicone rubber under different conditions (such as temperature), improving the efficiency and depth of material performance research.

[0007] The complete technical solution of this invention includes: A method for evaluating the properties of silicone rubber based on a hyperelastic and viscoplastic constitutive model includes the following steps: (1) Within the target temperature range, a uniaxial tensile test was conducted on the silicone rubber sample to obtain the engineering stress-strain curve. The experimental data were fitted to determine the constitutive model. (2) Based on the model determined by the fitting in step (1), the macroscopic shear modulus corresponding to each temperature point is obtained by fitting; and the shear modulus is defined as an interpolation function with temperature as the independent variable; (3) Create a three-dimensional geometric model and couple hyperelastic material nodes and viscoplastic material sub-nodes; (4) Mesh the three-dimensional geometric model, and use parametric scanning combined with transient research to perform simulation calculations. Extract the equivalent viscoplastic strain of the specimen after unloading and calculate the compressive permanent deformation rate.

[0008] Furthermore, the target temperature range is 25°C, 60°C, and 80°C.

[0009] Furthermore, in step (1), the experimental data is fitted using the least squares method to adjust the model parameters so that the sum of squared errors between the model prediction results and the experimental data points is minimized.

[0010] Furthermore, in step (2), the input point set of the interpolation function is the experimental temperature, and the output value is an array of shear modulus values ​​fitted at the corresponding temperature.

[0011] Furthermore, in step (3), the three-dimensional geometric model is a cylinder with a diameter and height of 10 mm.

[0012] Furthermore, the shear modulus of the hyperelastic material node calls the temperature interpolation function defined in step (2).

[0013] Furthermore, in step (3), a fixed constraint is applied to the bottom surface by specifying the displacement node. On the top surface, a piecewise function pw1(t) is defined to simulate the compression-holding-unloading process, and this function is assigned to the Z-direction displacement boundary condition of the top surface.

[0014] Furthermore, in step (4), the compressive permanent deformation rate is calculated using the following formula:

[0015] In the formula: ε is the compression permanent deformation rate; h0 is the height of the specimen before compression; h1 is the specified height of the specimen after uniform compression; h2 is the height of the specimen after the compression force is removed and no further deformation occurs.

[0016] The advantages of this invention over the prior art are: 1. The model selection is based on an objective system, and the results are reliable: A reasonable constitutive model was determined by fitting experimental data, which ensures that the simulation model is closer to the real material behavior from the root.

[0017] 2. The model has a complete physical mechanism and strong functionality. By coupling the hyperelastic model and the viscoplastic model at the node level, it fully reproduces the instantaneous elastic response and time-dependent plastic deformation of silicone rubber, making it possible to directly simulate and calculate the "compression permanent deformation rate", thus solving the lack of this function in the existing technology.

[0018] 3. By utilizing temperature interpolation functions and parametric scanning techniques, material parameters are dynamically correlated with temperature variables. Only a single finite element model is needed to automatically complete simulation calculations at multiple temperature points, achieving a leap from "one-to-one" to "one-to-many" analysis, significantly improving analysis efficiency. This is particularly suitable for studying the temperature sensitivity of material properties and performing operational condition scanning. By incorporating the temperature-dependent characteristics of material parameters into the constitutive model and combining node-level coupling and parametric scanning strategies, a leap from "single-point modeling" to "temperature domain simulation" is achieved. This method not only significantly improves simulation efficiency but also provides a standardized technical path for predicting material performance at various service temperatures.

[0019] 4. Standardized process, easy to promote: This invention provides a clear and repeatable operating procedure from experiment to simulation, which lowers the technical threshold for using advanced constitutive models to simulate complex aging behavior and is easy to promote and apply in engineering research and development and material evaluation. Attached Figure Description

[0020] Figure 1 This is a schematic diagram of the simulation process of the compression set rate of silicone rubber described in this invention.

[0021] Figure 2 The image shows the finite element model of the silicone rubber cylinder constructed in this embodiment.

[0022] Figure 3 This is a schematic diagram of the piecewise function pw1(t) called in the specified displacement boundary conditions used to simulate the compression-holding-unloading process.

[0023] Figure 4 The stress-strain curves are shown under different macroscopic shear moduli during compression.

[0024] Figure 5 This is a schematic diagram of the Bergstrom-Boyce model. Detailed Implementation

[0025] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0026] The core of this invention lies in constructing an integrated process of "experimental calibration - model coupling - multi-temperature prediction". The overall technical solution includes the following steps: (1) Experimental data acquisition and constitutive model fitting Uniaxial tensile tests were conducted on silicone rubber samples within the target temperature range (25℃, 60℃, 80℃) to obtain the actual engineering stress-strain curves.

[0027] The experimental data is fitted to determine the optimal constitutive model. The core of this data fitting process is to use the least squares method to continuously adjust the model parameters, minimizing the sum of squared errors between the model's predictions and the experimental data points. Common optimization algorithms, such as the Levenberg-Marquardt method, are used for iterative solutions until the preset optimization tolerance is reached.

[0028] The hyperelastic constitutive model that best fits the desired performance across all determined temperatures is:

[0029]

[0030] In the formula, W represents the strain energy density; The macroscopic shear modulus of a material is usually obtained by fitting experimental data. For rubber materials, it is generally between 0.1 and 10 MPa and can be selected based on the material hardness. λ represents the first strain invariant, whose value can be determined by experimental fitting; λ represents the tensile ratio, which can be obtained by fitting a uniaxial tensile experiment and has a value between 2 and 10. D This indicates the incompressibility parameter of the material. The elastic volume ratio can be determined by experimental fitting; P represents stress. N This represents the number of segments in the model. The coefficients represent the model expansion coefficients, derived from statistical mechanics, and are fixed constants.

[0031] Based on the data fitting, the determined N=8 segment model is well-suited for small deformation regions, requires only one parameter, and is simple and stable.

[0032] (2) Extraction and parameterization of constitutive parameters at multiple temperatures Based on the model determined by fitting in step (1), the macroscopic shear modulus μ corresponding to the above 8 segments of the model at each temperature point (25℃, 60℃, 80℃) is obtained by using an optimization solver. Subsequently, the shear modulus μ is defined as an interpolation function (int_μ(T)) with temperature as the independent variable. The input point set of this function is the experimental temperature [25, 60, 80], and the output value is an array of μ values ​​fitted at the corresponding temperature.

[0033] This step is crucial for achieving continuous temperature dependence of material parameters. By converting the originally fixed material parameters in commercial software into temperature-dependent interpolation functions, a single model can encompass material behavior under multiple temperature conditions. This not only avoids the complexity of modeling each temperature individually but also enables continuous interpolation and dynamic calling of material parameters across different temperatures. It provides a unified data interface for subsequent "parametric scanning + transient studies," significantly improving modeling efficiency and simulation analysis flexibility. This step also enhances the model's scalability and maintainability: if new research temperatures are needed, only uniaxial tensile experimental data at that temperature needs to be added, the corresponding macroscopic shear modulus fitted, and the input and output point sets of the interpolation function updated. There is no need to reconstruct the core settings of the basic model, such as geometry, physics, and boundary conditions, reducing the technical cost of model iteration.

[0034] (3) Construction of multiphysics model A three-dimensional geometric model is created, constructing a cylinder with a diameter and height of 10 mm to represent a standard compression specimen. In the solid mechanics physics field, hyperelastic material nodes and viscoplastic material sub-nodes are set sequentially, with the following key settings, which are the core operational points of this invention: Hyperelastic material nodes: The material model selected is model (1) above, and the number of segments is set to 8. In the model parameter settings, for the shear modulus μ, a fixed value is no longer entered, but the temperature interpolation function int_μ(T) defined in step (2) is called. At the same time, the global variable T is set to the temperature parameter to be studied.

[0035] Viscoplastic Material Nodes: For the aforementioned hyperelastic material nodes, polymer viscoplastic sub-nodes are added. The Bergstrom-Boyce model is selected for viscoplasticity. The parameters required for the BB model are set, including: energy factor, viscoplasticity coefficient, flow resistance, stress exponent, cutoff stress, strain exponent, etc. Since changes in these parameters have minimal impact on the calculation results, they can be set to fixed values ​​or preset values ​​from existing commercial software can be selected.

[0036] The core idea of ​​the Bergstrom-Boyce model is to decompose the total mechanical response of a material into two parallel networks, such as... Figure 5 As shown, network A (equilibrium network) represents the hyperelastic response of the material in a fully relaxed state and is a pure hyperelastic element; network B (non-equilibrium network) represents the rate-dependent response of the material and is composed of a hyperelastic element (spring) and a nonlinear viscous element (damper) connected in series.

[0037] The hyperelastic response of both networks can be described by the same strain energy density function W of the aforementioned hyperelastic material node model (1); the viscous element in network B controls the rate-dependent behavior of the material, and its effective creep strain rate Defined by the following formula:

[0038] In the formula, A represents the creep rate constant, which is typically 10 based on experimental fitting. -6 ~10 3 C represents the strain index, which is usually negative (-0.5 to -0.1); M is the stress index, M≥1; This represents the regularization parameter, typically 0.01 or 0.001; This represents the reference stress (usually taken as 1). Indicates the network stretch ratio. Indicates effective stress.

[0039] To simulate permanent deformation behavior, it is necessary to incorporate plastic deformation theory. This invention uses viscoplastic effects for description, where materials exhibit both plastic and viscous behavior under stress, including strain rate sensitivity, stress softening due to damage and failure, and stiffness degradation. Therefore, when considering viscoelastic and viscoplastic properties, this invention adds a viscoelastic and polymer viscoplastic Bergstrom-Boyce model to the child nodes of the hyperelastic node.

[0040] The coupling of the two nodes constitutes a two-layer coupled structure of the material constitutive model. The hyperelastic node serves as the base layer, describing the instantaneous elastic response of the material; the viscoplastic sub-node serves as the functional layer, introducing time-dependent irreversible deformation capability on top of the elastic response. Compared to a single hyperelastic material node, this coupled design provides a more complete physical mechanism, closely reflecting the actual mechanical behavior of the material. During the compression-holding-unloading process, silicone rubber exhibits both instantaneously recoverable elastic deformation and time-dependent irreversible viscoplastic deformation. A single hyperelastic node can only describe the former, failing to represent the latter; while the coupled structure can simultaneously characterize both deformation behaviors, fully reproducing the actual mechanical response characteristics of silicone rubber. Furthermore, it enables quantitative simulation of compression set: in the simulation results of a single hyperelastic node, the strain is zero when the stress is zero after unloading, making it impossible to obtain residual strain and thus unable to assess compression set; the coupled structure, through the viscoplastic node, captures the viscoplastic residual strain after unloading, providing core data support for quantitatively calculating the compression set rate, which is crucial for achieving compression set simulation.

[0041] Boundary and Load Conditions: By specifying displacement nodes, select an end face (such as the bottom face) and apply fixed constraints. On the other end face (top face), define a piecewise function pw1(t) to simulate the compression-holding-unloading process, and assign this function as the Z-direction displacement boundary condition for the top face.

[0042] (4) Grid generation and multi-condition research calculation The cylinder was meshed, with local refinement in areas prone to large deformation. A parametric sweep combined with transient analysis was employed. In the parametric sweep, the temperature parameter T was set to [25, 60, 80]. In the transient analysis, the time range covered the complete period of the piecewise function pw1(t). The solver employed geometric nonlinearity to accommodate large deformations. A schematic diagram of the simulation process for the compression set rate of silicone rubber is shown below. Figure 1 As shown, after the calculation is completed, the equivalent viscoplastic strain of the specimen after unloading is extracted using the 3D drawing group. The compressive permanent deformation rate is calculated using the following formula:

[0043] In the formula: ε is the compression permanent deformation rate; h0 is the height of the specimen before compression; h1 is the specified height of the specimen after uniform compression; h2 is the height of the specimen after the compression force is removed and no further deformation occurs.

[0044] Example 1 This embodiment uses a methyl vinyl silicone rubber (VMQ) as an example to evaluate its compression set properties at 25°C (room temperature), 60°C (medium operating temperature), and 80°C (high operating temperature).

[0045] S1: Experiment and Parameter Calibration Methyl vinyl silicone rubber (VMQ) dumbbell-shaped specimens were prepared according to national standards. Uniaxial tensile tests were conducted in a temperature-controlled chamber at 25℃, 60℃, and 80℃ until fracture, and the stress-strain curve data were recorded. The stress-strain data at different temperatures were fitted and optimized by comparing the current model predictions with experimental data. The Levenberg-Marquardt method was used for the solution, with an optimization tolerance of 0.01, and all parameters were combined using the least squares method. The aforementioned constitutive model was determined, and further fitting yielded the model parameters (macroscopic shear modulus μ) at each temperature: μ1 = 0.80 MPa at 25℃, μ2 = 0.77 MPa at 60℃, and μ3 = 0.86 MPa at 80℃.

[0046] Then, a global interpolation function mu_T(T) is defined.

[0047] S2: Model Building 1. Geometric Model: Create a cylinder with a diameter of 10mm and a height of 10mm as the compression specimen. (e.g.) Figure 2 As shown, Figure 2 The dimensions of the intermediate model are consistent with those of the cylindrical specimen used in the compression test. The specimen is a cylinder with a height of 10 mm and a diameter of 10 mm. Considering two-dimensional axisymmetry, the model is simplified to a rectangle. Considering the symmetry of the middle cross section, the model can be further simplified by half. The lower surface of the specimen is fixed, and a load is applied to the upper surface of the specimen, specifying the displacement.

[0048] 2. Define functions: Define a global piecewise function pw1(t) to simulate the compression-holding-unloading stage, and set the macroscopic shear modulus corresponding to each temperature obtained in step S1 as the interpolation function int1(mu) for subsequent parametric scanning.

[0049] 3. Materials: Select methyl vinyl silicone rubber (VMQ) and refine the relevant material properties.

[0050] 4. Physics Field Setup: Couple hyperelastic material nodes and viscoplastic material sub-nodes. The macroscopic shear modulus of the hyperelastic material node is set as the interpolation function int1(mu). Add polymer viscoplastic sub-nodes under the hyperelastic material node, using the Bergstrom-Boyce model, and define parameters such as energy factor and viscoplasticity coefficient.

[0051] 5. Boundary Conditions: Select the specified displacement method, choose the upper surface of the cylinder as the moving boundary, specify the displacement along the Z direction, and input the piecewise function pw1(t). See [link to documentation]. Figure 3 Select the lower surface of the cylinder as the fixed surface. Figure 3 China Figure 2The cylindrical sample was compressed, and the sample was compressed by 30% in 150s. The pressure was held for 0s, and the unloading was completed after 150s.

[0052] 6. Mesh Generation: The size of the mesh cells is set according to the calculation accuracy.

[0053] S3: Research and Analysis 1. Parametric scanning + transient study method: The parameter mu is scanned, with a value list [0,1,2], where "0", "1", and "2" represent the macroscopic shear modulus corresponding to 25℃, 60℃, and 80℃, respectively. The transient study time range covers the entire compression-holding-unloading stage, from 0s to 300s, with the time step interval set according to the required solution accuracy.

[0054] 2. Calculation and Solution: After the calculation is completed, the instantaneous and permanent deformations of the model under compression are plotted using the 3D plotting group based on the solution set. The permanent deformation is presented as "equivalent viscoplastic strain". The compressive stress-strain curve is plotted using the "1D plotting group". The curve shows that there is still residual strain when the stress is 0, which reflects the permanent deformation under compression. The permanent deformation rate under compression = abs(equivalent viscoplastic strain / 3*100%).

[0055] 3. Simulation Results: Stress-strain curves under different macroscopic shear moduli during compression are shown below. Figure 4 As shown, Figure 4 In the unloading process, the strain at which the stress is 0 is the permanent deformation. As the temperature increases, the value of the material's macroscopic shear modulus gradually decreases, the stress also gradually decreases, the point where the stress is 0 gradually shifts to the right, and the compressive permanent deformation gradually increases. The compressive permanent deformation rates of the material at 25℃, 60℃, and 80℃ obtained through model calculations are shown in Table 1.

[0056] The trend reflected by the model calculation conforms to the Arrhenius equation, indicating that temperature accelerates the aging (permanent deformation) of silicone rubber. The effectiveness and accuracy of this simulation method were verified by comparing the simulation results with measured aging data from the same batch of samples (assuming a deviation of <20%). Table 1 shows the simulation results of the compression set at different temperatures (25℃, 60℃, 80℃).

[0057] Table 1

[0058] Table 1 This embodiment demonstrates that the method provided by the present invention can systematically, efficiently and relatively accurately predict the compression set of silicone rubber at different temperatures, providing a valuable virtual testing method for engineering applications.

[0059] The above description is merely a preferred embodiment of the present invention and does not constitute any limitation on the present invention. Any simple modifications, alterations, or equivalent structural changes made to the above embodiments based on the technical essence of the present invention shall still fall within the protection scope of the present invention.

Claims

1. A method for evaluating the performance of a silicone rubber based on a hyperelastic and viscoplastic constitutive model, characterized by, Includes the following steps: (1) Within the target temperature range, a uniaxial tensile test was conducted on the silicone rubber sample to obtain the engineering stress-strain curve. The experimental data were fitted to determine the constitutive model. (2) Based on the model determined by the fitting in step (1), the macroscopic shear modulus corresponding to each temperature point is obtained by fitting; and the shear modulus is defined as an interpolation function with temperature as the independent variable; (3) Create a three-dimensional geometric model and couple hyperelastic material nodes and viscoplastic material sub-nodes; (4) Mesh the three-dimensional geometric model, and use parametric scanning combined with transient research to perform simulation calculations. Extract the equivalent viscoplastic strain of the specimen after unloading and calculate the compressive permanent deformation rate.

2. The method for evaluating the performance of silicone rubber based on the hyperelastic and viscoplastic constitutive model according to claim 1, characterized in that, The target temperature range is 25℃, 60℃, and 80℃.

3. The method for evaluating the performance of silicone rubber based on the hyperelastic and viscoplastic constitutive model according to claim 2, characterized in that, In step (1), the experimental data is fitted using the least squares method to adjust the model parameters so that the sum of squared errors between the model prediction results and the experimental data points is minimized.

4. The method for evaluating the performance of silicone rubber based on a hyperelastic and viscoplastic constitutive model according to claim 3, characterized in that, In step (2), the input point set of the interpolation function is the experimental temperature, and the output value is an array of shear modulus values ​​fitted at the corresponding temperature.

5. The method for evaluating the performance of silicone rubber based on a hyperelastic and viscoplastic constitutive model according to claim 4, characterized in that, In step (3), the three-dimensional geometric model is a cylinder with a diameter and height of 10 mm.

6. The method for evaluating the performance of silicone rubber based on a hyperelastic and viscoplastic constitutive model according to claim 5, characterized in that, The shear modulus of the hyperelastic material node calls the temperature interpolation function defined in step (2).

7. The method for evaluating the performance of silicone rubber based on a hyperelastic and viscoplastic constitutive model according to claim 6, characterized in that, In step (3), fixed constraints are applied to the bottom surface by specifying displacement nodes. On the top surface, a piecewise function pw1(t) is defined to simulate the compression-holding-unloading process, and this function is assigned to the Z-direction displacement boundary conditions of the top surface.

8. The method for evaluating the performance of silicone rubber based on a hyperelastic and viscoplastic constitutive model according to claim 7, characterized in that, In step (4), the compressive permanent deformation rate is calculated using the following formula: In the formula: ε is the compression permanent deformation rate; h0 is the height of the specimen before compression; h1 is the specified height of the specimen after uniform compression; h2 is the height of the specimen after the compression force is removed and no further deformation occurs.

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