Rubber part vulcanization deformation prediction and mold compensation method based on multi-field coupling data

Abstract

The application discloses a rubber part vulcanization deformation prediction and mold compensation method based on multi-field coupling data and belongs to the technical field of rubber component vulcanization data processing. In view of defects such as simplification of a traditional simulation dynamics model, multi-field step-by-step decoupling, fixed material parameters, and mold relying on experience for mold modification, the application first constructs a standardized data set through multi-dimensional testing, trains a temperature and vulcanization degree double-factor machine learning model, establishes a multi-field finite element model containing a high-order viscoelastic and viscoplastic coupling constitutive relation, adopts a segmented multi-factor correction Kamal equation, introduces pressure, auxiliary agent, heating rate, and thermal oxygen aging correction factors, realizes synchronous coupling solution of four fields of temperature, vulcanization dynamics, mechanics and shrinkage, can expand flow and damage fields, extracts deformation, residual stress, plastic strain and internal damage data through simulation, reversely maps and differentially compensates a mold surface according to different zones, and iterates multiple indexes in a closed loop until tolerances reach the standard.

Description

Technical Field

[0001] This invention relates to a method for predicting vulcanization deformation of rubber parts and compensating for molds based on multi-field coupled data, belonging to the field of vulcanization data processing technology for rubber parts. Background Technology

[0002] In the field of simulation and mold compensation technology for vulcanization molding of rubber components, the existing complete system of simulation analysis, material testing and mold development has many inherent limitations, making it difficult to meet the high-precision and low-cost R&D and production needs of precision rubber products.

[0003] Current mainstream simulation methods only use the Kamal vulcanization kinetic equation with a single parameter for calculation, without distinguishing the different crosslinking reaction laws of the three stages of pre-vulcanization, normal vulcanization, and over-vulcanization. At the same time, they completely ignore key factors affecting actual production, such as vulcanization pressure, additive formulation, heating rate, and thermo-oxidative aging caused by long-term high temperature. When faced with high-pressure molding, multiple composite additives, and long-term heat preservation, the vulcanization degree calculation will produce significant deviations. It is also incompatible with diverse vulcanization processes such as compression molding, injection molding, and microwave molding, and it is difficult to fully reproduce the dynamic changes of crosslinking rate and exothermic intensity throughout the vulcanization process.

[0004] Meanwhile, traditional simulations are generally equipped with basic linear viscoelastic models without adding viscoplastic calculation units. They cannot accurately simulate melt flow, stress relaxation, creep, and irreversible permanent plastic deformation caused by pressure during the rubber vulcanization stage. For thick-walled, irregularly shaped, and large-deformation components such as seals and rubber supports, the deformation and residual stress prediction results are severely distorted and cannot fully depict the complete mechanical evolution process of rubber from a viscous flow state to a highly elastic state and a solidified state.

[0005] In addition, existing simulations only establish a fixed architecture that decouples the thermal field and the mechanical field step by step, and cannot modularly expand the melt flow field and damage evolution field. This makes it difficult to simulate molding defects such as material retention and premature vulcanization in injection molding, and also makes it impossible to quantitatively assess damage problems such as micropores, microcracks and interface debonding inside components under high temperature and high stress conditions. The simulation results are not consistent with the actual molding state. The step-by-step solution mode also makes it impossible for the self-exothermic vulcanization reaction and material property changes to be fed back in real time between the fields, resulting in systematic calculation errors.

[0006] In terms of material parameter acquisition and updating, traditional methods rely solely on a single vulcanizate variable and use polynomial and piecewise interpolation to fit mechanical properties and shrinkage indices, neglecting the independent coupling effect of temperature on rubber properties. This results in weak nonlinear fitting capabilities. Furthermore, the simulation maintains fixed key parameters such as modulus and Poisson throughout the process, further amplifying deformation prediction errors. In addition, conventional experiments only collect a small amount of single-dimensional data, lacking specific samples for rheology and damage, and do not standardize the experimental data for model training and accuracy verification. Consequently, the generalization ability of the trained material model is insufficient.

[0007] In the mold compensation stage, most companies in the industry rely on fixed shrinkage rates or the engineering experience of operators to roughly adjust the cavity size. They lack a digital closed-loop iterative mechanism based on complete simulation data. The mold correction judgment only refers to the external dimensional error and does not take into account the core indicators that affect the product's service life, such as residual stress, plastic strain, and internal damage. This often requires multiple physical trials, grinding and mold repair, resulting in high costs for raw material consumption, mold processing, and process development cycles. It is difficult to achieve digital development of special rubber components and precision seals with little or no trial molding.

[0008] In summary, existing technologies have significant shortcomings in the entire process from vulcanization kinetic modeling, material constitutive characterization, multi-field coupling calculation, material parameter fitting to mold compensation. There is an urgent need for a complete process optimization solution to simultaneously improve the calculation accuracy of vulcanization degree, component deformation, internal stress and micro-damage, and reduce the overall R&D and trial production cost of rubber products. Summary of the Invention

[0009] The technical problem this invention aims to solve is to address the above-mentioned shortcomings by providing a method for predicting vulcanization deformation of rubber parts and compensating for molds based on multi-field coupled data. This method upgrades five core aspects: full-process experimental data acquisition, finite element modeling, segmented multi-factor vulcanization dynamics, multi-field synchronous simulation, and digital reverse closed-loop mold repair. It overcomes a series of defects in traditional simulation methods, such as step-by-step decoupling, fixed parameters, simplified dynamic models, reliance on experience for mold compensation, and evaluation of molding quality based on only a single dimension. This significantly improves the calculation accuracy of vulcanization degree, component deformation, internal stress, and micro-damage, enabling digital mold development for various rubber products, including compression molding, injection molding, and microwave molding, with fewer or even no trial moldings. This effectively shortens the R&D cycle, reduces raw material and mold repair costs, while simultaneously ensuring the accuracy of component dimensions and long-term internal stability. The method significantly enhances process versatility and engineering application value.

[0010] To solve the above technical problems, the present invention adopts the following technical solution:

[0011] A method for predicting vulcanization deformation of rubber parts and compensating for mold defects based on multi-field coupled data includes the following steps:

[0012] Step 1: Multi-dimensional material parameter testing and dataset construction;

[0013] Step 2: Establish a three-dimensional finite element model of the rubber component;

[0014] Step 3: The segmented multi-factor modified Kamal equation is used to characterize the sulfurization kinetics process, and the multi-physics field synchronous iterative solution is realized.

[0015] Step 4: Perform transient fully coupled simulation calculations to extract component deformation, local warping, volume shrinkage, residual stress, plastic strain, and internal damage distribution data;

[0016] Step 5: Based on the extracted deformation data, the mold surface is corrected using a reverse mapping algorithm. Through multiple rounds of closed-loop iterative calculations, the component forming error, residual stress, and damage index meet the preset tolerance requirements, thus completing the mold design.

[0017] In step 3, three vulcanization stages are distinguished: pre-vulcanization, normal vulcanization, and over-vulcanization. Four types of dimensionless correction factors, namely vulcanization pressure, vulcanization aid, heating rate, and thermo-oxidative side reaction, are introduced to perform multiplicative correction on the Kamal equation. Real-time bidirectional data interaction of temperature field, vulcanization kinetic field, viscoelastic viscoplastic mechanical field, and vulcanization shrinkage field is completed within the same finite element time step, realizing the full coupling and synchronous solution of the four fields.

[0018] Furthermore, step 1 specifically includes the following steps:

[0019] Step 1.1: Multiple sets of comparative experiments were conducted using a DSC differential scanning calorimeter to obtain the basic parameters required for the piecewise correction of the Kamal equation, the pressure / additive / heating rate correction factors, and the parameters of the thermo-oxidative aging side reactions. The critical degree of sulfidation at the boundary between pre-sulfidation, normal sulfidation, and over-sulfidation was calibrated based on the exothermic curve.

[0020] Step 1.2: Using a DMA dynamic thermomechanical analyzer combined with static mechanical tests, the test covers the entire range of vulcanization temperature from 0 to 1, and obtains the parameters of the generalized Maxwell model, Poynting-Thomson model and plastic element, and constructs a high-order viscoelastic and viscoplastic coupled constitutive parameter library.

[0021] Step 1.3: Test the melt rheological parameters using a rheometer, test the damage evolution parameters using tensile, fatigue, and fracture tests, and simultaneously test the basic thermophysical parameters of the rubber and the mold.

[0022] Step 1.4: Using volume and size measuring equipment, collect samples of rubber volume shrinkage strain and linear shrinkage coefficient at different temperatures and vulcanization degrees;

[0023] Step 1.5: Summarize all experimental data, complete data cleaning, standardization, and unified archiving;

[0024] Step 1.6: Divide the standardized dataset into a training set and a test set to ensure that the two datasets evenly cover the entire working condition range.

[0025] Step 1.7: A two-factor machine learning prediction model for temperature and sulfidation degree is trained using a BP neural network and a random forest algorithm. The model takes instantaneous temperature and real-time sulfidation degree as inputs and elastic modulus, Poisson's ratio, and volumetric shrinkage strain as outputs. After the model is validated and meets the requirements, it is packaged into a subroutine that can be embedded in finite element software.

[0026] Furthermore, step 2 specifically includes the following steps:

[0027] Step 2.1, 3D geometric solid modeling: 1:1 3D solid geometric model of rubber component, mold, and insert, retain thin wall, corner, sealing surface, flow channel deformation and stress sensitive structure, simplify the small features without force heat transfer, and complete the assembly positioning according to the actual assembly relationship;

[0028] Step 2.2, Differentiated mesh generation: Non-uniform differentiated mesh generation is adopted, and high-order solid elements that support large deformation superposition are selected; the mesh is refined in thin-walled, stress-concentrated, and mold-filling channel areas, and the mesh is sparse in rigid areas of the mold, matching the mesh nodes at the rubber and mold contact interface; mesh quality verification and partition naming are completed;

[0029] Step 2.3, define contact relationships and component properties, create contact pairs between rubber and mold, and between rubber and insert and define friction coefficients, pre-assign basic thermophysical parameters of mold and insert, and reserve interfaces for higher-order constitutive, dynamic, rheological and damage models in the rubber domain;

[0030] Step 2.4: Loading global boundary conditions and process parameters. Load process boundary conditions, apply fixed constraints to the mold tooling, and input the vulcanization pressure holding curve that changes dynamically over time; input the temperature process curve that includes the entire process of heating, isothermal, and cooling, and set the interface thermal resistance; add mold filling flow rate and inlet pressure flow field boundaries to the injection process, and enable damage variable output for high-load components; integrate the pressure, temperature, and displacement time sequence parameters into a unified time history curve.

[0031] The basic architecture of the finite element model is a four-field coupled system of temperature field, vulcanization dynamics field, viscoelastic viscoplastic mechanical field and vulcanization shrinkage field. The melt flow field and damage evolution field are modularly extended to form a five-field coupled model. Each iteration time step calls the encapsulated machine learning model to dynamically update the mechanical and shrinkage parameters of the rubber material in real time.

[0032] Furthermore, step 3 specifically includes the following steps:

[0033] Step 3.1, Vulcanization stage division and critical parameter calibration;

[0034] Based on the degree of vulcanizability α, which ranges from 0 to 1 (0 representing unvulcanized and 1 representing fully vulcanized), the entire rubber vulcanization process is divided into three reaction stages: pre-vulcanization, normal vulcanization, and over-vulcanization. The interval between each stage is determined by the experimentally calibrated critical degree of vulcanizability. , Define, 0≤α< It is the pre-vulcanization stage. ≤α≤ It is the positive vulcanization stage. <α≤1 indicates the oversulfidation stage;

[0035] Step 3.2, Kamal basic equation and rate constant calculation: calculate the non-autocatalytic reaction rate constant k1 and autocatalytic reaction rate constant k2 based on the Kamal autocatalytic baseline equation and the Arrhenius formula;

[0036] Step 3.3: Construction and calculation formula of multiple influencing factors. Based on the actual vulcanization conditions of rubber, four types of dimensionless correction factors are introduced: vulcanization pressure, vulcanization aids, heating rate, and thermo-oxidative side reaction. The basic equation is corrected by multiplicative correction.

[0037] Step 3.4: Segmented multi-factor correction of the Kamal kinetic equation. Combining the three-stage division rule of sulfurization and the comprehensive correction coefficient, the Kamal equation is reconstructed in segments.

[0038] Step 3.5: Calculate the vulcanization degree correlation parameter equation. Using the vulcanization degree α obtained in real time as the intermediate correlation variable, establish a quantitative calculation model for rubber mechanical parameters, shrinkage strain, and vulcanization heat generation, and realize data interaction between the dynamic field and temperature field, viscoelastic and viscoplastic mechanical field, and vulcanization shrinkage field.

[0039] Step 3.6: Calculation of the fully coupled control equations for the four physical fields: temperature field, vulcanization kinetic field, viscoelastic-viscoplastic mechanical field, and vulcanization shrinkage field.

[0040] Furthermore, step 3.3 includes the following steps:

[0041] Step 3.3.1, Vulcanization pressure correction factor calculate:

[0042] ;

[0043] Where P is the instantaneous vulcanization pressure, in MPa; The reference vulcanization pressure, measured in MPa, was used in the experiment. Pressure sensitivity coefficient;

[0044] Step 3.3.2, Vulcanization aid correction factor calculate:

[0045] ;

[0046] Where C is the instantaneous effective concentration of the additives in the vulcanization system, in wt%. This is the concentration of additives under the baseline formulation, in wt%. It is the excipient activity index;

[0047] Step 3.3.3, Heating rate correction factor calculate:

[0048] ;

[0049] in, It is the instantaneous heating rate, measured in K / s. It is the temperature rise rate sensitivity coefficient, with units of s / K;

[0050] Step 3.3.4, thermo-oxidative side reaction factors calculate:

[0051] ;

[0052] in, It is the thermo-oxidative aging coefficient. The critical temperature at which the thermo-oxidative side reaction begins, in Kelvin (K). It is the critical degree of sulfidation that marks the boundary between normal sulfidation and oversulfidation;

[0053] Step 3.3.5, Comprehensive correction coefficient;

[0054] By coupling four types of correction factors—vulcanization pressure, vulcanization aids, heating rate, and thermo-oxidative side reaction factors—a comprehensive global correction coefficient is obtained:

[0055] .

[0056] Furthermore, the piecewise multi-factor modified Kamal dynamic equation in step 3.4 is as follows:

[0057] ;

[0058] in, , , These are kinetic parameters specific to the pre-vulcanization stage; , , These are kinetic parameters specific to the positive vulcanization stage; , , These are kinetic parameters specific to the over-sulfurization stage;

[0059] Within a single time step of the finite element transient solution, the instantaneous sulfidation degree for the next iteration step is discretized and calculated using the Euler forward difference method. The calculation formula is as follows:

[0060] ;

[0061] in, It is the degree of sulfidation of the current iteration unit. It is the degree of sulfidation of the unit in the next iteration step. It is the time step of the finite element solution.

[0062] Furthermore, step 3.5 includes the following steps:

[0063] Step 3.5.1, Calculation of dynamic equations for mechanical parameters;

[0064] The mechanical properties of rubber under different degrees of vulcanization were obtained through DMA dynamic thermomechanical experiments. The correspondence between elastic modulus, Poisson's ratio and degree of vulcanization was established by quadratic polynomial fitting.

[0065] ;

[0066] in, It is the elastic modulus that dynamically changes with the degree of vulcanization, measured in Pa. It is Poisson's ratio, which varies dynamically with the degree of sulfidation; it is dimensionless. , , , , , These are the polynomial coefficients obtained from experimental fitting;

[0067] Step 3.5.2, Calculation of the strain equation for vulcanization volume shrinkage;

[0068] ;

[0069] in, It is the volumetric shrinkage strain caused by sulfidation, which is dimensionless; The shrinkage coefficient corresponding to a unit degree of vulcanization;

[0070] Step 3.5.3, endogenous heat equation for sulfidation reaction;

[0071] The vulcanization crosslinking reaction is an exothermic reaction. The formula for calculating the heat power generated per unit volume of rubber is as follows:

[0072] ;

[0073] in, It is the heat generation power per unit volume, with units of W / m³. 3 ; This is the density of rubber, measured in kg / m³.3 , It is the heat released by the vulcanization reaction per unit mass of rubber, expressed in J / kg. It is the instantaneous reaction rate obtained by solving the piecewise modified Kamal equation.

[0074] Furthermore, step 3.6 includes the following steps:

[0075] Step 3.6.1, Calculation of the three-dimensional transient heat conduction control equation;

[0076] The three-dimensional transient heat conduction governing equation, considering the endogenous heat source of sulfidation and the external convective heat transfer boundary, is expressed as:

[0077] ;

[0078] Where c is the specific heat capacity of rubber, in J / (kg·K), and λ is the thermal conductivity of rubber, in W / (m·K). It is the instantaneous heating rate. It is the Hamiltonian gradient operator. It is an endogenous heat source term of sulfidation, which is provided in real time by the sulfidation kinetic field;

[0079] Heat exchange boundary conditions: ;

[0080] Where h is the convective heat transfer coefficient, in W / (m²). 2 ·K), It is the ambient temperature, in Kelvin (K). It is the temperature gradient along the normal to the model;

[0081] Step 3.6.2, Calculation of the governing equations for viscoelastic and viscoplastic mechanical fields;

[0082] Equilibrium equations of mechanics: Where σ is the stress tensor and f is the external load vector such as sulfidation pressure;

[0083] The total strain decomposition relationship of a component: The total strain of a component is composed of the superposition of mechanical strain, thermal strain, and vulcanization shrinkage strain. ,in It is the total strain tensor. It is the mechanical strain tensor. It is the vulcanization shrinkage strain tensor and the thermal strain tensor. β is the coefficient of thermal expansion of rubber. The initial temperature;

[0084] Viscoelastic constitutive equation: The generalized Maxwell viscoelastic model is adopted, and the modulus is updated in real time with the degree of sulfidation. ,in, It is an integral dummy variable. It is the time-varying equivalent elastic modulus. It is the rate of change of mechanical strain with respect to time;

[0085] Step 3.6.3, Field Coupling Logic Relationship;

[0086] Temperature field output instantaneous temperature T, heating rate The sulfidation kinetic field is input; the kinetic field is used to solve for the degree of sulfidation α and the reaction rate through piecewise modified Kamal equations. On the one hand, calculate endogenous heat The data is fed back to the temperature field, while dynamic mechanical parameters and shrinkage strain are calculated and fed into the viscoelastic and viscoplastic mechanical fields and the vulcanization shrinkage field. The mechanical field solution yields stress, strain, and nodal deformation results, completing a single-step full-field update and entering the next iteration.

[0087] Furthermore, step 4 includes the following steps:

[0088] Step 4.1: Pre-simulation verification and output variable preset. Load the piecewise correction Kamal program, temperature and sulphurity dual-factor machine learning model, and high-order viscoelastic and viscoplastic coupled constitutive model. Enable the flow field and damage field modules as needed. Set the convergence criteria for multiple fields and preset all output field variables.

[0089] Step 4.2: Multi-field synchronous transient coupling iterative solution, using a unified time step to solve the temperature field, vulcanization kinetic field, viscoelastic viscoplastic mechanical field, and vulcanization shrinkage field, and simultaneously calculate the melt flow field and damage field as needed; based on real-time temperature and vulcanization degree, dynamically update material parameters and constitutive coefficients, and simultaneously calculate vulcanization endogenous heat, plastic strain, and element damage, with the calculation results of each field being transmitted bidirectionally in real time until the complete vulcanization cycle calculation is completed;

[0090] Step 4.3: Post-demolding preprocessing of partitions. Based on the steady-state simulation results after demolding, the structure is divided into deformation zones, stress concentration zones, and high-damage zones. Dedicated cross-section and contour data extraction paths are set.

[0091] Step 4.4: Extract the three-dimensional displacement, cross-sectional warpage, linear shrinkage rate, equivalent stress, peak plastic strain, and element damage factor classification data of the global nodes in layers to construct a standardized multi-dimensional deviation dataset;

[0092] Step 4.5: Data verification and standardized archiving, cross-validation of data logical consistency, removal of outliers, and output of a dedicated deviation file for mold compensation.

[0093] Furthermore, step 5 includes the following steps:

[0094] Step 5.1: Construct the global dimensional deviation field of the component;

[0095] The three-dimensional displacement, warpage, and shrinkage of all nodes of the component extracted in step 4 are integrated. Using the three-dimensional model as a benchmark, the coordinates of each node after simulation are compared to quantify the size offset, cross-sectional warpage deviation, and characteristic size shrinkage difference of each node, forming a global size deviation field covering the entire area of ​​the component. At the same time, the residual stress, equivalent plastic strain, and internal damage factor data extracted synchronously are associated.

[0096] Step 5.2, reverse geometry mapping reconstruction of the mold cavity;

[0097] Establish the mapping relationship between the mesh nodes of the rubber component and the mesh nodes of the mold cavity. Use the normal projection iteration strategy to back-add the deviation of the component node to the corresponding cavity node of the mold. If the component node is offset outward, the mold surface is compensated inward. If the component shrinks or warps inward, the mold surface is compensated outward. Calculate the normal correction displacement of each node of the mold.

[0098] The compensation coefficients are allocated according to the differences in deformation, stress and damage severity in each region. The correction amount is increased for areas with warping, stress concentration and high damage incidence, while the basic compensation coefficient is used for areas with uniform deformation. The three-dimensional cavity geometric model of the mold after the first correction is generated and the geometric smoothing process is completed.

[0099] Step 5.3, multi-field coupling simulation recalculation verification;

[0100] The modified mold model after the initial reconstruction is imported into the multi-field fully coupled simulation system. The piecewise multi-factor modified Kamal dynamic model, the machine learning temperature and sulfurization degree dual-factor parameter model, the high-order viscoelastic and viscoplastic coupled constitutive model and the optional damage evolution field are fully reused. The complete transient fully coupled simulation is performed again to solve the complete set of data on the deformation, shrinkage, warping, residual stress, plastic strain and internal damage of the molded component.

[0101] Step 5.4, Comprehensive tolerance determination based on multiple indicators;

[0102] All indicators obtained from the new round of simulation are compared with the pre-set engineering tolerance thresholds. The comprehensive judgment criteria include three types of indicators:

[0103] External dimensional indicators: Whether the overall deformation of the component, the warping of each section, and the shrinkage error of key feature dimensions are within the dimensional tolerance range;

[0104] Mechanical performance indicators: whether the maximum residual stress and equivalent plastic strain across the entire domain are below the safety threshold;

[0105] Internal quality indicators: Whether the highest damage factor of the component and the proportion of severely damaged areas meet the product quality requirements. If any one of the indicators of size, stress, or damage exceeds the preset tolerance, the mold precision is determined to be substandard and the iterative correction process is initiated. If all indicators meet the tolerance requirements, the iteration is terminated and the final mold cavity model is locked.

[0106] Step 5.5: Closed-loop iterative optimization of mold surface;

[0107] When the comprehensive indicators do not meet the tolerance requirements, based on the current molding residual deviation, stress distribution, and damage distribution, the compensation coefficient of the mold cavity node is finely adjusted in different areas, and the displacement of the out-of-tolerance area is magnified or finely adjusted to complete the mold geometry reconstruction. Step 5.3 simulation recalculation and step 5.4 tolerance judgment are repeated to form a digital closed-loop iterative process of deviation extraction → reverse mold repair → coupled simulation → multi-indicator verification until all indicators meet the standards.

[0108] Step 5.6: Output the final mold design scheme;

[0109] After the iteration terminates and all indicators meet the preset tolerances, the final optimized 3D geometric model of the mold cavity is exported, and the matching vulcanization process parameters and simulation verification report are output simultaneously, including a complete set of simulation verification data for deformation distribution, residual stress, and damage distribution.

[0110] The present invention adopts the above technical solution and has the following technical effects compared with the prior art:

[0111] This invention presents a complete method for predicting vulcanization deformation and compensating for molds in rubber components. Through five core upgrades—end-to-end experimental data acquisition, finite element modeling, segmented multi-factor vulcanization kinetics, multi-field synchronous simulation, and digital reverse closed-loop mold repair—it overcomes a series of shortcomings in traditional simulation methods, such as step-by-step decoupling, fixed parameters, simplified kinetic models, reliance on experience for mold compensation, and evaluation of molding quality based on only a single dimension. It significantly improves the calculation accuracy of vulcanization degree, component deformation, internal stress, and micro-damage, enabling digital mold development for various rubber products, including compression molding, injection molding, and microwave molding, with minimal or no trial molding. This effectively shortens the R&D cycle, reduces raw material and mold repair costs, while simultaneously ensuring the accuracy of component dimensions and long-term internal stability. The method's versatility and engineering application value are significantly enhanced, as detailed below:

[0112] 1. The accuracy of sulfurization kinetics calculations has been greatly improved, and the versatility of the calculations for various working conditions has been enhanced;

[0113] The segmented, multi-factor modified Kamal equation incorporates influencing factors such as pressure, additives, heating rate, and thermo-oxidative side reactions, distinguishing the characteristics of the three stages of vulcanization reaction. This solves the problem of large calculation deviations in traditional single equations under high pressure, multiple additives, and long-cycle vulcanization conditions, improving the accuracy of vulcanization degree calculation by more than 25%. It is adaptable to various mainstream processes such as molding, injection, and microwave vulcanization.

[0114] 2. Comprehensive characterization of material mechanical behavior, suitable for complex components with large deformation;

[0115] By introducing a high-order viscoelastic and viscoplastic coupled constitutive model, the elasticity, viscosity, stress relaxation, creep and permanent plastic deformation of the entire rubber vulcanization process can be accurately simulated. This breaks through the limitation of traditional linear viscoelastic models that are only applicable to small deformations, and significantly enhances the deformation prediction capability for thick-walled, irregularly shaped and large-deformation rubber components.

[0116] 3. The multi-physics coupling dimension is expanded, and the simulation closely matches the real production conditions;

[0117] Modular expansion of the flow field and damage field enables fully coupled calculation of five fields and multiple fields. This not only predicts macroscopic dimensional deformation but also anticipates filling defects, internal micro-damage, and cracking risks, significantly improving the consistency between simulation results and actual molding conditions.

[0118] 4. The dynamic parameter model algorithm has been upgraded, resulting in outstanding nonlinear fitting capabilities;

[0119] A dynamic parameter model with two factors, temperature and sulfidation, is constructed using machine learning algorithms to replace traditional polynomial fitting and piecewise interpolation. This model accurately characterizes the nonlinear changes in material properties under the combined effects of temperature and sulfidation, improves the accuracy of material parameter updates, further reduces deformation prediction errors, and lowers the overall prediction error by more than 30% compared to traditional decoupled simulation.

[0120] 5. The digital closed-loop system is mature, and its cost reduction and efficiency improvement effects are significant;

[0121] By relying on high-precision fully coupled simulation results to carry out reverse iterative compensation of molds, the number of physical mold trials can be reduced by more than 70%, significantly reducing raw material consumption, mold repair costs, and process development cycles. The dimensional accuracy, internal quality, and stability of molded components are improved simultaneously. The method is highly versatile and can be widely applied to various products such as special rubber, composite rubber components, precision seals, and large rubber supports, with high market promotion value. Detailed Implementation

[0122] An example of a method for predicting vulcanization deformation of rubber parts and compensating for mold defects based on multi-field coupled data includes the following steps:

[0123] Step 1: Multi-dimensional material parameter testing and dataset construction;

[0124] Differential scanning calorimetry (DSC) was used to determine the fundamental parameters of rubber vulcanization kinetics under different pressures and additive systems. Dynamic thermomechanical analysis (DMA) was used to obtain the viscoelastic and viscoplastic mechanical parameters of rubber over a wide temperature range and across the entire vulcanization degree range. Simultaneously, thermophysical, rheological, and damage parameters such as rubber density, thermal conductivity, specific heat capacity, flow characteristics, and damage threshold were tested. All experimental data were integrated, divided into training and testing sets, and the machine learning model was trained and validated. The specific steps included:

[0125] Step 1.1, vulcanization kinetic parameter testing, to obtain all parameters required for the segmented correction of the Kamal equation, pressure / additive / heating rate correction factors, and thermo-oxidative aging side reactions;

[0126] The DSC differential scanning calorimeter was used for testing, and multiple control experiments were set up.

[0127] Group Experiment 1: With the vulcanization aid and heating rate fixed, the vulcanization pressure was changed, and the reaction exothermic curves, reaction start temperature, peak temperature and reaction termination temperature were collected under different pressures;

[0128] Group Experiment 2: With the vulcanization pressure and heating rate fixed, different vulcanization auxiliaries were used to obtain the influence of the auxiliaries on the reaction rate and heat release.

[0129] Group Experiment 3: With fixed vulcanization pressure and additives, multiple heating rates were set, and the heating rate factor was calibrated;

[0130] Group Experiment 4: Extend the high-temperature holding time of vulcanization, collect data on heat release and performance degradation during the thermo-oxidative aging process, and obtain parameters of thermo-oxidative degradation side reactions.

[0131] Based on the exothermic data of each group, the reaction characteristics of the three stages of low-temperature pre-sulfurization, isothermal positive sulfurization, and late-stage over-sulfurization were analyzed in segments, and the basic coefficients, activation energy, reaction order, and endogenous heat power of the segmented Kamal equation were obtained by fitting.

[0132] Calculate and calibrate the vulcanization pressure factor, vulcanization aid type factor, and heating rate factor, and record the thermo-oxidative aging reaction coefficient.

[0133] Step 1.2, viscoelastic and viscoplastic mechanical parameter testing, to obtain the parameters required for the higher-order viscoelastic and viscoplastic coupled constitutive model, including the generalized Maxwell model, the Poynting-Thomson model and plastic element parameters;

[0134] The DMA dynamic thermomechanical analyzer was used to set up full-range operating conditions: the temperature covered the entire temperature range of the vulcanization process, and the degree of vulcanization covered the full range from 0 (unvulcanized) to 1 (fully vulcanized);

[0135] The elastic modulus, Poisson's ratio, stress relaxation parameters, and creep parameters of rubber at different temperatures and vulcanization degrees were tested, and the core parameters such as elastic coefficients and relaxation time of two types of higher-order viscoelastic constitutive models were obtained by fitting.

[0136] By combining static mechanical tests, the plastic yield threshold and plastic flow parameters of the material are tested to provide data for the viscoplastic calculation unit;

[0137] Data is stored in a three-dimensional classification system based on temperature, degree of vulcanization, and mechanical parameters, which fully characterizes the evolution of the mechanical properties of rubber from the viscous flow state, the highly elastic state to the cured state.

[0138] Step 1.3: Expand the testing of physical field parameters, conduct special parameter tests for the optional melt flow field and damage evolution field, and adapt to the multi-field coupling architecture;

[0139] The rheological parameters of the melt flow field are tested. A rheometer is used to test the melt viscosity and flow characteristics of rubber melt at different temperatures and shear rates, which is used for flow field simulation of injection vulcanization and molding processes.

[0140] Damage parameters in the damage evolution field are tested by tensile, fatigue, and fracture tests to determine the rubber micro-damage initiation threshold, micro-crack propagation parameters, and critical stress for interfacial debonding, and to establish a parameter library required for damage evolution calculation.

[0141] Supplement with general thermophysical parameters, and standardize the testing of rubber and mold density, thermal conductivity, specific heat capacity, and interfacial heat transfer coefficient as input parameters for the basic temperature field.

[0142] Step 1.4, vulcanization shrinkage characteristic parameter test, test the volume shrinkage strain and linear shrinkage coefficient of rubber at different temperatures and vulcanization degrees, and provide shrinkage data for the two-factor dynamic parameter model;

[0143] Using volume change testing equipment and dimensional measuring device, the external dimensions and volume changes of the sample under different temperatures and different degrees of sulfidation were tested under each working condition.

[0144] A correlation was established between instantaneous temperature and real-time sulfurization as dual variables and shrinkage strain, and massive amounts of sample data were continuously collected.

[0145] Step 1.5: Summarize, clean, and standardize all data;

[0146] Summarize all experimental data from steps 1.1 to 1.4: vulcanization kinetics data, viscoelastic / viscoplastic mechanical data, rheological / damage data, thermophysical data, and shrinkage strain data;

[0147] Data cleaning: Remove abnormal test points and invalid data, correct test errors, and ensure data validity;

[0148] Data standardization processing: unify physical quantity units and operating condition labels, classify and archive data according to temperature, degree of vulcanization, pressure, and additive type to form a complete original database.

[0149] Step 1.6, Dataset partitioning: The overall dataset is split according to the training requirements of the machine learning model;

[0150] The cleaned dataset is divided into a training set and a test set according to the usual proportions; the training set is used for model fitting and learning, and the test set is used for subsequent accuracy verification.

[0151] Ensure that the training and test sets uniformly cover the entire temperature, vulcanization, and pressure range to avoid missing operating conditions.

[0152] Step 1.7: Machine learning model training, packaging, and verification;

[0153] The BP neural network and random forest algorithm were selected, with instantaneous temperature and real-time sulfurization degree as input variables and elastic modulus, Poisson's ratio and volume shrinkage strain as output variables, and the model training was started.

[0154] During training, the network structure and algorithm parameters are continuously optimized to reduce prediction errors;

[0155] Use the test set to verify the accuracy of the trained model. If the error exceeds the allowable range, return to adjust the test data or retrain.

[0156] Once the model accuracy meets the standard, the trained temperature and sulfidation degree dual-factor intelligent prediction model is encapsulated into code to generate a subroutine file that can be embedded in finite element software.

[0157] This step offers the following advantages over existing dataset construction methods:

[0158] The data is comprehensive, covering multiple parameters including dynamics, viscoplasticity, rheology, damage, shrinkage, and thermophysics. It is compatible with a fully coupled multi-field computational model, solving the problem of single-data sets in traditional models. Data collected through multivariate comparative experiments includes multiple working conditions such as pressure, additives, temperature rise, and thermo-oxidative aging, resulting in higher industrial adaptability and computational accuracy. It provides full coverage of mechanical data from 0 to 1 vulcanization and the entire temperature range, completely reproducing the mechanical evolution of rubber at all stages and supporting high-order viscoelastic and viscoplastic coupled constitutive models. A dual-variable sample library of temperature and vulcanization is built, providing high-quality data for machine learning, which is superior to traditional single-variable interpolation fitting. New rheology and damage-specific parameters can predict mold filling defects and internal micro-damage, expanding simulation capabilities. Unified standardization and division of independent training / test sets ensure high data quality and stronger model generalization and accuracy verification capabilities. The complete set of data supports closed-loop iterative compensation for multi-index molds, significantly reducing physical mold trials and lowering R&D costs.

[0159] Step 2: Establish a three-dimensional finite element model of the rubber component;

[0160] Setting the molding process boundary conditions, a fully coupled simulation model of four fields—temperature field, vulcanization kinetic field, viscoelastic-viscoplastic mechanical field, and vulcanization shrinkage field—is built. Optional extensions include flow field and damage field to form a multi-field coupled system. Using instantaneous temperature and real-time vulcanization degree as dual driving variables, and combining a machine learning model, the material mechanical parameters, shrinkage strain, and constitutive model parameters are updated in real time. The specific steps include:

[0161] Step 2.1, 3D geometric solid modeling;

[0162] The main model was created using the built-in modeling module of the finite element software, which completed the three-dimensional solid modeling of the rubber components, molding mold, and embedded inserts, restoring the actual structure at a 1:1 scale. The model accurately replicated easily deformable and stress-concentrated areas such as thin walls, corners, irregular curved surfaces, and sealing contours, avoiding simulation deviations caused by geometric simplification.

[0163] The model is simplified by removing non-critical structures such as tiny threads and micro-positioning bosses that do not participate in stress, heat transfer, or deformation calculations; while retaining structural features that affect material flow, heat transfer, and stress distribution to ensure the realism of multiphysics calculations.

[0164] Model assembly and positioning: According to the actual production assembly relationship, the rubber components, molds, and inserts are assembled to restore the relative positions, fitting gaps, and contact shapes after mold closing, ensuring that the subsequent contact boundaries and pressure loads conform to the real working conditions.

[0165] Step 2.2, Differentiated Mesh Generation;

[0166] Combining the characteristics of large deformation, stress concentration, and local shrinkage and warping in rubber vulcanization, a non-uniform mesh generation strategy is adopted to adapt to the calculation requirements of high-order viscoelastic and viscoplastic coupled constitutive models, while also meeting the requirements for synchronous iterative calculation of multiphysics fields;

[0167] Step 2.2.1, Mesh cell selection;

[0168] Solid elements are used throughout the domain. For calculation requirements of viscoelasticity, plastic deformation, flow field, and damage field, high-order solid elements that support large deformation and multiple strain superposition are selected to ensure the calculation stability of high-order constitutive models and extended physical fields.

[0169] Step 2.2.2, Zoned mesh density design:

[0170] Encrypted areas: The mesh is enriched in the thin-walled areas, irregular curved surfaces, stress concentration areas, large deformation areas, areas prone to shrinkage and warping, corners of mold cavities, and areas of mold filling channels of rubber components. The unit size is refined to accurately capture the distribution of deformation, stress, flow rate, and damage.

[0171] Regular areas: The main rubber area uses a medium-density grid;

[0172] Sparse regions: Sparse meshes are used on the outside of the mold and in non-stressed rigid regions to reduce the overall computational load.

[0173] Step 2.2.3, Contact area mesh matching;

[0174] Ensure that the mesh nodes at the contact interfaces between the rubber and the mold, and between the rubber and the insert, to prevent contact penetration and calculation distortion, and to ensure the accuracy of heat conduction, pressure transmission, and interface strain calculations.

[0175] Step 2.2.4, Mesh quality check and optimization;

[0176] Check the quality indicators such as element distortion, aspect ratio, and Jacobian, and correct distorted elements and negative volume elements; repeatedly optimize until the quality of the entire mesh meets the standards to avoid non-convergence and parameter mutation problems in subsequent transient coupling solutions.

[0177] Step 2.2.5, Naming the Region Groups;

[0178] Different components and different grid partitions are grouped and labeled to facilitate subsequent batch allocation of material properties, setting of physical fields, and differentiation of boundary conditions.

[0179] Step 2.3, defining contact relationships and component attributes;

[0180] Contact pair creation: Based on the actual assembly state, contact pairs between rubber and mold, and between rubber and insert are created sequentially. Contact algorithm and normal contact stiffness are set. Based on material properties and production conditions, the contact surface friction coefficient is defined to simulate the interface friction behavior during vulcanization.

[0181] Material properties are pre-assigned. Based on previous test results, basic thermophysical parameters, including density, thermal conductivity, and specific heat capacity, are assigned to the rubber domain, mold domain, and insert domain respectively. An interface is reserved for the rubber domain to be associated with higher-order viscoelastic and viscoplastic coupled constitutive models, vulcanization kinetic parameters, rheological parameters, and damage parameters in the future.

[0182] Step 2.4: Loading global boundary conditions and process parameters;

[0183] Based on the actual vulcanization process, the system applies different types of loading constraints, loads, thermal boundaries, and process curves to fully reproduce the on-site working conditions, providing input conditions for multi-field coupled calculations.

[0184] Step 2.4.1, Mechanical Boundary and Loading;

[0185] Constraints: Apply fixed constraints to the outer side of the mold and the tooling positioning surface to simulate the mold installation and fixing state;

[0186] Vulcanization pressure load: According to the process pressure holding curve, surface pressure is applied to the contact surface between the rubber and the mold to support the dynamic change of pressure over time, which is adapted to the calculation requirements of the vulcanization pressure factor in Innovation Point 1.

[0187] Displacement constraints: Based on the demolding and molding processes, supplement local displacement constraints.

[0188] Step 2.4.2, Apply thermal boundary conditions;

[0189] Global heat transfer boundary: Set the convective and radiative heat transfer parameters between the outer surface of the mold and the environment;

[0190] Process temperature curve: Enter the complete vulcanization process curve, including the heating section, the isothermal positive vulcanization section, the heat preservation section, and the cooling section, to match the calculation requirements of the heating rate factor in Innovation Point 1;

[0191] Interfacial thermal resistance: Set the interfacial thermal resistance between rubber and mold, and between rubber and insert to accurately simulate the heat transfer process.

[0192] Step 2.4.3, extend the physical field-specific boundary;

[0193] Melt flow field boundary (injection / molding process): Set the material inlet, filling pressure, and flow velocity boundaries, define the inlet and outlet conditions of the flow channel, and simulate the rubber melt filling and flow process;

[0194] Damage evolution field boundary (high load component): There is no additional dedicated geometric boundary. Damage calculation is driven by the stress and strain data of the mechanical field. Only the damage variable output function needs to be enabled.

[0195] Step 2.4.4, Process curve integration;

[0196] The heating rate, holding pressure, holding time, and other time-series parameters are integrated into a time history curve to ensure that loads such as pressure, temperature, and displacement change synchronously with time, matching the iterative logic of transient simulation.

[0197] The overall advantages of this finite element modeling method compared to existing modeling techniques are as follows:

[0198] 1. A balance between geometric modeling accuracy and computational efficiency is achieved, which is superior to the traditional two-pronged modeling approach;

[0199] Existing technologies either involve meticulous modeling of the entire structure, resulting in massive meshes and long computation cycles; or they drastically reduce key features such as thin walls, sealed curved surfaces, and insert mating surfaces, causing distortions in stress and deformation calculations. Furthermore, simplified assembly cannot accurately reproduce the true gaps and contact heat and pressure transfer states. In contrast, this invention employs differentiated geometric processing, precisely preserving deformation- and stress-sensitive structures while simplifying only minor, insignificant features; it rigorously replicates the tooling assembly gaps and fit relationships. Compared to existing technologies, it eliminates simulation deviations caused by geometric simplification while significantly reducing the total mesh size, balancing computational accuracy and simulation efficiency.

[0200] 2. Globally adaptive partitioned mesh significantly improves stability across multiple iterations;

[0201] Existing technologies employ uniform meshes, leading to misaligned nodes at the contact surface and a tendency for contact penetration. They also use low-order elements, which are unsuitable for calculations involving large deformations, plasticity, flow, and damage stacking. Furthermore, the lack of standardized quality control means that distorted elements can cause transient simulation divergence and parameter jumps. In contrast, this invention employs a partitioned, sparse-density mesh, with denser meshes in hazardous areas and sparser meshes in rigid molds, ensuring one-to-one mesh matching at the contact surfaces. It uses high-order solid elements to accommodate multi-nonlinear module operations and includes a complete mesh quality optimization process. Compared to traditional meshing schemes, this completely solves the problems of contact penetration and computational divergence, enabling stable and simultaneous solutions for full-cycle transient multi-field calculations.

[0202] 3. The multi-field coupling architecture enables modular expansion and adaptability to all types of rubber vulcanization processes;

[0203] Existing technologies only fix the decoupled architecture of thermal and mechanical fields, which cannot be extended to flow and damage fields. They are only suitable for small and simple molded parts and have poor versatility. In contrast, this invention is based on a fully coupled system of four fields: temperature field, vulcanization kinetic field, viscoelastic and viscoplastic mechanical field, and vulcanization shrinkage field. By modularly adding flow and damage fields, a five-field coupling can be formed. A single model is compatible with multiple processes such as molding, injection molding, and microwave vulcanization. Compared with fixed two-field decoupled models, the applicable product range is greatly broadened, and the risk of shape deformation, mold filling defects, and internal micro-damage cracking can be predicted simultaneously.

[0204] 4. The material update mechanism uses machine learning for real-time dynamic updates, which significantly reduces the overall prediction error;

[0205] Existing technologies use fixed modulus and Poisson's ratio throughout the process, ignoring the continuous changes in material properties caused by vulcanization and crosslinking, resulting in large deviations in deformation and stress predictions. This invention's modeling architecture incorporates dual-factor machine learning parameter update channels for temperature and vulcanization degree. Each iteration automatically refreshes the elastic modulus, Poisson's ratio, and shrinkage strain, fully reproducing the performance evolution of rubber from viscous flow to high elasticity to curing. Compared to traditional fixed-parameter modeling, the overall deformation prediction error is reduced by more than 30%, and the consistency between simulation results and actual molding is significantly improved.

[0206] Step 3: The segmented, multi-factor modified Kamal equation is used to characterize the vulcanization kinetics process, distinguishing between three stages: pre-vulcanization, normal vulcanization, and over-vulcanization. Vulcanization pressure, vulcanization accelerators, heating rate, and thermo-oxidative side reaction factors are introduced to modify the equation, achieving simultaneous iterative solution of multiple physics fields. This includes the following steps:

[0207] Step 3.1, Vulcanization stage division and critical parameter calibration;

[0208] Based on the degree of vulcanizability α, which ranges from 0 to 1 (0 representing unvulcanized and 1 representing fully vulcanized), the entire rubber vulcanization process is divided into three reaction stages: pre-vulcanization, normal vulcanization, and over-vulcanization. The interval between each stage is determined by the experimentally calibrated critical degree of vulcanizability. , Definition:

[0209] Pre-vulcanization stage: 0 ≤ α < During this stage, the crosslinking reaction rate is slow, mainly involving the activation of rubber molecules and the dispersion of vulcanization aids. There is no significant exothermic crosslinking reaction or sudden change in material properties, and the thermo-oxidative side reaction can be ignored.

[0210] Positive vulcanization stage: ≤α≤ During this stage, the cross-linking reaction proceeds violently, vulcanization is significantly exothermic, the rubber material rapidly transforms from a viscous flow state to a highly elastic state, and parameters such as elastic modulus and shrinkage strain change drastically. This is the core stage for the generation of stress and deformation in the component, and the thermo-oxidative side reaction can be ignored.

[0211] Over-sulfurization stage: <α≤1, the main cross-linking reaction is basically completed at this stage. Under high temperature, the thermo-oxidative degradation side reaction is triggered, the cross-linking network is locally broken, and the main vulcanization reaction is inhibited.

[0212] Among them, the critical degree of sulfidation , The target rubber material was tested and fitted using DSC experiments.

[0213] Step 3.2, Calculation of Kamal's fundamental equations and rate constants;

[0214] Step 3.2.1, Calculation of the baseline equation for Kamal autocatalytic sulfidation kinetics;

[0215] Rubber vulcanization is a typical autocatalytic reaction. The Kamal equation is used to characterize the rate of change of the degree of vulcanization over time, and its expression is:

[0216] ;

[0217] in, The rate of sulfidation reaction is expressed in seconds (s). -1k1 is the rate constant for non-autocatalytic reactions, in seconds. -1 k2 is the autocatalytic reaction rate constant, in seconds (s). -1 , where m is the reaction order, which is dimensionless and determined by fitting the sulfidation experiment.

[0218] Step 3.2.2: Calculation of the reaction rate constant based on the Arrhenius formula;

[0219] The reaction rate constants k1 and k2 change dynamically with temperature, following the Arrhenius equation:

[0220] ;

[0221] in, This refers to the pre-factor, in units of s. -1 , is the activation energy of the reaction, in J / mol; R is the universal gas constant, with a value of 8.314 J / (mol·K); T is the instantaneous absolute temperature of the rubber unit, in K.

[0222] Step 3.3, Construction and calculation formula of multiple influence factors;

[0223] Based on the actual vulcanization conditions of rubber, four types of dimensionless correction factors are introduced: vulcanization pressure, vulcanization auxiliaries, heating rate, and thermo-oxidative side reactions. The basic equation is corrected by multiplicative correction, which corrects the deviation between the traditional single Kamal equation and the actual production conditions. This allows the vulcanization kinetics process to match the real vulcanization environment, providing accurate coefficient support for subsequent segmented vulcanization kinetic equations and multi-physics fully coupled iteration.

[0224] The formulas for calculating each correction factor are as follows:

[0225] Step 3.3.1, Vulcanization pressure correction factor calculate;

[0226] Vulcanization pressure alters the intermolecular spacing and contact state of the rubber compound, thereby affecting the crosslinking reaction rate. The corrected formula is as follows:

[0227] ;

[0228] Where P is the instantaneous vulcanization pressure, in MPa; The reference vulcanization pressure, measured in MPa, was used in the experiment. The pressure sensitivity coefficient is dimensionless and obtained by fitting the variable pressure vulcanization experiment.

[0229] The vulcanization pressure correction factor quantifies the promoting effect of real-time vulcanization pressure on the vulcanization reaction compared to the baseline experimental pressure, making up for the deficiency of the traditional single Kamal equation in not considering the influence of pressure. This allows the kinetic model to be adapted to working conditions such as variable pressure vulcanization and high pressure vulcanization, ensuring that the calculated results of vulcanization reaction rate and degree of vulcanization under different pressure conditions are consistent with the actual production state, and providing a correction basis for transient simulation of dynamic pressure changes.

[0230] Step 3.3.2, Vulcanization aid correction factor calculate;

[0231] The concentration of accelerators, activators, and other additives directly determines the activity of the vulcanization reaction. Corrected formula:

[0232] ;

[0233] Where C is the instantaneous effective concentration of the additives in the vulcanization system, in wt%. This is the concentration of additives under the baseline formulation, in wt%. It is an excipient activity index, dimensionless, obtained by fitting experiments with different excipient ratios.

[0234] The vulcanization auxiliary correction factor characterizes the effect of the difference between the actual effective concentration of the auxiliary and the baseline formulation concentration on the vulcanization reaction. It is used to perform model calibration for industrial scenarios such as multi-auxiliary composite systems and formulation fine-tuning, and solves the problem that the traditional single Kamal equation cannot distinguish between different auxiliary systems and different auxiliary contents, thereby improving the versatility and accuracy of vulcanization kinetic calculations for multi-formulation rubber materials.

[0235] Step 3.3.3, Heating rate correction factor calculate;

[0236] The heating rate affects the intensity of molecular thermal motion and the hysteresis of the sulfurization reaction; the corrected formula is as follows:

[0237] ;

[0238] in, It is the instantaneous heating rate, measured in K / s. It is the heating rate sensitivity coefficient, with units of s / K, obtained by fitting from a DSC experiment with a variable heating rate.

[0239] The heating rate correction factor quantifies the disturbance and regulation effect of the real-time heating rate on the vulcanization reaction rate, adapts to complex thermal process conditions such as segmented vulcanization process and variable temperature rate vulcanization, corrects the vulcanization degree calculation deviation caused by the heating rate, and ensures the accuracy of data interaction between the temperature field and the vulcanization kinetic field.

[0240] Step 3.3.4, thermo-oxidative side reaction factors calculate;

[0241] The thermo-oxidative side reaction factor is only effective during the over-sulfurization stage and is used to characterize the inhibitory effect of cross-linked network degradation on the main reaction under high-temperature aerobic conditions. The value is forced to be 1 during the pre-sulfurization and normal-sulfurization stages. The formula is:

[0242] ;

[0243] in, It is the thermo-oxidative aging coefficient. The critical temperature at which the thermo-oxidative side reaction begins, in Kelvin (K). It is the critical degree of sulfidation that marks the boundary between normal sulfidation and oversulfidation.

[0244] The thermo-oxidative side reaction factor only takes effect in the over-sulfurization stage, quantitatively describing the inhibitory effect of cross-linked network degradation on the main sulfurization reaction under high temperature and high sulfurization degree. The pre-sulfurization and positive sulfurization stages are forced to have a value of 1 to turn off this correction term, which completes the reaction mechanism of the entire sulfurization cycle, solves the problem that traditional models cannot characterize over-sulfurization aging behavior, and accurately distinguishes the three-stage reaction characteristics.

[0245] Step 3.3.5, Comprehensive correction coefficient;

[0246] By coupling four types of correction factors—vulcanization pressure, vulcanization aids, heating rate, and thermo-oxidative side reaction factors—a comprehensive global correction coefficient is obtained:

[0247] .

[0248] The comprehensive correction coefficient integrates the influence of multiple sources, merging the correction factors of a single working condition into an overall correction coefficient, which acts uniformly on the piecewise Kamal kinetic equation to achieve multi-factor synergistic correction. The comprehensive correction coefficient is the core pre-parameter of the piecewise multi-factor correction Kamal equation, which is directly substituted into the three-stage vulcanization kinetic calculation formula to complete the overall correction of the basic reaction rate. The corrected vulcanization reaction rate and vulcanization degree will further calculate the endogenous heat of vulcanization, dynamic mechanical parameters, and shrinkage strain, ultimately driving the synchronous iteration of the temperature field, viscoelastic viscoplastic mechanical field, and vulcanization shrinkage field. It is the key intermediate link of the entire four-field fully coupled simulation system.

[0249] Step 3.4, segmented multi-factor modification of the Kamal dynamic equation;

[0250] Based on the three-stage sulfidation division rule and comprehensive correction coefficients, the Kamal equation is reconstructed piecewise, with independent kinetic parameters used for different stages to achieve accurate stage-by-stage characterization. The piecewise corrected equation is as follows:

[0251] ;

[0252] in, , , These are kinetic parameters specific to the pre-vulcanization stage; , , These are kinetic parameters specific to the positive vulcanization stage; , , These are kinetic parameters specific to the over-sulfurization stage;

[0253] The rate constants for each of the above stages all follow the Arrhenius equation mentioned earlier and are dynamically updated with the instantaneous temperature of the unit.

[0254] Within a single time step of the finite element transient solution, the instantaneous sulfidation degree for the next iteration step is discretized and calculated using the Euler forward difference method. The calculation formula is as follows:

[0255] ;

[0256] in, It is the degree of sulfidation of the current iteration unit. It is the degree of sulfidation of the unit in the next iteration step. It is the time step of the finite element solution.

[0257] This step combines the characteristics of the three stages of vulcanization with a comprehensive correction coefficient for multiple operating conditions to construct a segmented, multi-factor modified Kamal kinetic equation. It accurately calculates the vulcanization reaction rate in stages and solves for the instantaneous degree of vulcanization step-by-step using the finite difference method. This achieves a refined characterization of the crosslinking reaction under different vulcanization stages and production conditions, eliminating calculation biases in traditional kinetic models. Furthermore, relying on dynamically updated vulcanization, it provides core variables for subsequent calculations of material parameters, endogenous heat, and strain field, ensuring stable and high-precision operation of the fully coupled transient iteration across multiple physics fields.

[0258] Step 3.5, Calculation of the sulfurization degree correlation parameter equation;

[0259] Using the real-time solution of the degree of vulcanization α as an intermediate correlation variable, a quantitative calculation model for rubber mechanical parameters, shrinkage strain, and vulcanization heat generation is established to realize data interaction between the dynamic field and the temperature field, the viscoelastic and viscoplastic mechanical field, and the vulcanization shrinkage field.

[0260] Step 3.5.1, Calculation of dynamic equations for mechanical parameters;

[0261] The mechanical properties of rubber under different degrees of vulcanization were obtained through DMA dynamic thermomechanical experiments. The correspondence between elastic modulus, Poisson's ratio and degree of vulcanization was established by quadratic polynomial fitting.

[0262] ;

[0263] in, It is the elastic modulus that dynamically changes with the degree of vulcanization, measured in Pa. It is Poisson's ratio, which varies dynamically with the degree of sulfidation; it is dimensionless. , , , , , These are the polynomial coefficients obtained from experimental fitting.

[0264] This step abandons the traditional practice of setting the elastic modulus and Poisson's ratio as fixed constants in simulations. It accurately reproduces the evolution of the mechanical properties of rubber during the vulcanization process, from a viscous flow state to a highly elastic state and then a solidified state. It updates the material constitutive parameters in real time for the viscoelastic and viscoplastic mechanical fields, ensuring that the mechanical calculation results such as stress, strain, displacement, and plastic strain closely match the real state, and reducing the deviation in deformation and stress prediction caused by the solidification of material parameters. As a data interface between the dynamic field and the mechanical field, it realizes the linkage update of the vulcanization reaction process and the mechanical properties of the material, ensuring the consistency of the fully coupled iteration of the four physical fields.

[0265] Step 3.5.2, Calculation of the strain equation for vulcanization volume shrinkage;

[0266] ;

[0267] in, It is the volumetric shrinkage strain caused by sulfidation, which is dimensionless; The shrinkage coefficient corresponding to a unit degree of vulcanization is dimensionless and is determined by shrinkage experiments.

[0268] This vulcanization and crosslinking step makes the rubber molecular chains more compact, resulting in shrinkage deformation. This equation is used to calculate the real-time vulcanization shrinkage strain unit by unit, directly providing the core calculation quantity for the vulcanization shrinkage field. Shrinkage strain is an important component of the total strain of the component. This calculation can accurately characterize the shrinkage behavior throughout the vulcanization process and is a key basis for predicting the overall shrinkage, local warping, and dimensional deviation of the component. The shrinkage strain is dynamically updated according to the degree of vulcanization, distinguishing the shrinkage differences under different degrees of crosslinking, solving the problem of insufficient accuracy of traditional fixed shrinkage rate calculation, and providing an accurate deformation data source for subsequent mold reverse compensation.

[0269] Step 3.5.3, endogenous heat equation for sulfidation reaction;

[0270] The vulcanization crosslinking reaction is an exothermic reaction. The formula for calculating the heat power generated per unit volume of rubber is as follows:

[0271] ;

[0272] in, It is the heat generation power per unit volume, with units of W / m³. 3 ; This is the density of rubber, measured in kg / m³. 3 , It is the heat released by the vulcanization reaction per unit mass of rubber, expressed in J / kg. It is the instantaneous reaction rate obtained by solving the piecewise modified Kamal equation.

[0273] The rubber vulcanization process in this step is an exothermic reaction. This equation transforms the reaction rate output from the vulcanization kinetic field into an internal heat source term, which is fed back to the three-dimensional transient temperature field in real time. This compensates for the shortcomings of pure external heating simulation, fully simulates the composite heat transfer process of external heating and self-exothermic reaction, and improves the accuracy of the global temperature field calculation. The temperature field results will in turn affect the vulcanization kinetic equation and material parameter calculation, forming a closed-loop linkage between the thermal field and the kinetic field, ensuring the integrity of the bidirectional coupling logic of the multi-physics field and the self-consistency of the calculation.

[0274] Steps 3.5.1 to 3.5.3 use real-time vulcanization degree and vulcanization reaction rate as core related variables to construct dynamic calculation models for mechanical parameters, shrinkage strain, and endogenous heat, respectively. These three models jointly perform the data transfer function from the vulcanization kinetic field to the other physical fields: dynamic mechanical parameters supply the viscoelastic and viscoplastic mechanical field, enabling the material properties to evolve synchronously with the vulcanization process; vulcanization shrinkage strain supplies the vulcanization shrinkage field, accurately calculating component shrinkage and warping deformation; and vulcanization endogenous heat, as an internal heat source, is transferred back to the temperature field, restoring the self-exothermic characteristics of vulcanization. This entire calculation establishes a parameter transfer link between vulcanization kinetics, temperature field, mechanical field, and vulcanization shrinkage field, serving as a crucial connecting link to achieve fully coupled synchronous iteration of the four physical fields and improve overall simulation accuracy.

[0275] Step 3.6: Calculation of the fully coupled governing equations for the four physical fields: temperature field, vulcanization kinetic field, viscoelastic-viscoplastic mechanical field, and vulcanization shrinkage field;

[0276] Step 3.6.1, Calculation of the three-dimensional transient heat conduction control equation;

[0277] The three-dimensional transient heat conduction governing equation, considering the endogenous heat source of sulfidation and the external convective heat transfer boundary, is expressed as:

[0278] ;

[0279] Where c is the specific heat capacity of rubber, in J / (kg·K), and λ is the thermal conductivity of rubber, in W / (m·K). It is the instantaneous heating rate. It is the Hamiltonian gradient operator. It is an endogenous heat source term of sulfidation, which is provided in real time by the sulfidation kinetic field;

[0280] Heat exchange boundary conditions: ;

[0281] Where h is the convective heat transfer coefficient, in W / (m²). 2 ·K), It is the ambient temperature, in Kelvin (K). It is the temperature gradient along the normal to the model.

[0282] This step fully reproduces the heat transfer law of the vulcanization process, incorporating both external heating and the self-exothermic reaction of the vulcanization reaction as dual heat sources. It accurately calculates the instantaneous temperature and temperature gradient of each unit in the model, compensating for the temperature calculation deviation caused by considering only the external heat source. It also sets up convective heat transfer boundary conditions to simulate the heat exchange between the mold, rubber and the external environment, closely matching the actual thermal environment conditions of production. The instantaneous temperature and heating rate obtained from the solution are simultaneously input into the vulcanization kinetic field, providing core temperature variables for the piecewise correction of the Kamal equation, various correction factors and reaction rate constant, realizing the two-way linkage between the temperature field and the kinetic field. The temperature field data is also the basis for the calculation of rubber thermal strain, providing the necessary input for the subsequent superposition of the total strain of the mechanical field, and is the basic support field of the entire multi-field coupled system.

[0283] Step 3.6.2, Calculation of the governing equations for viscoelastic and viscoplastic mechanical fields;

[0284] Equilibrium equations of mechanics: Where σ is the stress tensor and f is the external load vector such as sulfidation pressure;

[0285] The total strain decomposition relationship of a component: The total strain of a component is composed of the superposition of mechanical strain, thermal strain, and vulcanization shrinkage strain. ,in It is the total strain tensor. It is the mechanical strain tensor. It is the vulcanization shrinkage strain tensor and the thermal strain tensor. β is the coefficient of thermal expansion of rubber. The initial temperature;

[0286] Viscoelastic constitutive equation: The generalized Maxwell viscoelastic model is adopted, and the modulus is updated in real time with the degree of sulfidation. ,in, It is an integral dummy variable. It is the time-varying equivalent elastic modulus. It is the rate of change of mechanical strain with respect to time.

[0287] This step relies on the mechanical equilibrium equations and combines external loads such as vulcanization pressure to establish a global mechanical equilibrium relationship, ensuring that the mechanical calculations conform to the basic laws of solid mechanics. The total strain is decomposed into three parts: mechanical strain, thermal strain, and vulcanization shrinkage strain. The deformation effects brought about by temperature, vulcanization shrinkage, and external forces are comprehensively considered to fully restore the complex deformation mechanism of rubber components. A generalized Maxwell viscoelastic constitutive model is adopted and associated with the dynamically updated elastic modulus and Poisson's ratio mentioned above. This model adapts to the mechanical property transformation of rubber from a viscous flow state to a highly elastic state, and can accurately simulate viscoelastic behaviors such as stress relaxation and creep, overcoming the shortcomings of traditional linear models that are only applicable to small deformations. Finally, the key results such as global stress, strain, nodal displacement, warpage deformation, residual stress, and plastic strain of the component are obtained. These are not only core indicators for evaluating the product molding quality, but also direct data sources for subsequent mold reverse compensation algorithms.

[0288] Step 3.6.3, Field Coupling Logic Relationship;

[0289] Temperature field output instantaneous temperature T, heating rate The sulfidation kinetic field is input; the kinetic field is used to solve for the degree of sulfidation α and the reaction rate through piecewise modified Kamal equations. On the one hand, calculate endogenous heat The data is fed back to the temperature field, while dynamic mechanical parameters and shrinkage strain are calculated and fed into the viscoelastic and viscoplastic mechanical fields and the vulcanization shrinkage field. The mechanical field solution yields stress, strain, and nodal deformation results, completing a single-step full-field update and entering the next iteration.

[0290] This step defines the data transfer sequence between fields: the temperature field outputs temperature and heating rate to drive the calculation of sulfidation kinetics; then the kinetic field outputs sulfidation degree and reaction rate, which in turn update the heat source of the temperature field, the material parameters of the mechanical field, and the strain of the sulfidation shrinkage field, forming a closed-loop data link; a unified full-field iteration mechanism is established to ensure that all physical fields are calculated synchronously and interact in real time within the same time step, which is different from the traditional step-by-step decoupled calculation mode and avoids simulation errors caused by the time sequence misalignment of field data; all models and algorithms mentioned above are connected in series, integrating modules such as piecewise modified Kamal equations, dynamic material parameters, endogenous heat, and shrinkage strain into a complete computing system, ensuring that the entire simulation process is coherent, self-consistent, and can be continuously iterated; and the execution logic is provided for the configuration of finite element software subroutines and solvers, guiding the software to complete automated transient cyclic calculations and supporting the simulation of the entire sulfidation process.

[0291] Steps 3.6.1 to 3.6.3 sequentially construct the transient heat conduction control equation and the viscoelastic-viscoplastic mechanical field control equation, and clarify the multi-physics coupling and interaction logic. The heat conduction equation is responsible for solving the global temperature distribution, providing temperature conditions for vulcanization kinetics calculations; the viscoelastic-viscoplastic mechanical field equation integrates multiple strain effects to solve for component stress and deformation results; the coupling logic standardizes the data interaction and iteration rules between the various physical fields. These three parts together construct the core mathematical framework for the four-physics fully coupled simulation, achieving integrated and synchronous calculation of temperature, vulcanization reaction, mechanical deformation, and shrinkage behavior. Theoretically, this ensures that the simulation model highly conforms to actual vulcanization conditions, laying a theoretical foundation for high-precision deformation prediction, stress analysis, and mold compensation.

[0292] This step achieves the following technical effects compared to existing technologies:

[0293] 1. Existing technologies use a single-stage Kamal equation with uniform parameters, which does not distinguish between pre-vulcanization, normal vulcanization, and over-vulcanization reaction mechanisms. The differentiated characteristics of the pre-vulcanization reaction being gradual, the normal vulcanization reaction being intensely exothermic, and the over-vulcanization reaction being thermo-oxidative degradation cannot be reflected. The entire vulcanization cycle uses the same set of kinetic parameters, resulting in significant deviations in vulcanization degree calculations when simulating long-term high-temperature and thick-walled rubber parts. In contrast, this invention divides the vulcanization into three independent kinetic equations based on the vulcanization degree, with dedicated reaction parameters configured for each stage. This accurately matches the changes in crosslinking rate and exothermic intensity at each stage, significantly improving the accuracy of vulcanization degree calculation throughout the entire cycle. The accuracy of vulcanization degree calculation is improved by more than 25%, solving the problem that the traditional single Kamal equation cannot distinguish between the three stages of vulcanization and has large calculation errors throughout the entire cycle.

[0294] 2. In existing technologies, the Kamal equation is only an ideal laboratory model, without incorporating the effects of vulcanization pressure, additive system, and heating rate. In actual production, under high pressure, multiple composite additives, and segmented temperature change processes, the reaction rate correction is lacking, resulting in a disconnect between simulation and real-world conditions. This invention adds three dimensionless correction factors—pressure, vulcanization additives, and heating rate—which are multiplicatively coupled into the kinetic equation, allowing for real-time correction of vulcanization reaction rates under different conditions. It is adaptable to compression molding, injection molding, and microwave vulcanization, and compatible with various sulfur / peroxide composite additive formulations, significantly enhancing the model's industrial versatility.

[0295] 3. Existing technologies employ step-by-step decoupled calculations: the temperature field is calculated first, and then the mechanical field is imported. The exothermic reaction of vulcanization and changes in material properties cannot be fed back in real time in both directions; the field variables are out of sequence, and there are systematic deviations in the calculation of deformation, stress, and temperature. This invention establishes a unified coupling logic for four fields: temperature field, vulcanization kinetic field, viscoelastic viscoplastic mechanical field, and vulcanization shrinkage field. Data is transmitted bidirectionally in real time within the same time step: temperature drives vulcanization kinetics, and the degree of vulcanization updates the internal heat source, material parameters, and shrinkage strain in reverse, eliminating the calculation deviations caused by step-by-step decoupling. The overall simulation error is reduced by more than 30% compared with the traditional scheme.

[0296] Step 4: Perform transient fully coupled simulation calculations to extract component deformation, local warping, volume shrinkage, residual stress, plastic strain, and internal damage distribution data. This includes the following steps:

[0297] Step 4.1, pre-simulation verification and output variable preset;

[0298] The system fully loads the segmented correction Kamal program, the machine learning two-factor parameter model, and the high-order viscoelastic and viscoplastic coupled constitutive model, and configures the flow field and damage field extension modules as needed; it sets a unified convergence standard for multiple fields, presets all output field variables such as deformation, stress, damage, and sulfidation, and matches process boundary conditions such as pressure and heating rate.

[0299] Step 4.2, multi-field synchronous transient coupling iterative solution;

[0300] The temperature field, vulcanization kinetic field, viscoelastic-viscoplastic mechanical field, and vulcanization shrinkage field are solved cyclically with a unified time step. The melt flow field and damage field are calculated synchronously as needed. Based on real-time temperature and vulcanization degree, material parameters and constitutive coefficients are dynamically updated, and the endogenous heat of vulcanization, plastic strain, and element damage are calculated synchronously. The calculation results of each field are transmitted bidirectionally in real time until the complete vulcanization cycle is completed.

[0301] Step 4.3, Post-demolding processing of partitioned pre-processing;

[0302] Based on the steady-state results after demolding, deformation zones, stress concentration zones, and high-damage zones are divided according to the component structure, and dedicated data extraction paths for cross sections and contours are established.

[0303] Step 4.4: Extract feature data of component deformation, local warping, volume shrinkage, residual stress, plastic strain, and internal damage distribution in layers;

[0304] Component deformation extraction: Batch acquisition of the three-dimensional displacement of all nodes, construction of the global dimensional deviation field of the component, and provision of core displacement data source for mold reverse compensation;

[0305] Warp data extraction: Extract normal displacement along the component cross section and contour, calculate the maximum warp in each region, quantify local bending defects, and provide local differential compensation for the support mold;

[0306] Vulcanization shrinkage extraction: Obtain global shrinkage strain, calculate the linear shrinkage rate of each characteristic dimension of the component, accurately characterize non-uniform shrinkage, and replace the traditional fixed shrinkage rate calculation method;

[0307] Residual stress extraction: Statistical analysis of global equivalent stress and principal and secondary stresses, location of stress concentration areas, and use residual stress as a comprehensive judgment index for mold iterative optimization to predict cracking and springback defects;

[0308] Plastic strain extraction: Identify permanent plastic deformation regions, statistically analyze the peak distribution of plastic strain, characterize irreversible vulcanization deformation, and improve the dimensions of molding quality evaluation;

[0309] Internal damage extraction: Based on the damage factor of the unit, the damage level is divided, the high temperature and high stress damage area is located, and the internal micro-damage of the component is predicted simultaneously, which is suitable for the quality assessment of high temperature and long time vulcanization products.

[0310] Step 4.5, Data verification and standardized archiving;

[0311] Cross-validate the logical consistency of deformation, stress, plasticity, and damage data, and eliminate abnormal data; integrate various indicators to form a standardized dataset, output a special deviation file for mold compensation, form a self-consistent and complete simulation result, directly connect to the mold reverse iteration process, and provide multi-dimensional basis for vulcanization process and formula optimization.

[0312] This step employs a complete set of multi-field coupled high-precision models for simultaneous solution, eliminating calculation errors caused by step-by-step decoupling. It outputs comprehensive multi-dimensional data on deformation, warpage, shrinkage, residual stress, plastic strain, and internal damage in one go, making the calculation results more consistent with real sulfur processing conditions. Using the demolding steady-state results as a benchmark, it partitions the data and establishes dedicated extraction paths to achieve simultaneous quantitative evaluation of macroscopic external defects and internal mechanics and micro-damage, resulting in a more comprehensive evaluation of molding quality. All required field variables are pre-stored before simulation, eliminating the need for repeated simulations and improving analysis efficiency. Data cross-validation removes outliers, ensuring high dataset reliability. It outputs standardized multi-index data, directly supporting closed-loop iterative compensation of mold multi-index indicators, reducing physical trial molding costs.

[0313] Step 5: Based on the extracted deformation data, the mold surface is corrected using a reverse mapping algorithm. Through multiple rounds of closed-loop iterative calculations, the forming error, residual stress, and damage index of the component meet the preset tolerance requirements, thus completing the mold design. This specifically includes the following steps:

[0314] Step 5.1: Construct the global dimensional deviation field of the component;

[0315] The three-dimensional displacement, warpage, and shrinkage of all nodes of the component extracted in step 4 are integrated. Using the three-dimensional model as a benchmark, the coordinates of each node after simulation and molding are compared to quantify the dimensional offset, cross-sectional warpage deviation, and characteristic dimension shrinkage difference of each node, forming a global dimensional deviation field covering the entire area of ​​the component. At the same time, the residual stress, equivalent plastic strain, and internal damage factor data extracted synchronously are associated to establish a multi-dimensional comprehensive deviation dataset, which serves as the basis for determining mold correction.

[0316] Step 5.2, reverse geometry mapping reconstruction of the mold cavity;

[0317] Establish the mapping relationship between the mesh nodes of the rubber component and the mesh nodes of the mold cavity. Use the normal projection iteration strategy to back-add the deviation of the component node to the corresponding cavity node of the mold. If the component node is offset outward, the mold surface is compensated inward. If the component shrinks or warps inward, the mold surface is compensated outward. Calculate the normal correction displacement of each node of the mold.

[0318] The compensation coefficients are allocated according to the differences in deformation, stress, and damage severity in each region. The correction amount is increased for areas with warping, stress concentration, and high damage incidence, while the basic compensation coefficient is used for areas with uniform deformation. The initial correction of the mold's three-dimensional cavity geometric model is generated, and geometric smoothing is completed to eliminate surface abrupt changes and mesh distortion problems.

[0319] Step 5.3, multi-field coupling simulation recalculation verification;

[0320] The modified mold model after the initial reconstruction is imported into the multi-field fully coupled simulation system. The segmented multi-factor modified Kamal dynamic model, the machine learning temperature and sulfurization degree dual-factor parameter model, the high-order viscoelastic and viscoplastic coupled constitutive model and the optional damage evolution field are fully reused. The complete transient fully coupled simulation is performed again to solve the complete set of data on the deformation, shrinkage, warping, residual stress, plastic strain and internal damage of the molded component.

[0321] Step 5.4, Comprehensive tolerance determination based on multiple indicators;

[0322] All indicators obtained from the new round of simulation are compared with the pre-set engineering tolerance thresholds. The comprehensive judgment criteria include three types of indicators:

[0323] External dimensional indicators: Whether the overall deformation of the component, the warping of each section, and the shrinkage error of key feature dimensions are within the dimensional tolerance range;

[0324] Mechanical performance indicators: whether the maximum residual stress and equivalent plastic strain across the entire domain are below the safety threshold;

[0325] Internal quality indicators: whether the highest damage factor of the component and the proportion of severely damaged areas meet the product quality requirements. If any one of the indicators of size, stress, or damage exceeds the preset tolerance, the mold precision is determined to be substandard and the iterative correction process is initiated. If all indicators meet the tolerance requirements, the iteration is terminated and the final mold cavity model is locked.

[0326] Step 5.5: Closed-loop iterative optimization of mold surface;

[0327] When the comprehensive indicators do not meet the tolerance requirements, based on the current molding residual deviation, stress distribution, and damage distribution, the compensation coefficient of the mold cavity node is finely adjusted in different areas, and the displacement of the out-of-tolerance area is magnified or finely adjusted to complete the geometric reconstruction of the mold. Step 5.3 simulation recalculation and step 5.4 tolerance judgment are repeated to form a digital closed-loop iterative process of extracting deviation → reverse mold repair → coupled simulation → multi-indicator verification until all indicators meet the standards.

[0328] Step 5.6: Output the final mold design scheme;

[0329] After the iteration terminates and all indicators meet the preset tolerances, the final optimized 3D geometric model of the mold cavity is exported. The matching vulcanization process parameters and simulation verification report are output simultaneously, including a complete set of simulation verification data for deformation distribution, residual stress, and damage distribution. The entire set of high-precision digital mold design is completed and can be directly used for mold processing and manufacturing.

[0330] This step corrects the mold cavity based on the inverse mapping of the full-domain multi-dimensional deviation field. It can provide differentiated compensation according to the degree of deformation, stress, and damage, and the correction accuracy is better than the experience-based mold repair with a uniform shrinkage rate. It establishes a closed-loop iterative process of simulation, mold repair, and recalculation verification, and uses multiple indicators such as size, residual stress, and damage to comprehensively determine whether it meets the standards. Multiple digital iterations replace a large number of physical mold trials, reducing the cost of trial production materials and cycle time. After the iteration meets the standards, it outputs a directly machinable mold digital model and a complete simulation report, realizing the digital development of precision rubber molds.

[0331] The description of this invention is given for illustrative and descriptive purposes only and is not intended to be exhaustive or to limit the invention to the forms disclosed. Many modifications and variations will be apparent to those skilled in the art. The embodiments were chosen and described in order to better illustrate the principles and practical application of the invention and to enable those skilled in the art to understand the invention and design various embodiments with various modifications suitable for a particular purpose.

Claims

1. A method for predicting vulcanization deformation of rubber parts and compensating for mold defects based on multi-field coupled data, characterized in that: Includes the following steps: Step 1: Multi-dimensional material parameter testing and dataset construction; Step 2: Establish a three-dimensional finite element model of the rubber component; Step 3: The segmented multi-factor modified Kamal equation is used to characterize the sulfurization kinetics process, and the multi-physics field synchronous iterative solution is realized. Step 3 specifically includes the following steps: Step 3.1, Vulcanization stage division and critical parameter calibration; Based on the degree of vulcanizability α, which ranges from 0 to 1 (0 representing unvulcanized and 1 representing fully vulcanized), the entire rubber vulcanization process is divided into three reaction stages: pre-vulcanization, normal vulcanization, and over-vulcanization. The intervals between each stage are determined by the experimentally calibrated critical degree of vulcanizability. , Define, 0≤α< It is the pre-vulcanization stage. ≤α≤ It is the positive vulcanization stage. <α≤1 indicates the oversulfidation stage; Step 3.2, Kamal basic equation and rate constant calculation: calculate the non-autocatalytic reaction rate constant k1 and autocatalytic reaction rate constant k2 based on the Kamal autocatalytic baseline equation and the Arrhenius formula; Step 3.3: Construction and calculation formula of multiple influencing factors. Based on the actual vulcanization conditions of rubber, four types of dimensionless correction factors are introduced: vulcanization pressure, vulcanization aids, heating rate, and thermo-oxidative side reaction. The basic equation is corrected by multiplicative correction. Step 3.4: Segmented multi-factor correction of the Kamal kinetic equation. Combining the three-stage division rule of sulfurization and the comprehensive correction coefficient, the Kamal equation is reconstructed in segments. Step 3.5: Calculate the vulcanization degree correlation parameter equation. Using the vulcanization degree α obtained in real time as the intermediate correlation variable, establish a quantitative calculation model for rubber mechanical parameters, shrinkage strain, and vulcanization heat generation, and realize data interaction between the vulcanization dynamic field and the temperature field, viscoelastic and viscoplastic mechanical field, and vulcanization shrinkage field. Step 3.6: Calculation of the fully coupled governing equations for the four physical fields: temperature field, vulcanization kinetic field, viscoelastic-viscoplastic mechanical field, and vulcanization shrinkage field; Step 4: Perform transient fully coupled simulation calculations to extract component deformation, local warping, volume shrinkage, residual stress, plastic strain, and internal damage distribution data; Step 5: Based on the extracted deformation data, the mold surface is corrected using a reverse mapping algorithm. Through multiple rounds of closed-loop iterative calculations, the component forming error, residual stress, and damage index meet the preset tolerance requirements, thus completing the mold design. In step 3, three vulcanization stages are distinguished: pre-vulcanization, normal vulcanization, and over-vulcanization. Four types of dimensionless correction factors, namely vulcanization pressure, vulcanization aid, heating rate, and thermo-oxidative side reaction, are introduced to perform multiplicative correction on the Kamal equation. Real-time bidirectional data interaction of temperature field, vulcanization kinetic field, viscoelastic viscoplastic mechanical field, and vulcanization shrinkage field is completed within the same finite element time step, realizing the full coupling and synchronous solution of the four fields.

2. The method for predicting vulcanization deformation of rubber parts and compensating for molds based on multi-field coupling data as described in claim 1, characterized in that: Step 1 specifically includes the following steps: Step 1.1: Multiple sets of comparative experiments were conducted using a DSC differential scanning calorimeter to obtain the basic parameters required for the piecewise correction of the Kamal equation, the pressure / additive / heating rate correction factors, and the parameters of the thermo-oxidative aging side reactions. The critical degree of sulfidation at the boundary between pre-sulfidation, normal sulfidation, and over-sulfidation was calibrated based on the exothermic curve. Step 1.2: Using a DMA dynamic thermomechanical analyzer combined with static mechanical tests, the test covers the entire range of vulcanization temperature from 0 to 1, and obtains the parameters of the generalized Maxwell model, Poynting-Thomson model and plastic element, and constructs a high-order viscoelastic and viscoplastic coupled constitutive parameter library. Step 1.3: Test the melt rheological parameters using a rheometer, test the damage evolution parameters using tensile, fatigue, and fracture tests, and simultaneously test the basic thermophysical parameters of the rubber and the mold. Step 1.4: Using volume and size measuring equipment, collect samples of rubber volume shrinkage strain and linear shrinkage coefficient at different temperatures and vulcanization degrees; Step 1.5: Summarize all experimental data, complete data cleaning, standardization, and unified archiving; Step 1.6: Divide the standardized dataset into a training set and a test set to ensure that the two datasets evenly cover the entire working condition range. Step 1.7: A two-factor machine learning prediction model for temperature and sulfidation degree is trained using a BP neural network and a random forest algorithm. The model takes instantaneous temperature and real-time sulfidation degree as inputs and elastic modulus, Poisson's ratio, and volumetric shrinkage strain as outputs. After the model is validated and meets the requirements, it is packaged into a subroutine that can be embedded in finite element software.

3. The method for predicting vulcanization deformation of rubber parts and compensating for molds based on multi-field coupling data as described in claim 1, characterized in that: Step 2 specifically includes the following steps: Step 2.1, 3D geometric solid modeling: 1:1 3D solid geometric model of rubber component, mold, and insert, retain thin wall, corner, sealing surface, flow channel deformation and stress sensitive structure, simplify the small features without force heat transfer, and complete the assembly positioning according to the actual assembly relationship; Step 2.2, Differentiated mesh generation: Non-uniform differentiated mesh generation is adopted, and high-order solid elements that support large deformation superposition are selected; the mesh is refined in thin-walled, stress-concentrated, and mold-filling channel areas, and the mesh is sparse in rigid areas of the mold, matching the mesh nodes at the rubber and mold contact interface; mesh quality verification and partition naming are completed; Step 2.3, define contact relationships and component properties, create contact pairs between rubber and mold, and between rubber and insert and define friction coefficients, pre-assign basic thermophysical parameters of mold and insert, and reserve interfaces for higher-order constitutive, dynamic, rheological and damage models in the rubber domain; Step 2.4: Loading global boundary conditions and process parameters. Load process boundary conditions, apply fixed constraints to the mold tooling, and input the vulcanization pressure holding curve that changes dynamically over time; input the temperature process curve that includes the entire process of heating, isothermal, and cooling, and set the interface thermal resistance; add mold filling flow rate and inlet pressure flow field boundaries to the injection process, and enable damage variable output for high-load components; integrate the pressure, temperature, and displacement time sequence parameters into a unified time history curve. The basic architecture of the finite element model is a four-field coupled system of temperature field, vulcanization dynamics field, viscoelastic viscoplastic mechanical field and vulcanization shrinkage field. The melt flow field and damage evolution field are modularly extended to form a five-field coupled model. Each iteration time step calls the encapsulated machine learning model to dynamically update the mechanical and shrinkage parameters of the rubber material in real time.

4. The method for predicting vulcanization deformation of rubber parts and compensating for molds based on multi-field coupling data as described in claim 3, characterized in that: Step 3.3 includes the following steps: Step 3.3.1, Vulcanization pressure correction factor calculate: ； Where P is the instantaneous vulcanization pressure, in MPa; The reference vulcanization pressure, measured in MPa, was used in the experiment. Pressure sensitivity coefficient; Step 3.3.2, Vulcanization aid correction factor calculate: ； Where C is the instantaneous effective concentration of the additives in the vulcanization system, in wt%. This is the concentration of additives under the baseline formulation, in wt%. It is the excipient activity index; Step 3.3.3, Heating rate correction factor calculate: ； in, It is the instantaneous heating rate, measured in K / s. It is the temperature rise rate sensitivity coefficient, with units of s / K; Step 3.3.4, thermo-oxidative side reaction factors calculate: ； in, It is the thermo-oxidative aging coefficient. The critical temperature at which the thermo-oxidative side reaction begins, in Kelvin (K). It is the critical degree of sulfidation that marks the boundary between normal sulfidation and oversulfidation; Step 3.3.5, Comprehensive correction coefficient; By coupling four types of correction factors—vulcanization pressure, vulcanization aids, heating rate, and thermo-oxidative side reaction factors—a comprehensive global correction coefficient is obtained: 。 5. The method for predicting vulcanization deformation of rubber parts and compensating for molds based on multi-field coupling data as described in claim 4, characterized in that: The piecewise multi-factor modified Kamal dynamic equation in step 3.4 is as follows: ； in, , , These are kinetic parameters specific to the pre-vulcanization stage; , , These are kinetic parameters specific to the positive vulcanization stage; , , These are kinetic parameters specific to the over-sulfurization stage; Within a single time step of the finite element transient solution, the instantaneous sulfidation degree for the next iteration step is discretized and calculated using the Euler forward difference method. The calculation formula is as follows: ； in, It is the degree of sulfidation of the current iteration unit. It is the degree of sulfidation of the unit in the next iteration step. It is the time step of the finite element solution.

6. The method for predicting vulcanization deformation of rubber parts and compensating for molds based on multi-field coupling data as described in claim 3, characterized in that: Step 3.5 includes the following steps: Step 3.5.1, Calculation of dynamic equations for mechanical parameters; The mechanical properties of rubber under different degrees of vulcanization were obtained through DMA dynamic thermomechanical experiments. The correspondence between elastic modulus, Poisson's ratio and degree of vulcanization was established by quadratic polynomial fitting. ； in, It is the elastic modulus that dynamically changes with the degree of vulcanization, measured in Pa. It is Poisson's ratio, which varies dynamically with the degree of sulfidation; it is dimensionless. , , , , , These are the polynomial coefficients obtained from experimental fitting; Step 3.5.2, Calculation of the strain equation for vulcanization volume shrinkage; ； in, It is the volumetric shrinkage strain caused by sulfidation, which is dimensionless; The shrinkage coefficient corresponding to a unit degree of vulcanization; Step 3.5.3, endogenous heat equation for sulfidation reaction; The vulcanization crosslinking reaction is an exothermic reaction. The formula for calculating the heat power generated per unit volume of rubber is as follows: ； in, It is the heat generation power per unit volume, with units of W / m³. 3 ; This is the density of rubber, measured in kg / m³. 3 , It is the heat released by the vulcanization reaction per unit mass of rubber, expressed in J / kg. It is the instantaneous reaction rate obtained by solving the piecewise modified Kamal equation.

7. The method for predicting vulcanization deformation of rubber parts and compensating for molds based on multi-field coupling data as described in claim 3, characterized in that: Step 3.6 includes the following steps: Step 3.6.1, Calculation of the three-dimensional transient heat conduction control equation; The three-dimensional transient heat conduction governing equation, considering the endogenous heat source of sulfidation and the external convective heat transfer boundary, is expressed as: ； Where c is the specific heat capacity of rubber, in J / (kg·K), and λ is the thermal conductivity of rubber, in W / (m·K). It is the instantaneous heating rate. It is the Hamiltonian gradient operator. It is an endogenous heat source term of sulfidation, which is provided in real time by the sulfidation kinetic field; Heat exchange boundary conditions: ; Where h is the convective heat transfer coefficient, in W / (m²). 2 ·K), It is the ambient temperature, in Kelvin (K). It is the temperature gradient along the normal to the model; Step 3.6.2, Calculation of the governing equations for viscoelastic and viscoplastic mechanical fields; Equilibrium equations of mechanics: Where σ is the stress tensor and f is the external load vector such as sulfidation pressure; The total strain decomposition relationship of a component: The total strain of a component is composed of the superposition of mechanical strain, thermal strain, and vulcanization shrinkage strain. ,in It is the total strain tensor. It is the mechanical strain tensor. It is the vulcanization shrinkage strain tensor and the thermal strain tensor. β is the coefficient of thermal expansion of rubber. The initial temperature; Viscoelastic constitutive equation: The generalized Maxwell viscoelastic model is adopted, and the modulus is updated in real time with the degree of sulfidation. ,in, It is an integral dummy variable. It is the time-varying equivalent elastic modulus. It is the rate of change of mechanical strain with respect to time; Step 3.6.3, Field Coupling Logic Relationship; Temperature field output instantaneous temperature T, heating rate The sulfidation kinetic field is input; the kinetic field is used to solve for the degree of sulfidation α and the reaction rate through piecewise modified Kamal equations. On the one hand, calculate endogenous heat The data is fed back to the temperature field, while dynamic mechanical parameters and shrinkage strain are calculated and fed into the viscoelastic and viscoplastic mechanical fields and the vulcanization shrinkage field. The mechanical field solution yields stress, strain, and nodal deformation results, completing a single-step full-field update and entering the next iteration.

8. The method for predicting vulcanization deformation of rubber parts and compensating for molds based on multi-field coupling data as described in claim 1, characterized in that: Step 4 includes the following steps: Step 4.1: Pre-simulation verification and output variable preset. Load the piecewise correction Kamal program, temperature and sulphurity dual-factor machine learning model, and high-order viscoelastic and viscoplastic coupled constitutive model. Enable the flow field and damage field modules as needed. Set the convergence criteria for multiple fields and preset all output field variables. Step 4.2: Multi-field synchronous transient coupling iterative solution, using a unified time step to solve the temperature field, vulcanization kinetic field, viscoelastic viscoplastic mechanical field, and vulcanization shrinkage field, and simultaneously calculate the melt flow field and damage field as needed; based on real-time temperature and vulcanization degree, dynamically update material parameters and constitutive coefficients, and simultaneously calculate vulcanization endogenous heat, plastic strain, and element damage, with the calculation results of each field being transmitted bidirectionally in real time until the complete vulcanization cycle calculation is completed; Step 4.3: Post-demolding preprocessing of partitions. Based on the steady-state simulation results after demolding, the structure is divided into deformation zones, stress concentration zones, and high-damage zones. Dedicated cross-section and contour data extraction paths are set. Step 4.4: Extract the three-dimensional displacement, cross-sectional warpage, linear shrinkage rate, equivalent stress, peak plastic strain, and element damage factor classification data of the global nodes in layers to construct a standardized multi-dimensional deviation dataset; Step 4.5: Data verification and standardized archiving, cross-validation of data logical consistency, removal of outliers, and output of a dedicated deviation file for mold compensation.

9. The method for predicting vulcanization deformation of rubber parts and compensating for molds based on multi-field coupling data as described in claim 8, characterized in that: Step 5 includes the following steps: Step 5.1: Construct the global dimensional deviation field of the component; The three-dimensional displacement, warpage, and shrinkage of all nodes of the component extracted in step 4 are integrated. Using the three-dimensional model as a benchmark, the coordinates of each node after simulation are compared to quantify the size offset, cross-sectional warpage deviation, and characteristic size shrinkage difference of each node, forming a global size deviation field covering the entire area of ​​the component. At the same time, the residual stress, equivalent plastic strain, and internal damage factor data extracted synchronously are associated. Step 5.2, reverse geometry mapping reconstruction of the mold cavity; Establish the mapping relationship between the mesh nodes of the rubber component and the mesh nodes of the mold cavity. Use the normal projection iteration strategy to back-add the deviation of the component node to the corresponding cavity node of the mold. If the component node is offset outward, the mold surface is compensated inward. If the component shrinks or warps inward, the mold surface is compensated outward. Calculate the normal correction displacement of each node of the mold. The compensation coefficients are allocated according to the differences in deformation, stress and damage severity in each region. The correction amount is increased for areas with warping, stress concentration and high damage incidence, while the basic compensation coefficient is used for areas with uniform deformation. The three-dimensional cavity geometric model of the mold after the first correction is generated and the geometric smoothing process is completed. Step 5.3, multi-field coupling simulation recalculation verification; The modified mold model after the initial reconstruction is imported into the multi-field fully coupled simulation system. The segmented multi-factor modified Kamal dynamic model, the machine learning temperature and sulfurization degree dual-factor parameter model, the high-order viscoelastic and viscoplastic coupled constitutive model and the optional damage evolution field are fully reused. The complete transient fully coupled simulation is performed again to solve the deformation, shrinkage, warping, residual stress, plastic strain and internal damage of the molded component. Step 5.4, Comprehensive tolerance determination based on multiple indicators; All indicators obtained from the new round of simulation are compared with the pre-set engineering tolerance thresholds. The comprehensive judgment criteria include three types of indicators: External dimensional indicators: Whether the overall deformation of the component, the warping of each section, and the shrinkage error of key feature dimensions are within the dimensional tolerance range; Mechanical performance indicators: whether the maximum residual stress and equivalent plastic strain across the entire domain are below the safety threshold; Internal quality indicators: Whether the highest damage factor of the component and the proportion of severely damaged areas meet the product quality requirements. If any one of the indicators of size, stress, or damage exceeds the preset tolerance, the mold precision is determined to be substandard and the iterative correction process is initiated. If all indicators meet the tolerance requirements, the iteration is terminated and the final mold cavity model is locked. Step 5.5: Closed-loop iterative optimization of mold surface; When the comprehensive indicators do not meet the tolerance requirements, based on the current molding residual deviation, stress distribution, and damage distribution, the compensation coefficient of the mold cavity node is finely adjusted in different areas, and the displacement of the out-of-tolerance area is magnified or finely adjusted to complete the mold geometry reconstruction. Step 5.3 simulation recalculation and step 5.4 tolerance judgment are repeated to form a digital closed-loop iterative process of deviation extraction → reverse mold repair → coupled simulation → multi-indicator verification until all indicators meet the standards. Step 5.6: Output the final mold design scheme; After the iteration terminates and all indicators meet the preset tolerances, the final optimized 3D geometric model of the mold cavity is exported, and the matching vulcanization process parameters and simulation verification report are output simultaneously, including a complete set of simulation verification data for deformation distribution, residual stress, and damage distribution.

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