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\DOI{10.5802/crmeca.315}
\datereceived{2025-04-16}
\daterevised{2025-06-14}
\dateaccepted{2025-07-08}
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\dateposted{2025-08-07}
\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\title{Predicting stress--strain constitutive relationships: a deep
learning approach based on multi-head attention mechanism and long
short-term memory networks}

\alttitle{Pr\'{e}diction des relations constitutives
contrainte-d\'{e}formation : une approche d'apprentissage profond
bas\'{e}e sur le m\'{e}canisme d'attention multi-t\^{e}te et les
r\'{e}seaux \`{a} m\'{e}moire courte et longue (LSTM)}

\author{\firstname{Lei} \lastname{Han}\CDRorcid{0009-0007-7023-3693}\IsCorresp}
\address{School of Automobile and Transportation Engineering, 
Guangdong Polytechnic Normal University, Guangzhou, 510665, China}
\email[L. Han]{hanlei@gpnu.edu.cn}

\author{\firstname{Chunmei} \lastname{Duan}\CDRorcid{0009-0000-4997-1201}}
\address{School of Mechanical Engineering, 
Guangdong Ocean University, Zhanjiang, 524088, China}
\address{Guangdong Engineering Technology Research 
Center of Ocean Equipment and Manufacturing, Zhanjiang, 524088, China}

\author{\firstname{Taochuan} \lastname{Zhang}\CDRorcid{0000-0002-0489-8562}\IsCorresp}
\addressSameAs{2}{School of Mechanical Engineering, 
Guangdong Ocean University, Zhanjiang, 524088, China}
\addressSameAs{3}{Guangdong Engineering Technology Research Center of 
Ocean Equipment and Manufacturing, Zhanjiang, 524088, China}
\email[T. Zhang]{zhangtaochuan@hotmail.com}

\shortrunauthors

\keywords{\kwd{Deep learning}
\kwd{Stress--strain}
\kwd{Attention mechanism}
\kwd{Long short-term memory network}
\kwd{Material performance prediction}}

\altkeywords{\kwd{Apprentissage profond}
\kwd{Contrainte-d\'{e}formation}
\kwd{M\'{e}canisme d'attention}
\kwd{R\'{e}seau de m\'{e}moire courte et longue (LSTM)}
\kwd{Pr\'{e}diction des performances des mat\'{e}riaux}}

\thanks{Guangdong Polytechnic Normal University (Grant Number
2021SDKYA017),  Program for Scientific Research Start-up Funds of 
Guangdong Ocean University (Grant Numbers 060302062311, 060302062315), 
Collaborative Education Project of Industry University Cooperation  by
the Ministry of Education of China  (Grant numbers 230802784214645,
230803817223502,  230821384507208).}

\begin{abstract} 
The stress--strain constitutive relationship of materials is a critical
indicator for assessing their mechanical properties and serves as a
vital bridge connecting microstructures with macroscopic mechanical
behaviors. This study focuses on 106 metal alloy materials, employing
the finite element method to establish uniaxial tensile numerical
models, thereby simulating and obtaining a comprehensive dataset of
stress--strain relationship curves. Innovatively, this research proposes
a MA-LSTM-DLnet prediction model that integrates multi-head attention
mechanisms with long short-term memory networks, based on deep learning
methodologies. The model utilizes conventional key parameters such as
material strain, yield strength, tensile strength, elastic modulus,
density, strength coefficient, and stress as input features to
construct a multi-dimensional training dataset. It particularly
explores the learning capabilities and predictive performance of the
MA-LSTM-DLnet model under both complete and limited feature datasets.
The results demonstrate that the trained MA-LSTM-DLnet model can
accurately predict the stress--strain constitutive relationship curves
of new materials by merely acquiring conventional physical parameters
like yield strength, tensile strength, and elastic modulus, with a
similarity exceeding 95\% compared to test data. Compared to
traditional empirical formulas and numerical simulation methods, the
main innovation of this study lies in eliminating key parameters that
are difficult to determine in mathematical models (such as the strength
coefficient in the Ludwik isotropic hardening model), significantly
reducing repetitive work dependent on experiments to obtain
constitutive relationships, and achieving efficient and precise
material performance evaluation. 
\end{abstract}

\begin{altabstract} 
La relation constitutive contrainte-d\'{e}formation des mat\'{e}riaux
est un indicateur essentiel pour \'{e}valuer leurs propri\'{e}t\'{e}s
m\'{e}caniques, servant de lien crucial entre les microstructures et
les comportements m\'{e}caniques macroscopiques. Cette \'{e}tude porte
sur 106 alliages m\'{e}talliques, utilisant la m\'{e}thode des
\'{e}l\'{e}ments finis pour \'{e}tablir des mod\`{e}les num\'{e}riques
de traction uniaxiale, permettant ainsi de simuler et d'obtenir un
ensemble de donn\'{e}es complet de courbes contrainte-d\'{e}formation.
De mani\`{e}re innovante, cette recherche propose un mod\`{e}le de
pr\'{e}diction MA-LSTM-DLnet int\'{e}grant des m\'{e}canismes
d'attention multi-t\^{e}te avec des r\'{e}seaux de m\'{e}moire courte
et longue (LSTM), bas\'{e} sur des m\'{e}thodologies d'apprentissage
profond. Le mod\`{e}le utilise des param\`{e}tres cl\'{e}s
conventionnels tels que la d\'{e}formation du mat\'{e}riau, la limite
d'\'{e}lasticit\'{e}, la r\'{e}sistance \`{a} la traction, le module
d'\'{e}lasticit\'{e}, la densit\'{e}, le coefficient de r\'{e}sistance
et la contrainte comme caract\'{e}ristiques d'entr\'{e}e pour
construire un ensemble de donn\'{e}es d'entra\^{i}nement
multidimensionnel. L'\'{e}tude explore particuli\`{e}rement les
capacit\'{e}s d'apprentissage et les performances pr\'{e}dictives du
mod\`{e}le MA-LSTM-DLnet, aussi bien avec des ensembles de donn\'{e}es
complets qu'avec des donn\'{e}es limit\'{e}es. Les r\'{e}sultats
d\'{e}montrent que le mod\`{e}le entra\^{i}n\'{e} peut pr\'{e}dire avec
pr\'{e}cision les courbes de relations constitutives
contrainte-d\'{e}formation de nouveaux mat\'{e}riaux, en se basant
uniquement sur des param\`{e}tres physiques conventionnels comme la
limite d'\'{e}lasticit\'{e}, la r\'{e}sistance \`{a} la traction et le
module d'\'{e}lasticit\'{e}, avec une similarit\'{e} d\'{e}passant 
95 \% par rapport aux donn\'{e}es de test. Compar\'{e} aux formules
empiriques traditionnelles et aux m\'{e}thodes de simulation
num\'{e}rique, la principale innovation de cette \'{e}tude r\'{e}side
dans l'\'{e}limination de param\`{e}tres cl\'{e}s difficiles \`{a}
d\'{e}terminer dans les mod\`{e}les math\'{e}matiques (comme le
coefficient de r\'{e}sistance dans le mod\`{e}le d'\'{e}crouissage
isotrope de Ludwik), r\'{e}duisant ainsi consid\'{e}rablement le
travail exp\'{e}rimental r\'{e}p\'{e}titif n\'{e}cessaire pour obtenir
les relations constitutives, et permettant une \'{e}valuation efficace
et pr\'{e}cise des performances des mat\'{e}riaux.
\end{altabstract}

\maketitle

\end{noXML}

\section{Introduction}\label{sec1}
The constitutive relationship of materials is a crucial mathematical
model in the field of material mechanics, as it precisely describes the
correlation between internal stress and strain during the deformation
process. By providing a strain value, this model can accurately
calculate the corresponding stress, thereby offering a theoretical
foundation for predicting and analyzing the mechanical behavior of
materials under various working conditions. For instance, the
stress--strain relationship enables in-depth analysis of material
performance under different temperatures, loads, and time conditions,
which is of significant importance for engineering applications and
product  design~\cite{1,2,3,4,5}. Therefore, a thorough investigation
of the constitutive relationship not only enhances our understanding of
the mechanical responses of materials in practical applications but
also significantly improves the reliability and durability of products.

Currently, there are two primary approaches to obtaining the
stress--strain constitutive relationship of materials: experimental
testing and empirical formulas or numerical simulations. Common
experimental methods include tensile testing, which is applicable to a
wide range of materials such as metals, plastics, and 
elastomers~\cite{6,7,8,9,10}, however, these methods may not be
suitable for rare or expensive materials and are often time-consuming.
Among the classical empirical formulas, the Ramberg--Osgood formulation
is widely  used~\cite{11}. Under the assumption of isotropic materials,
this formula calculates stress and strain by incorporating parameters
such as elastic modulus, yield strength, tensile strength, and
hardening exponent. However, some key parameters in the model still
require experimental determination based on the material's 
properties~\cite{11}. Additionally, the Johnson-Cook and Ludwik models
are extensively employed to simulate the nonlinear constitutive
relationships of materials and are embedded in many commercial finite
element software packages. Nevertheless, these models also rely on
experimentally determined  parameters~\cite{12,13}. Many scholars have
combined experimental and simulation methods to establish or fit
constitutive relationships, but these models are often limited to
specific materials or working  conditions~\cite{14,15,16,17,18,19,20}.

Despite advancements in mathematical and physical theories, it remains
challenging to establish a universal mathematical formula or model for
the stress--strain relationship that applies to all material types or
even within a broad category, such as metal alloys. Currently, the only
approach is to conduct experiments for each alloy to obtain relevant
parameters, which are then used in empirical formulas or simulations,
or to directly perform tensile tests to acquire the corresponding
stress--strain curves. With the development of artificial intelligence
(AI) technologies, the field of materials science has gradually
embraced AI. In recent years, some researchers have utilized machine
learning or deep learning methods to predict material 
properties~\cite{21,22,23,24,25,26,27,28,29,30,31,32}. Unlike
traditional mathematical models that establish explicit mapping
relationships, deep learning prediction models are based on a
generalized, fuzzy mapping relationship between input and output
variables, mimicking the neural networks of the  brain~\cite{33}. This
fundamental difference makes it possible to develop a unified deep
learning model for predicting the constitutive relationships of
materials within a broad category.

Recently, a small number of scholars have begun to explore the use of
deep learning methods to predict stress--strain relationship 
curves~\cite{34,35,36,37,38,39,40}. For example,  El Fallaki Idrissi
et~al.~\cite{39} proposed an advanced meta-modeling framework that
integrates machine learning and Model Order Reduction (MOR) techniques.
They employed Proper Orthogonal Decomposition (POD) to decompose
stress--strain curves and used a Random Forest machine learning model
for interpolation. This enables the rapid prediction of macroscopic
nonlinear stress--strain curves under varying microstructural
parameters. Nikolic  et~al.~\cite{34} developed an automated
microstructure generation technique combined with finite element
analysis to obtain stress--strain curves under different
microstructures. They trained a convolutional neural network (CNN) on
microstructure image data to predict the stress--strain response of
materials at varying temperatures. Koenuma  et~al.~\cite{35}
synthesized material microtextures using numerical methods and computed
corresponding stress--strain curves, employing CNNs to capture the
features of synthetic textures and predict stress--strain curves under
different material textures. Zhang  et~al.~\cite{36} applied long
short-term memory (LSTM) deep learning methods to replicate the
stress--strain behavior of soils, investigating the ability of LSTM to
account for the stress history of soil behavior. Saha  et~al.~\cite{37}
utilized a deep learning (DL) framework, specifically a 3D U-Net
architecture, to map composite microstructures to corresponding stress
and strain fields under predefined loading paths, predicting the
evolution of microscale elastoplastic strain and stress in two-phase
media. Chen  et~al.~\cite{38} simulated material failure processes
using the phase-field method, generating composite microstructures and
corresponding stress--strain curves as training and testing data. They
employed a conditional generative adversarial network (cGAN) as the
deep learning model to predict stress--strain curves.  El Fallaki
Idrissi et~al.~\cite{40}  presented a novel method named Multiscale
Thermodynamics-Informed Neural Networks (MuTINN). It can quickly and
cost---effectively compute the inelastic behavior of woven composite
structures. On the premise of maintaining thermodynamic consistency,
MuTINN can substantially decrease the need for training data. Moreover,
it has been successfully incorporated into commercial finite element
software, accelerating the multiscale simulation of composite material
structures. 

In summary, although some researchers have begun to explore the
application of deep learning methods to material constitutive
relationships, most studies rely on classical and basic 
\mbox{neural} network
models trained on microstructure images. These approaches essentially
perform image classification to derive stress--strain relationship
curves. However, predicting material constitutive relationships
directly using conventional material properties---such as elastic
modulus, yield strength, tensile strength, and density---as inputs to
neural network models remains challenging for conventional
architectures. To address this limitation, this study proposes a novel
deep learning neural network model, MA-LSTM-DLnet, based on long
short-term memory (LSTM)  networks~\cite{41,42} and integrated into a
multi-head attention  mechanism~\cite{43}. This model incorporates a
built-in masking and decoding mechanism with a 4-head attention
mechanism. By training and validating the model on stress--strain data
from 106 alloy materials, we have developed a new and efficient method
for calculating material stress--strain relationships. This method
requires only conventional material properties---yield strength,
tensile strength, and elastic modulus---as input parameters to predict
the stress--strain relationship curve. Compared to traditional
mathematical models and tensile testing methods, this approach
eliminates the need for difficult-to-determine parameters that
typically require experimental or empirical derivation. Moreover, it
significantly reduces the repetitive work of obtaining constitutive
relationships through tensile tests, thereby greatly saving time and
labor costs.

\section{Method}\label{sec2} 
The research methodology and overall workflow employed in this study
are illustrated in  Figure~\ref{fig1}. First, the finite element method
(FEM) was utilized to establish uniaxial tensile simulation models for
106 alloy materials under room temperature conditions. Through
computational analysis, the stress--strain relationship curves for each
material at room temperature were obtained. These curves, combined with
conventional material properties such as yield strength, tensile
strength, elastic modulus, and density, were used as characteristic
parameters to represent the materials. This process resulted in the
creation of a comprehensive dataset comprising 106 time-series entries,
each containing complete stress--strain data, which was subsequently
used for training and validating the deep learning network model.
Second, the established deep learning model was trained on the dataset
to develop a high-precision predictive model for stress--strain
relationships. The predictive performance of the model was investigated
under both complete and limited material feature conditions. Finally,
the conventional material properties of the 106 alloy materials used in
this study, along with the data and instructions required for finite
element simulations, are provided in  Appendix~\ref{appA}.

\begin{figure}
\includegraphics{fig01} 
\caption{\label{fig1}Research method flowchart.}
\end{figure}

\subsection{FEM simulation for stress--strain}\label{sec2.1}
This study selected a total of 106 metal alloy materials from six major
categories, as detailed in  Appendix~\ref{appA}. To obtain the
stress--strain relationship curves for all alloy materials, a
mathematical model for material tensile behavior was established, and
the finite element method (FEM) was employed to simulate the tensile
behavior of each metal alloy material under specific heat treatment
conditions. Under the assumptions of isotropy, continuity, and small
strain rates during tensile deformation, the mathematical model for
material tensile behavior can be represented by the stress equilibrium
differential equation, geometric equation, and constitutive 
equation~\cite{44}.  Since the tensile deformation considers small
strain rates, a static tensile simulation was adopted. The stress
equilibrium differential equation and geometric equation are as
follows:
{\begin{equation}
\nabla \cdot \bsigma  + \mathbi{f}=0 \label{eq1}
\end{equation}}\unskip
where,  $\bsigma$ is the stress tensor, and $\mathbi{f}$ represents the
body force, which can be neglected for conventional static tensile
simulations.
{\begin{equation}
\bvarepsilon =
\tfrac{1}{2}(\nabla \mathbi{u}+(\nabla \mathbi{u})^{\mathbi{T}})
\label{eq2}
\end{equation}}\unskip
where $\bvarepsilon$ denotes the strain tensor, $\mathbi{u}$ represents
the displacement field, and $(\nabla \mathbi{u})^{\mathbi{T}}$  is the
transpose of the displacement gradient tensor. During the tensile
process, materials undergo both elastic and plastic deformation.
Therefore, the generalized Hooke's law was adopted to describe the
constitutive relationship. The Ludwik constitutive equation, an
empirical model based on experimental data, effectively describes the
relationship between flow stress and plastic strain during the plastic
deformation  stage~\cite{12}. Consequently, the Ludwik hardening model
was selected as the constitutive equation for the tensile simulation,
expressed as:
{\begin{equation}
\sigma =E(\varepsilon _{\mathrm{t}}-\varepsilon_{\mathrm{p}})+
K(\varepsilon_{\mathrm{p}})^{n}\label{eq3}
\end{equation}}\unskip
where $E$ is the elastic modulus, $\varepsilon _{\mathrm{t}}$ represents the
total strain, $\varepsilon_{\mathrm{p}}$ denotes the plastic strain, and $K$
and $n$ are parameters required for the Ludwik hardening model, with
$K$ representing the strength coefficient and $n$ the hardening
exponent. Additionally, the plastic flow rule, yield function, and
Karush--Kuhn--Tucker (KKT) conditions are given as follows:
{\begin{eqnarray}
&\displaystyle
\varepsilon_{\mathrm{p}}=\lambda 
\frac{\partial f}{\partial \sigma}
\label{eq4}\Seqnsplit 
&\displaystyle
f(\sigma ,\sigma _{y})=
\sqrt{\frac{3}{2}\sigma_{\mathrm{eq}}^{2}}-
\sigma _{y} \label{eq5}\Seqnsplit 
&\displaystyle
\lambda \geq 0,\quad f\leq 0,\quad \lambda f=0
\label{eq6}
\end{eqnarray}}\unskip
where $\lambda $ is the plastic multiplier, $f$ is the yield function,
$\sigma _{y}$ represents the yield stress, and $\sigma _{\mathrm{eq}}$ 
denotes the equivalent stress. Equations~(\ref{eq1})
through~(\ref{eq6}) collectively form the mathematical model for the
tensile  simulation. By incorporating the geometric shape of the
tensile specimen, corresponding boundary conditions, and material
property parameters, the finite element numerical algorithm was
employed to solve these equations, thereby obtaining the stress--strain
curve data for the respective materials.

In accordance with the Chinese National Standard (GB/T 228.1-2010),
this study employed rectangular cross-section tensile specimens for the
simulation of tensile behavior. The specific dimensions of the tensile
specimens are illustrated in Figure~\ref{fig2}. For the geometric
model, boundary conditions were applied as follows: one side of the
specimen was fixed, while the other side was subjected to a
displacement boundary condition with a uniform tensile length of 15 mm.
The remaining boundaries were set as free. By integrating these
boundary conditions with the aforementioned mathematical model, the
tensile behavior of various materials was simulated, enabling the
acquisition of true or equivalent stress--strain relationship curves.

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig2}Tensile test specimen geometry according to China
national standards.}
\end{figure}

Although the core focus of this study is on the deep learning-based
training and prediction of material stress--strain curve data,
experimental validation was conducted to ensure that the simulation
data closely approximates completeness and realism. Specifically, the
feasibility of using the finite element method to solve the
aforementioned model was verified. First, stress--strain tensile
experimental data for four materials---5052 and 7A04 Al alloys, duplex
stainless steel, and copper alloy---were obtained from the 
literature~\cite{45,46}. Subsequently, finite element models for these
four materials were established using the methodology described above.
The simulation results were then compared with the experimental data to
demonstrate the feasibility of obtaining stress--strain curves through
material tensile simulation. During the simulation process, seven key
material parameters were required, including yield strength $\sigma
_{y}$, tensile strength $\sigma _{u}$, elastic modulus $E$, density
$\rho $, Poisson's ratio $v$, strength coefficient $K$, and hardening
exponent $n$, as listed in Table~\ref{tab1}.

\begin{table}
\caption{\label{tab1}Material parameters for simulation}
\begin{tabular}{cccccccc}  
\thead
Name & $\sigma _{y}$~(MPa) & $\sigma _{u}$~(MPa) &
$E$~(GPa) & $\rho$~(kg/m$^{3}$) & $v$ & $K$ & $n$ \\ 
\endthead
5052 Al alloy & 220 & 267 & \070 & 2680 & 0.33 & 350 & 0.3 \\ 
7A04 Al alloy & 540 & 570 & \071 & 2800 & 0.33 & 630 & 0.3 \\ 
Duplex stainless steel & 652 & 820 & 208 & 7800 & 0.33 & 820 & 0.3 \\ 
Cu--Cr--Zr copper alloy & 254 & 328 & 100 & 8900 & 0.33 & 388 & 0.3
\botline
\end{tabular}
\end{table}

Using the parameters listed in  Table~\ref{tab1}, tensile simulations
were conducted for four materials: 5052 and 7A04 Al alloys, duplex
stainless steel, and copper alloy. A uniform tensile length of 15 mm
was applied to all specimens, a distance sufficient to prevent necking
during the tensile process. The simulated stress--strain curves were
compared with experimental data, as shown in  Figure~\ref{fig3}. From
the figure, it is evident that the simulated stress--strain data for
aluminum alloys and stainless steel exhibit excellent agreement with
the experimental data. For the copper alloy, the simulated
stress--strain data show slight deviations from the experimental
results, but the overall trend of the curves remains consistent. These
findings demonstrate the feasibility of using the aforementioned
numerical method to obtain stress--strain data from tensile
simulations.

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig3}Stress--strain curves: experimental data and
simulation results.}
\vspace*{5pt}
\end{figure}

When employing the Ludwik hardening model as the plastic constitutive 
equation, the key parameters influencing the simulation results are the
strength coefficient $K$ and the hardening exponent $n$. For different
materials, particularly the strength coefficient $K$, significant
variations can be observed. For the 106 materials selected in this
study, the strength coefficient $K$ was determined based on empirical
formulas provided by relevant  authors~\cite{47,48,49} and fitting
analysis of experimental data. Specifically, a compensation value
ranging from 0 to 100~MPa was added to the tensile strength of each
material to derive its strength coefficient. Furthermore, since the
core objective of this study is to obtain stress--strain data from
various materials for training a deep learning model to capture the
distribution patterns and trends of stress--strain curves, a certain
margin of error in the stress--strain data is permissible. For metal
alloys, the hardening exponent $n$ exhibits minimal variation across
materials, and its impact on simulation results is relatively small.
Therefore, a uniform value of 0.3 was adopted as the hardening exponent
for all simulations in this study.

Based on the aforementioned model and material parameters, tensile
simulations were conducted for 106 alloy materials. The obtained
stress--strain data were categorized into six major groups: aluminum
alloys, copper alloys, nickel alloys, tool steels, titanium alloys, and
other steels, as illustrated in  Figure~\ref{fig4}. From the figure, it
is evident that the stress--strain behavior of all materials initially
exhibits a linear relationship during the elastic stage, followed by
nonlinear changes during the elastoplastic and plastic hardening
stages. Furthermore, a comparison of the stress--strain curves for all
materials reveals that the yield points or hardening regions of the 106
alloy materials fall within a range of 100~MPa to 1400~MPa. The
majority of the materials exhibit yield points above 200~MPa, with only
a few materials below this threshold. Additionally, materials with
yield points around 400~MPa are relatively rare. For detailed
classification, such as the number of stress--strain curves within the
100~MPa to 300~MPa range, as well as the specific stress--strain curve
data for each material, please refer to  Appendix~\ref{appB}.

\begin{figure}
\includegraphics{fig04}
\caption{\label{fig4}Stress--strain curves of 106 materials.}
\end{figure}

\subsection{A deep learning approach}\label{sec2.2}
This study proposes a deep learning model based on a multi-head
attention mechanism and a long short-term memory (LSTM) network,
referred to as the MA-LSTM-DLnet model. This model belongs to the
category of time-series prediction models in neural networks and
consists of two main components: the attention mechanism and the LSTM
network. The attention mechanism employs a 4-head attention mechanism,
as multi-head attention allows the model to learn information in
parallel from different representation subspaces. For a detailed
explanation of the attention mechanism, refer to the work of Vaswani 
et~al.~\cite{50}. For the input sequence data $X$ used for learning or
prediction, the attention mechanism first applies three distinct linear
transformations to obtain the query $(Q)$, key $(K)$, and value $(V)$
matrices, as described by the following equations:
{\begin{eqnarray}
Q &=& XW^{Q}
\label{eq7}\Seqnsplit 
K &=& XW^{K} 
\label{eq8}\Seqnsplit 
V &=& XW^{V}
\label{eq9}
\end{eqnarray}}\unskip
Here, $W^{Q}$, $W^{K}$, $W^{V}$ are weight matrices learned by the
model, and $Q$,  $K$, and  $V$ represent the query, key, and value
matrices, respectively. These  equations map the input sequence $X$ to
the representation spaces of  $Q$, $K$, and  $V$, enabling the model to
capture features and dependencies from different perspectives. In the
case of multi-head attention, the transformed vectors are divided into
multiple heads, each assigned independent  $Q$,  $K$, and  $V$
matrices. For each head, the output is computed using the scaled
dot-product attention mechanism, as follows:
{\begin{equation}
\mathrm{Attention}\;(Q,K,V)= \mbox{Soft}\; \max 
\left(\frac{Q\cdot K^{\mathrm{T}}}{\sqrt{d_{\mathrm{k}}}}\right)\cdot V                      
\label{eq10}
\end{equation}}\unskip
Here, $d_{\mathrm{k}}$ is the dimensionality of the key matrix, and
$\sqrt{d_{\mathrm{k}}}$ serves as a scaling factor to prevent excessively large
dot-product results, which could lead to gradient vanishing. The term
$K^{\mathrm{T}}$ denotes the transpose of the key matrix.  
Equation~(\ref{eq10})
computes the attention weights for each head, which are then used to
calculate a weighted sum of the value matrix $V$. The outputs of all
heads are concatenated and passed through a linear layer to produce the
final attention output vector, as shown below:
{\begin{eqnarray}
\mbox{MutiHead}\;(X) &=& \mbox{Concat}\;
(H_{1},H_{2},\ldots,H_{h})W^{o}                     
\label{eq11} \Seqnsplit 
H_{i} &=& \mbox{Attention}\; (Q_{i},K_{i},V_{i})                              
\label{eq12}
\end{eqnarray}}\unskip
Here, $W^{o}$ is the output projection matrix.

The LSTM network is a specialized type of recurrent neural network
(RNN). Its core lies in the gating mechanism, which includes three
gates: the forget gate, input gate, and output  gate~\cite{51}. The
forget gate determines which information is retained or discarded from
the memory cell, using a sigmoid activation function to produce values
between 0 and 1, representing the proportion of information to retain.
Its expression is as follows:
{\begin{equation}
f_{t}= \sigma (W_{f}\cdot [h_{t-1},x_{t}]+b_{f})                              
\label{eq13} 
\end{equation}}\unskip
Here, $f_{t}$ is the output of the forget gate, $\sigma $ is the
sigmoid activation function, $W_{f}$ and $b_{f}$ are the weight matrix
and bias, $h_{t-1}$ is the hidden state from the previous time step,
and $x_{t}$ is the current input. The input gate determines which new
information is stored in the memory cell. It consists of two parts: a
sigmoid function to decide the update portion and a tanh function to
generate candidate values. Its expressions are as follows:
{\begin{eqnarray}
i_{t}  &=&
\sigma (W_{i}\cdot [h_{t-1},x_{t}]+b_{i})                               
\label{eq14}\Seqnsplit 
\widetilde{C_{t}} &=&
\tanh (W_{C}\cdot [h_{t-1},x_{t}]+b_{c})                             
\label{eq15}
\end{eqnarray}}\unskip
Here, $i_{t}$ is the output of the input gate, and  $\widetilde{C_{t}}$
is the new candidate value. The memory cell is a critical component of
the LSTM, responsible for storing long-term dependency information. Its
expression is as follows:
{\begin{equation}
\mathrm{c}_{t}=
f_{t}\cdot c_{t-1}+i_{t}\cdot 
\widetilde{C_{t}}                                 
\label{eq16}
\end{equation}}\unskip
Here, $\mathrm{c}_{t}$ is the current memory cell state, and $c_{t-1}$
is the memory cell state from the previous time step. Finally, the
output gate determines the output of the hidden state, using a sigmoid
function to decide which information is output. Its expressions are as
follows:
{\begin{eqnarray}
O_{t} &=&
\sigma (W_{o}\cdot [h_{t-1},x_{t}]+b_{o})                             
\label{eq17} \Seqnsplit 
h_{t} &=&
O_{t}\cdot \tanh (c_{t})                                  
\label{eq18}
\end{eqnarray}}\unskip
Here, $O_{t}$ is the output of the output gate, and $h_{t}$ is the
current hidden state.

Based on the aforementioned attention mechanism and LSTM network model,
this study proposes the MA-LSTM-DLnet model for predicting material
constitutive relationships. The model consists of three main
components, as illustrated in  Figure~\ref{fig5}. The first component
involves feeding preprocessed sequence data into the input layer,
accompanied by a positional encoding layer with a maximum dimension of
512. Through  Equations~(\ref{eq7})--(\ref{eq9}), linear
transformations are applied to initially obtain the $Q$, $K$, and $V$
matrices for the sequence data. Since the model employs four attention
heads, the initial $Q$, $K$, and $V$ matrices are divided into four parts,
each stored in a linear layer and corresponding to one attention head.
Using  Equation~(\ref{eq10}), the attention weights and outputs are
computed for each attention head based on its respective $Q$, $K$, and $V$
matrices. The outputs of all attention heads are then concatenated to
produce the final attention output. The second component takes the
features extracted and filtered by the attention mechanism as input to
the LSTM network. Here, the features undergo further training and
analysis through the forget gate, input gate, and output gate. In this
study, the selection of key model parameters was based on a combination
of empirical optimization and preliminary research. It was found that
with 800 hidden neurons, the model demonstrated satisfactory training
accuracy and generalization performance on the validation set.
Increasing the number beyond 800 did not significantly improve
performance but led to longer training times and higher computational
resource consumption. Thus, considering the balance between model
performance and computational efficiency, we ultimately selected 800
hidden neurons for the LSTM layer. The third component employs a
dropout layer to randomly zero out 50\% of the neuron outputs. The
remaining elements are then fed into a fully connected layer with a
single unit, which performs the final regression prediction output.

\begin{figure}
\includegraphics{fig05} 
\caption{\label{fig5}MA-LSTM-DLnet model.} 
\end{figure}

\subsection{Feature set configuration}\label{sec2.3} 
The essence of the prediction model is illustrated in 
Figure~\ref{fig6},  where the input consists of relevant feature
vectors or matrices $X$, and the output is the value $Y$. For the material
stress--strain relationship curve, stress $\sigma $ represents the
output value $Y$, while strain $\varepsilon$ is one of the features in
the input feature matrix $X$. However, since strain data alone is too
simplistic, the model cannot distinguish between different materials
during training. Therefore, strain data can only serve as one of the
features in the input feature matrix $X$. To enhance the model's ability
to differentiate between materials, additional features are
incorporated into the input feature matrix $X$, including material yield
strength $\sigma _{y}$, tensile strength $\sigma _{u}$, elastic modulus
$E$, density $\rho $, strength coefficient $K$, and the product of
elastic modulus and strain. By including these supplementary features,
the model can effectively distinguish between different materials
during the training process based on the input feature matrix $X$.

\begin{figure}
\includegraphics{fig06}
\caption{\label{fig6}Schematic diagram of the prediction process.} 
\end{figure}

To better distinguish and study the influence of different features in
the input feature matrix $X$, as well as to evaluate the accuracy of the
prediction model under various input feature configurations, the input
features were subdivided into different scenarios, as detailed in\break 
Table~\ref{tab2}.

\begin{table}
\caption{\label{tab2}Feature parameter configurations for network model
training}
\begin{tabular}{cccccccc}  
\thead
Name & $\sigma _{y}$ & $\sigma _{u}$ & $E$ & $\rho $ &
$K$ & $E\times \varepsilon $ & $\varepsilon $ \\ 
\endthead
Setting 1 & y & y & y & y & y & y & y \\ 
Setting 2 & y & y & y & y & n & y & y \\ 
Setting 3 & y & y & y & y & n & n & y \\ 
Setting 4 & y & y & y & n & n & n & y \\ 
Setting 5 & y & y & n & n & n & n & y
\botline 
\end{tabular}
\end{table}

Five distinct input feature matrices $X$ were established for training
based on different environmental configurations, with the corresponding
output matrix $Y$ representing stress. In the table, the letter  ``y''
indicates that the corresponding material feature is included in the
training, while the letter  ``n'' indicates that the feature is
excluded. For example, in \textit{Setting 1}, all seven material
feature parameters are included in the training, and the input feature
matrix $X$ and output $Y$ can be expressed as follows:
{\begin{equation}
\begin{cases} 
X=[\sigma _{y},\sigma _{u},E,\rho ,K,
E\times \varepsilon ,\varepsilon ]^{\mathrm{T}}\\
Y=[\sigma ]
\end{cases}
\label{eq19}
\end{equation}}\unskip

For \textit{Setting 2}, as shown in the table, the strength coefficient
$K$ is excluded from the input feature matrix $X$ and does not participate
in the training. In this case, the input feature matrix $X$ and output $Y$
are expressed as:
{\begin{equation}
\begin{cases} 
X=[\sigma _{y},\sigma _{u},E,\rho ,
E\times \varepsilon ,\varepsilon ]^{\mathrm{T}}\\
Y=[\sigma ]
\end{cases}
\label{eq20}
\end{equation}}\unskip

For \textit{Setting 3}, \textit{Setting 4}, and \textit{Setting 5},
specific feature parameters are sequentially excluded from the input
feature matrix $X$, as outlined in  Table~\ref{tab2}.

\subsection{Division of training and testing datasets}\label{sec2.4} 
The core dataset of this study consists of stress--strain data for 106
alloy materials obtained through FEM, as described in 
Section~\ref{sec2.1}. As outlined in  Section~\ref{sec2.3}, these data
were combined with material properties such as yield strength $\sigma
_{y}$, tensile strength $\sigma _{u}$, elastic modulus $E$, density
$\rho $, strength coefficient $K$, and the product of elastic modulus
and strain to form a comprehensive dataset of 106 entries, which is
suitable for training.

To evaluate the training performance of the MA-LSTM-DLnet model,  the
100 data entries were randomly divided into training set, validation
set and testing sets. As shown in Figure~\ref{fig7},  the dataset was
split into 80 entries for the training set, 10 entries for the
validation set, and 10 entries for the testing set. Specifically, 80
alloy materials were randomly selected for training to develop the
predictive model, while the remaining  10 alloy materials were reserved
as the testing set and excluded from training. These testing data were
used to compare with the model's predictions. To better demonstrate the
predictive accuracy, effectiveness, and generalization capability of
the trained model, the above process was repeated three times. This
means that the  100 material data entries were randomly divided three
times, resulting in a total of  30 alloy materials in the testing set.

\begin{figure}
\includegraphics{fig07}
\caption{\label{fig7}Dataset division: training, validation and testing
sets.}
\end{figure}

\subsection{Hyperparameter configuration}\label{sec2.5} 
Based on the data division strategy described in  Section~\ref{sec2.4},
the deep learning neural network model established in 
Section~\ref{sec2.2} was employed to train the dataset. To optimize
the model's training performance and convergence speed, the following
hyperparameters were tuned based on a combination of empirical
optimization methods and results from preliminary research:
\begin{enumerate}[(1)]
\item
Optimization Algorithm: Adaptive Moment Estimation (Adam)
\item 
Maximum Iterations: 100
\item
Mini-batch Size: 20
\item
Initial Learning Rate: 0.001
\item
Learning Rate Adjustment Strategy: Piecewise constant
\item
Learning Rate Decay Interval: 80
\item
Learning Rate Decay Factor: 0.1
\item
Gradient Threshold: 1
\end{enumerate}

Using the MA-LSTM-DLnet model with the above hyperparameter
configuration, the dataset was trained. The overall program execution
framework is illustrated in  Figure~\ref{fig8}.

\begin{figure}
\includegraphics{fig08}
\caption{\label{fig8}A program framework for MA-LSTM-DLnet training.}
\end{figure}

\section{Results and discussion}\label{sec3}

\subsection{Comprehensive feature analysis: training with complete
dataset}\label{sec3.1} 
As shown in  Equation~(\ref{eq19}), seven key feature parameters
representing material characteristics were selected to form the input
feature matrix $X$. These parameters include yield strength $\sigma
_{y}$, tensile strength $\sigma _{u}$, elastic modulus $E$, density
$\rho $, strength coefficient $K$, and the product of elastic modulus
and strain. Together, they constitute a 7-channel dataset used as
input, with stress $\sigma $ serving as the corresponding labeled
output. This mapping relationship dataset was used to train the model.
According to the feature configuration scheme presented in 
Table~\ref{tab2}, this  section investigates the training performance
of the model under \textit{Setting 1}, where all material feature
parameters are included in the training.

For the dataset of 100 alloy materials, the training, Validation and
testing sets were divided as described in  Section~\ref{sec2.4},
resulting in three groups with a total of 30 materials reserved for
testing and validation. The first group of test materials included 10
randomly selected alloys: M2 tool steel, M42 tool steel, 3105 aluminum
alloy, C10300 copper alloy, C10500 copper alloy, N06333 nickel alloy,
200 nickel alloy, 1151 steel, C110M titanium alloy, C130AM titanium
alloy. These materials were excluded from training and used to compare
with the predictions of the trained model. The remaining 90 alloy
materials were processed into the training dataset for the
MA-LSTM-DLnet model to develop the stress--strain prediction model.
After training the MA-LSTM-DLnet model, the corresponding feature
parameters of the first group of test materials were input, and the
prediction results are shown in Figure~\ref{fig9}. From the figure, it
is evident that the MA-LSTM-DLnet model performs well in predicting the
stress--strain relationship curves. Notably, the predicted
stress--strain curves for eight materials align closely with the test
data in both the linear and nonlinear stages, which is a surprising and
intriguing result.  A further detailed observation of the comparative
test data for the 10 materials in the figure reveals that the predicted
stress--strain curve of the 200 nickel alloy in Figure~\ref{fig9}(f)
exhibits an abrupt change near the yield point. Additionally, while the
overall trend of the prediction results for the 3105 aluminum alloy in 
Figure~\ref{fig9}(c) is consistent with the test data, the prediction
accuracy is slightly lower.

\begin{figure}
\includegraphics{fig09}
\caption{\label{fig9}MA-LSTM-DLnet model for predicting stress--strain
curves.}
\end{figure}

The deviation in the prediction for 200 nickel alloy and 3105
aluminum alloy, contrasted with the accurate predictions for the other
eight alloys, cannot be fully explained by the comparison of these
ten materials alone. However, some distinctions can be observed.
Comparing 
\mbox{Figure~\ref{fig9}(a--j),} the most notable difference lies in
the varying yield strengths of the materials. However, yield strength
is not the fundamental reason for prediction accuracy. The primary
factor is the size of the training dataset. As shown in 
Figure~\ref{fig4} and detailed in  Appendix~\ref{appB}, the number of
stress--strain curves for materials with yield strengths of around
100~MPa in the training dataset is relatively small. This scarcity
leads to lower prediction accuracy for stress--strain curves within
this yield strength range. This observation also indicates that the
material's physical properties and types do not significantly affect
the model's prediction accuracy.

Since the prediction results are material stress--strain constitutive
relationship curves, it is not feasible to compare the accuracy or
calculate errors by selecting individual or multiple points on the
curves. Instead, statistical calculations are required for the entire
curves. To provide a clearer and more scientific comparison between the
model's predictions and the test data, this study employs the
coefficient of determination method to evaluate the similarity between
the two curves. The formula for the coefficient of determination is as
follows:
\begin{equation}
R=1-\|Y_{\mathrm{P}}-Y_{\mathrm{T}}\|^{2}/\|Y_{\mathrm{T}}-
\overline{Y_{\mathrm{T}}}\|^{2} \label{eq21}
\end{equation}
Here, $Y_{\mathrm{T}}$ represents the test values, $Y_{\mathrm{P}}$
denotes the predicted values, $\overline{Y_{\mathrm{T}}}$ is the mean
of the test data, and $\|\cdot \|^{2}$ represents the squared norm.
Using the above  equation, the
similarity between the stress--strain curves predicted by the
MA-LSTM-DLnet model and the given test curves for the first group of
materials is shown  in  Figure~\ref{fig10}.  In the figure, the
horizontal axis labels A1, B1, C1, D1, E1, F1, G1, H1, J1 and K1
correspond to the ten materials in the first group: M2 tool steel, M42
tool steel, 3105 aluminum alloy, C10300 copper alloy, C10500 copper
alloy, N06333 nickel alloy, 200 nickel alloy, 1151 steel, C110M
titanium alloy, C130AM titanium  alloy. The vertical axis represents
the yield strength of the corresponding materials. From 
Figure~\ref{fig10}, it can be observed that the similarity between the
stress--strain curves predicted by the MA-LSTM-DLnet model and the test
data exceeds 97\% for all materials except the 200 nickel alloy.

\begin{figure}
\includegraphics{fig10}
\caption{\label{fig10}Similarity of predicted and test stress--strain
curves.}
\end{figure}

To further demonstrate the predictive performance of the MA-LSTM-DLnet
model, a second group of test materials was randomly selected for
prediction and comparison with the original data. Similar to the first
group, the second group of test materials consists of ten alloys: S2
Tool steel, T1 Tool steel, 7050 aluminum alloy, C35300 copper alloy,
C36200 copper alloy, 718SPF nickel alloy, MA754 nickel alloy, 1110
steel, R54520 titanium alloy and 21S titanium alloy. The feature data
representing the second group of test materials were input into the
MA-LSTM-DLnet model, and the resulting stress--strain curves are shown
in  Figure~\ref{fig11}. 

\begin{figure}
\includegraphics{fig11}
\caption{\label{fig11}MA-LSTM-DLnet model for predicting stress--strain
curves in group 2 materials.}
\end{figure}

From the figure, it is evident that the ten stress--strain curves
predicted by the MA-LSTM-DLnet model align very well with the original
test data curves. Compared to the first group of materials, the
predictions for the second group are nearly perfect. Upon closer
examination, slight discrepancies are observed in the linear regions of
the predicted stress--strain curves for 7050 aluminum alloy and 21S
titanium alloy. Additionally, the predicted stress--strain curve for
C35300 copper alloy and R54520 Ti alloy exhibits minor deviations in
the hardening region.

Similarly, using  Equation~(\ref{eq21}), the similarity between the
stress--strain curves predicted by the model for the second group of
materials and the original test data was calculated and analyzed.  As
shown in Figure~\ref{fig12}, the horizontal axis labels A2, B2, C2, D2,
E2, F2, G2, H2, J2 and K2 correspond to the ten materials in the second
group: S2 Tool steel, T1 Tool steel, 7050 aluminum alloy, C35300 copper
alloy, C36200 copper alloy, 718SPF nickel alloy, MA754 nickel alloy,
1110 steel, R54520 titanium alloy and 21S titanium alloy. The vertical
axis represents the yield strength of the corresponding materials. From
Figure~\ref{fig12}, it can be observed that, compared to the first
group of test materials, the similarity between the predicted values
and the test values for the second group of materials exceeds 96\% in
all cases.

\begin{figure}
\includegraphics{fig12}
\caption{\label{fig12}Similarity of predicted and test stress--strain
curves in group 2 materials.}
\end{figure}

The third group of test materials, similar to the previous two groups,
consists of ten randomly selected alloys: 5754 aluminum alloy, C43000
copper alloy, C61400 copper alloy, 718 nickel alloy, AF955 nickel
alloy, 5115 steel, 15-Mo-0.5Si titanium alloy, 5621 titanium alloy, FX
tool steel and H10 tool steel. The MA-LSTM-DLnet model was used to
predict the stress--strain curves for the third group of test
materials, and the results are shown in Figure~\ref{fig13}. From the
figure, it can be observed that the predicted stress--strain curves
align well with the original test data for all materials except the
5754 aluminum alloy. This further demonstrates that the prediction
accuracy is highly dependent on the quantity of training data. The
yield strength of 5754 aluminum alloy is around 100~MPa, and the
scarcity of training data within this range results in lower prediction
accuracy for this material.

\begin{figure}
\includegraphics{fig13}
\caption{\label{fig13}MA-LSTM-DLnet model for predicting stress--strain
curves in group 3 materials.} 
\end{figure}

Using Equation~(\ref{eq21}), the similarity between the stress--strain
curves predicted for the third group of materials and the original test
data was calculated and analyzed.  As shown in Figure~\ref{fig14}, the
horizontal axis labels A3, B3, C3, D3, E3, F3, G3, H3, J3 and K3
correspond to the ten materials in the third group: 5754 aluminum
alloy, C43000 copper alloy, C61400 copper alloy, 718 nickel alloy,
AF955 nickel alloy, 5115 steel, 15-Mo-0.5Si titanium alloy, 5621
titanium alloy, FX tool steel and H10 tool steel. The vertical axis
represents the yield strength of the corresponding materials. From the 
figure, it can be observed that the similarity for 5754 aluminum alloy
is 87.8\%, while the similarity for the other five materials exceeds
96\%. By  comprehensively comparing Figure~\ref{fig10}.
Figures~\ref{fig12} and~\ref{fig14},  it is evident that the similarity
is relatively low for two aluminum alloys {and 200 nickel alloy} with
lower yield strengths, resulting in suboptimal prediction performance.
As previously discussed, this can be explained by referring to the
original stress--strain data in  Figure~\ref{fig4} or the training data
in  Appendix~\ref{appB}. The scarcity of original training data for
stress--strain curves with yield strengths of around 100~MPa leads to
inadequate model training and, consequently, poorer prediction
performance. Overall, the prediction results for the three groups of
test materials demonstrate that, given sufficient training data, the
MA-LSTM-DLnet model exhibits stable performance and strong
generalization capabilities in predicting material stress--strain
relationships. Under certain conditions, the MA-LSTM-DLnet model can
serve as a viable alternative to existing theoretical calculations and
tensile testing methods for predicting the constitutive relationships
of alloy materials.

\begin{figure}
\includegraphics{fig14}
\caption{\label{fig14}Similarity of predicted and test stress--strain
curves in group 3 materials.}
\end{figure}

\subsection{Exclusion of strength coefficient: impact on training
performance}\label{sec3.2} 
In  Section~\ref{sec3.1}, the training dataset under \textit{Setting
1}, which includes all features, was investigated. The analysis results
demonstrate that the deep learning approach and the proposed
MA-LSTM-DLnet model can effectively predict material constitutive
relationships. However, in practical scenarios, such as when using the
Ludwik isotropic hardening model to calculate stress--strain
relationships, key parameters like the strength coefficient $K$ are
often difficult to obtain. Moreover, the representation of $K$ may vary
under different working conditions. If alternative models are used,
similar challenges arise with other key parameters, which are equally
difficult to determine. Therefore, this  section aims to explore the
predictive performance of the MA-LSTM-DLnet model by removing these
hard-to-obtain key parameters and utilizing only conventional material
properties to construct the feature training dataset. Following
\textit{Setting 2} in Table~\ref{tab2}, the strength coefficient $K$ was
excluded, and the training dataset was composed of six features: yield
strength $\sigma _{y}$, tensile strength $\sigma _{u}$, elastic modulus
$E$, density $\rho $, the product of elastic modulus and strain, and
strain $\varepsilon $. 

To facilitate comparison with the results in  Section~\ref{sec3.1}, the
same three groups of 30 materials were used as test materials. The
prediction results of the MA-LSTM-DLnet model for the first group of
test materials are shown in  Figure~\ref{fig15}. Here,
\textit{Predicted Data1} represents the predictions using the
\textit{Setting 1} dataset, where all seven material features were
included in the training, as discussed in  Section~\ref{sec3.1}.
\textit{Predicted Data2} represents the predictions using the
\textit{Setting 2} configuration, where the strength coefficient $K$
was excluded from the training features. 

\begin{figure}
\includegraphics{fig15}
\caption{\label{fig15}MA-LSTM-DLnet model for predicting stress--strain
curves: \textit{Predicted Data1} which incorporates a full feature set
and \textit{Predicted Data2} excluding intensity coefficients.}
\end{figure}

From  Figure~\ref{fig15}, it is evident that the predictions under both
\textit{Setting 1} and \textit{Setting 2} are nearly identical for nine
of the ten materials, aligning well with the test data and showing
excellent consistency.  The only significant deviation occurs for the
200 Ni alloys as in  Figure~\ref{fig15}(g), as explained in detail in 
Section~\ref{sec3.1}, which is unrelated to the model's predictive
performance. These results clearly demonstrate that the proposed
MA-LSTM-DLnet model and methodology can replace traditional theoretical
and numerical calculation methods. Furthermore, they eliminate the need
for hard-to-obtain key parameters, simplifying the process of
determining material stress--strain constitutive relationships.
Additionally, this study highlights that deep learning methods can
leverage simple, conventional material properties to predict
stress--strain constitutive relationships, thereby reducing the
repetitive work associated with traditional tensile testing methods and
significantly lowering labor costs.

Using Equation~(\ref{eq21}), the similarity between the predicted
values (\textit{Predicted Data2}) and the original test data was
calculated.  The results are shown in Figure~\ref{fig16}, where the
horizontal axis labels A1, B1, C1, D1, E1, F1, G1, H1, J1 and K1
correspond to the ten materials in the first group: M2 tool steel, M42
tool steel, 3105 aluminum alloy, C10300 copper alloy, C10500 copper
alloy, N06333 nickel alloy, 200 nickel alloy, 1151 steel, C110M
titanium alloy, C130AM titanium alloy. The vertical axis represents
the yield strength of the corresponding materials. From the  figure, it
can be observed that, except for the 3105 aluminum alloy and the 200
nickel alloy, the similarity between the predicted values and the
original test data exceeds 93\% for all other materials.

\begin{figure}
\includegraphics{fig16}
\caption{\label{fig16}Similarity of predicted data2 and test
stress--strain curves.}
\vspace*{6pt}
\end{figure}

The MA-LSTM-DLnet model was used to predict the stress--strain
relationships for the second group of test materials, which includes S2
Tool steel, T1 Tool steel, 7050 aluminum alloy, C35300 copper alloy,
C36200 copper alloy, 718SPF nickel alloy, MA754 nickel alloy, 1110
steel, R54520 titanium alloy and 21S titanium alloy. The prediction
results are shown in Figure~\ref{fig17}. When the strength coefficient
$K$ is excluded from the material features, the model's predictions for
this group are even more accurate than those for the first group of
test materials. As shown in  Figure~\ref{fig17}, the stress--strain
curves of \textit{Predicted Data1} (with $K$ included) and
\textit{Predicted Data2} (with $K$ excluded) are nearly identical for
five of the six materials, with only minor deviations observed for
C35300 copper alloy. This further demonstrates that, when using deep
learning methods to predict material constitutive relationships, it is
unnecessary to consider key parameters that are difficult to obtain in
traditional methods.

\begin{figure}
\includegraphics{fig17}
\caption{\label{fig17}MA-LSTM-DLnet model for predicting stress--strain
curves in group 2 materials: \textit{Predicted Data1} which
incorporates a full feature set and \textit{Predicted Data2} excluding
intensity coefficients.}
\end{figure}

Using Equation~(\ref{eq21}), the similarity between \textit{Predicted
Data2} and the original test data for the second group of materials was
calculated. The results are shown in  Figure~\ref{fig18}, where the
horizontal axis labels A2, B2, C2, D2, E2, F2, G2, H2, J2 and K2
correspond to the ten materials in the second group: S2 Tool steel, T1
Tool steel, 7050 aluminum alloy, C35300 copper alloy, C36200 copper
alloy, 718SPF nickel alloy, MA754 nickel alloy, 1110 steel, R54520
titanium alloy and 21S titanium alloy. The vertical axis represents the
yield strength of the corresponding materials. From the  figure, it can
be observed that the similarity to C35300 copper alloy is 86.9\%, while
the similarity to the other materials exceeds 96\%. This aligns with
the trends demonstrated in  Figure~\ref{fig17}.

\begin{figure}
\includegraphics{fig18}
\caption{\label{fig18}Similarity of predicted data2 and test
stress--strain curves in group 2 materials.}
\end{figure}

The third group of test materials includes 5754 aluminum alloy, C43000
copper alloy, C61400 copper alloy, 718 nickel alloy, AF955 nickel
alloy, 5115 steel, 15-Mo-0.5Si titanium alloy, 5621 titanium alloy, FX
tool steel and H10 tool steel. The MA-LSTM-DLnet model was used to
predict their stress--strain relationships, and the results are shown
in  Figure~\ref{fig19}. From the figure, it is evident that
\textit{Predicted Data1}, \textit{Predicted Data2}, and the original
test data align well for all materials except 5754 aluminum alloy,
which exhibits some deviation. By comprehensively comparing the
prediction results for the three groups of test materials in 
Figures~\ref{fig15},~\ref{fig17} and~\ref{fig19}, it is
further demonstrated that the predictive performance of the deep
learning model is not highly sensitive to the strength coefficient $K$.
Removing the strength coefficient as a feature has a negligible impact
on the predicted stress--strain curves.

\begin{figure}
\vspace*{4pt}
\includegraphics{fig19}
\vspace*{6pt}
\caption{\label{fig19}MA-LSTM-DLnet model for predicting stress--strain
curves in group 3 materials: \textit{Predicted Data1} which
incorporates a full feature set and \textit{Predicted Data2} excluding
intensity coefficients.}
\end{figure}

Using  Equation~(\ref{eq21}), the similarity between \textit{Predicted
Data2} and the original test data for the third group of materials was
calculated. The results are shown in  Figure~\ref{fig20}, where the
horizontal axis labels A3, B3, C3, D3, E3, F3, G3, H3, J3 and K3
correspond to the ten materials in the third group: 5754 aluminum
alloy, C43000 copper alloy, C61400 copper alloy, 718 nickel alloy,
AF955 nickel alloy, 5115 steel, 15-Mo-0.5Si titanium alloy, 5621
titanium alloy, FX tool steel and H10 tool  steel. The vertical axis
represents the yield strength of the corresponding materials. From the
figure, it is evident that the similarity for 5754 aluminum alloy is
64.4\%, while the similarity to the other materials {exceeds 95\%}. By
comparing the similarity calculations in  Figures~\ref{fig16}, 
\ref{fig18} and~\ref{fig20}, it is clear that the lower
similarity for aluminum alloys is not related to the deep learning
model itself. As explained in  Section~\ref{sec3.1}, the primary reason
is the scarcity of training data for materials with yield strengths of
around 100~MPa.

\begin{figure}
\includegraphics{fig20}
\vspace*{4pt}
\caption{\label{fig20}Similarity of predicted data2 and test
stress--strain curves in group 3 materials.}
\end{figure}

\subsection{Sequential feature omission: evaluating training
outcomes}\label{sec3.3} 
As shown in  Table~\ref{tab2},  Sections~\ref{sec3.1} and~\ref{sec3.2}
correspond totraining performance under \textit{Setting 1} and
\textit{Setting 2}, respectively. The analysis above demonstrates that
the MA-LSTM-DLnet model is fully capable of being adopted for studying
or predicting material constitutive relationships. To further
investigate the influence of material features, this  section analyzes
the predictive performance of the MA-LSTM-DLnet model under
\textit{Setting 3}, \textit{Setting 4}, and \textit{Setting 5} as
outlined in  Table~\ref{tab2}. Given the extensive scope and the
training performance observed under \textit{Setting 1} and
\textit{Setting 2}, the second group of test materials---S2 Tool
steel, T1 Tool steel, 7050 aluminum alloy, C35300 copper alloy, C36200
copper alloy, 718SPF nickel alloy, MA754 nickel alloy, 1110 steel,
R54520 titanium alloy and 21S titanium alloy---was selected for
prediction. For \textit{Setting 3}, the feature configuration includes
yield strength $\sigma _{y}$, tensile strength $\sigma _{u}$, elastic
modulus $E$, density $\rho $, and strain $\varepsilon $, forming a
5-channel feature dataset as the input matrix $X$. The strength
coefficient $K$ and the product of elastic modulus and strain were
excluded. As shown in  Figure~\ref{fig21}, \textit{Predicted Data1}
corresponds to \textit{Setting 1}, \textit{Predicted Data2} corresponds
to \textit{Setting 2}, and \textit{Predicted Data3} corresponds to
\textit{Setting 3}. From the  figure, it is evident that the
predictions under \textit{Setting 3} exhibit slightly lower agreement
with the test results compared to \textit{Setting 1} and
\textit{Setting 2}.

\begin{figure}
\includegraphics{fig21}
\caption{\label{fig21}MA-LSTM-DLnet model for predicting stress--strain
curves in group 2 materials: \textit{Predicted Data1} which
incorporates a full feature set, \textit{Predicted Data2} excluding
intensity coefficients, \textit{Predicted Data3} excluding intensity
coefficients and $E\times \varepsilon $.}
\end{figure}

For \textit{Setting 4}, the feature configuration includes yield
strength $\sigma _{y}$, tensile strength $\sigma _{u}$, elastic modulus
$E$, and strain $\varepsilon $, forming a 4-channel feature dataset as
the input matrix $X$. The density $\rho $, strength coefficient $K$, and
the product of elastic modulus and strain were excluded. As shown in 
Figure~\ref{fig22}, \textit{Predicted Data4} corresponds to
\textit{Setting 4}. The overall prediction results are slightly lower
than the actual test results, but no significant fluctuations are
observed.

\begin{figure}
\includegraphics{fig22}
\caption{\label{fig22}MA-LSTM-DLnet model for predicting stress--strain
curves in group 2 materials: \textit{Predicted Data1} which
incorporates a full feature set, \textit{Predicted Data2} excluding
intensity coefficients, \textit{Predicted Data3} excluding intensity
coefficients and $E\times \varepsilon $, \textit{Predicted Data4}
excluding density, intensity coefficients and 
$E\times \varepsilon$.}
\end{figure}

For \textit{Setting 5}, the feature configuration includes yield
strength $\sigma _{y}$, tensile strength $\sigma _{u}$, and strain
$\varepsilon $, forming a 3-channel dataset. The elastic modulus $E$,
density $\rho $, strength coefficient $K$, and the product of elastic
modulus and strain were excluded. As shown in  Figure~\ref{fig23},
\textit{Predicted Data5} corresponds to \textit{Setting 5}. The overall
prediction performance is similar to that of \textit{Setting 4}. 

\begin{figure}
\vspace*{-2pt}
\includegraphics{fig23}
\caption{\label{fig23}MA-LSTM-DLnet model for predicting stress--strain
curves in group 2 materials: \textit{Predicted Data1} which
incorporates a full feature set, \textit{Predicted Data2} excluding
intensity coefficients, \textit{Predicted Data3} excluding intensity
coefficients and $E\times \varepsilon $, \textit{Predicted Data4}
excluding density, intensity coefficients and $E\times \varepsilon $,
\textit{Predicted Data5} excluding elastic modulus, density, intensity
coefficients and $E\times \varepsilon $.}
\vspace*{-2pt}
\end{figure}

In summary, the results indicate that reducing the number of feature
parameters representing the material has a minor impact on the model's
prediction performance. However, the most significant factor
influencing the results is the size of the training dataset. Based on a
comprehensive analysis of the model's predictions under all conditions,
it is evident that while reducing the number of features or the absence
of key parameters may affect prediction accuracy, the impact is not
substantial if the error range remains stable. This can be mitigated by
increasing the size of the original training dataset.

Using  Equation~(\ref{eq21}), the similarity between \textit{Predicted
Data3}, \textit{Predicted Data4}, \textit{Predicted Data5} (under
\textit{Setting 3}, \textit{Setting 4}, and \textit{Setting 5},
respectively) and the original test data for the second group of
materials was calculated. This allows for a clearer observation of the
errors between the model's predictions and the test data. As shown in 
Figure~\ref{fig24}, the horizontal axis labels A2, B2, C2, D2, E2, F2,
G2, H2, J2 and K2 correspond to the ten materials in the second group:
S2 Tool steel, T1 Tool steel, 7050 aluminum alloy, C35300 copper alloy,
C36200 copper alloy, 718SPF nickel alloy, MA754 nickel alloy, 1110
steel, R54520 titanium alloy and 21S titanium alloy. The vertical axis
represents the yield strength of the corresponding materials. From the
figure, it can be observed that, under different feature input
configurations, the similarity to C35300 copper alloy ranges from 80\%
to 93\%, which is the lowest among the materials. For the other
materials, the similarity exceeds 91\%. Comparing the similarities
under different conditions, it is evident that the influence of
different feature parameters on prediction accuracy is minimal and
lacks a clear pattern. This contrasts with traditional mathematical
methods, where feature parameters are critical, and the absence of any
single parameter can significantly impact the calculation results. 

\begin{figure}
\includegraphics{fig24}
\vspace*{-2pt}
\caption{\label{fig24}Similarity of predicted data3, predicted data4,
predicted data5 and test stress--strain curves in group 2
materials.}
\vspace*{-2pt}
\end{figure}

Therefore, given a sufficient amount of material data, it is possible
to train a prediction model with limited parameter dependency. Such a
model can not only replace traditional complex mathematical
calculations but also establish a unified predictive framework for
material constitutive relationships, eliminating the need for
repetitive tensile experiments when switching materials.

\section{Conclusions}\label{sec4}
This study systematically investigated the stress--strain constitutive
relationships of metal alloy materials using deep learning methods,
achieving the following innovative results:
\begin{enumerate}[(1)]
\item
\textbf{High-Quality Dataset Construction}: 
The finite element method (FEM) was employed to numerically simulate
the uniaxial tensile behavior of 106 common alloy 
\mbox{materials,}
successfully constructing a multi-feature dataset containing complete
stress--strain curves. This dataset integrates fundamental mechanical
properties and various physical 
\mbox{characteristics,} providing a
high-quality data foundation for training deep learning models.
\item
\textbf{Innovative Deep Learning Model}: 
A novel deep learning time-series model, MA-LSTM-DLnet, was proposed,
combining multi-head attention mechanisms and long short-term memory
(LSTM) networks. By incorporating the attention mechanism, the model
effectively captures key features in stress--strain curves, while
leveraging the LSTM network's robust capability for processing
sequential data to achieve precise predictions of material mechanical
behavior. The results demonstrate that the similarity between the
stress--strain curves predicted by the MA-LSTM-DLnet model and the test
data exceeds 95\%, significantly outperforming traditional methods.
\item
\textbf{Practical Advantages of MA-LSTM-DLnet}: 
A key advantage of the MA-LSTM-DLnet model is its reliance solely on
conventional material properties (e.g., yield strength, tensile
strength, and elastic modulus) for high-precision predictions,
eliminating the need for hard-to-obtain parameters such as strength
coefficients and hardening exponents 
\mbox{required} in traditional
elastoplastic theories. This feature not only simplifies the
\mbox{computational} process but also significantly enhances the model's
practicality and \mbox{reliability.}
\item
\textbf{Limitations}: 
This study combines FEM with deep learning to predict the 
stress--strain constitutive relationships of materials, achieving good
results. However, there are some limitations. Firstly, the model's
accuracy depends on the training dataset's quality and quantity.
Insufficient or unrepresentative data may lead to poor prediction.
Secondly, the model has no physical significance and cannot provide
physical explanations for the stress--strain relationship. It merely
offers empirical predictions based on data. Lastly, the model's current
applicability is limited to the 106 metal alloys studied and may not
generalise to other materials.
\item
\textbf{Future Directions}: 
Future research should focus on expanding and diversifying the training
dataset, collecting data for more materials with different properties
and behaviours to enhance accuracy and generalisation. Also, hybrid
models combining deep learning with physical models could be explored
to integrate the advantages of both. Furthermore, the model can be
improved by adding more relevant features and optimizing its structure
and hyperparameters. This would make it a more powerful tool for
material constitutive relationship prediction.
\end{enumerate}

\vspace*{-2pt}

\section*{Data availability}

\vspace*{-2pt}

The datasets generated during and/or analyzed during the current study
are available from the corresponding author on a reasonable request.
All data used in this study are fully available without any
restrictions. The data that support the findings of this study are
available from the corresponding author upon request. 

\vspace*{-2pt}

\section*{Acknowledgments}

\vspace*{-2pt}

We extend our sincere appreciation to the reviewers for their
constructive comments and suggestions, which have significantly
contributed to the improvement of this manuscript.

\vspace*{-2pt}

\section*{Declaration of interests}

\vspace*{-2pt}

The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.

\section*{Funding}
This work was supported by the Start-up Funding Project from 
Guangdong Polytechnic Normal University (Grant Number 2021SDKYA017), 
the Program for Scientific Research Start-up Funds of 
Guangdong Ocean University (Grant Numbers 060302062311 and 060302062315), 
the Collaborative Education Project of Industry University Cooperation 
by the Ministry of Education of China 
(Grant numbers 230802784214645, 230803817223502 and
230821384507208). The authors are grateful for the financial support
provided by these funding bodies, which made this research possible.

\CDRGrant[GPNU]{2021SDKYA017}
\CDRGrant[GDOU]{060302062311}
\CDRGrant[GDOU]{060302062315} 
\CDRGrant[MOE (China)]{230802784214645}
\CDRGrant[MOE (China)]{230803817223502}
\CDRGrant[MOE (China)]{230821384507208}

\back{}

\appendix{}

\gdef\appendixlabel{Appendix A}

\setcounter{table}{0}
\renewcommand\thetable{A\arabic{table}}

\section{}\label{appA}
Appendix~\ref{appA} provides a list of the 106 materials required for
this study, along with the corresponding computational parameters
needed for tensile simulations. Since the core focus of this study is
on exploring the predictive capabilities of deep learning
methods---specifically, whether deep learning can learn and predict
material stress--strain relationship curves---we prioritized the
overall trend of the stress--strain curves over achieving zero-error
alignment with experimental data for each material. Given the time
constraints, our primary concern was whether the stress--strain curve
trends could be accurately predicted using deep learning methods,
rather than achieving perfect experimental accuracy.

For this reason, we did not conduct extensive experiments to obtain
stress--strain curves for each material. Instead, we utilized the
finite element method (FEM) to simulate and generate the material data.
Consequently, the material parameters provided in the appendix are
primarily sourced from online databases, literature, and other
references. These parameters have not been rigorously verified for
every material. If you intend to use the material parameters provided
in the appendix, please evaluate their suitability for your specific
research needs and determine whether further rigorous verification is
necessary. The materials are categorized as follows:

\textbf{Entries 1--21}: Aluminum alloys

\textbf{Entries 22--41}: Copper alloys

\textbf{Entries 42--60}: Tool steels

\textbf{Entries 61--80}: Nickel alloys

\textbf{Entries 81--93}: Titanium alloys

\textbf{Entries 94--106}: Other alloy steels

\vspace*{1pc}

\noindent\begin{inftab}
\begin{tabular}{ccccccccc}  
\thead
No. & Name & Treatment & $\sigma _{y}$ & 
$\sigma _{u}$ & $E$ & $\rho $ & $\nu $ & $K$ \\ 
\endthead
\0\01 & 5052 & H36 & \0220 & \0267 & 70 & 2680 & 0.33 & \0350 \\ 
\0\02 & 7A04 & Solid & \0540 & \0570 & 71 & 2800 & 0.33 & \0630 \\ 
\0\03 & 2219 & T81 temper & \0350 & \0455 & 73 & 2839 & 0.33 & \0515 \\ 
\0\04 & 2224 & T351tempr & \0330 & \0460 & 71 & 2800 & 0.33 & \0520 \\ 
\0\05 & 2017 & T4 temper & \0245 & \0390 & 73 & 2800 & 0.33 & \0550 \\ 
\0\06 & 2020 & T6 temper & \0275 & \0310 & 73 & 2780 & 0.32 & \0370 \\ 
\0\07 & 3003 & Solid, F temper & \0115 & \0142 & 69 & 2730 & 0.33 & \0242 \\ 
\0\08 & 3105 & Solid & \0100 & \0140 & 70 & 2700 & 0.33 & \0240 \\ 	
\0\09 & 4032 & T6 temper & \0290 & \0380 & 70 & 2700 & 0.33 & \0440 \\
\010 & 5005 & H18 temper & \0160 & \0200 & 69 & 2700 & 0.32 & \0300 \\
\011 & 5182 & Annealed & \0100 & \0175 & 70 & 2700 & 0.32 & \0275 \\
\012 & 5754 & O temper & \0120 & \0165 & 69 & 2670 & 0.32 & \0265 \\
\013 & 6013 & T4 temper & \0180 & \0317 & 70 & 2700 & 0.33 & \0377 \\
\014 & 6110 & T6 temper & \0380 & \0480 & 69 & 2700 & 0.32 & \0540 \\\inftabbreak  
\015 & 7050 & T73 temper & \0608 & \0650 & 73 & 2820 & 0.33 & \0710 \\
\016 & 7249 & T76511 temper & \0562 & \0656 & 73 & 2820 & 0.33 & \0716 \\
\017 & 7475 & T61 temper & \0469 & \0524 & 71 & 2810 & 0.33 & \0584 \\
\018 & 201.0 & T7 castings & \0415 & \0470 & 70 & 2710 & 0.33 & \0530 \\
\019 & 332.0 & T5 temper & \0276 & \0310 & 70 & 2700 & 0.33 & \0370 \\
\020 & 355.0 & T6 temper & \0275 & \0295 & 70 & 2700 & 0.32 & \0355 \\
\021 & A201.0 & T7 temper & \0442 & \0480 & 70 & 2700 & 0.32 & \0540 \\
\022 & C10100 & R220 & \0140 & \0220 & 130 & 8930 & 0.34 & \0320 \\
\023 & C10300 & H01 temper & \0275 & \0320 & 120 & 8930 & 0.35 & \0380 \\
\024 & C10500 & Cold-rolled strip & \0410 & \0510 & 120 & 8900 & 0.35 & \0570 \\
\025 & C10700 & Cold-rolled strip & \0230 & \0320 & 120 & 9000 & 0.33 & \0380 \\
\026 & C12210 & Annealed & \0392 & \0617 & 120 & 8900 & 0.35 & \0677 \\
\027 & C13000 & Annealed & \0221 & \0300 & 120 & 8900 & 0.34 & \0360 \\
\028 & C14310 & Annealed & \0200 & \0260 & 120 & 8930 & 0.34 & \0320 \\
\029 & C14520 & Annealed & \0397 & \0936 & 120 & 8930 & 0.33 & \0936 \\
\030 & C15720 & Annealed & \0365 & \0462 & 117 & 8940 & 0.34 & \0522 \\
\031 & C18070 & TR02 temper & \0170 & \0460 & 120 & 8930 & 0.34 & \0520 \\
\032 & C19720 & HR02 temper & \0330 & \0400 & 120 & 8840 & 0.34 & \0460 \\
\033 & C22600 & HR01 temper & \0295 & \0345 & 120 & 8780 & 0.34 & \0405 \\
\034 & C26800 & HR01 temper & \0275 & \0369 & 103 & 8470 & 0.34 & \0429 \\
\035 & C33500 & Annealed & \0317 & \0420 & 105 & 8470 & 0.35 & \0480 \\
\036 & C35300 & Hard & \0414 & \0515 & 105 & 8470 & 0.35 & \0675 \\
\037 & C36200 & Annealed & \0303 & \0338 & 100 & 8500 & 0.35 & \0398 \\
\038 & C43000 & Hard & 495 & \0530 & \0110 & 8400 & 0.24 & \0590 \\
\039 & C61400 & Annealed & \0300 & \0540 & 117 & 7890 & 0.34 & \0600 \\
\040 & C71500 & H01 temper & \0310 & \0460 & 152 & 8490 & 0.34 & \0520 \\
\041 & Cu--Cr--Zr  & Solid, aging & \0254 & \0328 & 100 & 8900 & 0.33 & \0388 \\
\042 & A8 & Q\&DT & \0620 & 1035 & 200 & 7870 & 0.3\0 & 1035 \\
\043 & A9 & HDT & \0635 & 1110 & 131 & 7870 & 0.3\0 & 1110 \\
\044 & D2 & HRC 37 & \0320 & \0780 & 180 & 7700 & 0.28 & \0780 \\
\045 & H10 & QT & \0981 & 1280 & 206 & 7800 & 0.3\0 & 1280 \\
\046 & CBS-223  & Solid, annealed & 1138 & 1310 & 216 & 7750 & 0.3\0 & 1310 \\
\047 & H12 & QT & \0868 & 1000 & 207 & 7840 & 0.32 & 1000 \\
\048 & H21 & QT & \0966 & 1100 & 207 & 7800 & 0.3\0 & 1100 \\
\049 & M2 & HRC 65 & 1140 & 1356 & 220 & 8100 & 0.3\0 & 1356 \\
\050 & M42 & Cast, annealed  & \0610 & \0925 & 220 & 8100 & 0.3\0 & \0925 \\
\051 & S2 & Annealed & \0786 & \0890 & 210 & 7800 & 0.3\0 & \0890 \\
\052 & T1 & Annealed & \0710 & \0886 & 210 & 7850 & 0.3\0 & \0886 \\
\053 & 6F3 & Solid, annealed & \0835 & 1035 & 210 & 7800 & 0.3\0 & 1035 \\
\054 & CX & Solid, annealed & \0835 & 1151 & 210 & 7800 & 0.3\0 & 1151 \\
\055 & MD Xtra & Annealed & \0827 & \0979 & 210 & 7680 & 0.3\0 & \0979 \\
\056 & FX & Annealed & \0835 & 1414 & 200 & 7800 & 0.3\0 & 1414 \\
\057 & PX & HRC 48 & \0986 & 1225 & 200 & 7800 & 0.3\0 & 1225 \\
\058 & WF & H, 444-477 BHN & 1448 & 1621 & 200 & 7800 & 0.3\0 & 1621 \\
\059 & DIEVAR & HRC 44-46 & 1210 & 1480 & 210 & 7700 & 0.32 & 1480 \\
\060 & 2101 & Solid, annealed & \0617 & \0860 & 215 & 7850 & 0.3\0 & \0860 \\
\061 & Nickel200 & Annealed & \0148 & \0370 & 206 & 8890 & 0.3\0 & \0430 \\
\062 & Nickel201 & Annealed & \0100 & \0380 & 206 & 8890 & 0.3\0 & \0440 \\
\063 & N03301 & Age hardened & \0621 & \0780 & 193 & 8030 & 0.3\0 & \0840 \\\inftabbreak
\064 & PM 1000 & CR-A & \0345 & \0450 & 210 & 8900 & 0.3\0 & \0510 \\
\065 & Monel 400 & Annealed & \0320 & \0482 & 176 & 8800 & 0.32 & \0542 \\
\066 & N06333 & HRA & \0620 & \0930 & 201 & 8200 & 0.3\0 & \0930 \\
\067 & 20Cb-3 & Annealed & \0380 & \0510 & 193 & 8080 & 0.31 & \0570 \\
\068 & 20Mo-4 & Annealed & \0510 & \0700 & 206 & 8050 & 0.3\0 & \0760 \\
\069 & N08031 & SAA & \0347 & \0573 & 206 & 8100 & 0.3\0 & \0633 \\
\070 & 625 Plus & Double-aged & \0827 & 1065 & 206 & 8840 & 0.3\0 & 1065 \\
\071 & Inconel 22 & SHT sheet & \0310 & \0690 & 207 & 8700 & 0.3\0 & \0750 \\
\072 & Inconel706 & AR20 & \0650 & \0965 & 210 & 8050 & 0.32 & \0965 \\
\073 & 718SPF & AA20 & 1034 & 1241 & 206 & 8840 & 0.3\0 & 1241 \\
\074 & N07750 & AHRT20 & \0550 & \0910 & 206 & 8250 & 0.3\0 & \0910 \\
\075 & MA754 & AS20 & \0570 & \0950 & 203 & 8300 & 0.3\0 & \0950 \\
\076 & CMSX-4 & HTA400 & \0860 & \0950 & 206 & 8700 & 0.33 & \0950 \\
\077 & Udimet 500 & Aged, annealed & \0720 & 1340 & 206 & 8050 & 0.3\0 & 1340 \\
\078 & 718 Plus & Aged, annealed & 1034 & 1241 & 206 & 8250 & 0.3\0 & 1241 \\
\079 & AF955 & Aged, annealed & \0827 & \0965 & 206 & 8750 & 0.3\0 & \0965 \\
\080 & Waspaloy & Mill annealed & \0910 & 1335 & 213 & 8200 & 0.3\0 & 1335 \\
\081 & Ti Grade 1 & Solid, annealed & \0200 & \0290 & 110 & 4510 & 0.33 & \0350 \\
\082 & TC4 & Solid, annealed & \0830 & 1200 & 110 & 4440 & 0.34 & 1200 \\
\083 & 5 Al - 2.5 Sn & Solid & \0685 & \0930 & 110 & 4420 & 0.33 & \0930 \\
\084 & GR9 & Mill annealed & \0665 & \0735 & 110 & 4470 & 0.39 & \0795 \\
\085 & ATI 425 & Mill annealed & \0772 & \0924 & 110 & 4450 & 0.33 & \0924 \\
\086 & UNS R54520 & Annealed & \0360 & \0860 & 120 & 4480 & 0.33 & \0920 \\
\087 & C-110M & Solid, longitudinal & \0810 & \0900 & 120 & 4730 & 0.33 & \0900 \\
\088 & C-130AM & Mill annealed & \0900 & 1000 & 120 & 4510 & 0.33 & 1000 \\
\089 & 5621-S & SSTPF & \0985 & 1200 & 120 & 4510 & 0.33 & 1200 \\
\090 & Ti-6242 & AP-T-L & \0965 & 1100 & 120 & 4510 & 0.33 & 1100 \\
\091 & Timetal 21S & AP-T-L-SA & \0917 & \0946 & 110 & 4510 & 0.33 & \0946 \\ 
\092 & Ti - 1 Ge  & Solid & \0860 & 1200 & 110 & 4510 & 0.33 & 1200 \\
\093 & Ti15 Mo0.5 Si  & Solid, 5~K/min & \0830 & \0900 & 110 & 4510 & 0.33 & \0900 \\
\094 & 1008 steel & Solid, oxidized & \0275 & \0405 & 200 & 7872 & 0.33 & \0465 \\
\095 & 1018 steel & Solid, oxidized & \0560 & \0780 & 200 & 7872 & 0.33 & \0840 \\
\096 & EN08 steel & QT & \0530 & \0660 & 200 & 7850 & 0.33 & \0720 \\
\097 & LZ50 steel & Solid, annealed & \0744 & \0973 & 200 & 7850 & 0.33 & \0973 \\
\098 & G4802  & Solid & \0375 & \0489 & 200 & 7850 & 0.33 & \0549 \\
\099 & 22MnB5 steel & Solid & \0262 & \0395 & 210 & 7850 & 0.33 & \0455 \\
100 & 1110 steel & Solid & \0365 & \0749 & 210 & 7850 & 0.33 & \0809 \\ 
101 & 1141 steel & Solid & \0583 & \0843 & 210 & 7850 & 0.33 & \0903 \\ 
102 & 1151 steel & Solid & \0426 & \0698 & 210 & 7850 & 0.33 & \0758 \\ 
103 & 1522 steel & Solid & \0286 & \0433 & 200 & 7850 & 0.33 & \0493 \\ 
104 & 3140 steel & Solid & \0284 & \0953 & 210 & 7850 & 0.33 & \0953 \\ 
105 & 5115 steel & Solid & \0379 & \0651 & 210 & 7850 & 0.33 & \0711 \\ 
106 & Stainless steel & Solid, annealed & 
\0652 & \0820 & 208 & 7800 & 0.33 & \0820
\botline
\end{tabular}
\end{inftab}

\section{}\label{appB}
In this study, the finite element method (FEM) was employed to simulate
the tensile behavior of 106 alloy materials. The resulting data were
primarily used for model training to demonstrate the feasibility of
deep learning methods for predicting material constitutive
relationships. It is important to note that the stress--strain curves
for each alloy material provided in the appendix have not been
rigorously validated. If you intend to use the data provided in the
appendix, please assess its suitability for your specific research
needs and determine whether further rigorous verification is necessary.

For ease of presentation, the stress--strain curve data for each alloy
material are displayed in image format as follows:

\xfigure{fx01}{Figures~1--21 correspond to aluminum alloys}

\xfigure{fx02}{Figures~22--41 correspond to copper alloys}

\xfigure{fx03}{Figures~42--60 correspond to tool steels}

\xfigure{fx04}{Figures~61--80 correspond to nickel alloys}

\xfigure{fx05}{Figures~81--93 correspond to titanium alloys}

\xfigure{fx06}{Figures~94--106 correspond to other alloy steels}

\printbibliography
\refinput{crmeca20250325-reference.tex} 

\end{document}
