Method and device for realizing stress mapping of elastoplastic material and storage medium

Abstract

The invention relates to the technical field of machine learning, in particular to a method and device for achieving stress mapping of an elastoplastic material and a storage medium. The method comprises the steps that training data are acquired, the training data comprise a plurality of sample data sets, and each sample data set comprises strain data and corresponding external force; according to the training data, a target neural network model is trained in an unsupervised learning mode, the target neural network model is used for indicating the mapping relation between strain data and stress of the elastoplastic material sample under the action of external force, the target neural network model meets the constraint of an equilibrium equation, and the equilibrium equation is used for indicating the stress of the elastoplastic material sample at any given moment. The stress distribution of the elastoplastic material sample meets the balance condition of the external force. According to the embodiment, the model meeting the equilibrium equation constraint can be trained in an unsupervised learning mode without label data (namely actual stress), and stress mapping of the elastoplastic material is achieved.

Description

Technical Field

[0001] The present disclosure relates to the field of machine learning technologies, and in particular, to a method, apparatus, and storage medium for implementing stress mapping of elastoplastic materials. Background Art

[0002] The plastic constitutive model is the core of describing the stress-strain relationship of materials in material mechanics, and it consists of two major parts: a physical model and a mathematical model. The physical model is based on plastic phenomenology and reflects the internal characteristics of materials by introducing internal variables such as equivalent plastic strain and damage factor. The mathematical model uses classical yield criteria, such as Tresca and von Mises criteria, and evolution equations such as isotropic hardening and kinematic hardening to express material behavior.

[0003] In the construction of the plastic constitutive model of new materials, obtaining sufficient experimental data is a major challenge. Although strain data can be relatively easily obtained through digital image correlation technology, stress measurement requires specimens with specific geometric structures and dimensions, which increases the complexity and cost of experiments. Numerical simulation provides another possibility for generating multi-axial stress-strain curves, but the simulation depends on known or assumed material constitutive behaviors and has strict requirements on the composition and interaction of materials. In addition, the calculation accuracy and time overhead of numerical methods are also significant limitations. Machine learning technologies, especially neural networks, provide new ideas for establishing constitutive models with their powerful non-linear fitting capabilities. Neural networks can replace traditional complex constitutive calculations through forward propagation, but the stress-strain data required for training neural networks usually comes from simulations of different loading paths of known constitutive models. When facing new materials, due to the lack of experimental data, especially insufficient stress, it is difficult to construct a dataset of sufficient scale to support the application of machine learning methods.

[0004] In the related art, a reasonable and effective method for implementing stress mapping of elastoplastic materials has not been provided. Summary of the Invention

[0005] In view of this, the present disclosure proposes a method, apparatus, and storage medium for implementing stress mapping of elastoplastic materials.

[0006] According to one aspect of the present disclosure, a method for implementing stress mapping of elastoplastic materials is provided, and the method includes:

[0007] Obtain training data, where the training data includes multiple groups of sample data groups, and each group of the sample data groups includes strain data and corresponding external forces, the strain data is used to indicate the degree of deformation of an elastoplastic material sample under the action of an external force, and the external force is used to indicate the total external force acting on the elastoplastic material sample;

[0008] Based on the training data, a target neural network model is trained by an unsupervised learning method. The target neural network model is used to indicate the mapping relationship between the strain data and the stress of the elastoplastic material sample under the action of an external force, and the target neural network model satisfies the constraints of the equilibrium equation, which is used to indicate that at any given moment, the stress distribution of the elastoplastic material sample satisfies the equilibrium condition of the external force. The stress is used to indicate the internal force per unit area of the elastoplastic material sample.

[0009] In a possible implementation manner, the training of the target neural network model by an unsupervised learning method according to the training data includes:

[0010] For each group of the sample data groups, according to the strain data, the predicted stress is determined through the target neural network model;

[0011] According to the predicted stress and the external force, the loss value is calculated through the equilibrium equation. The loss value is used to indicate the difference between the predicted stress and the stress that satisfies the constraints of the equilibrium equation;

[0012] By minimizing the loss value, the parameters of the target neural network model are adjusted.

[0013] In another possible implementation manner, the strain data includes strain increments corresponding to multiple time steps. The determining of the predicted stress according to the strain data through the target neural network model includes:

[0014] For each time step, based on the strain increment corresponding to the time step, the stress increment of the next time step is output through the target neural network model;

[0015] Repeat the above process at each time step until a preset model convergence condition is satisfied, and the predicted stress is determined.

[0016] In another possible implementation manner, the outputting of the stress increment of the next time step through the target neural network model based on the strain increment corresponding to the time step for each time step includes:

[0017] For each time step, the strain increment, stress, plastic strain, and internal variables corresponding to the time step are input into the target neural network model, and the stress increment, plastic strain increment, and internal variable increment of the next time step are output;

[0018] Among them, the initial values of the stress, the plastic strain, and the internal variable are all zero. The plastic strain is used to indicate the deformation history of irreversible deformation that occurs in the elastoplastic material sample under the action of an external force, and the internal variable is used to indicate the deformation history and hardening state of the internal state variables of the elastoplastic material sample under the action of an external force.

[0019] In another possible implementation, the target neural network model includes a first neural network, a second neural network, and a third neural network. For each time step, inputting the strain increment, stress, plastic strain, and internal variable corresponding to the time step into the target neural network model, and outputting the stress increment, plastic strain increment, and internal variable increment of the next time step, including:

[0020] For each time step, determine a trial stress according to the strain increment and the elastic stiffness of the elastoplastic material sample;

[0021] Output a first stress through the first neural network according to the trial stress;

[0022] Output a second stress through the second neural network according to the plastic strain and the internal variable;

[0023] In the case where the first stress and the second stress do not satisfy the model convergence condition, output the stress increment, the plastic strain increment, and the internal variable increment of the next time step through the third neural network according to the stress, the plastic strain, and the internal variable.

[0024] In another possible implementation, the method further includes:

[0025] In the case where the first stress and the second stress satisfy the model convergence condition, determine the trial stress as the predicted stress.

[0026] In another possible implementation, the model convergence condition includes that the difference between the first stress and the second stress is less than a preset threshold.

[0027] According to another aspect of the present disclosure, there is provided a device for realizing stress mapping of an elastoplastic material, the device including:

[0028] An acquisition module, configured to acquire training data, the training data including multiple groups of sample data groups, each group of the sample data groups including strain data and corresponding external forces, the strain data being used to indicate the degree of deformation that occurs in the elastoplastic material sample under the action of an external force, and the external force being used to indicate the total external force acting on the elastoplastic material sample;

[0029] A training module for training a target neural network model by unsupervised learning according to the training data. The target neural network model is used to indicate the mapping relationship between the strain data and the stress of the elastoplastic material sample under the action of an external force, and the target neural network model satisfies the constraint of the equilibrium equation. The equilibrium equation is used to indicate that at any given moment, the stress of the elastoplastic material sample satisfies the equilibrium condition of the external force, and the stress is used to indicate the internal force per unit area of the elastoplastic material sample.

[0030] In a possible implementation, the training module is further configured to:

[0031] For each group of the sample data groups, determine the predicted stress according to the strain data through the target neural network model;

[0032] Calculate a loss value according to the predicted stress and the external force through the equilibrium equation. The loss value is used to indicate the difference between the predicted stress and the stress that satisfies the constraint of the equilibrium equation;

[0033] Adjust the parameters of the target neural network model by minimizing the loss value.

[0034] In another possible implementation, the strain data includes strain increments corresponding to multiple time steps, and the training module is further configured to:

[0035] For each time step, output the stress increment of the next time step through the target neural network model based on the strain increment corresponding to the time step;

[0036] Repeat the above process at each time step until a preset model convergence condition is satisfied to determine the predicted stress.

[0037] In another possible implementation, the training module is further configured to:

[0038] For each time step, input the strain increment, stress, plastic strain, and internal variable corresponding to the time step into the target neural network model, and output the stress increment, plastic strain increment, and internal variable increment of the next time step;

[0039] Wherein, the initial values of the stress, the plastic strain, and the internal variable are all zero. The plastic strain is used to indicate the deformation history of the irreversible deformation of the elastoplastic material sample under the action of an external force, and the internal variable is used to indicate the deformation history and hardening state of the internal state variable of the elastoplastic material sample under the action of an external force.

[0040] In another possible implementation, the target neural network model includes a first neural network, a second neural network, and a third neural network. The training module is further configured to:

[0041] For each time step, determine a trial stress according to the strain increment and the elastic stiffness of the elastoplastic material sample.

[0042] Output a first stress through the first neural network according to the trial stress.

[0043] Output a second stress through the second neural network according to the plastic strain and the internal variables.

[0044] In the case where the first stress and the second stress do not satisfy the model convergence condition, output the stress increment, the plastic strain increment, and the internal variable increment at the next time step through the third neural network according to the stress, the plastic strain, and the internal variables.

[0045] In another possible implementation, the device further includes:

[0046] The training module is further configured to determine the trial stress as the predicted stress in the case where the first stress and the second stress satisfy the model convergence condition.

[0047] In another possible implementation, the model convergence condition includes that the difference between the first stress and the second stress is less than a preset threshold.

[0048] According to another aspect of the present disclosure, there is provided a computing device including a memory, a processor, and a computer program stored on the memory. The processor executes the computer program to implement the steps of the above method.

[0049] According to another aspect of the present disclosure, there is provided a non-volatile computer-readable storage medium having a computer program stored thereon. The computer program, when executed by a processor, implements the steps of the above method.

[0050] According to another aspect of the present disclosure, there is provided a computer program product including a computer program or a non-volatile computer-readable storage medium carrying the computer program. The computer program, when executed by a processor, implements the steps of the above method.

[0051] Embodiments of the present disclosure propose a method for realizing stress mapping of elastoplastic materials. By obtaining training data including strain data and corresponding external forces, and using this data to train a target neural network model through an unsupervised learning method. This model can reveal the mapping relationship between the strain data and stress of an elastoplastic material sample under the action of an external force, and follow the constraints of the equilibrium equation to ensure that at any specific moment, the stress of the material sample is in balance with the external force. The significant advantage of this method is that it does not rely on traditional stress, complex yield functions, the number of internal variables, or evolution functions, but directly trains the model based on strain and external forces. This process not only simplifies the data requirements, avoids the challenges of constructing complex equations, but also guides the model learning through the constraints of the equilibrium equation, thus ensuring the physical consistency and accuracy of the model output.

[0052] Other features and aspects of the present disclosure will become clear from the following detailed description of exemplary embodiments with reference to the accompanying drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] The accompanying drawings, which are included in and constitute a part of this specification, illustrate exemplary embodiments, features, and aspects of the present disclosure together with the specification and are used to explain the principles of the present disclosure.

[0054] Figure 1 The structural schematic diagram of a computing device provided by an exemplary embodiment of the present disclosure is shown.

[0055] Figure 2 The flowchart of a method for realizing stress mapping of elastoplastic materials provided by an exemplary embodiment of the present disclosure is shown.

[0056] Figure 3 The schematic diagram of the principle of a method for realizing stress mapping of elastoplastic materials provided by another exemplary embodiment of the present disclosure is shown.

[0057] Figure 4 The schematic diagram of the ICNN structure provided by an exemplary embodiment of the present disclosure is shown.

[0058] Figure 5 The schematic diagram of the results of the training set and test set of finite element verification provided by an exemplary embodiment of the present disclosure is shown.

[0059] Figure 6 The schematic diagram of the loading path of finite element verification provided by an exemplary embodiment of the present disclosure is shown.

[0060] Figure 7 The schematic diagram of model application, experimental verification method, and experimental verification results provided by an exemplary embodiment of the present disclosure is shown.

[0061] Figure 8A schematic diagram showing the test results of an experimentally verified uniaxial tensile test provided by an exemplary embodiment of the present disclosure.

[0062] Figure 9 A schematic diagram showing the evaluation of experimentally verified plastic strain and stress fields provided by an exemplary embodiment of the present disclosure.

[0063] Figure 10 A block diagram of a device shown according to an exemplary embodiment. Detailed implementation manners

[0064] Various exemplary embodiments, features, and aspects of the present disclosure will be described in detail below with reference to the accompanying drawings. The same reference numerals in the drawings denote elements having the same or similar functions. Although various aspects of the embodiments are shown in the drawings, the drawings are not necessarily drawn to scale unless otherwise specified.

[0065] As used herein, the terms "comprising", "including", "having", or variations thereof are open-ended and include one or more stated features, wholes, elements, steps, components, or functions, but do not exclude the existence or addition of one or more other features, wholes, elements, steps, components, functions, or groups thereof.

[0066] When an element is referred to as being "connected", "coupled", "responsive" or variations thereof to another element, it can be directly connected, coupled, or responsive to the other element, or intervening elements may be present.

[0067] Although terms such as first, second, third, etc. may be used herein to describe various elements / operations, these elements / operations should not be limited by these terms. These terms are only used to distinguish one element / operation from another. Thus, without departing from the teachings of the inventive concept, a first element / operation in some embodiments may be referred to as a second element / operation in other embodiments.

[0068] The term "exemplary" as used herein means "serving as an example, embodiment, or illustration". Any embodiment illustrated herein as "exemplary" should not necessarily be construed as superior to or better than other embodiments.

[0069] In addition, for a better illustration of the present disclosure, numerous specific details are given in the following detailed implementation manners. Those skilled in the art should understand that the present disclosure can be implemented without some specific details. In some instances, methods, means, elements, and circuits well-known to those skilled in the art are not described in detail so as to highlight the gist of the present disclosure.

[0070] The method provided by the embodiments of the present disclosure mainly relates to digital speckle technology, machine learning technology, and stress measurement technology. This solution particularly focuses on the application of machine learning algorithms, aiming to achieve accurate identification of the mechanical properties of materials, especially for elastoplastic materials such as metals. Further, this method can realize stress mapping of these materials and effective identification of constitutive models. Through the integrated application of these technologies, this method can provide strong technical support for the research and application of material mechanical properties.

[0071] First, some terms related to the embodiments of the present disclosure are introduced.

[0072] 1. Digital Image Correlation (DIC) technology is a non-contact full-field measurement method based on digital image processing, which is widely used in fields such as material mechanics, structural engineering, and experimental mechanics. This technology sprays random speckles on the surface of an object, takes speckle images before and after loading, and uses image correlation algorithms to calculate the gray-scale changes between the two images, thereby obtaining displacement field and strain field information on the surface of the object. DIC technology has advantages such as high precision, strong adaptability, and wide measurement range, and can realize the analysis of material strain. To determine the stress of a material, it is usually necessary to measure the strain of the material with the help of strain gauges or DIC technology, and calculate the corresponding stress according to the constitutive model determined by material parameters such as Young's modulus of the material. Or the distribution state of the material can be determined according to photoelastic experiments. And through multi-scale simulations, a microscopic model of the material is established, and the stress state of the material is determined through numerical calculations.

[0073] 2. A neural network (English: Neural Network) is a computational model that simulates the working mechanism of human brain neurons and is one of the important methods in machine learning. It consists of multiple layers of artificial neurons, including an input layer, a hidden layer, and an output layer. Each neuron realizes non-linear transformation of the input data through weights and activation functions, so as to capture complex features and patterns. Neural networks perform outstandingly in processing unstructured data such as images, speech, and text, and are widely used in tasks in deep learning, such as image classification, speech recognition, natural language processing, and autonomous driving. Its powerful learning ability stems from the support of large-scale data and the drive of high-performance computing.

[0074] 3. Elastoplastic materials refer to materials that can exhibit both elastic deformation and plastic deformation under external forces. In the elastic stage, the deformation is reversible and the material can return to its original state when the external force is removed. In the plastic stage, the material undergoes permanent deformation and cannot fully recover even after the external force is removed. Common elastoplastic materials include metals and some polymers, and their mechanical behavior is usually affected by the internal microstructure of the material and the external stress state. Such materials are widely used in engineering and manufacturing fields. For example, in scenarios such as building structures, mechanical components, and the automotive industry, it is necessary to comprehensively consider their load-bearing capacity and deformation performance.

[0075] Secondly, the application scenarios related to the present disclosure are introduced. Please refer to Figure 1 , which shows a schematic structural diagram of a computing device provided by an exemplary embodiment of the present disclosure.

[0076] The computing device can be a terminal or a server. The terminal includes a mobile terminal or a fixed terminal. For example, the terminal can be a mobile phone, a tablet computer, a laptop computer, a desktop computer, etc. The server can be a single server, or a server cluster composed of several servers, or a cloud computing service center.

[0077] The computing device includes a processor 10, a memory 20, and a communication interface 30. Those skilled in the art can understand that Figure 1 the structure shown in

[0078] does not constitute a limitation on the computing device, and it may include more or fewer components than shown in the figure, or combine certain components, or have different component arrangements. Among them:

[0079] The memory 20 can be used to store software programs and modules. The processor 10 executes various functional applications and data processing by running the software programs and modules stored in the memory 20. The memory 20 may mainly include a program storage area and a data storage area. Among them, the program storage area may store the operating system 21, the acquisition module 22, the training module 23, and application programs 24 required for at least one function, etc.; the data storage area may store data created according to the use of the computing device, etc. The memory 20 can be implemented by any type of volatile or non-volatile storage device or a combination thereof, such as Static Random Access Memory (SRAM), Electrically Erasable Programmable Read-Only Memory (EEPROM), Erasable Programmable Read Only Memory (EPROM), Programmable Read-Only Memory (PROM), Read Only Memory (ROM), magnetic memory, flash memory, magnetic disk or optical disk. Accordingly, the memory 20 may further include a memory controller to provide the processor 10 with access to the memory 20.

[0080] Among them, the processor 10 executes the following functions by running the acquisition module 22: acquiring training data, where the training data includes multiple groups of sample data groups, and each group of sample data groups includes strain data and corresponding external forces. The strain data is used to indicate the degree of deformation of the elastoplastic material sample under the action of the external force, and the external force is used to indicate the total external force acting on the elastoplastic material sample. The processor 10 executes the following functions by running the training module 23: training a target neural network model based on the training data through an unsupervised learning method. The target neural network model is used to indicate the mapping relationship between the strain data and the stress of the elastoplastic material sample under the action of the external force, and the target neural network model satisfies the constraints of the equilibrium equation. The equilibrium equation is used to indicate that at any given moment, the stress distribution of the elastoplastic material sample satisfies the equilibrium condition of the external force, and the stress is used to indicate the internal force per unit area of the elastoplastic material sample.

[0081] That is to say, in the face of the challenges of stress monitoring technology for elastoplastic materials, the embodiments of the present disclosure propose an innovative method. This method trains a target neural network model through unsupervised learning to achieve stress mapping of elastoplastic materials. The core of this method lies in that it does not require traditional stress, yield function form, the number of internal variables, or evolution function, but directly uses strain data and the corresponding external force to train the target neural network model. This process not only simplifies the data requirements but also avoids the difficulties in establishing an appropriate equation form because the model is guided by the constraints of the equilibrium equation for learning without stress as input.

[0082] This target neural network model, which is an Equilibrium-based Neural Network (ENN) model, can indicate the mapping relationship between strain data and stress when an elastoplastic material sample is subjected to an external force. The innovation of the target neural network model lies in that it can train a model that satisfies the constraints of the equilibrium equation through unsupervised learning, that is, without label data. The advantage of this method is that it can generate a large amount of data through non-uniformly deformed specimens during a single loading process, effectively solving the problem of data scarcity in deep learning.

[0083] The trained target neural network model adopts the incremental method, which means that the model can calculate the corresponding stress increment based on the current stress-strain state and the input strain increment. The application of this incremental method enables the target neural network model to be adapted to the finite element method, providing a new technical path for the constitutive modeling of elastoplastic materials. In this way, the target neural network model not only improves the efficiency of data processing but also enhances the applicability and flexibility of the model in practical engineering applications.

[0084] Next, several exemplary embodiments are used to introduce the method provided by the embodiments of the present disclosure.

[0085] Please refer to Figure 2 , which shows a flowchart of a method for realizing stress mapping of elastoplastic materials provided by an exemplary embodiment of the present disclosure. This embodiment is illustrated by using this method in the Figure 1 shown computing device. The method includes the following steps.

[0086] Step 201, obtain training data. The training data includes multiple groups of sample data groups, and each group of sample data groups includes strain data and the corresponding external force. The strain data is used to indicate the degree of deformation of the elastoplastic material sample under the action of the external force, and the external force is used to indicate the total external force acting on the elastoplastic material sample.

[0087] When training the target neural network model, training data including multiple groups of sample data sets can be used to train the model so that the model can learn and understand the behavioral characteristics of elastoplastic materials under different conditions. These sample data sets can be derived from the same elastoplastic material sample or from multiple different elastoplastic material samples. Using multiple groups of sample data sets from the same elastoplastic material sample helps to deeply capture and understand the response characteristics of the material under diverse conditions, thereby improving the model's accurate prediction ability of the material behavior. Introducing multiple groups of sample data sets from different elastoplastic material samples, on the other hand, helps to enhance the model's generalization ability, enabling it to adapt to and predict the elastoplastic behavior of different materials, and thus play a role in a wider range of application scenarios. Through this diversified training data set, the model can not only accurately simulate the physical properties of a specific material, but also maintain high adaptability and prediction accuracy when facing new materials or unknown conditions.

[0088] Strain refers to the relative local deformation of an object under the action of an external force. In an elastoplastic material sample, strain data is used to indicate the degree of deformation that occurs under the action of an external force. Strain data can be obtained in various ways. For example, a vibrating wire strain gauge can be used. This method calculates the strain based on the frequency change of the vibrating wire. Another method is to obtain strain data through digital image correlation technology. For example, the digital image correlation technology is the DIC technology. This technology sprays speckles on the surface of the elastoplastic material sample and uses a high-speed camera to collect images. The collected images are imported into DIC analysis software to detect and analyze the strain data of the elastoplastic material sample. Optionally, through the DIC technology, the specimen (i.e., the elastoplastic material sample) can be divided into many small units. These units are discrete in space and form a grid. Each unit has its own nodes, which are the reference points for calculating the strain. The DIC technology calculates the displacement of each node by comparing the images of the specimen surface before and after loading. Using these displacement data, the DIC technology can calculate the strain components of each node. In two dimensions, there are usually three strain components: two normal strain components and one shear strain component.

[0089] In some embodiments, the external force can be directly measured by a loading device, which is a device that can accurately measure the external force in an experiment, such as a universal testing machine. The external force includes the resultant external force acting on the elastoplastic material sample, which indicates the magnitude and direction of the total external force received by the elastoplastic material sample.

[0090] In some embodiments, to increase the diversity of training data, a non-uniform strain field can be introduced, for example, by designing geometric defects such as circular holes or elliptical holes, or by non-uniform loading (i.e., applying forces of different magnitudes or directions in different regions), so that different nodes of the material experience different stress and strain states. These factors will cause the stress distribution inside the material to be non-uniform, thus generating more complex strain patterns, which helps to simulate various complex situations that materials may encounter in the real world. Compared with uniform deformation, this method can significantly increase the diversity of data in the training set, thereby improving the generalization ability of the target neural network model after training. The generalization ability refers to the performance ability of the model on unseen data. If the model is trained only under single or limited conditions, it may not be able to generalize well to new and different conditions. By increasing the diversity of training data, the model can learn a wider range of stress-strain relationships, thereby improving its prediction ability in new situations.

[0091] In some embodiments, during the material loading process, multiple different loading stages (time steps) will be experienced. At each time step, a large amount of stress-strain data will be generated. In traditional supervised learning, it may be difficult to obtain sufficient stress-strain data, especially in the plastic stage of the material. By collecting data at multiple time steps, the amount of data in the training set can be significantly increased, thereby providing more information for the model, enabling it to learn and predict the behavior of the material more accurately, ensuring the sufficiency of the data volume in the training set, and solving the problem of insufficient stress-strain data in supervised learning.

[0092] Step 202, according to the training data, train the target neural network model by unsupervised learning. The target neural network model is used to indicate the mapping relationship between the strain data and the stress of the elastoplastic material sample under the action of external forces, and the target neural network model satisfies the constraints of the equilibrium equation. The equilibrium equation is used to indicate that at any given moment, the stress distribution of the elastoplastic material sample satisfies the equilibrium condition of the external forces, and the stress is used to indicate the internal force per unit area of the elastoplastic material sample.

[0093] Unsupervised learning is a method of machine learning, suitable for data mining and pattern discovery, and does not require a large amount of labeled data.

[0094] The target neural network model is used to indicate the mapping relationship between the strain data and the stress of the elastoplastic material sample under the action of external forces and satisfies the constraints of the equilibrium equation. Such a model can be an ENN model. The target neural network model realizes the accurate prediction of the stress-strain relationship by embedding the equilibrium equation, that is, the general physical law, into the training process. The operating mechanism of the target neural network model is as follows:

[0095] 1. Input and Output: The model receives strain data as input and outputs the corresponding stress accordingly. This input-output relationship is guided and optimized by the equilibrium equations embedded during the training process.

[0096] 2. Weak Form of the Equilibrium Equation: The model establishes the connection between stress and external forces through the weak form of the quasi-static equilibrium equation. This approach transforms the traditional constitutive model problem into a problem of satisfying the equilibrium equation, thereby improving the physical consistency and prediction accuracy of the model.

[0097] 3. Strain Increment and Stress Increment: The model can calculate the stress increment corresponding to the strain increment based on the current stress-strain state. This ability enables the model to adapt to the finite element method and provides a new tool for engineering analysis.

[0098] 4. Dynamic Update: The model allows for dynamic updating of stress predictions when new strain data is input. This dynamic update mechanism enables the model to reflect the response of materials under complex loading paths in real time, enhancing the applicability and flexibility of the model.

[0099] Through this method, the target neural network model can not only provide in-depth understanding of the elastoplastic material behavior but also play a key role in practical engineering applications.

[0100] Based on the neural network training process of the equilibrium equation, for unknown elastoplastic materials, it is possible to directly measure the stress state of the material under non-uniform deformation and achieve stress mapping. The obtained neural network can further be used as a constitutive model of the material for finite element simulation calculations.

[0101] In some embodiments, for each set of sample data groups, according to the strain data, the predicted stress is determined through the target neural network model; according to the predicted stress and the external force, the loss value is calculated through the equilibrium equation, and the loss value is used to indicate the difference between the predicted stress and the stress that satisfies the constraints of the equilibrium equation; by minimizing the loss value, the parameters of the target neural network model are adjusted. Optionally, the strain data includes strain increments corresponding to multiple time steps. According to the strain data, determining the predicted stress through the target neural network model includes: for each time step, based on the strain increment corresponding to the time step, the stress increment for the next time step is output through the target neural network model; repeating the above process at each time step until the preset model convergence condition is met, and the predicted stress is determined. It should be noted that the details of the training process of the target neural network model can refer to the relevant descriptions in the following embodiments and will not be introduced here.

[0102] In some embodiments, after training the target neural network model, the method steps of applying the target neural network model include but are not limited to the following steps:

[0103] 1. Model Validation: Before applying the model in practice, it is necessary to validate the trained neural network model to ensure that it can accurately predict the stress-strain relationship of elastoplastic materials under different external forces. This usually involves using a portion of the data that was not involved in training (the validation set) to test the generalization ability of the model.

[0104] 2. Model Prediction: Input the strain data and external force into the trained neural network model, and the model will output the corresponding stress.

[0105] 3. Result Analysis: Analyze the stress output by the neural network model to verify whether it meets the constraints of the equilibrium equation. The equilibrium equation is used to indicate that at any given moment, the stress distribution of the elastoplastic material sample satisfies the equilibrium condition of the external force.

[0106] 4. Model Adjustment: If the results predicted by the model do not meet the constraints of the equilibrium equation or the prediction accuracy is not high enough, it may be necessary to adjust the model. This may include retraining the model, adjusting the network structure, increasing the training data, or adjusting the model parameters, etc.

[0107] 5. Practical Application: After the model has been fully validated and adjusted, it can be applied to actual elastoplastic material stress mapping problems to predict the stress-strain behavior of the material under external forces, thereby guiding engineering design and material selection.

[0108] 6. Continuous Monitoring and Update: During the actual application process of the model, it is necessary to continuously monitor the prediction performance of the model and update and optimize the model based on new data and feedback to maintain the accuracy and reliability of the model.

[0109] In summary, traditional stress mapping methods usually require a large amount of stress, the form of the yield function, the number of internal variables, or evolution functions. The method provided by the embodiments of the present disclosure directly uses strain data and the corresponding external forces to train the target neural network model, thereby simplifying the data requirements. Moreover, in traditional elastoplastic material analysis, establishing an appropriate equation form is a challenge. This method trains the neural network model through an unsupervised learning method, avoiding this difficulty because the model is guided by the constraints of the equilibrium equation for learning without stress as input. And by embedding the constraints of the equilibrium equation into the training process of the neural network, this method ensures the physical consistency of the model output and can maintain the prediction accuracy even without explicit stress. And this method allows the model to dynamically update stress predictions when new strain data is input, and this dynamic update mechanism enables the model to reflect the response of the material under complex loading paths in real time, enhancing the applicability and flexibility of the model. Therefore, through the above method, efficient and accurate prediction of elastoplastic material stress mapping can be achieved, while reducing the experimental cost and improving the flexibility of engineering applications.

[0110] Please refer to Figure 3 , which shows a schematic diagram of the principle of a method for implementing stress mapping of elastoplastic materials provided by another exemplary embodiment of the present disclosure. In this embodiment, the method is used for Figure 1 the computing device shown in the figure to illustrate by way of example.

[0111] Figure 3 (a) shows the process of data collection and experimental setup.

[0112] 1. Experimental setup: Use an experimental setup of the DIC method to apply an external force and collect data.

[0113] 2. Data type: In the experiment, the data collected is non-uniform strain field data, which can include strain data and external force. The strain data can include three main strain components: ε 11 , ε 22 and ε 12， The external force can be F b . These data describe the deformation of the material during the force application process, but do not include stress. The DIC technique calculates the displacement of each node by comparing the images of the specimen surface before and after loading, and then obtains the full-field strain data.

[0114] Figure 3 (b) shows the unsupervised learning process of the target neural network model. Unsupervised learning means that no labeled data (i.e., actual stress) is used during the training process. Instead, the network self-supervises through internal mechanisms (such as loss functions) to ensure the physical reasonableness of the prediction results. This method allows the network to be trained without actual stress, thus having broad application potential in the field of material mechanics.

[0115] 1. Network structure of the target neural network model: Construct an unsupervised learning network with strain data as the input and predicted stress as the output. For the t-th time step, the input is three strain components: ε 11 (t), ε 22 (t) and ε 12 (t), and the output is three stress components: σ 11 (t), σ 22 (t), σ 12 (t).

[0116] 2. Loss function of the target neural network model: The loss function is a crucial part in the training of neural networks. It measures the difference between the stress predicted by the model and the stress that actually satisfies the physical equilibrium equation. In the mechanical behavior of elastoplastic materials, stress must satisfy the equilibrium equation, which is a fundamental physical law describing the internal force balance of materials. The loss function calculates the loss value by comparing the predicted stress with the stress that satisfies these physical constraints. During the training process, the goal is to minimize the loss function. By adjusting the parameters of the network (such as weights and biases), the neural network learns how to reduce the difference between stresses, thereby better satisfying the constraints of the equilibrium equation.

[0117] The unsupervised learning process of the target neural network model includes: The training input of the model is a strain sequence, that is, the strain state of the material at different time points. The training output of the model is the corresponding stress sequence, that is, the stress state of the material at the same time points. During the training process, the equilibrium equation, which is a general physical law, is embedded into the training process of the target neural network model. Through the weak form of the quasi-static equilibrium equation, the relationship between stress and external forces can be established. This method usually involves transforming the equilibrium equation from the strong form (differential form) to the weak form (integral form) for use in finite element analysis.

[0118] Traditional constitutive models describe the stress-strain relationship of materials. Here, the problem is transformed into ensuring that the predicted stress satisfies the equilibrium equation. This means that the model not only has to learn the stress-strain relationship of materials but also ensure that these relationships are physically reasonable.

[0119] After meshing the specimen, the stress can be integrated through the element shape function and in-plane components. This process involves concepts in the finite element method, where the element shape function is a shape function used to describe the displacement at any point within the element, and the in-plane components are the strain-displacement matrices used to relate displacement to strain. Schematically, the following formula is used to calculate the nodal force:

[0120]

[0121] where, represents the nodal force vector at time step t; Ω represents the element domain; σ t represents the stress tensor at time step t. The stress tensor is a second-order tensor that contains normal stress and shear stress components; ▽N α (x) represents the element shape function N α(x) gradient; x represents the x - direction in the spatial coordinates, y represents the y - direction in the spatial coordinates, α represents the index of the in - plane component, usually taking values 1 and 2, corresponding to two spatial directions (i.e., the x - direction and the y - direction) respectively. The physical meaning of this formula is to calculate the force at each node within the element by integrating the product of the stress tensor and the gradient of the shape function. This is one of the basic steps in the finite - element method, used to calculate the nodal forces from the stress field and further analyze the mechanical response of the structure.

[0122] The SmoothL1 loss function can be used to measure the loss value, which has good robustness to outliers. Schematically, the loss value s(x) is calculated using the following formula:

[0123]

[0124] where x represents the difference between the predicted value and the true value.

[0125] During the model training process provided by the embodiments of the present disclosure, since there is no stress, that is, no true value, the difference between the predicted value and the true value cannot be directly used to calculate the loss function. However, physical laws and equilibrium equations can be utilized to construct the loss function, thereby training the model in an unsupervised learning framework. In some embodiments, the node types of the specimen can be divided into internal nodes and boundary nodes. Internal nodes refer to the nodes located inside the specimen, which are not directly affected by external boundary conditions. Boundary nodes refer to the nodes located on the boundary of the specimen, which are directly affected by external conditions such as applied forces, displacements, pressures, etc. Therefore, the loss function provided by the embodiments of the present disclosure can consist of two parts:

[0126] 1. Equilibrium of internal - node forces: Under quasi - static loading, the internal nodes of the specimen are not subject to external forces and should be in an equilibrium state. The first part L in of the loss function can be expressed as:

[0127]

[0128] where n t is the total number of time steps; n in is the total number of internal nodes; s is the SmoothL1 loss function; represents the resultant external force of the i - th node at time step t.

[0129] 2. Equilibrium of boundary - node forces: For the nodes under force / displacement boundary conditions, although the magnitude of the individual node forces cannot be obtained, their resultant force is known. The second part L b of the loss function can be expressed as:

[0130]

[0131] where, n t is the total number of time steps; n b is the total number of nodes in the set of boundary nodes; n b,j is the total number of nodes in boundary set j; s is the SmoothL1 loss function; is the force on the k-th node in boundary node set j at time step t; is the known resultant force on boundary node set j at time step t. Here, the set of boundary nodes refers to the set of all nodes located on the boundary of the specimen. Boundary set j is a subset of the set of boundary nodes, which is used to more specifically define and handle specific boundary regions or boundary condition types in the model.

[0132] The total loss function can be the sum of the above two parts or a weighted sum of the above two parts. By minimizing this loss function, the model can learn the stress distribution under given strain and external force conditions, even without direct stress as training labels. The advantage of this method is that it can use physical laws to guide the learning process, thereby improving the generalization ability and interpretability of the model.

[0133] Figure 3 (c) shows the network architecture of the target neural network model. This model can be constructed based on the internal variable theory and is designed to simulate the elastoplastic behavior of materials. The model evaluates whether the material enters the plastic deformation state through the yield function and uses the augmented space of internal variables and their evolution equations to describe the plastic deformation process in detail. Among them, the internal variable theory is a material mechanics theory that describes the plastic deformation and hardening behavior of materials by introducing internal state variables (internal variables). In plasticity mechanics, the yield function is used to judge whether the material enters the plastic deformation state. When the stress state satisfies the conditions of the yield function, the material begins to undergo plastic deformation. In traditional methods, determining the specific forms of these related functions and internal variables is a complex task. However, this model cleverly uses the excellent nonlinear fitting ability of neural networks to replace the forms of these complex functions, thus combining the advantages of data-driven methods and constitutive theories.

[0134] In some embodiments, the model predicts the stress increment at the next time step based on the current strain increment at each time step. This process includes, for each time step, inputting the strain increment, stress, plastic strain, and internal variables corresponding to the time step into the target neural network model, and outputting the stress increment, plastic strain increment, and internal variable increment at the next time step. It should be noted that the initial values of stress, plastic strain, and internal variables are all zero. Plastic strain is used to indicate the deformation history of irreversible deformation that occurs in the elastoplastic material sample under the action of external forces, and internal variables are used to indicate the deformation history and hardening state of the internal state variables of the elastoplastic material sample under the action of external forces.

[0135] That is to say, for each time step t, the input of the target neural network model is the strain increment Δε, stress σ t , plastic strain and internal variable q t , and the output is the stress increment Δσ, plastic strain increment Δε p and internal variable increment Δq at time step t + 1. This process can be expressed by the following formula:

[0136]

[0137] The initial condition is The strain ε can be decomposed into elastic strain ε e and plastic strain ε p The sum is ε = ε e + ε p . For each time step, the trial stress can be determined according to the strain increment and the elastic stiffness of the elastoplastic material sample. The trial stress refers to the stress when the material is completely in the elastic state under the given strain increment. This process can be expressed as: σ trial = C:Δε, where σ trial is the trial stress, C is the elastic stiffness of the elastoplastic material sample, and Δε is the strain increment.

[0138] In some embodiments, the target neural network model includes three sub-networks: the first neural network, the second neural network, and the third neural network. For each time step, according to the trial stress, the first stress is output through the first neural network; according to the plastic strain and internal variable, the second stress is output through the second neural network; in the case where the first stress and the second stress do not meet the model convergence condition, according to the stress, plastic strain, and internal variable, the stress increment, plastic strain increment, and internal variable increment of the next time step are output through the third neural network. For example, the first neural network is an input convex neural network (ICNN), the second neural network is a neural network (NN)-q, and the third neural network is an NN-h. The following further introduces the three sub-networks respectively.

[0139] The input parameters of the first neural network include the trial stress, and the output parameters include the first stress (also known as the equivalent stress). Schematically, as Figure 4 described, the first neural network is an ICNN, and the ICNN can be expressed as a function f ic :

[0140]

[0141] Among them, is the first stress, σtrial It is the trial stress.

[0142] ICNN can ensure that the first stress output is a convex function with respect to the input parameters. This is achieved by forcing some of the weight parameters to be non - negative and using a convex and non - decreasing activation function. Schematically, the activation function selected by ICNN can be the Exponential Linear Unit (ELU) function:

[0143]

[0144] where ELU(x) is the output value of the ELU function and x is the input value of the ELU function.

[0145] Consider an ICNN with n layers, where n is a positive integer. The k - th layer can be expressed as:

[0146]

[0147] where, σ (k+1) represents the output of the (k + 1)-th layer, that is, the result after being processed by the ELU activation function; W k represents the weight matrix of the k - th layer, which is used to linearly transform the input data; σ (k) represents the output of the k - th layer; σ (0) represents the output of the input layer, that is, the trial stress; b k represents the bias vector of the k - th layer. Taking the absolute value of all the weight matrices for k = 1, …, n - 1 can ensure convexity. This structure ensures the convexity of the yield surface and improves the computational stability.

[0148] Figure 3 (d) shows the input and output parameters of the second neural network. The input parameters of the second neural network include plastic strain and internal variables, and the output parameters include the second stress (also called the yield stress). Schematically, the second neural network is NN - q, and NN - q can be expressed as a function f y :

[0149]

[0150] where, is the second stress, is the plastic strain at the t - th time step, and q t is the internal variable at the t - th time step.

[0151] It should be noted that when using a neural network to simulate and predict the stress-strain behavior of materials, there is no need to deeply understand the number of internal variables and their physical meanings. Therefore, the output of the neural network is not limited to the equivalent stress or yield stress with clear physical meanings. These outputs should be regarded as a formal yield criterion and mainly used for reference and analysis. Such a method allows for the effective evaluation and prediction of material behavior without fully understanding the complex mechanisms inside the material.

[0152] Figure 3 (e) shows the input and output parameters of the third neural network. The input parameters of the third neural network include stress, plastic strain, and internal variables, which are used to represent the evolution equation, and the output parameters include the plastic strain increment and the internal variable increment. Schematically, the third neural network is NN-h, and NN-h can be represented by the following function h:

[0153]

[0154] Wherein, is the internal variable increment; is the non-negative plastic flow rate, which controls the change rate of plastic strain; ε p is the plastic strain; q is the internal variable; σ is the stress; h can be regarded as a vector, differing from the evolution of the internal variable by a coefficient.

[0155] In some embodiments, when the first stress and the second stress satisfy the model convergence condition, the trial stress is determined as the predicted stress. The model convergence condition may include that the difference between the first stress and the second stress is less than a preset threshold. Schematically, by comparing the first stress with the second stress to determine whether the current state is a yielding state:

[0156]

[0157] If the difference f between the first stress and the second stress is less than the preset threshold Tol, that is, indicating that the yielding state has not been reached, then the trial stress is used as the result of the model output, that is, σ n+1 = σ trial , and the plastic strain and the internal variable remain unchanged, that is, q n+1 = q n . Otherwise, the stress and internal variable at the next time step can be calculated through the implicit graphical return algorithm.

[0158] In some embodiments, the associated plasticity of the elastoplastic material sample can be expressed as:

[0159]

[0160] Among them, r is the plastic flow direction, f is the yield function, and σ is the stress.

[0161] The Newton method is used for iterative solution. The iterative process from time step t = m to t = m + 1, where m is an integer. For the initial iterative step, the initial conditions σ (0) = σ trial , q (0) = q m are set, and the plastic increment parameter Δλ = 0. The residuals a (0) , b (0) and f (0) can be expressed as:

[0162]

[0163] b (0) = -q (0) + q m + Δλ (0) h (0)

[0164]

[0165] Among them, f ic and f y are the functions of the first neural network and the second neural network, respectively.

[0166] The linearized equation is:

[0167] a (k) + C -1 :Δσ (k) + Δλ (k) Δr (k) + δλ (k) r (k) = 0

[0168] b (k) -Δq (k) + Δλ (k) Δh (k) + δλ (k) h (k) = 0

[0169]

[0170] Among them, that is, Δr (k) represents the product of the partial derivative of r (k) with respect to the stress σ (k) and the internal variable q (k) and the increment, and Δh (k) represents h(k) The partial derivative of the stress σ (k) and the internal variable q (k) multiplied by the increment. C is the stiffness matrix of the material, calculated from the elastic model and Poisson's ratio, and can be obtained through a standard uniaxial tensile test. Through the above linearized equation, δλ (k) can be solved, that is, the increment of Δλ in the k-th iteration:

[0171]

[0172] where:

[0173]

[0174] where, is the partial derivative of f (k) with respect to σ (k) and q (k) ; A (k) is a matrix containing the partial derivatives of Δλ (k) with respect to r (k) and q (k) ; and are vectors.

[0175] Thus, the stress increment Δσ (k) and the internal variable increment Δq (k) are obtained:

[0176]

[0177] Use the increment to update the stress, plastic strain, and internal variable:

[0178]

[0179] The iterative process continues until the preset model convergence condition is met, that is where Tol is the preset threshold.

[0180] In a schematic example, the effectiveness of the solution provided by the embodiments of the present disclosure is verified by the finite element method. First, data is generated by the finite element method for numerical verification. A remarkable feature of the embodiments of the present disclosure is the introduction of the concept of a non-uniform deformation field. Further, a square plate specimen with a circular hole (side length is a specific value H) is considered, as Figure 5 shown respectively represent the displacement vector and the force vector in the x1 direction (horizontal direction); respectively represent the displacement vector and the force vector in the x2 direction (vertical direction). A biaxial displacement loading is applied to the specimen, and the loading path is as Figure 6(a) As shown, the corresponding resultant external force is obtained. In this example, the widely used von Mises yield function and isotropic hardening criterion are first adopted as the material model, and the isotropic hardening material parameters shown in Table 1 are set. Among them, the parameter E is the Young's modulus, which represents the stiffness of the material within the elastic range, that is, the ratio of stress to strain data. υ is the Poisson's ratio, which describes the ratio of the strain in the perpendicular direction to the strain in a certain direction when the material is compressed in one direction. The Poisson's ratio is one of the geometric properties of the material. are the three invariants of the plastic strain, and these parameters are used to describe the degree and characteristics of the plastic deformation of the material. are the three invariants of the yield stress, and these parameters define the stress level at which the material begins to undergo plastic deformation.

[0181] During the finite element calculation process, the specimen was finely meshed, and a total of 226 nodes and 392 three-node triangular elements were generated. This mesh generation method shows extremely high efficiency without the need for stress information.

[0182] Table 1

[0183]

[0184] The input strain field and external force are used to train the target neural network model, that is, the ENN model. Another remarkable feature of the embodiments of the present disclosure is that the stress field can be extracted and the constitutive model can be learned. The strain data is input into the trained ENN, and the predicted stress is obtained and compared with the stress σ ij output by the finite element method. The results are as Figure 5 (b). To examine the stress prediction results of the entire specimen at each time step, the relative error Δ ij is used as the evaluation index:

[0185]

[0186] where n e is the total number of elements. The results show that the comparison of the stress components is concentrated around y = x, and the errors are Δ 11 = 2.19%, Δ 22 = 1.76%, Δ 12 = 5.64%. This shows that the ENN model can accurately identify the stress field and replace the traditional constitutive model. The ENN model has obtained excellent prediction results for the stress components at all time steps, which proves that the ENN model can capture the characteristics of the yield function and the evolution law of the internal variables.

[0187] The ENN model is trained with known strain fields and external forces, so there is a certain basis for the accuracy of the stress prediction results of the training set. To test the generalization ability of the trained ENN, the elliptical hole in Figure 5 (a) is replaced with Figure 5 the three circular holes in (c), changing the topological structure of the model. respectively represent the displacement vector and the force vector along the x1 direction (horizontal direction); Figure 6 (b). These changes make the data of the test set very different from that of the training set. In the test set, a total of 442 nodes and 800 elements are divided, with 36 time steps, and 29,600 sets of stress-strain data (including the initial state) can be generated. The strain is input into the trained ENN model to obtain the prediction results of the test set. Similarly, the output results of the ENN are compared with the results obtained by the finite element method, as shown in Figure 5 (d). The errors of the stress components are Δ 11 = 2.07%, Δ 22 = 1.67% and Δ 22 = 8.03%. The reason for the larger shear stress error may be that it is smaller than the normal stress and more difficult to identify. The results of the training set and the test set show that the ENN can effectively extract the constitutive model through unsupervised learning.

[0188] In another illustrative example, the effectiveness of the solution provided by the embodiments of the present disclosure is demonstrated through experiments. Given that the actual stress field is usually not directly measurable, an experimental verification method is proposed and the corresponding applications are discussed, as shown in Figure 7 (a). The strain data is collected by the DIC method to train the ENN model. Then, the trained ENN model is incorporated into the finite element software as the constitutive model. In this process, the tangent stiffness matrix also needs to be obtained to ensure the accuracy and reliability of the model. The tangent stiffness matrix can be expressed as:

[0189]

[0190] where C ep represents the elastic stiffness matrix, which describes the stiffness characteristics of the material within the elastic range. r represents the plastic flow direction, which is a vector related to the yield function of the material and indicates the direction in which plastic deformation occurs. f y is the yield function, which is a scalar function used to describe the condition for the material to start plastic deformation. When the value of f y exceeds a certain critical value, the material starts to undergo plastic deformation. ε p$\varepsilon_p$ is the plastic strain, representing the permanent deformation part experienced by the material. $q$ is an internal variable, which is a variable describing the internal state of the material. $h$ is a vector related to the internal variable $q$, which may represent the rate of change of the internal variable or other physical quantities related to the internal variable.

[0191] Through these steps, the ENN model has laid the foundation for two major applications: 1. Material model library: The ENN model can serve as a material model library for describing the constitutive behavior of new materials. Under given geometric conditions and external excitations, the deformation field and stress field can be solved by the finite element method. This provides a powerful tool for the design and evaluation of new materials, helping engineers and researchers better understand and predict the performance of materials in practical applications. 2. Stress identifier: The ENN model can also act as a stress identifier. By regularly photographing the surface of the new material structure and using DIC technology to obtain strain history data, the ENN model can quickly and accurately calculate the corresponding stress field, providing an efficient method for structural health monitoring. This ability is crucial for predicting the fatigue life of materials, detecting early damage, and ensuring structural safety.

[0192] Generally speaking, the ENN model can not only accurately simulate and predict the constitutive behavior of materials, but also play an important role in structural health monitoring, providing valuable support for the development and application of new materials.

[0193] In this experiment, the aluminum alloy AlSi10mg was selected as the research object. It is an elastoplastic material prepared by 3D printing technology, containing silicon and magnesium elements, and possessing excellent strength and hardness characteristics. Through uniaxial tensile tests and DIC technology, the Young's modulus and Poisson's ratio of this material were measured, and the results are as Figure 8 shown. The figure contains two test results and their fitting curves. Figure 8 (a) shows the relationship between stress ($\sigma$) and strain ($\varepsilon$ 22 ). The black curve in the figure represents the fitting result, and the red and blue curves represent the results of Test 1 and Test 2 respectively. It can be seen that as the strain increases, the stress also increases, showing the elastic behavior of the material. A DIC analysis image of the material is also inserted in the figure, showing the deformation of the material under stress. Figure 8 (b) shows the relationship between the transverse strain ($\varepsilon$ 11 ) and strain ($\varepsilon$ 22 ). Similarly, the black curve is the fitting result, and the red and blue curves represent the results of Test 1 and Test 2 respectively. The figure shows that as the strain increases, the transverse strain gradually decreases, which is related to the Poisson's ratio of the material.

[0194] To verify the effectiveness of the solution provided by the embodiments of the present disclosure, specimens with elliptical holes were used as the training set, as Figure 7(as shown in (b)). By comparing the strain fields measured experimentally with those predicted by the ENN model, the accuracy of the ENN model was evaluated. In particular, the strain components in the tensile direction of the material were focused on, and the good similarity confirmed the prediction accuracy of the ENN model.

[0195] Furthermore, to test the generalization ability of the ENN model, the elliptical hole in the test set was replaced with three circular holes, as Figure 7 (as shown in (c)). Again, the strain field obtained experimentally was compared with the results calculated by the trained ENN model, and the results also showed high accuracy, demonstrating the robustness of the ENN model.

[0196] In addition, after incorporating the ENN model into the material library, it can be used for finite element calculations to display the distributions of the strain field and stress field. Although the DIC technique can provide information on the strain field of materials, the direct measurement of plastic strain and stress fields is still challenging. Thanks to the interpretability of the ENN model, these two key physical quantities can be evaluated in experiments, as Figure 9 shown. Figure 9 Parts (a) and (b) in represent different experimental conditions or material states. Through these figures, the strain and stress distributions in different regions of the material can be visually seen, and this function helps to solve the problems of structural detection and evaluation in engineering. This method starting from experimental data and then verifying with experimental data achieves a closed loop.

[0197] Generally speaking, the embodiments of the present disclosure achieve closed-loop verification through the method of starting from experimental data and then verifying with experimental data. This not only proves the effectiveness of the ENN model in predicting the behavior of elastoplastic materials, but also provides a new tool for structural health monitoring and material property evaluation.

[0198] However, it should be noted that embedding more constraints may affect the generalization ability of the ENN model, especially when dealing with new materials lacking mature knowledge. Nevertheless, considering the general applicability of the equilibrium equation in mechanics, it is believed that the concept of the ENN model is not limited to stress mapping in elastoplastic materials and has the potential to be extended to the modeling of other types of material behaviors such as viscoelastic and viscoplastic materials.

[0199] The core of the embodiments of the present disclosure lies in the neural network training process based on the equilibrium equation, which enables the direct determination of the stress state of unknown elastoplastic materials even under non-uniform deformation conditions and achieves accurate stress mapping. The obtained neural network model can further be used as a constitutive model of the material for finite element simulation calculations.

[0200] In summary, the embodiments of the present disclosure provide an end-to-end stress mapping method based on a neural network of balance equations, demonstrating its potential as an alternative for calculating the stress of elastoplastic solids. This method is not only applicable to two-dimensional cases but can also be easily extended to three-dimensional scenarios. By adopting balance constraints and introducing additional physical constraints, the embodiments of the present disclosure are expected to reduce the scale of the required training dataset while improving the calculation accuracy on elastoplastic solids, especially in fields with established theoretical support.

[0201] The following are device embodiments of the present disclosure. For parts not elaborated in detail in the device embodiments, reference may be made to the technical details disclosed in the above method embodiments.

[0202] The embodiments of the present disclosure provide a device for realizing stress mapping of elastoplastic materials. This device can be implemented in whole or in part as a computing device through software, hardware, and a combination of both. The device includes: an acquisition module and a training module.

[0203] The acquisition module is used to acquire training data. The training data includes multiple groups of sample data groups, and each group of sample data groups includes strain data and corresponding external forces. The strain data is used to indicate the degree of deformation of the elastoplastic material sample under the action of external forces, and the external force is used to indicate the total external force acting on the elastoplastic material sample.

[0204] The training module is used to train a target neural network model by unsupervised learning according to the training data. The target neural network model is used to indicate the mapping relationship between the strain data and the stress of the elastoplastic material sample under the action of external forces, and the target neural network model satisfies the constraints of the balance equation. The balance equation is used to indicate that at any given moment, the stress of the elastoplastic material sample satisfies the equilibrium condition of the external force, and the stress is used to indicate the internal force per unit area of the elastoplastic material sample.

[0205] In a possible implementation manner, the training module is further used to:

[0206] For each group of sample data groups, determine the predicted stress according to the strain data through the target neural network model;

[0207] Calculate a loss value according to the predicted stress and the external force through the balance equation. The loss value is used to indicate the difference between the predicted stress and the stress that satisfies the constraints of the balance equation;

[0208] Adjust the parameters of the target neural network model by minimizing the loss value.

[0209] In another possible implementation manner, the strain data includes strain increments corresponding to multiple time steps. The training module is further used to:

[0210] For each time step, based on the strain increment corresponding to the time step, the stress increment for the next time step is output through the target neural network model;

[0211] Repeat the above process at each time step until the preset model convergence condition is satisfied, and determine the predicted stress.

[0212] In another possible implementation, the training module is further configured to:

[0213] For each time step, the strain increment, stress, plastic strain, and internal variables corresponding to the time step are input into the target neural network model, and the stress increment, plastic strain increment, and internal variable increment for the next time step are output;

[0214] Among them, the initial values of the stress, plastic strain, and internal variables are all zero. The plastic strain is used to indicate the deformation history of the irreversible deformation that occurs in the elastoplastic material sample under the action of an external force, and the internal variable is used to indicate the deformation history and hardening state of the internal state variables of the elastoplastic material sample under the action of an external force.

[0215] In another possible implementation, the target neural network model includes a first neural network, a second neural network, and a third neural network. The training module is further configured to:

[0216] For each time step, determine the trial stress according to the strain increment and the elastic stiffness of the elastoplastic material sample;

[0217] Output the first stress through the first neural network according to the trial stress;

[0218] Output the second stress through the second neural network according to the plastic strain and the internal variables;

[0219] In the case where the first stress and the second stress do not satisfy the model convergence condition, output the stress increment, plastic strain increment, and internal variable increment for the next time step through the third neural network according to the stress, plastic strain, and internal variables.

[0220] In another possible implementation, the device further includes:

[0221] The training module is further configured to determine the trial stress as the predicted stress in the case where the first stress and the second stress satisfy the model convergence condition.

[0222] In another possible implementation, the model convergence condition includes that the difference between the first stress and the second stress is less than a preset threshold.

[0223] It should be noted that when the device provided in the above embodiments realizes its functions, only the division of the above-mentioned respective function modules is used for illustration. In actual applications, the above functions can be allocated to different function modules according to actual needs, that is, the content structure of the device is divided into different function modules to complete all or part of the functions described above.

[0224] Regarding the device in the above embodiments, the specific manners in which each module performs operations have been described in detail in the embodiments related to the method, and will not be elaborated here.

[0225] Embodiments of the present disclosure also provide a computing device, including a memory, a processor, and a computer program stored on the memory, where the processor executes the computer program to implement the steps of the above method.

[0226] Embodiments of the present disclosure also provide a non-volatile computer-readable storage medium, on which a computer program is stored, and when the computer program is executed by a processor, the steps of the above method are implemented.

[0227] Embodiments of the present disclosure also provide a computer program product, including a computer program, or a non-volatile computer-readable storage medium carrying the computer program, and when the computer program is executed by a processor, the steps of the above method are implemented.

[0228] Figure 10 It is a block diagram of a device 1900 shown according to an exemplary embodiment. For example, the device 1900 can be provided as a computing device for implementing stress mapping of elastoplastic materials. Refer to Figure 10 , the device 1900 includes a processing component 1922, which further includes one or more processors, and memory resources represented by a memory 1932 for storing instructions executable by the processing component 1922, such as application programs. The application programs stored in the memory 1932 can include one or more modules each corresponding to a set of instructions. In addition, the processing component 1922 is configured to execute instructions to perform the above method.

[0229] The device 1900 may also include a power supply component 1926 configured to perform power management of the device 1900, a wired or wireless network interface 1950 configured to connect the device 1900 to a network, and an input / output interface 1958 (I / O interface). The device 1900 can operate based on an operating system stored in the memory 1932, such as Windows ServerTM, MacOS XTM, UnixTM, LinuxTM, FreeBSDTM or the like.

[0230] In an exemplary embodiment, a non - volatile computer - readable storage medium is also provided, such as a memory 1932 including computer program instructions, and the computer program instructions can be executed by a processing component 1922 of the device 1900 to complete the above - mentioned method.

[0231] A computer - readable storage medium can be a tangible device that can hold and store programs / instructions used by an instruction - execution device. A computer - readable storage medium can be, for example, but is not limited to, an electrical storage device, a magnetic storage device, an optical storage device, an electromagnetic storage device, a semiconductor storage device, or any suitable combination of the foregoing. More specific examples (a non - exhaustive list) of the computer - readable storage medium include: a portable computer disk, a hard disk, a random access memory (RAM), a read - only memory (ROM), an erasable programmable read - only memory (EPROM or flash memory), a static random access memory (SRAM), a portable compact disk read - only memory (CD - ROM), a digital versatile disk (DVD), a memory stick, a floppy disk, a mechanically encoded device, such as a punched card or raised structures in grooves storing instructions thereon, and any suitable combination of the foregoing. The computer - readable storage medium used herein is not construed as an instantaneous signal itself, such as a radio wave or other freely propagating electromagnetic wave, an electromagnetic wave propagated through a waveguide or other transmission medium (e.g., an optical pulse through an optical fiber cable), or an electrical signal transmitted through a wire.

[0232] The computer programs (or computer - readable program instructions) described herein can be downloaded from a computer - readable storage medium to various computing / processing devices, or downloaded to an external computer or external storage device through a network, such as the Internet, a local area network, a wide area network, and / or a wireless network. The network can include a copper transmission cable, an optical fiber transmission, a wireless transmission, a router, a firewall, a switch, a gateway computer, and / or an edge server. A network adapter or network interface in each computing / processing device receives the computer - readable program instructions from the network and forwards the computer - readable program instructions for storage in the computer - readable storage medium in each computing / processing device.

[0233] A computer program (or computer program instructions) for performing the operations of the present disclosure may be assembly instructions, instruction set architecture (ISA) instructions, machine instructions, machine - related instructions, microcode, firmware instructions, state - setting data, or source code or object code written in any combination of one or more programming languages, including object - oriented programming languages such as Smalltalk, C++, etc., and conventional procedural programming languages such as the "C" language or similar programming languages. The computer - readable program instructions may be executed entirely on a user's computer, partially on the user's computer, executed as a stand - alone software package, partially on the user's computer and partially on a remote computer, or entirely on a remote computer or server. In the case of a remote computer, the remote computer may be connected to the user's computer through any type of network, including a local area network (LAN) or a wide area network (WAN), or may be connected to an external computer (e.g., through the Internet using an Internet service provider). In some embodiments, by using the state information of the computer - readable program instructions to customize an electronic circuit, such as a programmable logic circuit, a field - programmable gate array (FPGA), or a programmable logic array (PLA), the electronic circuit can execute the computer - readable program instructions to implement various aspects of the present disclosure.

[0234] Aspects of the present disclosure are described herein with reference to the flowcharts and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the present disclosure. It should be understood that each block of the flowcharts and / or block diagrams, and combinations of blocks in the flowcharts and / or block diagrams, can be implemented by computer - readable program instructions.

[0235] These computer - readable program instructions can be provided to a processor of a general - purpose computer, a special - purpose computer, or other programmable data - processing apparatus to produce a machine such that the instructions, when executed by the processor of the computer or other programmable data - processing apparatus, create a means for implementing the functions / actions specified in one or more blocks of the flowchart and / or block diagram. The computer - readable program instructions can also be stored in a computer - readable storage medium, and these instructions cause a computer, a programmable data - processing apparatus, and / or other devices to work in a particular manner. Thus, the computer - readable medium storing the instructions includes a manufacture, which includes instructions for implementing various aspects of the functions / actions specified in one or more blocks of the flowchart and / or block diagram.

[0236] Computer-readable program instructions may also be loaded onto a computer, other programmable data processing apparatus, or other device, causing a series of operational steps to be performed on the computer, other programmable data processing apparatus, or other device to produce a computer-implemented process such that the instructions executed on the computer, other programmable data processing apparatus, or other device implement the functions / acts specified in one or more boxes of the flowchart and / or block diagram.

[0237] The flowcharts and block diagrams in the figures illustrate the architecture, functionality, and operation of possible implementations of systems, methods, and computer program products according to various embodiments of the present disclosure. In this regard, each block in the flowchart or block diagram may represent a module, a segment of code, or a portion of an instruction, which contains one or more executable instructions for implementing the specified logical function. In some alternative implementations, the functions noted in the blocks may occur out of the order noted in the figures. For example, two consecutive blocks may in fact be executed substantially in parallel, or they may sometimes be executed in the reverse order, depending upon the functionality involved. It should also be noted that each block of the block diagrams and / or flowcharts, and combinations of blocks in the block diagrams and / or flowcharts, can be implemented by a dedicated hardware-based system that performs the specified functions or acts, or by a combination of dedicated hardware and computer instructions.

[0238] The embodiments of the present disclosure have been described above. The above description is exemplary, not exhaustive, and is not limited to the disclosed embodiments. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the described embodiments. The choice of terms used herein is intended to best explain the principles of the embodiments, the practical application, or improvements made to the technology in the marketplace, or to enable other ordinary skilled artisans in the art to understand the embodiments disclosed herein.

Claims

1. A method for realizing stress mapping of elastoplastic materials, characterized in that The method includes: Obtaining training data, where the training data includes multiple groups of sample data groups, and each group of the sample data groups includes strain data and corresponding external forces. The strain data is used to indicate the degree of deformation of the elastoplastic material sample under the action of the external force, and the external force is used to indicate the total external force acting on the elastoplastic material sample; Training a target neural network model by an unsupervised learning method according to the training data. The target neural network model is used to indicate the mapping relationship between the strain data and the stress of the elastoplastic material sample under the action of the external force, and the target neural network model satisfies the constraint of the equilibrium equation. The equilibrium equation is used to indicate that at any given moment, the stress distribution of the elastoplastic material sample satisfies the equilibrium condition of the external force, and the stress is used to indicate the internal force per unit area of the elastoplastic material sample.

2. The method according to claim 1, characterized in that The training of the target neural network model by an unsupervised learning method according to the training data includes: For each group of the sample data groups, determining the predicted stress through the target neural network model according to the strain data; Calculating a loss value through the equilibrium equation according to the predicted stress and the external force. The loss value is used to indicate the difference between the predicted stress and the stress that satisfies the constraint of the equilibrium equation; Adjusting the parameters of the target neural network model by minimizing the loss value.

3. The method according to claim 2, wherein The strain data includes strain increments corresponding to multiple time steps. The determining of the predicted stress through the target neural network model according to the strain data includes: For each time step, outputting the stress increment of the next time step through the target neural network model based on the strain increment corresponding to the time step; Repeating the above process at each time step until a preset model convergence condition is satisfied, and determining the predicted stress.

4. The method according to claim 3, characterized in that The outputting of the stress increment of the next time step through the target neural network model based on the strain increment corresponding to the time step for each time step includes: For each time step, inputting the strain increment, stress, plastic strain, and internal variable corresponding to the time step into the target neural network model, and outputting the stress increment, plastic strain increment, and internal variable increment of the next time step; Wherein, the initial values of the stress, the plastic strain, and the internal variable are all zero. The plastic strain is used to indicate the deformation history of the irreversible deformation of the elastoplastic material sample under the action of the external force, and the internal variable is used to indicate the deformation history and hardening state of the internal state variables of the elastoplastic material sample under the action of the external force.

5. The method according to claim 4, wherein The target neural network model includes a first neural network, a second neural network, and a third neural network. The inputting of the strain increment, stress, plastic strain, and internal variable corresponding to the time step into the target neural network model for each time step and outputting the stress increment, plastic strain increment, and internal variable increment of the next time step includes: For each time step, determining a trial stress according to the strain increment and the elastic stiffness of the elastoplastic material sample; Output a first stress through the first neural network according to the trial stress; Output a second stress through the second neural network according to the plastic strain and the internal variable; In the case that the first stress and the second stress do not satisfy the model convergence condition, output the stress increment, the plastic strain increment, and the internal variable increment at the next time step through the third neural network according to the stress, the plastic strain, and the internal variable.

6. The method according to claim 5, wherein The method further includes: In the case that the first stress and the second stress satisfy the model convergence condition, determine the trial stress as the predicted stress.

7. The method according to claim 5 or 6, characterized in that, The model convergence condition includes that the difference between the first stress and the second stress is less than a preset threshold.

8. A device for realizing stress mapping of elastoplastic materials, characterized in that, The device includes: An acquisition module, configured to acquire training data, where the training data includes multiple groups of sample data groups, and each group of the sample data groups includes strain data and corresponding external forces. The strain data is used to indicate the deformation degree of the elastoplastic material sample under the action of the external force, and the external force is used to indicate the total external force acting on the elastoplastic material sample; A training module, configured to train a target neural network model in an unsupervised learning manner according to the training data. The target neural network model is used to indicate the mapping relationship between the strain data and the stress of the elastoplastic material sample under the action of the external force, and the target neural network model satisfies the constraint of the equilibrium equation. The equilibrium equation is used to indicate that at any given moment, the stress of the elastoplastic material sample satisfies the equilibrium condition of the external force, and the stress is used to indicate the internal force per unit area of the elastoplastic material sample.

9. A computing device, comprising a memory, a processor, and a computer program stored on the memory, characterized in that, The processor executes the computer program to implement the steps of the method according to any one of claims 1 to 7.

10. A non-volatile computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, the steps of the method according to any one of claims 1 to 7 are implemented.

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