Metal rubber multi-temperature field hysteresis loop prediction method with model constraint parameter mapping

Abstract

The application provides a metal-rubber multi-temperature field hysteresis loop prediction method based on model constraint parameter mapping, wherein the multi-temperature field comprises a uniform constant temperature field and a non-uniform temperature field, and the non-uniform temperature field is a temperature gradient field distributed along the thickness direction; the prediction method is based on cyclic compression experimental data at multiple discrete constant temperature points, a hysteresis loop parameterization model is established, and hysteresis parameter vectors under different constant temperature conditions are identified; a temperature state feature is constructed, a weighted kernel ridge regression method with temperature zone feature enhancement is adopted to establish a temperature-parameter continuous mapping model; for a target constant temperature field, the target temperature is input into the mapping model and a complete hysteresis loop is reconstructed; for a target temperature gradient field, the temperature field distributed along the thickness direction is discretized into multiple local temperature layers, local hysteresis parameters and local hysteresis loops of each layer are predicted, and then weighted integration along the thickness direction is performed to obtain an overall equivalent hysteresis loop; and the application realizes rapid prediction of a complete hysteresis loop of metal-rubber under multi-temperature field conditions.

Description

Technical Field

[0001] This invention relates to the technical field of methods for predicting the mechanical properties of metal rubber materials, and in particular to a method for predicting hysteresis loops in multiple temperature fields of metal rubber by mapping model constraint parameters. Background Technology

[0002] Metal rubber is a disordered, porous, entangled material formed by spirally winding, laying, and cold-pressing metal wires. It combines the high-temperature resistance, corrosion resistance, and high strength of metallic materials with the high elasticity, damping energy dissipation, and vibration isolation capabilities of rubber-like materials. It has attracted widespread attention and is gradually being applied in complex service environments such as aerospace, ship equipment, and nuclear power plant supports. Due to its internal structure consisting of a large number of interwoven, entangled, and contacting metal wires, metal rubber exhibits complex responses such as contact compaction, frictional slippage, local rearrangement, and path-dependent hysteresis under pressure, macroscopically displaying a significant nonlinear stress-strain hysteresis loop.

[0003] In practical engineering applications, metal-rubber vibration isolation or support components are often located in complex thermal environments. Their temperature state may exhibit a generally isothermal field, or it may present as a temperature gradient field with heating at one end and heat dissipation at the other, or a temperature difference along the thickness direction. Temperature changes alter the degree of thermal expansion between the metal wires, the contact preload level, the interfacial friction state, and the load-bearing path, thus affecting the peak load-bearing capacity, loading stiffening characteristics, unloading opening, and hysteresis energy dissipation capacity of the metal-rubber. If only mechanical parameters at room temperature or a single isothermal assumption are used for prediction, it is difficult to accurately describe the evolution of the hysteresis loop of metal-rubber under different temperature field conditions.

[0004] Currently, methods for predicting the temperature-related mechanical properties of rubber mainly include experimental testing, empirical fitting, constitutive modeling, and finite element simulation. Experimental testing can directly obtain the mechanical response under specific temperature conditions, but it requires separate experiments for each target temperature or temperature gradient boundary, resulting in long cycles, high costs, and difficulty in covering the numerous intermediate temperature points and non-uniform temperature fields that may exist in engineering projects. Empirical fitting methods are relatively simple in form, but they often lack clear physical parameter meanings and struggle to explain the coordinated changes in the main loop skeleton, unloading opening, and hysteresis morphology caused by temperature variations. Finite element simulation can analyze local contact states, but the number of contact points and the complexity of contact relationships within rubber make directly establishing a complete and detailed model computationally expensive, hindering rapid prediction in engineering projects.

[0005] Furthermore, existing data-driven methods often directly use macroscopic responses such as the entire hysteresis loop, average stiffness, or loss factor as prediction objects, easily leading to black-box regression relationships. While these methods can achieve a certain fitting effect within a specific data range, they lack constraints on physical characteristics such as the metal-rubber master loop skeleton, loading-unloading openings, and path hysteresis corrections. When the temperature changes, especially when extending from discrete isothermal points to untested isothermal points or temperature gradient conditions, direct regression methods are prone to problems such as unstable predictions, discontinuous parameter evolution, or unclear physical meanings. Therefore, there is an urgent need for a method that can establish a physically constrained continuous temperature-parameter mapping relationship based on finite isothermal experimental data, and further apply it to the prediction of complete hysteresis loops under multi-temperature field conditions. Summary of the Invention

[0006] This invention proposes a method for predicting hysteresis loops in multi-temperature fields of metal-rubber based on model-constrained parameter mapping. It can establish a physically constrained continuous temperature-parameter mapping relationship based on finite isothermal experimental data, and is applicable to the prediction of complete hysteresis loops under multi-temperature field conditions. This invention achieves rapid prediction of complete hysteresis loops in metal-rubber under multi-temperature field conditions, providing technical support for the performance evaluation and parameter design of metal-rubber components in complex thermal environments.

[0007] The present invention adopts the following technical solution.

[0008] A method for predicting hysteresis loops in multi-temperature fields of metal rubber using model-constrained parameter mapping is proposed. The multi-temperature fields include a uniform isothermal field and a non-uniform temperature field, where the non-uniform temperature field can be a temperature gradient field distributed along the thickness direction of the metal rubber sample. The prediction method first establishes a physically meaningful parameterized model of the hysteresis loop based on cyclic compression experimental data at multiple discrete isothermal points and identifies the hysteresis parameter vectors under different isothermal conditions. Then, temperature state features are constructed, and a temperature-parameter continuous mapping model is established using a weighted kernel ridge regression method with enhanced temperature region features. For the isothermal field as the prediction target, the target temperature is input into the mapping model, and the complete hysteresis loop is reconstructed. For the temperature gradient field as the prediction target, the temperature field distributed along the thickness direction is discretized into multiple local temperature layers, and the local hysteresis parameters and local hysteresis loops of each layer are predicted respectively. Finally, a weighted integral along the thickness direction is used to obtain the overall equivalent hysteresis loop.

[0009] The prediction method includes the following steps;

[0010] Step S1: Obtain hysteresis loop data of cyclic compression of metal rubber at multiple isothermal points, and perform stress-strain conversion and steady-cycle extraction on the load-displacement data;

[0011] Step S2: Establish a parameterized model of the metal rubber hysteresis loop, decompose the macroscopic hysteresis response into the main loop skeleton, unloading opening and hysteresis correction component, and identify the hysteresis parameter vector at each isothermal point.

[0012] Step S3: Based on the identification parameters under discrete isothermal points, construct temperature state features, and establish a temperature-parameter continuous mapping model using the weighted kernel ridge regression method with enhanced temperature region features.

[0013] Step S4: For the target uniform isothermal field, input the target temperature into the temperature-parameter continuous mapping model to obtain the predicted parameter vector at that temperature, and substitute it into the hysteresis loop parameterization model to obtain the complete hysteresis loop under the target isothermal condition.

[0014] Step S5: For the target non-uniform temperature field, establish the local temperature distribution in the thickness direction of the sample based on the temperature boundary or temperature field distribution, input the local temperature of each layer into the temperature-parameter continuous mapping model, and obtain the local hysteresis parameter field in the thickness direction.

[0015] Step S6: Substitute the local hysteresis parameters of each thickness layer into the hysteresis loop parameterization model, reconstruct the local loading and unloading branches of each layer respectively, and perform thickness-weighted integration on the local response of each layer based on the same nominal strain history to obtain the overall equivalent hysteresis loop under non-uniform temperature field.

[0016] Step S7: Output indicators such as peak stress, hysteresis loop area, mid-section secant stiffness, root mean square error, and mean absolute error to evaluate the accuracy of the predicted hysteresis loop under multiple temperature fields.

[0017] The specific steps of step S1 are as follows:

[0018] Step S11: Prepare a metal rubber sample; specifically, the metal rubber sample is obtained by spiral winding, fixed pitch treatment, laying and entanglement and cold pressing of metal wire, so that the metal rubber sample forms an entangled network structure composed of multi-point contact, spatial interlocking and multiple pores inside.

[0019] Step S12: Conduct a constant temperature cyclic compression test; specifically: place the metal rubber sample in multiple constant temperature environments for sufficient heat preservation, and apply displacement-controlled cyclic compression load after the thermal state stabilizes, and collect the load and displacement signals during the loading process.

[0020] Step S13: Perform data preprocessing; specifically: convert the load-displacement data into stress-strain data based on the cross-sectional area and molding height of the specimen; eliminate cycles with significant influence from the initial clamping gap, and select the loading and unloading branches in the stable cycles as hysteresis loop modeling data;

[0021] Step S14: Obtain the isothermal sample library; specifically, the mean value of multiple sets of repeated test hysteresis loops at the same isothermal point is processed to obtain the average hysteresis loop at that temperature, and the repeatability error range can be obtained at the same time, providing basic data for subsequent parameter identification and model verification.

[0022] Step S1: Obtain isothermal hysteresis loop data of the metal rubber. Specifically, a cylindrical metal rubber sample is selected. The sample is obtained by spiral winding, stretching to a fixed pitch, laying and entanglement, and cold pressing of 304 stainless steel wire. The sample's interior consists of a large number of interwoven spiral coil units forming a multi-point contact network. The sample is placed in the target isothermal environment for sufficient heat preservation. After the thermal expansion displacement tends to stabilize, a cyclic compression load is applied using a displacement control method, and load and displacement signals are acquired simultaneously.

[0023] In one specific embodiment, 22℃, 50℃, 100℃, 150℃, 200℃, 250℃, and 300℃ can be selected as discrete isothermal test points. Multiple sets of repeated tests are conducted at each temperature point, and each set of tests continuously executes multiple loading-unloading cycles. Data with significant influence from clamping gaps and initial contact adjustments in the first cycle are discarded, and the loading and unloading branches in the stable cycle are selected as the hysteresis loop modeling objects. Based on the sample cross-sectional area and height, the load-displacement data are converted into stress-strain data, and the mean value of the repeated test hysteresis loops at the same temperature is calculated to obtain an isothermal hysteresis loop sample library, such as... Figure 2 As shown.

[0024] In step S2, to avoid ambiguity in the formula symbols, the following conventions are uniformly defined: In the formulas for the relevant model of the metal-rubber sample, the horizontal line above the symbol represents the offset coordinate quantity after transformation relative to the offset reference point; the "^" above the symbol represents the predicted value obtained from the mapping model; the superscript "T" represents the transpose of a vector or matrix; the subscript p represents the offset reference state; the subscript s represents the main loop skeleton; the subscripts L and U represent the loading branch and unloading branch, respectively; the superscript g represents the temperature gradient condition; the subscripts i and j represent the isothermal training sample numbers; the subscript l represents the temperature layer number in the thickness direction; and the subscript q represents the parameter component number in the hysteresis parameter vector. Unless otherwise specified, engineering strain and all proportional parameters are dimensionless quantities; the units of stress, stiffness parameters, and hysteresis correction amplitude are MPa; and the unit of temperature is °C.

[0025] The specific steps for establishing a parameterized model of the hysteresis loop of metal-rubber include:

[0026] Step S21: Establish a hysteresis response decomposition model; under one-dimensional axial compression conditions, the macroscopic deformation is represented by engineering strain ε and the macroscopic response is represented by stress σ. The hysteresis loop of the metal rubber is decomposed into the main loop skeleton component, the unloading opening component and the hysteresis correction component.

[0027] The main loop skeleton component is used to describe the nonlinear load-bearing capacity during compression; the unloading opening component is used to describe the opening degree between the loading and unloading branches; and the hysteresis correction component is used to describe the path-related deviation during unloading.

[0028] Step S22: Introduce a dual-bias reference point; the metal-rubber sample exhibits unstable responses such as clamping gap elimination, weak contact compaction, and local contact rearrangement during the initial stage of small strain, thus introducing a bias reference point. The offset coordinates are used to characterize the equivalent offset of the stable principal loop relative to the geometric origin; the offset coordinates are expressed as:

[0029] Formula 1;

[0030] in, The strain after biasing. The stress after biasing, For bias reference strain, This is the bias reference stress;

[0031] Step S23: Establish the main loop skeleton model; the metal-rubber sample contact network exhibits characteristics such as initial bearing, gradual compaction, and nonlinear reinforcement during compression. The main loop skeleton is represented as a superposition of linear terms and power function reinforcement terms:

[0032] Formula 2

[0033] in, The stress in the principal loop skeleton in the offset coordinate system at temperature T; These are the basic bearing stiffness parameters for the small deformation stage. For nonlinear enhancement amplitude parameters, is a nonlinear exponential parameter, and is a dimensionless positive number; This indicates that the model is used for the stable bearing range after the offset reference point;

[0034] Step S24: Establish the unloading opening model: Introduce unloading opening parameters or equivalent frictional opening coefficient This is used to characterize the degree of opening between the loading branch and the unloading branch; This can be identified by the proportional relationship between the experimental unloading branch and the ideal loading reference branch within the unloading stable region. can be The conversion yields:

[0035] Formula 3;

[0036] in, The scaling factor of the ideal unloading branch relative to the ideal loading reference branch at temperature T; The equivalent friction opening coefficient is obtained by converting the hysteresis loop opening; the two are two equivalent expressions of the same unloading opening feature, and are not independent hysteresis parameters.

[0037] Step S25: Establish a hysteresis correction model; for the path deviation of the actual unloading branch relative to the ideal unloading branch, introduce a hysteresis correction term. The hysteresis correction term can be described by both the amplitude parameter and the shape control parameter, and the relevant formula is:

[0038] Formula 4;

[0039] And there are:

[0040] Formula 5;

[0041] Formula 6;

[0042] in, The hysteresis correction stress is the actual unloading branch at temperature T relative to the ideal unloading reference branch. To normalize the unloading path variable; This represents the peak strain corresponding to the unloading start point. The minimum strain corresponding to the unloading endpoint; when The time indicates the unloading start point. The time indicates the end point of the unloading process; and The hysteresis correction amplitude parameter; For the initial hysteresis bias of unloading, the exponential decay function is used. The attenuation control parameters for unloading hysteresis bias; For the shape control parameters of the middle and later stages of unloading;

[0043] Step S26: Output hysteresis parameter vector; integrate the bias reference state, main loop skeleton, unloading opening, and hysteresis correction parameters at each isothermal point. The following is a set of hysteresis parameter vectors:

[0044]

[0045] Formula 7;

[0046] in, This is the column vector of hysteresis parameters corresponding to the i-th isothermal sample; For the first The temperature of a constant-temperature sample; , This represents the total number of isothermal samples used for parameter mapping modeling; the parameters in the vector are arranged in a fixed order to ensure consistency of fields in the parameter sample library, training output, and prediction results.

[0047] Step S2: Establish a parameterized model of the hysteresis loop and identify the isothermal parameters.

[0048] During quasi-static compression, the stress response of the metal-rubber can be considered as a combination of a nonlinear elastic skeleton, a loading-unloading opening, and path-dependent hysteresis correction. First, a bias reference point is introduced. Addressing the reference point offset issue during the initial stage of small strain. This offset reference point is used to distinguish between the initial contact adjustment section and the stable bearing section, avoiding misidentification of unstable responses such as clamping gap elimination, weak contact compaction, and residual friction lock release as main loop skeleton features.

[0049] In offset coordinates, the principal loop skeleton is described by superimposing linear and power-law reinforcement terms. This principal loop skeleton reflects the continuous reinforcement response of the metal-rubber during compression, caused by the gradual participation of contact points in load bearing, local contact compaction, and the expansion of the load bearing path. Subsequently, the unloading opening parameters are identified based on the proportional relationship between the ideal loading reference branch and the ideal unloading reference branch. and according to Conversion of equivalent frictional opening coefficient Finally, to address the path deviation between the actual unloading branch and the ideal unloading branch, a hysteresis correction term is established and identified. Parameters such as these.

[0050] Thus, a set of hysteresis parameter vectors is obtained at each isothermal point. The different parameters in this parameter vector correspond to the bias reference state, the main loop skeleton, the unloading opening, and the unloading hysteresis correction feature, respectively. This allows for the characterization of the main morphology of the complete hysteresis loop with fewer parameters, such as... Figure 3 As shown.

[0051] Step S3 includes the following steps;

[0052] Step S31: Construct temperature state characteristics; assuming the lowest temperature covered by the isothermal experimental data is... The highest temperature is Then, the temperature T is normalized:

[0053] Formula 8;

[0054] in, This is the normalized temperature.

[0055] Step S32: Introduce temperature range gate functions; to describe the stage-specific characteristics of parameter evolution in the intermediate temperature transition region and the high temperature enhancement region, a continuous gate function is introduced. :

[0056] when hour, ;

[0057] when hour, , ;

[0058] when hour, ;

[0059] in, The starting temperature for the temperature transition zone. The temperature at which the temperature transition is fully activated is defined; this continuous gate function is used to describe the smooth change process of temperature effect from inactive to fully activated.

[0060] Step S33: Construct temperature region enhancement feature vector; Based on normalized temperature and temperature region gate function, construct temperature state feature vector:

[0061] Formula 9;

[0062] in, Used to describe the continuous changes in the mesothermal transition zone. This feature is used to describe the activation level of the high-temperature enhancement region; this feature construction enables the regression model to learn the parameter variation trends in different temperature regions more stably under small sample conditions.

[0063] Step S34: Perform parameter scaling transformation; for parameters that are always positive and change significantly with temperature, perform the transformation using a logarithmic form relative to the reference temperature parameter value:

[0064] Formula 10;

[0065] For bias parameters or parameters with small variations, the transformation is performed using the difference relative to the reference temperature parameter value:

[0066] Formula 11;

[0067] in, Represents the first element in the hysteresis parameter vector. The values ​​of the parameters to be predicted at temperature T; This indicates that the parameter is at the reference temperature T. ref The following reference values; For the first The regression output of each parameter after scaling transformation; after prediction, the actual physical parameters are obtained through corresponding inverse transformation; this process can reduce the impact of differences in the dimensions and numerical scales of different parameters on the regression process and reduce non-physical prediction values.

[0068] Step S35: Establish a weighted kernel ridge regression model. The transformation amount for any objective parameter... Weighted kernel ridge regression was used to establish its relationship with temperature state characteristics. Mapping relationship between them:

[0069] Formula 12;

[0070] in, Target temperature state characteristics Next Predicted values ​​of the parameter transformation quantities; The total number of training samples; Number the training samples; For the first Temperature state feature vector of each training sample; For the first The parameter mapping model corresponds to the first parameter mapping model. The regression coefficients of each training sample; For target features and the first Kernel function values ​​between features of each training sample; , The parameter dimension of the hysteresis parameter vector; the kernel function uses the radial basis function:

[0071] Formula 13;

[0072] in, The kernel width parameter of the basis kernel function is used to control how smoothly the model smooths changes in temperature characteristics;

[0073] Step S36: Determine the regression coefficients through weighted regularization optimization; the objective function of weighted kernel ridge regression includes a sample weighting error term and a regularization term. The sample weights are used to enhance the influence of samples in the intermediate temperature transition region and the high temperature enhancement region on the mapping relationship, and the regularization term is used to suppress overfitting and non-physical oscillations in small sample regression. The kernel width parameter and regularization coefficient are determined through cross-validation within the training set;

[0074] Step S37: Adopt a core parameter-auxiliary parameter hierarchical prediction strategy; First, predict the core parameters that have a significant impact on the morphology of the loop body, including offset stress, foundation bearing stiffness, nonlinear strengthening amplitude, unloading opening parameters, and main hysteresis correction amplitude; Then, use the temperature state characteristics and the core parameter prediction results as input to further predict auxiliary parameters such as offset strain, nonlinear exponent, and unloading shape control parameters; This hierarchical prediction strategy can utilize the constraint relationship between core parameters and auxiliary parameters to reduce the instability caused by simultaneous regression of multiple parameters.

[0075] In step S3, the hysteresis parameter vectors identified at different isothermal points in step S2 are used as output samples, and the temperature state features are used as input samples to establish a continuous temperature-parameter mapping relationship. To avoid excessive smoothing of parameter evolution in the intermediate temperature transition region and the high temperature enhancement region due to using only temperature as a single input, a temperature region gate function is introduced based on the normalized temperature to construct a temperature state feature vector including the normalized temperature, intermediate temperature transition features, high temperature enhancement features, and their interaction terms.

[0076] In the implementation, a weighted kernel ridge regression method is used to establish a nonlinear mapping relationship between temperature state characteristics and hysteresis parameters. For parameters with different dimensions and varying amplitudes, a logarithmic transformation or interpolation transformation is performed first, followed by regression prediction. After prediction, an inverse transformation is performed to recover the physical parameters. The radial basis function is used as the kernel function, and the kernel width and regularization coefficient are determined through cross-validation within the training set. Sample weights are used to enhance the influence of samples in the intermediate temperature transition region and the high temperature enhancement region on the model, thereby improving the stability of parameter prediction under small sample conditions.

[0077] To further reduce the instability of simultaneous multi-parameter prediction, a hierarchical prediction strategy of core parameters and auxiliary parameters is adopted. First, core parameters such as offset stress, foundation bearing stiffness, nonlinear hardening amplitude, unloading opening parameters, and main hysteresis correction amplitudes are predicted. Then, temperature state characteristics and the core parameter prediction results are used as input to predict auxiliary parameters such as offset strain, nonlinear exponent, and unloading shape control parameters. This yields the final temperature-parameter continuous mapping model, and its overall prediction process is as follows: Figure 1 As shown.

[0078] The specific steps of step S4 are as follows:

[0079] Step S41: Set the target uniform constant temperature field temperature The It can be a tested constant temperature point, or it can be an untested intermediate temperature point within the calibrated temperature range;

[0080] Step S42: Set the target temperature Input the temperature state feature construction module to obtain the temperature state features corresponding to the target temperature. ;

[0081] Step S43, Input the temperature-parameter continuous mapping model established in step S3 to obtain the predicted hysteresis parameter vector under the target isothermal field, expressed by the formula:

[0082] Formula 14;

[0083] in, This is a temperature-parameter mapping model obtained through training with isothermal samples.

[0084] Step S44: Convert the prediction parameter vector Substitute the parameterized model of the hysteresis loop established in step S2, and reconstruct the loading branch, ideal unloading branch and hysteresis correction branch under the target isothermal field respectively to obtain the complete stress-strain hysteresis loop;

[0085] Step S45: Output engineering parameters such as peak stress, hysteresis loop area, and mid-section secant stiffness under the target isothermal field. This enables rapid prediction of the hysteresis loop of the metal-rubber under untested target isothermal points within the calibration temperature range.

[0086] In step S4, for the target uniform isothermal field, let the target temperature be... .Will Input the temperature state feature construction module established in step S3 to obtain the target temperature state feature. Subsequently, Input a temperature-parameter continuous mapping model to obtain the prediction hysteresis parameter vector at the target temperature. Substituting this parameter vector into the hysteresis loop parameterized model established in step S2, the loading branch, ideal unloading branch, and hysteresis correction branch can be reconstructed respectively, ultimately yielding the complete stress-strain hysteresis loop under the target isothermal field, as shown below. Figure 4 As shown.

[0087] When the target temperature When the isothermal point has been tested, the accuracy of the mapping model can be verified by comparing the predicted hysteresis loop with the experimental hysteresis loop; when the target temperature... When dealing with untested intermediate temperature points within the calibration temperature range, this method can be used to predict the hysteresis loop of metal-rubber under intermediate isothermal conditions. Therefore, this method avoids conducting separate cyclic compression tests for each intermediate temperature point.

[0088] The specific steps of step S5 are as follows:

[0089] Step S51: Set the target non-uniform temperature field; the non-uniform temperature field can be determined by steady-state heat transfer analysis, experimental temperature measurement results, or boundary temperature conditions; when the target temperature field is a unidirectional steady-state temperature gradient field, the high-temperature boundary temperature is set to... The low-temperature boundary temperature is Normalized thickness coordinates are ,in Indicates the high temperature boundary. If we denote a low-temperature boundary, then the local temperature along the thickness direction can be expressed as:

[0090] Formula 15;

[0091] in, Thickness location under temperature gradient conditions Local temperature at; subscript Indicates temperature gradient operating conditions; The temperature is the high-temperature boundary temperature, indicated by the subscript. Indicates the high-temperature boundary; Low temperature boundary temperature, subscript Indicates the low-temperature boundary; For normalized thickness coordinates, which are dimensionless quantities; Corresponding to the high temperature boundary, Corresponding to the low temperature boundary;

[0092] Step S52: Discretize the sample thickness direction into The temperature layer; the first The normalized thickness coordinates, local temperature, and local parameter vectors of the layer are respectively expressed as: ,in:

[0093] Formula 16;

[0094] in, Let be the total number of discrete temperature layers in the thickness direction, and , Number the temperature layers; For the first Normalized thickness coordinates of the layer; For the first Local temperature of the layer; This refers to the temperature state characteristics corresponding to this local temperature; For the first The prediction hysteresis parameter vector of the layer;

[0095] Step S53: For any nonlinear or experimentally measured temperature field, the local temperature values ​​of each measuring point or layer node can be directly input into the temperature-parameter mapping model to obtain the local parameter vector at the corresponding location. Therefore, this method is not only applicable to linear temperature gradient fields, but also to other non-uniform temperature fields that can be discretized.

[0096] In step S5, for the temperature gradient condition where one end is heated and the other end is cooled, let the high-temperature boundary temperature of the sample be... The low-temperature boundary temperature is Based on steady-state heat transfer analysis or experimental temperature measurements, a temperature field along the sample thickness is established. If the temperature field can be approximated as a linear distribution, then... Describe the local temperature along the thickness direction; if the temperature field is nonlinearly distributed, the local temperature of each layer can be obtained from actual temperature measurement points or numerical thermal analysis results. The temperature gradient loading device and the temperature field distribution along the thickness direction are shown below. Figure 5 As shown.

[0097] Discretize the sample thickness direction as Each temperature layer is divided into several temperature zones, and the local temperature is calculated for each temperature layer. Subsequently, the local temperature of each layer was controlled. By inputting the temperature-parameter continuous mapping model, the local hysteresis parameters of the corresponding layers can be obtained. This allows us to obtain a local parameter field along the thickness direction, which reflects the differences in the main loop skeleton, unloading opening, and hysteresis correction parameters at different height locations due to varying local temperatures. The local response distribution under the temperature gradient field is shown below. Figure 6 As shown.

[0098] The specific steps of step S6 are as follows:

[0099] Step S61, Local loop reconstruction; The first... Layer local hysteresis parameters Substituting into the hysteresis loop parameterized model, we obtain the local loading branch of this layer. and partial unload branch ;

[0100] Step S62, Nominal Strain Treatment: Considering that the metal wires inside the metal rubber interweave with each other in the thickness direction to form a continuous entangled contact network, the thickness-direction layering is mainly used to describe the differences in local temperature and local parameter states, and does not simplify the sample into an independent regular layered structure. During macroscopic compression, it is assumed that each temperature layer has the same nominal strain history, and the local stress response of each layer is calculated at the same strain point;

[0101] Step S63: Thickness-weighted integration; Perform thickness-weighted integration on the local loading and unloading branches of each layer to obtain the overall equivalent loading curve and overall equivalent unloading curve under the target non-uniform temperature field:

[0102] Formula 17;

[0103] Formula 18;

[0104] in, and The equivalent loading curve and equivalent unloading curve are shown respectively under temperature gradient conditions. and The first A temperature layer at nominal strain Predicted loading stress and predicted unloading stress at the location; For the first Dimensionless thickness integral weight; The total number of temperature layers, and satisfying the following conditions: .

[0105] Step S64: Determine the integral weight; the integral weight in the thickness direction can be a trapezoidal integral weight, in which the weights of the two end layers are smaller and the weights of the middle layers are equal; or the weights can be determined according to the actual temperature layer thickness or non-uniform layer spacing.

[0106] Step S65: Obtain the complete equivalent hysteresis loop; from the overall equivalent loading curve and overall equivalent unloading curve Together they form the complete equivalent stress-strain hysteresis loop of metallic rubber under a non-uniform temperature field.

[0107] Step S6: Reconstruct the local hysteresis loop and integrate to obtain the global equivalent loop. Specifically, this involves converting the local parameters of each temperature layer... Substituting into the hysteresis loop parameterized model, the local loading branches of this layer are obtained respectively. and partial unload branch It should be noted that the thickness-direction layering is only used to describe local temperature and parameter differences in a continuous entangled network, and does not simplify the metal-rubber sample into a regular layered structure that is independent of each other. Considering that the metal wires interweave with each other in the thickness direction and form a continuous contact network, this embodiment assumes that each temperature layer has the same nominal strain history during macroscopic compression.

[0108] At the same nominal strain point, thickness-weighted integrals are performed on the local loading and unloading branches of each temperature layer to obtain the overall equivalent loading and unloading curves under the temperature gradient field. The integration weights can be trapezoidal or set according to the layer thickness or the actual discretization method. Finally, the overall equivalent loading and unloading curves form the complete equivalent hysteresis loop under the target temperature gradient field, as shown below. Figure 7 As shown.

[0109] The specific steps of step S7 are as follows:

[0110] Step S71: Calculate the peak stress; the peak stress is used to characterize the overall load-bearing capacity of the metal rubber at the point of maximum compressive strain under the target temperature field.

[0111] Step S72: Calculate the hysteresis loop area; the hysteresis loop area is obtained by the area integral between the overall equivalent loading curve and the overall equivalent unloading curve, and is used to characterize the energy dissipation capacity during the cyclic compression process.

[0112] Step S73: Calculate the mid-section secant stiffness; The mid-section secant stiffness is calculated based on the stress difference and strain difference in the strain interval of the loading branch, and is used to characterize the stiffening characteristics of the main loading branch.

[0113] Step S74: Calculate the curve prediction error; compare the predicted hysteresis loop with the experimental hysteresis loop or the reference hysteresis loop, use the normalized root mean square error of peak stress and the normalized mean absolute error of peak stress to evaluate the stress amplitude prediction deviation, and use the hysteresis loop area error to evaluate the hysteresis energy consumption prediction deviation.

[0114] Step S75: Output the prediction results; the final output includes the complete hysteresis loop, peak stress, hysteresis loop area, mid-section secant stiffness, and error evaluation results under the target temperature field, providing a basis for the performance evaluation and structural parameter design of metal-rubber components under complex thermal environments.

[0115] Step S7, when evaluating the prediction results under multiple temperature fields, calculates engineering indices such as peak stress, hysteresis loop area, and secant stiffness for the predicted hysteresis loops obtained under uniform isothermal and non-uniform temperature fields. Peak stress characterizes the overall load-bearing capacity of the metal-rubber under the target temperature field; hysteresis loop area characterizes the energy dissipation capacity during cyclic loading; and secant stiffness characterizes the stiffening characteristics of the main loading branch. When experimental loops are available for comparison, the normalized root mean square error of peak stress, the normalized mean absolute error of peak stress, and the loop area error are further calculated. These indices can be used to evaluate the accuracy of the prediction model in terms of overall stress amplitude, loop morphology, and hysteresis energy dissipation. The prediction errors under isothermal fields and the prediction accuracy of engineering indices such as peak stress, hysteresis loop area, and secant stiffness under temperature gradient fields are as follows: Figure 8 As shown.

[0116] In one specific embodiment, isothermal cyclic compression data at 22℃, 50℃, 100℃, 150℃, 200℃, 250℃, and 300℃ were used as calibration samples to identify hysteresis parameter vectors at each temperature point and establish a temperature-parameter continuous mapping model with enhanced temperature zone characteristics. For untested isothermal points, the target temperature was directly input into the mapping model to predict the corresponding hysteresis parameters and reconstruct the complete hysteresis loop. For temperature gradient conditions such as room temperature to 100℃, room temperature to 200℃, and room temperature to 300℃, the temperature field in the thickness direction was discretized into multiple local temperature layers, and the local parameters and local loops of each layer were predicted respectively. The overall equivalent hysteresis loop was obtained by thickness direction integration. Verification results show that this method can predict the main load-bearing response, loading stiffening characteristics, and unloading return morphology of the hysteresis loop of metal rubber under multi-temperature field conditions.

[0117] In summary, this invention, based on isothermal experimental data, obtains hysteresis parameters under different temperature conditions through physical parameter identification and model-constrained temperature-parameter mapping. For a uniform isothermal field, the complete hysteresis loop at the target temperature can be directly reconstructed. For a temperature gradient field, a local parameter field in the thickness direction can be further constructed, and the overall equivalent hysteresis loop is obtained through local loop reconstruction and thickness direction integration. Therefore, this invention achieves rapid prediction of the complete hysteresis loop of metal rubber under multiple temperature field conditions, providing technical support for the performance evaluation and parameter design of metal rubber components in complex thermal environments.

[0118] Compared with the prior art, the present invention has the following beneficial technical effects:

[0119] (1) The present invention can be applied to the prediction of hysteresis loops of metal rubber under uniform isothermal field and non-uniform temperature field. It can predict not only the tested isothermal point, but also the untested intermediate isothermal point within the calibrated temperature range. It can be further extended to the prediction of the overall equivalent hysteresis loop under temperature gradient conditions, and has a wider range of applications.

[0120] (2) This invention differs from the method of directly fitting the entire hysteresis loop to a black box. Instead, it first establishes a parameterized model of the hysteresis loop with physical meaning, and then establishes a continuous temperature-parameter mapping relationship in the parameter space. This makes the prediction results subject to physical components such as the main loop skeleton, unloading opening and hysteresis correction, and has better interpretability and stability.

[0121] (3) By introducing temperature zone gate function and temperature zone enhancement feature, this invention can improve the ability of the model to express the parameter evolution law of the medium temperature transition zone and high temperature enhancement zone under small sample conditions, and avoid the problem of excessive parameter smoothing caused by using only temperature as a single input.

[0122] (4) The present invention uses the weighted kernel ridge regression method to establish a nonlinear mapping relationship between temperature state and hysteresis parameter, which can realize continuous parameter prediction under limited isothermal experimental sample conditions, reduce the dependence on a large number of repeated experiments, and improve the prediction efficiency of mechanical properties of metal rubber in multiple temperature fields.

[0123] (5) The present invention uses a core parameter-auxiliary parameter hierarchical prediction strategy to prioritize the prediction stability of key features such as peak load, main loop skeleton, unloading opening and main hysteresis deviation, and further corrects local bias and unloading shape parameters, which can reduce the instability of simultaneous regression of multiple parameters.

[0124] (6) This invention proposes a local parameter field construction and thickness direction hysteresis integration method for non-uniform temperature fields, which can preserve the differences in stiffness, unloading opening and energy consumption characteristics of different temperature layers, and avoid information loss caused by simply using equivalent temperature or average parameters.

[0125] (7) The present invention can output complete stress-strain hysteresis loops as well as engineering indicators such as peak stress, hysteresis loop area and mid-section secant stiffness, which can provide technical support for performance prediction, parameter design, structural optimization and service evaluation of metal rubber vibration isolation components under complex thermal environments. Attached Figure Description

[0126] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments:

[0127] Appendix Figure 1 This is a schematic diagram of a method for predicting hysteresis loops in multiple temperature fields of metal rubber based on model constraint parameter mapping in an embodiment of the present invention.

[0128] Appendix Figure 2 This is a schematic diagram of the isothermal cyclic compression test and hysteresis data acquisition of metal rubber in an embodiment of the present invention;

[0129] Appendix Figure 3 This is a schematic diagram of the parameterization and parameter identification process for metal-rubber hysteresis loops in an embodiment of the present invention.

[0130] Appendix Figure 4 This is a schematic diagram of the temperature hysteresis loop prediction verification under a constant temperature field in an embodiment of the present invention.

[0131] Appendix Figure 5 This is a schematic diagram of the temperature gradient loading device and the temperature field distribution in the thickness direction in an embodiment of the present invention;

[0132] Appendix Figure 6 This is a schematic diagram of the local response distribution under a temperature gradient field in an embodiment of the present invention;

[0133] Appendix Figure 7 This is a schematic diagram of the prediction and verification of the equivalent hysteresis loop under a temperature gradient field in an embodiment of the present invention.

[0134] Appendix Figure 8 This is a schematic diagram illustrating the prediction accuracy evaluation under multiple temperature fields in an embodiment of the present invention. Detailed Implementation

[0135] As shown in the figure, a method for predicting hysteresis loops in multi-temperature fields of metal rubber using model-constrained parameter mapping is presented. The multi-temperature fields include a uniform isothermal field and a non-uniform temperature field, where the non-uniform temperature field can be a temperature gradient field distributed along the thickness direction of the metal rubber sample. The prediction method first establishes a physically meaningful parameterized model of the hysteresis loop based on cyclic compression experimental data at multiple discrete isothermal points, and identifies the hysteresis parameter vectors under different isothermal conditions. Then, temperature state features are constructed, and a temperature-parameter continuous mapping model is established using a weighted kernel ridge regression method with enhanced temperature region features. For the isothermal field as the prediction target, the target temperature is input into the mapping model, and the complete hysteresis loop is reconstructed. For the temperature gradient field as the prediction target, the temperature field distributed along the thickness direction is discretized into multiple local temperature layers, and the local hysteresis parameters and local hysteresis loops of each layer are predicted respectively. Finally, a weighted integral along the thickness direction is used to obtain the overall equivalent hysteresis loop.

[0136] The prediction method includes the following steps;

[0137] Step S1: Obtain hysteresis loop data of cyclic compression of metal rubber at multiple isothermal points, and perform stress-strain conversion and steady-cycle extraction on the load-displacement data;

[0138] Step S2: Establish a parameterized model of the metal rubber hysteresis loop, decompose the macroscopic hysteresis response into the main loop skeleton, unloading opening and hysteresis correction component, and identify the hysteresis parameter vector at each isothermal point.

[0139] Step S3: Based on the identification parameters under discrete isothermal points, construct temperature state features, and establish a temperature-parameter continuous mapping model using the weighted kernel ridge regression method with enhanced temperature region features.

[0140] Step S4: For the target uniform isothermal field, input the target temperature into the temperature-parameter continuous mapping model to obtain the predicted parameter vector at that temperature, and substitute it into the hysteresis loop parameterization model to obtain the complete hysteresis loop under the target isothermal condition.

[0141] Step S5: For the target non-uniform temperature field, establish the local temperature distribution in the thickness direction of the sample based on the temperature boundary or temperature field distribution, input the local temperature of each layer into the temperature-parameter continuous mapping model, and obtain the local hysteresis parameter field in the thickness direction.

[0142] Step S6: Substitute the local hysteresis parameters of each thickness layer into the hysteresis loop parameterization model, reconstruct the local loading and unloading branches of each layer respectively, and perform thickness-weighted integration on the local response of each layer based on the same nominal strain history to obtain the overall equivalent hysteresis loop under non-uniform temperature field.

[0143] Step S7: Output indicators such as peak stress, hysteresis loop area, mid-section secant stiffness, root mean square error, and mean absolute error to evaluate the accuracy of the predicted hysteresis loop under multiple temperature fields.

[0144] The specific steps of step S1 are as follows:

[0145] Step S11: Prepare a metal rubber sample; specifically, the metal rubber sample is obtained by spiral winding, fixed pitch treatment, laying and entanglement and cold pressing of metal wire, so that the metal rubber sample forms an entangled network structure composed of multi-point contact, spatial interlocking and multiple pores inside.

[0146] Step S12: Conduct a constant temperature cyclic compression test; specifically: place the metal rubber sample in multiple constant temperature environments for sufficient heat preservation, and apply displacement-controlled cyclic compression load after the thermal state stabilizes, and collect the load and displacement signals during the loading process.

[0147] Step S13: Perform data preprocessing; specifically: convert the load-displacement data into stress-strain data based on the cross-sectional area and molding height of the specimen; eliminate cycles with significant influence from the initial clamping gap, and select the loading and unloading branches in the stable cycles as hysteresis loop modeling data;

[0148] Step S14: Obtain the isothermal sample library; specifically, the mean value of multiple sets of repeated test hysteresis loops at the same isothermal point is processed to obtain the average hysteresis loop at that temperature, and the repeatability error range can be obtained at the same time, providing basic data for subsequent parameter identification and model verification.

[0149] Step S1: Obtain isothermal hysteresis loop data of the metal rubber. Specifically, a cylindrical metal rubber sample is selected. The sample is obtained by spiral winding, stretching to a fixed pitch, laying and entanglement, and cold pressing of 304 stainless steel wire. The sample's interior consists of a large number of interwoven spiral coil units forming a multi-point contact network. The sample is placed in the target isothermal environment for sufficient heat preservation. After the thermal expansion displacement tends to stabilize, a cyclic compression load is applied using a displacement control method, and load and displacement signals are acquired simultaneously.

[0150] In one specific embodiment, 22℃, 50℃, 100℃, 150℃, 200℃, 250℃, and 300℃ can be selected as discrete isothermal test points. Multiple sets of repeated tests are conducted at each temperature point, and each set of tests continuously executes multiple loading-unloading cycles. Data with significant influence from clamping gaps and initial contact adjustments in the first cycle are discarded, and the loading and unloading branches in the stable cycle are selected as the hysteresis loop modeling objects. Based on the sample cross-sectional area and height, the load-displacement data are converted into stress-strain data, and the mean value of the repeated test hysteresis loops at the same temperature is calculated to obtain an isothermal hysteresis loop sample library, such as... Figure 2 As shown.

[0151] In step S2, to avoid ambiguity in the formula symbols, the following conventions are uniformly defined: In the formulas for the relevant model of the metal-rubber sample, the horizontal line above the symbol represents the offset coordinate quantity after transformation relative to the offset reference point; the "^" above the symbol represents the predicted value obtained from the mapping model; the superscript "T" represents the transpose of a vector or matrix; the subscript p represents the offset reference state; the subscript s represents the main loop skeleton; the subscripts L and U represent the loading branch and unloading branch, respectively; the superscript g represents the temperature gradient condition; the subscripts i and j represent the isothermal training sample numbers; the subscript l represents the temperature layer number in the thickness direction; and the subscript q represents the parameter component number in the hysteresis parameter vector. Unless otherwise specified, engineering strain and all proportional parameters are dimensionless quantities; the units of stress, stiffness parameters, and hysteresis correction amplitude are MPa; and the unit of temperature is °C.

[0152] The specific steps for establishing a parameterized model of the hysteresis loop of metal-rubber include:

[0153] Step S21: Establish a hysteresis response decomposition model; under one-dimensional axial compression conditions, the macroscopic deformation is represented by engineering strain ε and the macroscopic response is represented by stress σ. The hysteresis loop of the metal rubber is decomposed into the main loop skeleton component, the unloading opening component and the hysteresis correction component.

[0154] The main loop skeleton component is used to describe the nonlinear load-bearing capacity during compression; the unloading opening component is used to describe the opening degree between the loading and unloading branches; and the hysteresis correction component is used to describe the path-related deviation during unloading.

[0155] Step S22: Introduce a dual-bias reference point; the metal-rubber sample exhibits unstable responses such as clamping gap elimination, weak contact compaction, and local contact rearrangement during the initial stage of small strain, thus introducing a bias reference point. This is used to characterize the equivalent offset of the stable principal loop relative to the geometric origin; the offset coordinates are expressed as:

[0156] Formula 1;

[0157] in, The strain after biasing. The stress after biasing, For bias reference strain, This is the bias reference stress;

[0158] Step S23: Establish the main loop skeleton model; the metal-rubber sample contact network exhibits characteristics such as initial bearing, gradual compaction, and nonlinear reinforcement during compression. The main loop skeleton is represented as a superposition of linear terms and power function reinforcement terms:

[0159] Formula 2

[0160] in, The stress in the principal loop skeleton in the offset coordinate system at temperature T; These are the basic bearing stiffness parameters for the small deformation stage. For nonlinear enhancement amplitude parameters, is a nonlinear exponential parameter, and is a dimensionless positive number; This indicates that the model is used for the stable bearing range after the offset reference point;

[0161] Step S24: Establish the unloading opening model: Introduce unloading opening parameters or equivalent frictional opening coefficient This is used to characterize the degree of opening between the loading branch and the unloading branch; This can be identified by the proportional relationship between the experimental unloading branch and the ideal loading reference branch within the unloading stable region. can be The conversion yields:

[0162] Formula 3;

[0163] in, The scaling factor of the ideal unloading branch relative to the ideal loading reference branch at temperature T; The equivalent friction opening coefficient is obtained by converting the hysteresis loop opening; the two are two equivalent expressions of the same unloading opening feature, and are not independent hysteresis parameters.

[0164] Step S25: Establish a hysteresis correction model; for the path deviation of the actual unloading branch relative to the ideal unloading branch, introduce a hysteresis correction term. The hysteresis correction term can be described by both the amplitude parameter and the shape control parameter, and the relevant formula is:

[0165] Formula 4;

[0166] And there are:

[0167] Formula 5;

[0168] Formula 6;

[0169] in, The hysteresis correction stress is the actual unloading branch at temperature T relative to the ideal unloading reference branch. To normalize the unloading path variable; This represents the peak strain corresponding to the unloading start point. The minimum strain corresponding to the unloading endpoint; when The time indicates the unloading start point. The time indicates the end point of the unloading process; and The hysteresis correction amplitude parameter; For the initial hysteresis bias of unloading, the exponential decay function is used. The attenuation control parameters for unloading hysteresis bias; For the shape control parameters of the middle and later stages of unloading;

[0170] Step S26: Output hysteresis parameter vector; integrate the bias reference state, main loop skeleton, unloading opening, and hysteresis correction parameters at each isothermal point. The following is a set of hysteresis parameter vectors:

[0171]

[0172] Formula 7;

[0173] in, This is the column vector of hysteresis parameters corresponding to the i-th isothermal sample; For the first The temperature of a constant-temperature sample; , This represents the total number of isothermal samples used for parameter mapping modeling; the parameters in the vector are arranged in a fixed order to ensure consistency of fields in the parameter sample library, training output, and prediction results.

[0174] Step S2: Establish a parameterized model of the hysteresis loop and identify the isothermal parameters.

[0175] During quasi-static compression, the stress response of the metal-rubber can be considered as a combination of a nonlinear elastic skeleton, a loading-unloading opening, and path-dependent hysteresis correction. First, a bias reference point is introduced. Addressing the reference point offset issue during the initial stage of small strain. This offset reference point is used to distinguish between the initial contact adjustment section and the stable bearing section, avoiding misidentification of unstable responses such as clamping gap elimination, weak contact compaction, and residual friction lock release as main loop skeleton features.

[0176] In offset coordinates, the principal loop skeleton is described by superimposing linear and power-law reinforcement terms. This principal loop skeleton reflects the continuous reinforcement response of the metal-rubber during compression, caused by the gradual participation of contact points in load bearing, local contact compaction, and the expansion of the load bearing path. Subsequently, the unloading opening parameters are identified based on the proportional relationship between the ideal loading reference branch and the ideal unloading reference branch. and according to Conversion of equivalent frictional opening coefficient Finally, to address the path deviation between the actual unloading branch and the ideal unloading branch, a hysteresis correction term is established and identified. Parameters such as these.

[0177] Thus, a set of hysteresis parameter vectors is obtained at each isothermal point. The different parameters in this parameter vector correspond to the bias reference state, the main loop skeleton, the unloading opening, and the unloading hysteresis correction feature, respectively. This allows for the characterization of the main morphology of the complete hysteresis loop with fewer parameters, such as... Figure 3 As shown.

[0178] Step S3 includes the following steps;

[0179] Step S31: Construct temperature state characteristics; assuming the lowest temperature covered by the isothermal experimental data is... The highest temperature is Then, the temperature T is normalized:

[0180] Formula 8;

[0181] in, This is the normalized temperature.

[0182] Step S32: Introduce temperature range gate functions; to describe the stage-specific characteristics of parameter evolution in the intermediate temperature transition region and the high temperature enhancement region, a continuous gate function is introduced. :

[0183] when hour, ;

[0184] when hour, , ;

[0185] when hour, ;

[0186] in, The starting temperature for the temperature transition zone. The temperature at which the temperature transition is fully activated is defined; this continuous gate function is used to describe the smooth change process of temperature effect from inactive to fully activated.

[0187] Step S33: Construct temperature region enhancement feature vector; Based on normalized temperature and temperature region gate function, construct temperature state feature vector:

[0188] Formula 9;

[0189] in, Used to describe the continuous changes in the mesothermal transition zone. This feature is used to describe the activation level of the high-temperature enhancement region; this feature construction enables the regression model to learn the parameter variation trends in different temperature regions more stably under small sample conditions.

[0190] Step S34: Perform parameter scaling transformation; for parameters that are always positive and change significantly with temperature, perform the transformation using a logarithmic form relative to the reference temperature parameter value:

[0191] Formula 10;

[0192] For bias parameters or parameters with small variations, the transformation is performed using the difference relative to the reference temperature parameter value:

[0193] Formula 11;

[0194] in, Represents the first element in the hysteresis parameter vector. The values ​​of the parameters to be predicted at temperature T; This indicates that the parameter is at the reference temperature T. ref The following reference values; For the first The regression output of each parameter after scaling transformation; after prediction, the actual physical parameters are obtained through corresponding inverse transformation; this process can reduce the impact of differences in the dimensions and numerical scales of different parameters on the regression process and reduce non-physical prediction values.

[0195] Step S35: Establish a weighted kernel ridge regression model. The transformation amount for any objective parameter... Weighted kernel ridge regression was used to establish its relationship with temperature state characteristics. Mapping relationship between them:

[0196] Formula 12;

[0197] in, Target temperature state characteristics Next Predicted values ​​of the parameter transformation quantities; The total number of training samples; Number the training samples; For the first Temperature state feature vector of each training sample; For the first The parameter mapping model corresponds to the first parameter mapping model. The regression coefficients of each training sample; For target features and the first Kernel function values ​​between features of each training sample; , The parameter dimension of the hysteresis parameter vector; the kernel function uses the radial basis function:

[0198] Formula 13;

[0199] in, The kernel width parameter of the basis kernel function is used to control how smoothly the model smooths changes in temperature characteristics;

[0200] Step S36: Determine the regression coefficients through weighted regularization optimization; the objective function of weighted kernel ridge regression includes a sample weighting error term and a regularization term. The sample weights are used to enhance the influence of samples in the intermediate temperature transition region and the high temperature enhancement region on the mapping relationship, and the regularization term is used to suppress overfitting and non-physical oscillations in small sample regression. The kernel width parameter and regularization coefficient are determined through cross-validation within the training set;

[0201] Step S37: Adopt a core parameter-auxiliary parameter hierarchical prediction strategy; First, predict the core parameters that have a significant impact on the morphology of the loop body, including offset stress, foundation bearing stiffness, nonlinear strengthening amplitude, unloading opening parameters, and main hysteresis correction amplitude; Then, use the temperature state characteristics and the core parameter prediction results as input to further predict auxiliary parameters such as offset strain, nonlinear exponent, and unloading shape control parameters; This hierarchical prediction strategy can utilize the constraint relationship between core parameters and auxiliary parameters to reduce the instability caused by simultaneous regression of multiple parameters.

[0202] In step S3, the hysteresis parameter vectors identified at different isothermal points in step S2 are used as output samples, and the temperature state features are used as input samples to establish a continuous temperature-parameter mapping relationship. To avoid excessive smoothing of parameter evolution in the intermediate temperature transition region and the high temperature enhancement region due to using only temperature as a single input, a temperature region gate function is introduced based on the normalized temperature to construct a temperature state feature vector including the normalized temperature, intermediate temperature transition features, high temperature enhancement features, and their interaction terms.

[0203] In the implementation, a weighted kernel ridge regression method is used to establish a nonlinear mapping relationship between temperature state characteristics and hysteresis parameters. For parameters with different dimensions and varying amplitudes, a logarithmic transformation or interpolation transformation is performed first, followed by regression prediction. After prediction, an inverse transformation is performed to recover the physical parameters. The radial basis function is used as the kernel function, and the kernel width and regularization coefficient are determined through cross-validation within the training set. Sample weights are used to enhance the influence of samples in the intermediate temperature transition region and the high temperature enhancement region on the model, thereby improving the stability of parameter prediction under small sample conditions.

[0204] To further reduce the instability of simultaneous multi-parameter prediction, a hierarchical prediction strategy of core parameters and auxiliary parameters is adopted. First, core parameters such as offset stress, foundation bearing stiffness, nonlinear hardening amplitude, unloading opening parameters, and main hysteresis correction amplitudes are predicted. Then, temperature state characteristics and the core parameter prediction results are used as input to predict auxiliary parameters such as offset strain, nonlinear exponent, and unloading shape control parameters. This yields the final temperature-parameter continuous mapping model, and its overall prediction process is as follows: Figure 1 As shown.

[0205] The specific steps of step S4 are as follows:

[0206] Step S41: Set the target uniform constant temperature field temperature The It can be a tested constant temperature point, or it can be an untested intermediate temperature point within the calibrated temperature range;

[0207] Step S42: Set the target temperature Input the temperature state feature construction module to obtain the temperature state features corresponding to the target temperature. ;

[0208] Step S43, Input the temperature-parameter continuous mapping model established in step S3 to obtain the predicted hysteresis parameter vector under the target isothermal field, expressed by the formula:

[0209] Formula 14;

[0210] in, This is a temperature-parameter mapping model obtained through training with isothermal samples.

[0211] Step S44: Convert the prediction parameter vector Substitute the parameterized model of the hysteresis loop established in step S2, and reconstruct the loading branch, ideal unloading branch and hysteresis correction branch under the target isothermal field respectively to obtain the complete stress-strain hysteresis loop;

[0212] Step S45: Output engineering parameters such as peak stress, hysteresis loop area, and mid-section secant stiffness under the target isothermal field. This enables rapid prediction of the hysteresis loop of the metal-rubber under untested target isothermal points within the calibrated temperature range.

[0213] In step S4, for the target uniform isothermal field, let the target temperature be... .Will Input the temperature state feature construction module established in step S3 to obtain the target temperature state feature. Subsequently, Input a temperature-parameter continuous mapping model to obtain the prediction hysteresis parameter vector at the target temperature. Substituting this parameter vector into the hysteresis loop parameterized model established in step S2, the loading branch, ideal unloading branch, and hysteresis correction branch can be reconstructed respectively, ultimately yielding the complete stress-strain hysteresis loop under the target isothermal field, as shown below. Figure 4 As shown.

[0214] When the target temperature When the isothermal point has been tested, the accuracy of the mapping model can be verified by comparing the predicted hysteresis loop with the experimental hysteresis loop; when the target temperature... When dealing with untested intermediate temperature points within the calibration temperature range, this method can be used to predict the hysteresis loop of metal-rubber under intermediate isothermal conditions. Therefore, this method avoids conducting separate cyclic compression tests for each intermediate temperature point.

[0215] The specific steps of step S5 are as follows:

[0216] Step S51: Set the target non-uniform temperature field; the non-uniform temperature field can be determined by steady-state heat transfer analysis, experimental temperature measurement results, or boundary temperature conditions; when the target temperature field is a unidirectional steady-state temperature gradient field, the high-temperature boundary temperature is set to... The low-temperature boundary temperature is Normalized thickness coordinates are ,in Indicates the high temperature boundary. If we denote a low-temperature boundary, then the local temperature along the thickness direction can be expressed as:

[0217] Formula 15;

[0218] in, Thickness location under temperature gradient conditions Local temperature at; subscript Indicates temperature gradient operating conditions; The temperature is the high-temperature boundary temperature, indicated by the subscript. Indicates the high-temperature boundary; Low temperature boundary temperature, subscript Indicates the low-temperature boundary; For normalized thickness coordinates, which are dimensionless quantities; Corresponding to the high temperature boundary, Corresponding to the low temperature boundary;

[0219] Step S52: Discretize the sample thickness direction into The temperature layer; the first The normalized thickness coordinates, local temperature, and local parameter vectors of the layer are respectively expressed as: ,in:

[0220] Formula 16;

[0221] in, Let be the total number of discrete temperature layers in the thickness direction, and , Number the temperature layers; For the first Normalized thickness coordinates of the layer; For the first Local temperature of the layer; This refers to the temperature state characteristics corresponding to this local temperature; For the first The prediction hysteresis parameter vector of the layer;

[0222] Step S53: For any nonlinear or experimentally measured temperature field, the local temperature values ​​of each measuring point or layer node can be directly input into the temperature-parameter mapping model to obtain the local parameter vector at the corresponding location. Therefore, this method is not only applicable to linear temperature gradient fields, but also to other non-uniform temperature fields that can be discretized.

[0223] In step S5, for the temperature gradient condition where one end is heated and the other end is cooled, let the high-temperature boundary temperature of the sample be... The low-temperature boundary temperature is Based on steady-state heat transfer analysis or experimental temperature measurements, a temperature field along the sample thickness is established. If the temperature field can be approximated as a linear distribution, then... Describe the local temperature along the thickness direction; if the temperature field is nonlinearly distributed, the local temperature of each layer can be obtained from actual temperature measurement points or numerical thermal analysis results. The temperature gradient loading device and the temperature field distribution along the thickness direction are shown below. Figure 5 As shown.

[0224] Discretize the sample thickness direction as Each temperature layer is divided into several temperature zones, and the local temperature is calculated for each temperature layer. Subsequently, the local temperature of each layer was controlled. By inputting the temperature-parameter continuous mapping model, the local hysteresis parameters of the corresponding layers can be obtained. This allows us to obtain a local parameter field along the thickness direction, which reflects the differences in the main loop skeleton, unloading opening, and hysteresis correction parameters at different height locations due to varying local temperatures. The local response distribution under the temperature gradient field is shown below. Figure 6 As shown.

[0225] The specific steps of step S6 are as follows:

[0226] Step S61, Local loop reconstruction; The first... Layer local hysteresis parameters Substituting into the hysteresis loop parameterized model, we obtain the local loading branch of this layer. and partial unload branch ;

[0227] Step S62, Nominal Strain Treatment: Considering that the metal wires inside the metal rubber interweave with each other in the thickness direction to form a continuous entangled contact network, the thickness-direction layering is mainly used to describe the differences in local temperature and local parameter states, and does not simplify the sample into an independent regular layered structure. During macroscopic compression, it is assumed that each temperature layer has the same nominal strain history, and the local stress response of each layer is calculated at the same strain point;

[0228] Step S63: Thickness-weighted integration; Perform thickness-weighted integration on the local loading and unloading branches of each layer to obtain the overall equivalent loading curve and overall equivalent unloading curve under the target non-uniform temperature field:

[0229] Formula 17;

[0230] Formula 18;

[0231] in, and The equivalent loading curve and equivalent unloading curve are shown respectively under temperature gradient conditions. and The first A temperature layer at nominal strain Predicted loading stress and predicted unloading stress at the location; For the first Dimensionless thickness integral weight; The total number of temperature layers, and satisfying the following conditions: .

[0232] Step S64: Determine the integral weight; the integral weight in the thickness direction can be a trapezoidal integral weight, in which the weights of the two end layers are smaller and the weights of the middle layers are equal; or the weights can be determined according to the actual temperature layer thickness or non-uniform layer spacing.

[0233] Step S65: Obtain the complete equivalent hysteresis loop; from the overall equivalent loading curve and overall equivalent unloading curve Together they form the complete equivalent stress-strain hysteresis loop of metallic rubber under a non-uniform temperature field.

[0234] Step S6: Reconstruct the local hysteresis loop and integrate to obtain the global equivalent loop. Specifically, this involves converting the local parameters of each temperature layer... Substituting into the hysteresis loop parameterized model, the local loading branches of this layer are obtained respectively. and partial unload branch It should be noted that the thickness-direction layering is only used to describe local temperature and parameter differences in a continuous entangled network, and does not simplify the metal-rubber sample into a regular layered structure that is independent of each other. Considering that the metal wires interweave with each other in the thickness direction and form a continuous contact network, this embodiment assumes that each temperature layer has the same nominal strain history during macroscopic compression.

[0235] At the same nominal strain point, thickness-weighted integrals are performed on the local loading and unloading branches of each temperature layer to obtain the overall equivalent loading and unloading curves under the temperature gradient field. The integration weights can be trapezoidal or set according to the layer thickness or the actual discretization method. Finally, the overall equivalent loading and unloading curves form the complete equivalent hysteresis loop under the target temperature gradient field, as shown below. Figure 7 As shown.

[0236] The specific steps of step S7 are as follows:

[0237] Step S71: Calculate the peak stress; the peak stress is used to characterize the overall load-bearing capacity of the metal rubber at the point of maximum compressive strain under the target temperature field.

[0238] Step S72: Calculate the hysteresis loop area; the hysteresis loop area is obtained by the area integral between the overall equivalent loading curve and the overall equivalent unloading curve, and is used to characterize the energy dissipation capacity during the cyclic compression process.

[0239] Step S73: Calculate the mid-section secant stiffness; The mid-section secant stiffness is calculated based on the stress difference and strain difference in the strain interval of the loading branch, and is used to characterize the stiffening characteristics of the main loading branch.

[0240] Step S74: Calculate the curve prediction error; compare the predicted hysteresis loop with the experimental hysteresis loop or the reference hysteresis loop, use the normalized root mean square error of peak stress and the normalized mean absolute error of peak stress to evaluate the stress amplitude prediction deviation, and use the hysteresis loop area error to evaluate the hysteresis energy consumption prediction deviation.

[0241] Step S75: Output the prediction results; the final output includes the complete hysteresis loop, peak stress, hysteresis loop area, mid-section secant stiffness, and error evaluation results under the target temperature field, providing a basis for the performance evaluation and structural parameter design of metal-rubber components under complex thermal environments.

[0242] Step S7, when evaluating the prediction results under multiple temperature fields, calculates engineering indices such as peak stress, hysteresis loop area, and secant stiffness for the predicted hysteresis loops obtained under uniform isothermal and non-uniform temperature fields. Peak stress characterizes the overall load-bearing capacity of the metal-rubber under the target temperature field; hysteresis loop area characterizes the energy dissipation capacity during cyclic loading; and secant stiffness characterizes the stiffening characteristics of the main loading branch. When experimental loops are available for comparison, the normalized root mean square error of peak stress, the normalized mean absolute error of peak stress, and the loop area error are further calculated. These indices can be used to evaluate the accuracy of the prediction model in terms of overall stress amplitude, loop morphology, and hysteresis energy dissipation. The prediction errors under isothermal fields and the prediction accuracy of engineering indices such as peak stress, hysteresis loop area, and secant stiffness under temperature gradient fields are as follows: Figure 8 As shown.

[0243] In one specific embodiment, isothermal cyclic compression data at 22℃, 50℃, 100℃, 150℃, 200℃, 250℃, and 300℃ were used as calibration samples to identify hysteresis parameter vectors at each temperature point and establish a temperature-parameter continuous mapping model with enhanced temperature zone characteristics. For untested isothermal points, the target temperature was directly input into the mapping model to predict the corresponding hysteresis parameters and reconstruct the complete hysteresis loop. For temperature gradient conditions such as room temperature to 100℃, room temperature to 200℃, and room temperature to 300℃, the temperature field in the thickness direction was discretized into multiple local temperature layers, and the local parameters and local loops of each layer were predicted respectively. The overall equivalent hysteresis loop was obtained by thickness direction integration. Verification results show that this method can predict the main load-bearing response, loading stiffening characteristics, and unloading return morphology of the hysteresis loop of metal rubber under multi-temperature field conditions.

Claims

1. A method for predicting hysteresis loops in multi-temperature fields of metal-rubber based on model constraint parameter mapping, characterized in that: The multiple temperature fields include a uniform isothermal field and a non-uniform temperature field, wherein the non-uniform temperature field is a temperature gradient field distributed along the thickness direction of the metal rubber sample. The prediction method first establishes a hysteresis loop parameterized model based on cyclic compression experimental data at multiple discrete isothermal points and identifies the hysteresis parameter vectors under different isothermal conditions. Then, temperature state features are constructed, and a temperature-parameter continuous mapping model is established using a weighted kernel ridge regression method with enhanced temperature region features. For the isothermal field as the prediction target, the target temperature is input into the mapping model and the complete hysteresis loop is reconstructed. For the temperature gradient field as the prediction target, the temperature field distributed along the thickness direction is discretized into multiple local temperature layers, and the local hysteresis parameters and local hysteresis loops of each layer are predicted respectively. Then, the overall equivalent hysteresis loop is obtained by weighted integration along the thickness direction.

2. The method for predicting hysteresis loops in multiple temperature fields of metal-rubber by model constraint parameter mapping according to claim 1, characterized in that: The prediction method Includes the following steps; Step S1: Obtain hysteresis loop data of cyclic compression of metal rubber at multiple isothermal points, and perform stress-strain conversion and steady-cycle extraction on the load-displacement data; Step S2: Establish a parameterized model of the metal rubber hysteresis loop, decompose the macroscopic hysteresis response into the main loop skeleton, unloading opening and hysteresis correction component, and identify the hysteresis parameter vector at each isothermal point. Step S3: Based on the identification parameters under discrete isothermal points, construct temperature state features, and establish a temperature-parameter continuous mapping model using the weighted kernel ridge regression method with enhanced temperature region features. Step S4: For the target uniform isothermal field, input the target temperature into the temperature-parameter continuous mapping model to obtain the predicted parameter vector at that temperature, and substitute it into the hysteresis loop parameterization model to obtain the complete hysteresis loop under the target isothermal condition. Step S5: For the target non-uniform temperature field, establish the local temperature distribution in the thickness direction of the sample based on the temperature boundary or temperature field distribution, input the local temperature of each layer into the temperature-parameter continuous mapping model, and obtain the local hysteresis parameter field in the thickness direction. Step S6: Substitute the local hysteresis parameters of each thickness layer into the hysteresis loop parameterization model, reconstruct the local loading and unloading branches of each layer respectively, and perform thickness-weighted integration on the local response of each layer based on the same nominal strain history to obtain the overall equivalent hysteresis loop under non-uniform temperature field. Step S7: Output peak stress, hysteresis loop area, mid-section secant stiffness, root mean square error, and mean absolute error indices to evaluate the accuracy of the predicted hysteresis loop under multiple temperature fields.

3. The method for predicting hysteresis loops in multiple temperature fields of metal-rubber based on model constraint parameter mapping according to claim 2, characterized in that: The specific steps of step S1 are as follows: Step S11: Prepare a metal rubber sample; specifically, the metal rubber sample is obtained by spiral winding, fixed pitch treatment, laying and entanglement and cold pressing of metal wire, so that the metal rubber sample forms an entangled network structure composed of multi-point contact, spatial interlocking and multiple pores inside. Step S12: Conduct a constant temperature cyclic compression test; specifically: place the metal rubber sample in multiple constant temperature environments for sufficient heat preservation, and apply displacement-controlled cyclic compression load after the thermal state stabilizes, and collect the load and displacement signals during the loading process. Step S13: Perform data preprocessing; Specifically, based on the cross-sectional area and molding height of the specimen, the load-displacement data are converted into stress-strain data; cycles with significant initial clamping gap effects are eliminated, and the loading and unloading branches in the stable cycles are selected as hysteresis loop modeling data; Step S14: Obtain the isothermal sample library; specifically, the mean value of multiple sets of repeated test hysteresis loops at the same isothermal point is processed to obtain the average hysteresis loop at that temperature, and the repeatability error range can be obtained at the same time, providing basic data for subsequent parameter identification and model verification.

4. The method for predicting hysteresis loops in multiple temperature fields of metal-rubber by model constraint parameter mapping according to claim 2, characterized in that: In step S2, in the formula of the metal-rubber sample related model, the horizontal line above the symbol represents the offset coordinate after transformation relative to the offset reference point; the "^" above the symbol represents the predicted value obtained by the mapping model; the superscript "T" represents the transpose of a vector or matrix; the subscript p represents the offset reference state; the subscript s represents the main loop skeleton; the subscripts L and U represent the loading branch and unloading branch respectively; the superscript g represents the temperature gradient condition; the subscripts i and j represent the isothermal training sample number; the subscript l represents the temperature layer number in the thickness direction; and the subscript q represents the parameter component number in the hysteresis parameter vector. Unless otherwise specified, the engineering strain and each proportional parameter are dimensionless quantities, and the units of stress, stiffness parameters and hysteresis correction amplitude are MPa, and the unit of temperature is °C. The specific steps for establishing a parameterized model of the hysteresis loop of metal-rubber include: Step S21: Establish a hysteresis response decomposition model; under one-dimensional axial compression conditions, the macroscopic deformation is represented by engineering strain ε and the macroscopic response is represented by stress σ. The hysteresis loop of the metal rubber is decomposed into the main loop skeleton component, the unloading opening component and the hysteresis correction component. The main loop skeleton component is used to describe the nonlinear load-bearing capacity during compression; the unloading opening component is used to describe the opening degree between the loading branch and the unloading branch; and the hysteresis correction component is used to describe the path-related deviation during unloading. Step S22: Introduce a dual-bias reference point; the metal-rubber sample exhibits an unstable response in the initial stage of small strain, so a bias reference point is introduced. The offset coordinates are used to characterize the equivalent offset of the stable principal loop relative to the geometric origin; the offset coordinates are expressed as: Official 1; in, The strain after biasing. The stress after biasing, For bias reference strain, This is the bias reference stress; Step S23: Establish the main loop skeleton model; the metal-rubber sample contact network exhibits initial bearing, gradual compaction, and nonlinear reinforcement characteristics during compression. The main loop skeleton is represented as a superposition of linear terms and power function reinforcement terms: Official 2 in, The stress in the principal loop skeleton in the offset coordinate system at temperature T; These are the basic bearing stiffness parameters for the small deformation stage. For nonlinear enhancement amplitude parameters, is a nonlinear exponential parameter, and is a dimensionless positive number; This indicates that the model is used for the stable bearing range after the offset reference point; Step S24: Establish the unloading opening model: Introduce unloading opening parameters or equivalent frictional opening coefficient This is used to characterize the degree of opening between the loading branch and the unloading branch; The relationship between the experimental unloading branch and the ideal loading reference branch within the unloading stable region was identified. Depend on The conversion yields: Official 3; in, The scaling factor of the ideal unloading branch relative to the ideal loading reference branch at temperature T; The equivalent friction opening coefficient is obtained by converting the hysteresis loop opening; the two are two equivalent expressions of the same unloading opening feature, and are not independent hysteresis parameters. Step S25: Establish a hysteresis correction model; for the path deviation of the actual unloading branch relative to the ideal unloading branch, introduce a hysteresis correction term. The hysteresis correction term is described by both the amplitude parameter and the shape control parameter, and the relevant formula is: Official 4; And there are: Official 5; Official 6; in, The hysteresis correction stress is the actual unloading branch at temperature T relative to the ideal unloading reference branch. To normalize the unloading path variable; This represents the peak strain corresponding to the unloading start point. The minimum strain corresponding to the unloading endpoint; when The time indicates the unloading start point. The time indicates the end point of the unloading process; and The hysteresis correction amplitude parameter; For the initial hysteresis bias of unloading, the exponential decay function is used. The attenuation control parameters for unloading hysteresis bias; For the shape control parameters of the middle and later stages of unloading; Step S26: Output hysteresis parameter vector; integrate the bias reference state, main loop skeleton, unloading opening, and hysteresis correction parameters at each isothermal point. The hysteresis parameter vector is obtained below: Formula 7; in, This is the column vector of hysteresis parameters corresponding to the i-th isothermal sample; For the first The temperature of a constant-temperature sample; , This represents the total number of isothermal samples used for parameter mapping modeling; the parameters in the vector are arranged in a fixed order to ensure consistency of fields in the parameter sample library, training output, and prediction results.

5. The method for predicting hysteresis loops in multiple temperature fields of metal-rubber by model constraint parameter mapping according to claim 2, characterized in that: Step S3 includes the following steps; Step S31: Construct temperature state characteristics; assuming The lowest temperature covered by the isothermal experiment data is The highest temperature is Then, the temperature T is normalized: Official 8; in, Normalized temperature; Step S32: Introduce temperature range gate functions; to describe the stage-specific characteristics of parameter evolution in the intermediate temperature transition region and the high temperature enhancement region, a continuous gate function is introduced. : when hour, ; when hour, , ; when hour, ; in, The starting temperature for the temperature transition zone, The temperature at which the temperature transition is fully activated is defined; a continuous gate function is used to describe the smooth transition of the temperature effect from inactive to fully activated. Step S33: Construct temperature region enhancement feature vector; Based on normalized temperature and temperature region gate function, construct temperature state feature vector: Official 9; in, Used to describe the continuous changes in the intermediate temperature transition zone. Used to describe the activation level of the high-temperature enhancement region; Step S34: Perform parameter scaling transformation; for parameters that are always positive and change significantly with temperature, perform the transformation using a logarithmic form relative to the reference temperature parameter value: Official 10; For bias parameters or parameters with small variations, the transformation is performed using the difference relative to the reference temperature parameter value: Official 11; in, Represents the first element in the hysteresis parameter vector. The values ​​of the parameters to be predicted at temperature T; This indicates that the parameter is at the reference temperature T. ref The following reference values; For the first The regression output of each parameter after scaling transformation; after prediction, the actual physical parameters are obtained through corresponding inverse transformation. Step S35: Establish a weighted kernel ridge regression model; for any objective parameter, the transformation amount... Weighted kernel ridge regression was used to establish its relationship with temperature state characteristics. Mapping relationship between them: Official 12; in, Target temperature state characteristics Next Predicted values ​​of the parameter transformation quantities; The total number of training samples; Number the training samples; For the first Temperature state feature vector of each training sample; For the first The parameter mapping model corresponds to the first parameter mapping model. The regression coefficients of each training sample; For target features and the first Kernel function values ​​between features of each training sample; , The parameter dimension of the hysteresis parameter vector; the kernel function uses the radial basis function: Official 13; in, The kernel width parameter of the basis kernel function is used to control how smoothly the model smooths changes in temperature characteristics; Step S36: Determine the regression coefficients through weighted regularization optimization; the objective function of weighted kernel ridge regression includes a sample weighting error term and a regularization term. The sample weights are used to enhance the influence of samples in the intermediate temperature transition zone and the high temperature enhancement zone on the mapping relationship, and the regularization term is used to suppress overfitting and non-physical oscillations in small sample regression; the kernel width parameter and the regularization coefficient are determined through cross-validation within the training set. Step S37: Adopt a core parameter-auxiliary parameter hierarchical prediction strategy; First, predict the core parameters that have a significant impact on the morphology of the loop body, including offset stress, foundation bearing stiffness, nonlinear strengthening amplitude, unloading opening parameters, and main hysteresis correction amplitude; Then, use the temperature state characteristics and the core parameter prediction results as input to further predict the offset strain, nonlinear exponent, and unloading shape control parameters.

6. The method for predicting hysteresis loops in multiple temperature fields of metal-rubber by model constraint parameter mapping according to claim 2, characterized in that: The specific steps of step S4 are as follows: Step S41: Set the target uniform constant temperature field temperature The This refers to either a tested constant temperature point or an untested intermediate temperature point within the calibrated temperature range. Step S42: Set the target temperature Input the temperature state feature construction module to obtain the temperature state features corresponding to the target temperature. ; Step S43, Input the temperature-parameter continuous mapping model established in step S3 to obtain the predicted hysteresis parameter vector under the target isothermal field, expressed by the formula: Official 14; in, This is a temperature-parameter mapping model obtained through training with isothermal samples; Step S44: Convert the prediction parameter vector Substitute the parameterized model of the hysteresis loop established in step S2, and reconstruct the loading branch, ideal unloading branch and hysteresis correction branch under the target isothermal field respectively to obtain the complete stress-strain hysteresis loop; Step S45: Output the peak stress, hysteresis loop area, and mid-section secant stiffness engineering parameters under the target isothermal field.

7. The method for predicting hysteresis loops in multiple temperature fields of metal-rubber by model constraint parameter mapping according to claim 2, characterized in that: The specific steps of step S5 are as follows: Step S51: Set the target non-uniform temperature field; the non-uniform temperature field is determined by steady-state heat transfer analysis, experimental temperature measurement results, or boundary temperature conditions; when the target temperature field is a unidirectional steady-state temperature gradient field, the high-temperature boundary temperature is set to... The low-temperature boundary temperature is Normalized thickness coordinates are ,in Indicates the high temperature boundary. To represent a low-temperature boundary, the local temperature along the thickness direction is expressed as: Official 15; in, Thickness location under temperature gradient conditions Local temperature at; subscript Indicates temperature gradient operating conditions; The temperature is the high-temperature boundary temperature, indicated by the subscript. Indicates the high-temperature boundary; Low temperature boundary temperature, subscript Indicates the low-temperature boundary; For normalized thickness coordinates, which are dimensionless quantities; Corresponding to the high temperature boundary, Corresponding to the low temperature boundary; Step S52: Discretize the sample thickness direction into The temperature layer; the first The normalized thickness coordinates, local temperature, and local parameter vectors of the layer are respectively expressed as: ,in: Official 16; in, Let be the total number of discrete temperature layers in the thickness direction, and , Number the temperature layers; For the first Normalized thickness coordinates of the layer; For the first Local temperature of the layer; This represents the temperature state characteristics corresponding to this local temperature. For the first The prediction hysteresis parameter vector of the layer; Step S53: For any nonlinear or experimentally measured temperature field, the local temperature values ​​of each measuring point or each layer node can be directly input into the temperature-parameter mapping model to obtain the local parameter vector at the corresponding location.

8. The method for predicting hysteresis loops in multiple temperature fields of metal-rubber by model constraint parameter mapping according to claim 2, characterized in that: The specific steps of step S6 are as follows: Step S61, Local loop reconstruction; The first... Layer local hysteresis parameters Substituting into the hysteresis loop parameterized model, we obtain the local loading branch of this layer. and partial unload branch ; Step S62: Same nominal strain treatment; During the macroscopic compression process, it is assumed that each temperature layer has the same nominal strain history, and the local stress response of each layer is calculated at the same strain point; Step S63: Thickness-weighted integration; Perform thickness-weighted integration on the local loading and unloading branches of each layer to obtain the overall equivalent loading curve and overall equivalent unloading curve under the target non-uniform temperature field: Official 17; Official 18; in, and The equivalent loading curve and equivalent unloading curve are shown respectively under temperature gradient conditions. and The first A temperature layer at nominal strain Predicted loading stress and predicted unloading stress at the location; For the first Dimensionless thickness integral weight; The total number of temperature layers, and satisfying the following conditions: ; Step S64: Determine the integral weight; The integral weight in the thickness direction adopts a trapezoidal integral weight, in which the weights of the two end layers are smaller and the weights of the middle layers are equal; The weights can also be determined according to the actual temperature layer thickness or non-uniform layer spacing. Step S65: Obtain the complete equivalent hysteresis loop; from the overall equivalent loading curve and overall equivalent unloading curve Together they form the complete equivalent stress-strain hysteresis loop of metallic rubber under a non-uniform temperature field.

9. The method for predicting hysteresis loops in multiple temperature fields of metal-rubber by model constraint parameter mapping according to claim 2, characterized in that: The specific steps of step S7 are as follows: Step S71: Calculate the peak stress; Peak stress is used to characterize the overall load-bearing capacity of rubber-metal at the point of maximum compressive strain under a target temperature field. Step S72: Calculate the hysteresis loop area; the hysteresis loop area is obtained by the area integral between the overall equivalent loading curve and the overall equivalent unloading curve, and is used to characterize the energy dissipation capacity during the cyclic compression process. Step S73: Calculate the mid-section secant stiffness; The mid-section secant stiffness is calculated based on the stress difference and strain difference in the strain interval of the loading branch, and is used to characterize the stiffening characteristics of the main loading branch. Step S74: Calculate the curve prediction error; compare the predicted hysteresis loop with the experimental hysteresis loop or the reference hysteresis loop, use the normalized root mean square error of peak stress and the normalized mean absolute error of peak stress to evaluate the stress amplitude prediction deviation, and use the hysteresis loop area error to evaluate the hysteresis energy consumption prediction deviation. Step S75: Output the prediction results; the final output includes the complete hysteresis loop, peak stress, hysteresis loop area, mid-section secant stiffness, and error evaluation results under the target temperature field, providing a basis for the performance evaluation and structural parameter design of metal-rubber components under complex thermal environments.

10. The method for predicting hysteresis loops in multiple temperature fields of metal-rubber by model constraint parameter mapping according to claim 9, characterized in that: Step S7, when evaluating the prediction results under multiple temperature fields, calculates the peak stress, hysteresis loop area, and mid-section secant stiffness engineering indices for the predicted hysteresis loops obtained under uniform isothermal and non-uniform temperature fields, respectively. The peak stress characterizes the overall load-bearing capacity of the metal rubber under the target temperature field; the hysteresis loop area characterizes the energy dissipation capacity during cyclic loading; and the mid-section secant stiffness characterizes the stiffening characteristics of the main loading branch. Furthermore, when experimental loops are available for comparison, the normalized root mean square error of the peak stress, the normalized average absolute error of the peak stress, and the loop area error are calculated.

Images