A cyclic plasticity constitutive stress updating method based on a learnable hardening function

Abstract

The application discloses a cyclic plastic constitutive stress updating method based on a learnable hardening function, relates to the technical field of material analysis of ships and ocean engineering, and comprises the following steps: step 1, strain-stress hysteresis data of a material under cyclic loading is obtained; step 2, the strain-stress data is preprocessed; step 3, a neural hardening function module is constructed in a cyclic plastic constitutive model; step 4, the neural hardening function module is combined with a cyclic plastic stress updating process; step 5, a loss function is constructed by comparing a complete predicted stress sequence with a real stress sequence in training data; and step 6, the gradient of the loss function to trainable parameters is calculated by using an automatic differentiation method. The method solves the problems that the fixed hardening function form and the limited empirical parameters in a traditional cyclic elastic-plastic constitutive model are not flexible enough in expressing the complex cyclic hardening evolution law and the parameter calibration is easily affected by initial values and parameter coupling.

Description

Technical Field

[0001] This invention relates to the field of ship and marine engineering materials analysis technology, and in particular to a method for updating cyclic plastic constitutive stress based on a learnable hardening function. Background Technology

[0002] The elastoplastic response of metallic materials under cyclic loading is a crucial foundation for structural strength verification, fatigue life assessment, and finite element numerical simulation. For ship and marine engineering structures, materials typically exhibit significant hysteresis behavior, Bauschinger effect, cyclic hardening, or softening path-related characteristics during repeated tension / compression, loading-unloading, and reverse loading processes. These phenomena are closely related to the evolution of the material's yield surface, isotropic hardening, kinematic hardening, and cumulative plastic deformation. Therefore, establishing an elastoplastic constitutive model that accurately describes the cyclic stress-strain response is essential.

[0003] Existing constitutive models of cyclic plasticity are typically built upon the classical elastoplastic theory framework comprised of yield criteria, flow rules, and hardening rules. Among these, the von Mises yield criterion and the Chaboche nonlinear kinematic hardening model are widely used in numerical simulations of cyclic plasticity in metals due to their clear mathematical forms and physical meanings. These models usually employ one or more back stress components to describe the translation of the yield surface and combine them with isotropic hardening terms to characterize the expansion or contraction of the yield surface, thereby simulating the cyclic hysteresis behavior of materials.

[0004] However, the hardening laws in traditional Chaboche-type models often rely on pre-assumed functional forms. For example, isotropic hardening is often expressed using exponential or Voce forms, and the hardening modulus and dynamic hardening restitution coefficient in kinematic hardening are usually set as constants or a few empirical parameters. For metallic materials under complex cyclic loading, fixed-form hardening functions are difficult to accurately characterize the differences in hardening evolution under different plastic accumulation stages, different loading amplitudes, and different reverse loading states. Although the fitting ability can be improved by adding back stress components, this further exacerbates problems such as parameter coupling, sensitivity to initial values, non-uniqueness of identification results, and difficulty in optimization convergence, thus limiting the applicability of the model in predicting complex cyclic plastic responses.

[0005] With the development of machine learning methods, data-driven constitutive models that directly utilize neural networks to establish the mapping relationship between strain history and stress response have gradually gained attention. Such methods can, to some extent, avoid the manual setting of complex analytical functions and possess strong nonlinear fitting capabilities. However, completely black-box stress mapping typically relies on a large number of data samples covering different stress states and loading paths. When training data comes only from conventional uniaxial cyclic tests or a limited number of numerical loading paths, the model is prone to exhibiting good fitting performance for known loading conditions but insufficient predictive ability for unseen loading paths. Furthermore, if the model does not explicitly introduce yield conditions, plastic flow, hardening variables, and stress update mechanisms, its prediction results may not satisfy the fundamental physical constraints of elastoplastic theory, thereby weakening the stability and consistency of the stress update process. Therefore, under finite cyclic stress-strain data conditions, a metal elastoplastic constitutive modeling method is still needed that can enhance the expressive power of the hardening function within the framework of elastoplastic theory while taking into account physical constraints and stress update stability.

[0006] The above content is only used to help understand the technical solution of the present invention and does not represent an admission that the above content is prior art. Summary of the Invention

[0007] The purpose of this invention is to address the shortcomings of existing technologies by proposing a cyclic plastic constitutive stress update method based on a learnable hardening function. This method combines elastoplastic theory with neural networks to overcome the limitations of traditional fixed analytical hardening functions. It solves the problems of insufficient flexibility in expressing complex cyclic hardening evolution laws by fixed hardening function forms and finite empirical parameters in traditional cyclic elastoplastic constitutive models, and the susceptibility of parameter calibration to the influence of initial values ​​and parameter coupling.

[0008] To achieve the above objectives, the present invention adopts the following technical solution: A method for updating cyclic plastic constitutive stress based on a learnable hardening function includes the following steps: Step 1: Obtain strain-stress hysteresis data of the material under cyclic loading conditions and construct a sample database for model training; Step 2: Preprocess the strain-stress data, remove invalid and duplicate data points, and downsample data points that exceed a preset threshold for non-critical features; Step 3: Construct a neural hardening function module in the cyclic plastic constitutive model so that isotropic hardening, kinematic hardening growth, and kinematic hardening recovery can adaptively evolve according to the plastic state of the material; Step 4: Combine the neural sclerosis function module with the cyclic plastic stress update process to establish a return mapping stress update module that includes elastic prediction, yield judgment, plastic multiplier solution, back stress component update and stress correction. Step 5: Compare the complete predicted stress sequence with the real stress sequence in the training data, and use the normalized weighted mean square error as the loss function; Step 6: Calculate the gradient of the loss function with respect to the trainable parameters using the automatic differentiation method, and update the learnable parameters through the optimizer; Step 7: Repeat the stress integral, loss calculation and parameter update process within the preset number of training rounds, and record the loss value, prediction error and fitting accuracy after each training round; when the loss of the current round is lower than that of all previous rounds, save the current model parameters as the optimal model parameters, and restore the model with the minimum loss after training as the final neural sclerosis constitutive model.

[0009] Furthermore, in step 2, key features include changes in the peak value, valley value, and loading direction in the cyclic hysteresis curve.

[0010] Furthermore, step 3 includes the following steps: Step 301: Based on cumulative plastic strain Construct an isotropic hardening function The formula is as follows: ; in, These are non-negative trainable weights. It is a set of cumulative plastic strain change smoothing basis functions; Step 302: Based on the absolute value of each back stress component and cumulative plastic strain The growth factor of the back stress component under the current state is calculated using the following formula: ; in, It is the first The kinematic hardening growth factor of each back stress component under the current state It is the corresponding learnable motion hardening growth function; Step 303: Based on the absolute value of each back stress component and cumulative plastic strain The dynamic hardening recovery coefficient of the back stress component under the current state is calculated using the following formula: ; in, For the first The dynamic hardening restitution coefficient of each back stress component under the current state. This is the corresponding learnable motion hardening recovery function.

[0011] Furthermore, in step 301, The formula is as follows: ; in, It is a smoothing parameter. It is the first The node positions of each basis function.

[0012] Furthermore, in step 302, The formula is as follows: ; in, It is the first The basic kinematic hardening growth value of each back stress component It is the bounded amplification factor of the learnable motion hardening growth function. It is by and The residual term is formed by the smoothing basis functions; in, The formula is as follows: ; ; ; in, It is the first The absolute value of each back stress component is a correction value for the kinematic hardening growth function. It is control Follow The changing residual weights are based on the first... Absolute value of each back stress component For the input of the first A smooth basis function, L It is the number of smooth basis functions for the back stress; It is the cumulative plastic strain For the first A correction term for the motion hardening growth function, It is control Follow Changing residual weights It is based on the cumulative plastic strain For the input of the first A smooth basis function, M It is the number of smooth basis functions for cumulative plastic strain.

[0013] Furthermore, in step 303, The formula is as follows: ; in, It is the first The basic dynamic hardening recovery value of each back stress component It is the bounded amplification factor of the learnable motion hardening recovery function. It is by and The residual term is formed by the smoothing basis functions; in, The formula is as follows: ; ; ; in, It is the first The absolute value of each back stress component is the correction value of the kinematic hardening restitution function. It is control Follow Changing residual weights Therefore, the first Absolute value of each back stress component For the input of the first A smooth basis function, L It is the number of smooth basis functions for the back stress; It is the cumulative plastic strain For the The correction term for the motion hardening recovery function, It is control Follow Changing residual weights It is based on the cumulative plastic strain For the input of the first A smooth basis function, M It is the number of smooth basis functions for cumulative plastic strain.

[0014] Furthermore, in step 4, a return-mapped stress update module is established, which includes elastic prediction, yield judgment, plastic multiplier solution, back stress component update, and stress correction, enabling the learning of isotropic hardening functions. Learnable motion hardening growth function and learnable motion hardening recovery function It is invoked during the stepwise stress integration process and outputs the current neural isotropic sclerosis value. Growth coefficient of exercise hardening and dynamic hardening recovery coefficient ;in, Used for yield strength calculation and Used for semi-implicit updates of each back stress component, and further involved in the plastic return mapping consistency solution.

[0015] Furthermore, step 4 includes the following steps: Step 401: Before stepwise integration of the cyclic strain path, the initial plastic strain is... Initial cumulative plastic strain and the initial values ​​of each back stress component , and All are set to zero, and the initial back stress resultant is set to zero. Set to zero as the starting state for subsequent elastic prediction, yield judgment, plastic multiplier solution, back stress component update and stress correction return mapping stress update; Step 402: In the (n+1)th loading step, read the total strain of the current loading step. And call the plastic strain updated in the previous loading step. Cumulative plastic strain and each back stress component , and This serves as the historical state input for the current loading step elasticity prediction and yield judgment; Step 403: In each loading step, the difference between the total strain of the current loading step and the updated plastic strain of the previous loading step is taken as the current elastic test strain, and the test stress is calculated based on the elastic modulus, as follows: ; in, This is the elastic predicted stress obtained assuming no new plastic deformation occurs in the current loading step. It is the elastic modulus. For the total strain of the current loading step, This represents the plastic strain updated in the previous loading step. Step 404: Read the back stress components updated in the previous loading step. , and And these are superimposed to form the resultant back stress of the previous moment. ,Right now ;Will and The difference between them is used as the relative stress of the test. ,Right now The cumulative plastic strain updated in the previous loading step will be used to adjust the cumulative plastic strain. Input can learnable isotropic hardening function To obtain the isotropic sclerosis value of the nerve required for the current trial yield judgment. and the initial yield stress and Together as the yield strength of the current loading step test ; Step 405: When When the value is not greater than zero, the current loading step is determined as an elastic step; when... When the value is greater than zero, the current loading step is determined to be a plastic step, and the plastic return mapping correction process is initiated. Step 406: When the current loading step is an elastic step, Directly used as the stress of the current loading step Output, while maintaining plastic strain Cumulative plastic strain and each back stress component , and The result remains unchanged, and then the calculation proceeds to the next loading step. Step 407: When the current loading step is the plastic step, according to The sign determines the direction of plastic flow. ;when When positive, the direction of plastic flow is positive; when... When the value is negative, the direction of plastic flow is negative; Step 408: Combine the back stress components from the previous loading step The absolute value and the cumulative plastic strain of the previous loading step Input a learnable motion hardening growth function Get the current loading step number kinematic hardening growth coefficient of each back stress component ; The back stress components of the previous loading step The absolute value and the cumulative plastic strain at the previous moment Input learnable motion hardening recovery function Get the current loading step number Dynamic hardening restitution coefficient of each back stress component ; Step 409: Using plastic multipliers As unknowns, a consistent relationship is established between the cumulative plastic strain, stress, back stress components, and yield strength after plastic correction; where the cumulative plastic strain after plastic correction is... The stress after plastic correction is obtained by reverting the test stress along the direction of plastic flow. Each back stress component is composed of , , , The yield strength after plastic correction is determined by the direction of plastic flow and is determined by the initial yield stress. and To be determined jointly; Step 410: Solve for the plastic multipliers using the bisection method. Establish about The yield uniformity residual function is given by the following formula: ; in, It is the direction of plastic flow. It is the candidate stress after plastic correction. It is the candidate back stress resultant after plastic correction, and it is the candidate isotropic hardening value output by the learnable isotropic hardening function. Step 411: Based on plastic multipliers and plastic flow direction Update plastic strain, i.e. Update the accumulated plastic strain, that is ; Step 412: For the first Each back stress component, based on the back stress component of the previous loading step. Growth coefficient of exercise hardening Dynamic hardening recovery coefficient Plastic multipliers and plastic flow direction Perform a semi-implicit update to obtain the back stress components after the current loading step ends. ,Right now ; Step 413: Superimpose the updated back stress components to obtain the resultant back stress after the current loading step ends. ,Right now This represents the state of the yield surface movement after the current loading step ends, and serves as a historical state variable in the yield test judgment of the next loading step. Step 414: Based on the total strain of the current loading step and the updated plastic strain Recalculate the stress of the current loading step. ,Right now Thus, the stress response after plastic return mapping correction is obtained; Step 415: Accumulate the plastic strain of the current loading step Input can learnable isotropic hardening function The current neural isotropic sclerosis value is obtained. And according to the current step stress Current step back stress resultant Initial yield stress as well as The yield residual for the current step is calculated using the following formula: ; in, It is the final stress returned by the current loading step after mapping correction. It is the back stress after the current loading step ends. It is the initial yield stress. It is the current cumulative plastic strain The isotropic hardening value is obtained after inputting a learnable isotropic hardening function; Step 416: Record the final stress of the current loading step. Plastic strain Cumulative plastic strain Each back stress component , , Back stress resultant Isotropic sclerosis value Growth coefficient of exercise hardening Dynamic hardening recovery coefficient Plastic multipliers The yield residual serves as the data basis for subsequent training evaluation, state variable analysis, and result output. Step 417: Repeat steps 402 to 416 along the complete cyclic strain path until the stress integration of all loading steps is completed, and obtain the complete predicted stress sequence and the corresponding material state variable evolution sequence.

[0016] Furthermore, in step 410, when When this occurs, it indicates that the stress state after plastic correction has returned to the yield surface, at which point... As the lower bound of the bisection interval, and to estimate the upper bound of the bisection interval. The formula is as follows: ; in, It is a positive bias used to ensure that the bisection interval covers the root value of the yield surface. E It is the elastic modulus; like If it is greater than zero, then gradually increase the value. until satisfaction is achieved. and The solution interval; subsequently, in the interval Take the midpoint inside and calculate ;when When this occurs, it indicates that the current amount of plastic correction is insufficient, causing... ;when When this occurs, it indicates that the current plasticity correction has reached or exceeded the yield level, causing... Repeat the above binary search iteration process. After a preset number of iterations, take: As the plastic multiplier of the current loading step, it ensures that the superposition of the stress, back stress components, and yield strength after plastic correction meet the yield consistency condition.

[0017] Furthermore, in step 5, the formula for the loss function is as follows: ; in, It is the number of loading steps. For the first The model predicts stress in each loading step. For the first The actual stress of each loading step The stress normalization scale is derived from the absolute maximum value of the true stress in the training path.

[0018] Compared with the prior art, the beneficial effects of this invention are as follows: By introducing learnable isotropic hardening functions, learnable kinematic hardening growth functions, and learnable kinematic hardening recovery functions into the cyclic plastic constitutive model, adaptive learning of yield strength evolution, back stress growth, and dynamic recovery laws is achieved, reducing the dependence of traditional models on fixed hardening function forms and empirical parameters. This invention does not directly utilize neural networks to predict stress, but rather embeds the neural hardening function into the processes of elasticity prediction, yield judgment, plasticity multiplier solution, back stress update, and return-mapped stress correction. While maintaining the cyclic plasticity mechanical mechanism and the evolution law of state variables, it achieves accurate prediction of cyclic stress-strain response. Through the backpropagation mechanism of the loss function, the weights of the neural hardening function can be optimized simultaneously, thereby gradually approximating the prediction results to the true response data. It combines physical interpretability with data-driven learning capabilities, improving the model stability and engineering application capability of hysteretic responses under complex cyclic loading conditions. Attached Figure Description

[0019] Figure 1 This is a flowchart of a cyclic plastic constitutive stress update method based on a learnable hardening function; Figure 2 A comparison of the engineering stress-strain hysteresis curves of the traditional elastoplastic constitutive model and the neurosclerotic constitutive model under a loading amplitude of 0.2%; Figure 3 A comparison of the engineering stress-strain hysteresis curves of the traditional elastoplastic constitutive model and the neurosclerotic constitutive model under a loading amplitude of 0.4%; Figure 4 A comparison of the engineering stress-strain hysteresis curves of the traditional elastoplastic constitutive model and the neurosclerotic constitutive model under a loading amplitude of 0.6%; Figure 5 A comparison of the engineering stress-strain hysteresis curves of the traditional elastoplastic constitutive model and the neurosclerotic constitutive model under a loading amplitude of 0.8%; Figure 6A comparison of the engineering stress-strain hysteresis curves of the traditional elastoplastic constitutive model and the neurosclerotic constitutive model under a variable amplitude loading path (0.2%→0.4%→0.6%→0.8%). Figure 7 The curve represents the evolution of the loss function. Figure 8 A comparison of the engineering stress-strain hysteresis curves of the traditional elastoplastic constitutive model and the neurosclerotic constitutive model under a loading amplitude of 0.7%; Figure 9 A comparison of the engineering stress-strain hysteresis curves of the traditional elastoplastic constitutive model and the neurosclerotic constitutive model under a loading amplitude of 0.9%. Detailed Implementation

[0020] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments.

[0021] Example 1: A method for updating cyclic plastic constitutive stress based on a learnable hardening function, such as... Figure 1 As shown, it includes the following steps: Step 1: Obtain strain-stress hysteresis data of the material under cyclic loading conditions and construct a sample database for model training.

[0022] Step 2: Preprocess the strain-stress data to remove invalid and duplicate data points; downsample data points that exceed a preset threshold for non-critical features to reduce the training scale and improve training efficiency.

[0023] In this embodiment, the key features in step 2 include: changes in the peak value, valley value, and loading direction in the cyclic hysteresis curve.

[0024] Step 3: Construct a neural hardening function module in the cyclic plastic constitutive model so that isotropic hardening, kinematic hardening growth, and kinematic hardening recovery can adaptively evolve according to the plastic state of the material.

[0025] In this embodiment, in step 3, the neural hardening function module is used to replace the preset hardening parameters in the traditional cyclic plasticity model, thereby enabling isotropic hardening, kinematic hardening growth, and kinematic hardening recovery to evolve adaptively according to the plastic state of the material.

[0026] In this embodiment, step 3 includes the following steps: Step 301: Based on cumulative plastic strain Construct an isotropic hardening function The formula is as follows: ; in, These are non-negative trainable weights. It is a set of cumulative plastic strain change smoothing basis functions.

[0027] In this embodiment, in step 301, the cumulative plastic strain change smoothing basis function is zero in the initial state and changes monotonically with the cumulative plastic strain. This ensures that the material does not undergo additional isotropic hardening in the initial state and that the hardening evolution conforms to the basic physical law of gradual development of cumulative plastic deformation.

[0028] In this embodiment, in step 301, The formula is as follows: ; in, It is a smoothing parameter. It is the first The node positions of each basis function.

[0029] In this embodiment, in step 301, ,and Since they are non-negative trainable weights, ,therefore Capable of adapting to accumulated plastic strain The evolution becomes smoother with the increase of stress. During stress integration, the current accumulated plastic strain is... enter The isotropic hardening value of the nerve was obtained and compared with the initial yield stress. The current yield strength is determined jointly, that is, the current yield strength is... .

[0030] Step 302: Based on the absolute value of each back stress component and cumulative plastic strain The growth factor of the back stress component under the current state is calculated using the following formula: ; in, It is the first The kinematic hardening growth factor of each back stress component under the current state It is the corresponding learnable motion hardening growth function.

[0031] In this embodiment, in step 302, compared with the traditional Chaboche model, The settings are different from the constants; this embodiment will... It is used to replace the fixed motion hardening growth parameters in the traditional Chaboche model, so that the back stress growth capability can adapt to the cyclic loading process.

[0032] In this embodiment, in step 302, This function describes the growth capacity of the back stress component under different back stress amplitudes and different plastic accumulation states. The function uses the basic kinematic hardening growth value... Centered on the back stress amplitude correction and the plastic cumulative correction, it is bounded.

[0033] In this embodiment, in step 302, The formula is as follows: ; in, It is the first The basic kinematic hardening growth value of each back stress component It is the bounded amplification factor of the learnable motion hardening growth function. It is by and The residual term is formed by the smoothing basis functions.

[0034] in, The formula is as follows: ; ; ; in, It is the first The absolute value of each back stress component is a correction value for the kinematic hardening growth function. It is control Follow Changing residual weights Therefore, the first Absolute value of each back stress component For the input of the first A smooth basis function, L It is the number of smooth basis functions for the back stress; It is the cumulative plastic strain For the A correction term for the motion hardening growth function, It is control Follow Changing residual weights It is based on the cumulative plastic strain For the input of the first A smooth basis function, M It is the number of smooth basis functions for cumulative plastic strain.

[0035] Step 303: Based on the absolute value of each back stress component and cumulative plastic strain The dynamic hardening recovery coefficient of the back stress component under the current state is calculated using the following formula: ; in, It is the first The dynamic hardening restitution coefficient of each back stress component under the current state. It is the corresponding learnable motion hardening recovery function.

[0036] In this embodiment, step 303 differs from the traditional Chaboche model where the dynamic hardening recovery coefficient is set to a constant. In this embodiment, step 303... It is used to replace the constant recovery parameter in the traditional Chaboche model, enabling the model to more flexibly characterize back stress saturation, reverse loading, and hysteresis behavior under cyclic loading.

[0037] In this embodiment, in step 303, This function describes the dynamic recovery capability of the back stress component under different back stress amplitudes and different plastic accumulation states. The function uses the basic dynamic hardening recovery value... Centered on the back stress amplitude recovery correction and the plastic cumulative recovery correction, it is bounded.

[0038] In this embodiment, in step 303, The formula is as follows: ; in, It is the first The basic dynamic hardening recovery value of each back stress component It is the bounded amplification factor of the learnable motion hardening recovery function. It is by and The residual term is formed by the smoothing basis functions.

[0039] in, The formula is as follows: ; ; ; in, It is the first The absolute value of each back stress component is the correction value of the kinematic hardening restitution function. It is control Follow Changing residual weights Therefore, the first Absolute value of each back stress component For the input of the first A smooth basis function, L It is the number of smooth basis functions for the back stress; It is the cumulative plastic strain For the The correction term for the motion hardening recovery function, It is control Follow Changing residual weights It is based on the cumulative plastic strain For the input of the first A smooth basis function, M It is the number of smooth basis functions for cumulative plastic strain.

[0040] Step 4: Combine the neural sclerosis function module with the cyclic plastic stress update process to establish a return mapping stress update module that includes elastic prediction, yield judgment, plastic multiplier solution, back stress component update and stress correction.

[0041] In this embodiment, step 4 establishes a return-mapped stress update module that includes elastic prediction, yield determination, plastic multiplier solution, back stress component update, and stress correction, enabling the learning of isotropic hardening functions. Learnable motion hardening growth function and learnable motion hardening recovery function It is invoked during the stepwise stress integration process and outputs the current neural isotropic sclerosis value. Growth coefficient of exercise hardening and dynamic hardening recovery coefficient ;in, Used for yield strength calculation and Used for semi-implicit updates of each back stress component, and further involved in the plastic return mapping consistency solution.

[0042] In this embodiment, step 4 includes the following steps: Step 401: Before stepwise integration of the cyclic strain path, the initial plastic strain is... Initial cumulative plastic strain and the initial values ​​of each back stress component , and All are set to zero, and the initial back stress resultant is set to zero. Set to zero as the starting state for subsequent elastic prediction, yield determination, plastic multiplier solution, back stress component update, and stress correction, returning to the mapped stress update.

[0043] Step 402: In the (n+1)th loading step, read the total strain of the current loading step. And call the plastic strain updated in the previous loading step. Cumulative plastic strain and each back stress component , and This serves as the historical state input for the current loading step's elasticity prediction and yield judgment.

[0044] Step 403: In each loading step, the difference between the total strain of the current loading step and the updated plastic strain of the previous loading step is taken as the current elastic test strain, and the test stress is calculated based on the elastic modulus, as shown in the following formula: ; in, This is the elastic predicted stress obtained assuming no new plastic deformation occurs in the current loading step. It is the elastic modulus. For the total strain of the current loading step, This is the plastic strain updated after the previous loading step.

[0045] Step 404: Read the back stress components updated in the previous loading step. , and And these are superimposed to form the resultant back stress of the previous moment. ,Right now ;Will and The difference between them is used as the relative stress of the test. ,Right now The cumulative plastic strain updated in the previous loading step will be used to adjust the cumulative plastic strain. Input can learnable isotropic hardening function To obtain the isotropic sclerosis value of the nerve required for the current trial yield judgment. and the initial yield stress and Together as the yield strength of the current loading step test .

[0046] Step 405: When When the value is not greater than zero, the current loading step is determined as an elastic step; when... When the value is greater than zero, the current loading step is determined to be a plastic step, and the plastic return mapping correction process is initiated.

[0047] Step 406: When the current loading step is an elastic step, Directly used as the stress of the current loading step Output, while maintaining plastic strain Cumulative plastic strain and each back stress component , and If it remains unchanged, proceed to the next loading step for calculation.

[0048] Step 407: When the current loading step is the plastic step, according to The sign determines the direction of plastic flow. ;when When positive, the direction of plastic flow is positive; when... When the value is negative, the direction of plastic flow is negative.

[0049] Step 408: Combine the back stress components from the previous loading step The absolute value and the cumulative plastic strain of the previous loading step Input a learnable motion hardening growth function Get the current loading step number Kinematic hardening growth coefficient of each back stress component ; The back stress components of the previous loading step The absolute value and the cumulative plastic strain at the previous moment Input learnable motion hardening recovery function Get the current loading step number Dynamic hardening restitution coefficient of each back stress component .

[0050] Step 409: Using plastic multipliers As unknowns, a consistent relationship is established between the cumulative plastic strain, stress, back stress components, and yield strength after plastic correction; where the cumulative plastic strain after plastic correction is... The stress after plastic correction is obtained by reverting the test stress along the direction of plastic flow. Each back stress component is composed of , , , The yield strength after plastic correction is determined by the direction of plastic flow and is determined by the initial yield stress. and To be determined jointly.

[0051] Step 410: Solve for the plastic multipliers using the bisection method. Establish about The yield uniformity residual function is given by the following formula: ; in, It represents the direction of plastic flow and the candidate stress after plastic correction. It is the candidate back stress resultant after plastic correction. It is a candidate isotropic hardening value for the output of the learnable isotropic hardening function.

[0052] In this embodiment, in step 410, when When this occurs, it indicates that the stress state after plastic correction has returned to the yield surface, at which point... As the lower bound of the bisection interval, and to estimate the upper bound of the bisection interval. The formula is as follows: ; in, It is a positive bias used to ensure that the bisection interval covers the root value of the yield surface. E It is the elastic modulus.

[0053] like If it is greater than zero, then gradually increase the value. until satisfaction is achieved. and The solution interval; subsequently, in the interval Take the midpoint inside and calculate ;when When this occurs, it indicates that the current amount of plastic correction is insufficient, causing... ;when When this occurs, it indicates that the current plasticity correction has reached or exceeded the yield level, causing... Repeat the above binary search iteration process. After a preset number of iterations, take: As the plastic multiplier of the current loading step, it ensures that the superposition of the stress, back stress components, and yield strength after plastic correction meet the yield consistency condition.

[0054] Step 411: Based on plastic multipliers and plastic flow direction Update plastic strain, i.e. Update the accumulated plastic strain, that is .

[0055] Step 412: For the first Each back stress component, based on the back stress component of the previous loading step. Growth coefficient of exercise hardening Dynamic hardening recovery coefficient Plastic multipliers and plastic flow direction Perform a semi-implicit update to obtain the back stress components after the current loading step ends. ,Right now .

[0056] Step 413: Superimpose the updated back stress components to obtain the resultant back stress after the current loading step ends. ,Right now It represents the state of the yield surface movement after the current loading step ends, and serves as a historical state variable in the yield test judgment of the next loading step.

[0057] Step 414: Based on the total strain of the current loading step and the updated plastic strain Recalculate the stress of the current loading step. ,Right now Thus, the stress response after plastic return mapping correction is obtained.

[0058] Step 415: Accumulate the plastic strain of the current loading step Input can learnable isotropic hardening function The current neural isotropic sclerosis value is obtained. And according to the current step stress Current step back stress resultant Initial yield stress as well as The yield residual for the current step is calculated using the following formula: ; in, It is the final stress returned by the current loading step after mapping correction. It is the back stress after the current loading step ends. It is the initial yield stress. It is the current cumulative plastic strain The isotropic hardening value is obtained by inputting a learnable isotropic hardening function.

[0059] In this embodiment, the yield residual of the current step in step 415 is used to characterize whether the stress state after the return mapping meets the yield consistency requirement.

[0060] Step 416: Record the final stress of the current loading step. Plastic strain Cumulative plastic strain Each back stress component , , Back stress resultant Isotropic sclerosis value Growth coefficient of exercise hardening Dynamic hardening recovery coefficient Plastic multipliers The yield residual serves as the data basis for subsequent training evaluation, state variable analysis, and result output.

[0061] Step 417: Repeat steps 402 to 416 along the complete cyclic strain path until the stress integration of all loading steps is completed, and obtain the complete predicted stress sequence and the corresponding material state variable evolution sequence.

[0062] Step 5: Compare the complete predicted stress sequence with the real stress sequence in the training data, and use the normalized weighted mean square error as the loss function.

[0063] In this embodiment, the formula for the loss function in step 5 is as follows: ; in, It's the number of loading steps. For the first The model predicts stress in each loading step. For the first The actual stress of each loading step The stress normalization scale is derived from the absolute maximum value of the true stress in the training path and is used for stress error normalization in the loss function.

[0064] Step 6: Calculate the gradient of the loss function with respect to the trainable parameters using the automatic differentiation method, and update the learnable parameters (learnable isotropic hardening function parameters, learnable motion hardening growth function parameters, and learnable motion hardening recovery function parameters) through the optimizer. Gradient clipping and learning rate adjustment are performed during the parameter update process to improve training stability and convergence.

[0065] Step 7: Repeat the stress integration, loss calculation, and parameter update process within the preset number of training rounds, and record the loss value, prediction error, and fitting accuracy after each training round. When the loss in the current round is lower than that in all previous rounds, save the current model parameters as the optimal model parameters. After training, restore the model with the minimum loss as the final neural sclerosis constitutive model.

[0066] The cyclic plastic constitutive stress update method based on a learnable hardening function in this embodiment has the following beneficial effects: Compared with existing technologies, this embodiment introduces learnable isotropic hardening functions, learnable kinematic hardening growth functions, and learnable kinematic hardening recovery functions into the cyclic plastic constitutive model. This enables adaptive learning of yield strength evolution, back stress growth, and dynamic recovery laws, reducing the dependence of traditional models on fixed hardening function forms and empirical parameters. Instead of directly using neural networks to predict stress, this embodiment embeds the neural hardening function into the processes of elasticity prediction, yield judgment, plasticity multiplier solution, back stress update, and back-mapped stress correction. While maintaining the cyclic plasticity mechanical mechanism and the evolution law of state variables, it achieves accurate prediction of cyclic stress-strain response. Through the backpropagation mechanism of the loss function, the weights of the neural hardening function can be optimized simultaneously, thereby gradually approximating the prediction results to the true response data. It combines physical interpretability with data-driven learning capabilities, improving the model stability and engineering application capabilities of hysteretic responses under complex cyclic loading conditions.

[0067] Example 2: To make the technical solution, model operation process and technical effects of Example 1 clearer, the training process of the neural sclerosis constitutive model of Example 1 will be further explained below with reference to specific training data.

[0068] Engineering strain-stress data under uniaxial cyclic tensile loading conditions were used as training data. The training data included one variable-amplitude cyclic loading path and multiple single-amplitude cyclic loading paths. Specifically, the variable-amplitude cyclic loading path (0.2%→0.4%→0.6%→0.8%) and the single-amplitude cyclic loading paths corresponding to strain amplitudes of 0.2%, 0.4%, 0.6%, and 0.8% were used as training data to jointly train the neural sclerosis constitutive model of Example 1. Cyclic stress-strain hysteresis data could be obtained by simulating dog-bone shaped specimens under uniaxial tensile and compressive loads with different cyclic strain amplitudes using ABAQUS finite element software, or directly measured through corresponding tensile and compressive fatigue tests.

[0069] Save the variable amplitude value cyclic loading data to an Excel file. The table should contain at least three columns: the first column, named "step," stores the loading step data; the second column, named "strain," stores the strain data arranged by loading step; and the third column, named "stress," stores the stress data corresponding to the strain of each loading step. After reading, clean the data by deleting null values, non-numerical points, and duplicate data points.

[0070] Save the cyclic loading data for single strain amplitudes of 0.2%, 0.4%, 0.6%, and 0.8% to an Excel file. Each single-amplitude cyclic curve in the file is stored as a repeating three-column data block, with each data block including the loading step column, strain column, and stress column in sequence. The first row of each data block is used to indicate the strain amplitude corresponding to the data block, such as 0.2%, 0.4%, 0.6%, and 0.8%, followed by the loading step, strain, and stress data for that amplitude starting from the third row.

[0071] Three back stress branches are set, namely elastic modulus Take a fixed value of 210,000 MPa, initial yield stress Set to 260 MPa. Number of smoothing basis functions. Set it to 28.

[0072] The number of training epochs was set to 1200, and the initial learning rate was set to 4.0 × 10⁻⁶. -3 The weight decay factor is set to 1.0 × 10. -6 The gradient clipping threshold was set to 10.0, the number of iterations for the plasticity multiplier bisection method was set to 14, and the random seed was set to 42. The AdamW optimizer was used to update the trainable parameters of the model, and a cosine annealing learning rate scheduling strategy was used to gradually adjust the training step size. In each round of training, the variable amplitude cyclic strain sequence was first input into the neural sclerosis constitutive model, and stress integration was performed stepwise along the strain path.

[0073] The training method of Example 1 is repeated within a preset number of training rounds. After each round of training, the training loss of the current round is recorded. When the training loss of the current round is less than the previously saved minimum loss, the model parameters of that round are saved as the current optimal model parameters. After training, the model parameters corresponding to the round with the minimum training loss are restored to obtain the trained neural sclerosis constitutive model.

[0074] The training results are shown in Table 1 and Figure 2-7 As shown, the neural sclerosis constitutive model can reproduce the reference stress response well at amplitudes of 0.2%, 0.4%, 0.6%, 0.8%, and variable amplitude training values. Specifically, at an amplitude of 0.2%, the MAE is 5.34 MPa, the RMSE is 6.74 MPa, and the relative RMSE is 2.23%. 2 The MAE is 0.9991; at a 0.4% amplitude, the MAE is 3.09 MPa, the RMSE is 3.80 MPa, and the relative RMSE is 0.92%. 2 The MAE is 0.9998; at a 0.6% amplitude, the MAE is 5.87 MPa, the RMSE is 7.00 MPa, and the relative RMSE is 1.53%. 2The MAE is 0.9996; at a 0.8% amplitude, the MAE is 12.62 MPa, the RMSE is 14.41 MPa, and the relative RMSE is 3.00%. 2 The value is 0.9985; under variable amplitude loading, the MAE is 7.38 MPa, the RMSE is 10.43 MPa, and the relative RMSE is 2.15%. 2 The value is 0.9989. The above results indicate that the proposed neurosclerosis constitutive model can not only fit the cyclic stress response under a single strain amplitude, but also maintain high prediction accuracy under loading conditions with progressively varying strain amplitudes. Combined with... Figure 2 As shown in Figure 7, under the training path, the stress-strain curves output by the neural sclerosis constitutive model have good consistency with the reference curves in the hysteresis loop opening, loading-unloading path, reverse loading segment, and peak-valley region. This indicates that by learning the isotropic hardening function, the kinematic hardening growth function, and the kinematic hardening recovery function, the model can effectively learn the cyclic hardening evolution law and back stress evolution characteristics contained in the training data.

[0075] Table 1 Training Indicators

[0076] Example 3: This embodiment illustrates the stress prediction process of the neural sclerosis constitutive model trained in Embodiment 2 under a single strain amplitude cyclic loading path that was not involved in the training. The uniaxial cyclic strain history corresponding to strain amplitudes of 0.7% and 0.9% is selected as input. The neural sclerosis constitutive model.pth file is called, and the corresponding stress response is output, forming a stress-strain hysteresis curve. This curve is then compared with the stress-strain hysteresis curve obtained from a traditional elastoplastic model to evaluate the predictive performance of the neural sclerosis model.

[0077] Stress-strain hysteresis data were obtained by simulating dog bone-shaped specimens under cyclic tensile and compressive loads with strain amplitudes of 0.7% and 0.9% using ABAQUS finite element software. Alternatively, these data can be directly obtained through uniaxial cyclic tensile and compressive fatigue tests. The strain data serves as the prediction input for the trained neurosclerosis constitutive model, while the stress data is only used for subsequent prediction result comparison and evaluation and does not participate in model parameter updates. The cyclic strain data were saved to an Excel file.

[0078] Read the data blocks corresponding to the 0.7% and 0.9% strain amplitudes, and identify the corresponding amplitudes based on the amplitude identifier in the first line. For each curve, read its strain sequence and corresponding true stress sequence, and delete null values, non-numerical points, and duplicate data points. When the data volume of a single validation curve is large, downsample the curve and prioritize retaining loading reversal points, peak points, valley points, and path endpoints to ensure that key inflection features in the cyclic hysteresis curve are not lost.

[0079] Call the neural sclerosis constitutive model .pth file that was trained and saved in Example 1.

[0080] During prediction validation, the learnable isotropic hardening function is maintained after training. Learnable motion hardening growth function and learnable motion hardening recovery function The parameters remain unchanged.

[0081] The initial plastic strain Initial cumulative plastic strain and the initial values ​​of the three back stress branches , and All are set to zero, and the initial back stress resultant is set to zero. Set to zero. This ensures that both the 0.7% and 0.9% prediction paths start stress prediction from the same initial material state.

[0082] The cyclic strain sequence corresponding to the 0.7% strain amplitude is input into the trained neural sclerosis constitutive model according to the loading steps. In each loading step, the model performs elastic prediction, trial yield judgment, elastic update or plastic return mapping correction, and gradually outputs the predicted stress and state variables.

[0083] The loading process is repeated along the complete cyclic path of 0.7% and 0.9% strain amplitude until all loading steps are calculated. Predicted stress-strain hysteresis curves and state variable evolution results are generated at 0.7% and 0.9% strain amplitudes.

[0084] Prediction results are as follows Figure 8-9 As shown, the model's predicted curve and the reference curve exhibit good consistency in the hysteresis loop opening, loading-unloading path, reverse loading segment, and peak-valley region. This indicates that the trained neural sclerosis constitutive model, without retraining, can directly output a high-precision stress response based on the new cyclic strain history, demonstrating good predictive ability. The above description is merely a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the technical scope disclosed in this invention, based on the technical solution and inventive concept of the present invention, should be covered within the scope of protection of this invention.

Claims

1. A method for updating cyclic plastic constitutive stress based on a learnable hardening function, characterized in that, Includes the following steps: Step 1: Obtain strain-stress hysteresis data of the material under cyclic loading conditions and construct a sample database for model training; Step 2: Preprocess the strain-stress data, remove invalid and duplicate data points, and downsample data points that exceed a preset threshold for non-critical features; Step 3: Construct a neural hardening function module in the cyclic plastic constitutive model, enabling isotropic hardening, kinetic hardening growth, and kinetic hardening recovery to adaptively evolve according to the material's plastic state; this includes the following steps: Step 301: Based on cumulative plastic strain Construct an isotropic hardening function The formula is as follows: ； in, These are non-negative trainable weights. It is a set of cumulative plastic strain change smoothing basis functions; Step 302: Based on the absolute value of each back stress component and cumulative plastic strain The growth factor of the back stress component under the current state is calculated using the following formula: ； in, It is the first The kinematic hardening growth factor of each back stress component under the current state It is the corresponding learnable motion hardening growth function; The formula is as follows: ； in, It is the first The basic kinematic hardening growth value of each back stress component It is the bounded amplification factor of the learnable motion hardening growth function. It is by and The residual term is formed by the smoothing basis functions; in, The formula is as follows: ； ； ； in, It is the first The absolute value of each back stress component is a correction value for the kinematic hardening growth function. It is control Follow Changing residual weights Therefore, the first Absolute value of each back stress component For the input of the first A smooth basis function, L It is the number of smooth basis functions for the back stress; It is the cumulative plastic strain For the A correction term for the motion hardening growth function, It is control Follow Changing residual weights It is based on the cumulative plastic strain For the input of the first A smooth basis function, M It is the number of smooth basis functions for cumulative plastic strain; Step 303: Based on the absolute value of each back stress component and cumulative plastic strain The dynamic hardening recovery coefficient of the back stress component under the current state is calculated using the following formula: ； in, For the first The dynamic hardening restitution coefficient of each back stress component under the current state. This is the corresponding learnable motion hardening recovery function; The formula is as follows: ； in, It is the first The basic dynamic hardening recovery value of each back stress component It is the bounded amplification factor of the learnable motion hardening recovery function. It is by and The residual term is formed by the smoothing basis functions; in, The formula is as follows: ； ； ； in, It is the first The absolute value of each back stress component is the correction value of the kinematic hardening restitution function. It is control Follow Changing residual weights Therefore, the first Absolute value of each back stress component For the input of the first A smooth basis function, L It is the number of smooth basis functions for the back stress; It is the cumulative plastic strain For the first The correction term of the motion hardening recovery function, It is control Follow Changing residual weights It is based on the cumulative plastic strain For the input of the first A smooth basis function, M It is the number of smooth basis functions for cumulative plastic strain; Step 4: Combine the neural sclerosis function module with the cyclic plastic stress update process to establish a return mapping stress update module that includes elastic prediction, yield judgment, plastic multiplier solution, back stress component update and stress correction. Step 5: Compare the complete predicted stress sequence with the real stress sequence in the training data, and use the normalized weighted mean square error as the loss function; Step 6: Calculate the gradient of the loss function with respect to the trainable parameters using the automatic differentiation method, and update the learnable parameters through the optimizer; Step 7: Repeat the stress integral, loss calculation and parameter update process within the preset number of training rounds, and record the loss value, prediction error and fitting accuracy after each training round; when the loss of the current round is lower than that of all previous rounds, save the current model parameters as the optimal model parameters, and restore the model with the minimum loss after training as the final neural sclerosis constitutive model.

2. The method for updating cyclic plastic constitutive stress based on a learnable hardening function according to claim 1, characterized in that, In step 2, key features include: changes in the peak and valley values ​​and loading direction in the cyclic hysteresis curve.

3. The cyclic plastic constitutive stress update method based on a learnable hardening function according to claim 1, characterized in that, In step 301, The formula is as follows: ； in, It is a smoothing parameter. It is the first The node positions of each basis function.

4. The cyclic plastic constitutive stress update method based on a learnable hardening function according to claim 1, characterized in that, In step 4, a return-mapped stress update module is established, which includes elastic prediction, yield judgment, plastic multiplier solution, back stress component update, and stress correction, enabling the learning of isotropic hardening functions. Learnable motion hardening growth function and learnable motion hardening recovery function It is invoked during the stepwise stress integration process and outputs the current neural isotropic sclerosis value. Growth coefficient of exercise hardening and dynamic hardening recovery coefficient ;in, Used for yield strength calculation and Used for semi-implicit updates of each back stress component, and further involved in the plastic return mapping consistency solution.

5. The cyclic plastic constitutive stress update method based on a learnable hardening function according to claim 4, characterized in that, Step 4 includes the following steps: Step 401: Before stepwise integration of the cyclic strain path, the initial plastic strain is... Initial cumulative plastic strain and the initial values ​​of each back stress component , and All are set to zero, and the initial back stress resultant is set to zero. Set to zero as the starting state for subsequent elastic prediction, yield judgment, plastic multiplier solution, back stress component update and stress correction return mapping stress update; Step 402: In the (n+1)th loading step, read the total strain of the current loading step. And call the plastic strain updated in the previous loading step. Cumulative plastic strain and each back stress component , and This serves as the historical state input for the current loading step elasticity prediction and yield judgment; Step 403: In each loading step, the difference between the total strain of the current loading step and the updated plastic strain of the previous loading step is taken as the current elastic test strain, and the test stress is calculated based on the elastic modulus, as shown in the following formula: ； in, This is the elastic predicted stress obtained assuming no new plastic deformation occurs in the current loading step. It is the elastic modulus. For the total strain of the current loading step, This represents the plastic strain updated in the previous loading step. Step 404: Read the back stress components updated in the previous loading step. , and And these are superimposed to form the resultant back stress of the previous moment. ,Right now ;Will and The difference between them is used as the relative stress of the test. ,Right now The cumulative plastic strain updated in the previous loading step will be used to adjust the cumulative plastic strain. Input can learnable isotropic hardening function To obtain the isotropic sclerosis value of the nerve required for the current trial yield judgment. and the initial yield stress and Together as the yield strength of the current loading step test ; Step 405: When When the value is not greater than zero, the current loading step is determined as an elastic step; when... When the value is greater than zero, the current loading step is determined to be a plastic step, and the plastic return mapping correction process is initiated. Step 406: When the current loading step is an elastic step, Directly used as the stress of the current loading step Output, while maintaining plastic strain Cumulative plastic strain and each back stress component , and The result remains unchanged, and then the calculation proceeds to the next loading step. Step 407: When the current loading step is the plastic step, according to The sign determines the direction of plastic flow. ;when When positive, the direction of plastic flow is positive; when... When the value is negative, the direction of plastic flow is negative; Step 408: Combine the back stress components from the previous loading step The absolute value and the cumulative plastic strain of the previous loading step Input a learnable motion hardening growth function Get the current loading step number Kinematic hardening growth coefficient of each back stress component ; the back stress components of the previous loading step The absolute value and the cumulative plastic strain at the previous moment Input learnable motion hardening recovery function Get the current loading step number Dynamic hardening restitution coefficient of each back stress component ; Step 409: Using plastic multipliers As unknowns, a consistent relationship is established between the cumulative plastic strain, stress, back stress components, and yield strength after plastic correction; where the cumulative plastic strain after plastic correction is... The stress after plastic correction is obtained by reverting the test stress along the direction of plastic flow. Each back stress component is composed of , , , The yield strength after plastic correction is determined by the direction of plastic flow and is determined by the initial yield stress. and To be determined jointly; Step 410: Solve for the plastic multipliers using the bisection method. Establish about The yield uniformity residual function is given by the following formula: ； in, It is the direction of plastic flow. It is the candidate stress after plastic correction. It is the candidate back stress resultant after plastic correction. It is a candidate isotropic hardening value for the output of the learnable isotropic hardening function; Step 411: Based on plastic multipliers and plastic flow direction Update plastic strain, i.e. Update the accumulated plastic strain, that is ; Step 412: For the first Each back stress component, based on the back stress component of the previous loading step. Growth coefficient of exercise hardening Dynamic hardening recovery coefficient Plastic multipliers and plastic flow direction Perform a semi-implicit update to obtain the back stress components after the current loading step ends. ,Right now ; Step 413: Superimpose the updated back stress components to obtain the resultant back stress after the current loading step ends. ,Right now This represents the state of the yield surface movement after the current loading step ends, and serves as a historical state variable in the yield test judgment of the next loading step. Step 414: Based on the total strain of the current loading step and updated plastic strain Recalculate the stress of the current loading step. ,Right now Thus, the stress response after plastic return mapping correction is obtained; Step 415: Accumulate the plastic strain of the current loading step Input can learnable isotropic hardening function The current neural isotropic sclerosis value is obtained. And according to the current step stress Current step back stress resultant Initial yield stress as well as The yield residual for the current step is calculated using the following formula: ； in, It is the final stress returned by the current loading step after mapping correction. It is the back stress after the current loading step ends. It is the initial yield stress. It is the current cumulative plastic strain The isotropic hardening value is obtained after inputting a learnable isotropic hardening function; Step 416: Record the final stress of the current loading step. Plastic strain Cumulative plastic strain Each back stress component , , Back stress resultant Isotropic sclerosis value Growth coefficient of exercise hardening Dynamic hardening recovery coefficient Plastic multipliers The yield residual serves as the data basis for subsequent training evaluation, state variable analysis, and result output. Step 417: Repeat steps 402 to 416 along the complete cyclic strain path until the stress integration of all loading steps is completed, and obtain the complete predicted stress sequence and the corresponding material state variable evolution sequence.

6. The cyclic plastic constitutive stress update method based on a learnable hardening function according to claim 5, characterized in that, In step 410, when When this occurs, it indicates that the stress state after plastic correction has returned to the yield surface, at which point... As the lower bound of the bisection interval, and to estimate the upper bound of the bisection interval. The formula is as follows: ； in, It is a positive bias used to ensure that the bisection interval covers the root value of the yield surface. E It is the elastic modulus; like If it is greater than zero, then gradually increase the value. until satisfaction is achieved. and The solution interval; subsequently, in the interval Take the midpoint inside and calculate ;when When this occurs, it indicates that the current amount of plastic correction is insufficient, causing... ;when When this occurs, it indicates that the current plasticity correction has reached or exceeded the yield level, causing... Repeat the above binary search iteration process. After a preset number of iterations, take: As the plastic multiplier of the current loading step, it ensures that the superposition of the stress, back stress components, and yield strength after plastic correction meet the yield consistency condition.

7. The cyclic plastic constitutive stress update method based on a learnable hardening function according to claim 1, characterized in that, In step 5, the formula for the loss function is as follows: ； in, It's the number of loading steps. For the first The model predicts stress in each loading step. For the first The actual stress of each loading step The stress normalization scale is derived from the absolute maximum value of the true stress in the training path.

Images