A Novel Approach for Modeling Strain Hardening in Plasticity and Its Material Parameter Identification by Bayesian Optimization for Automotive Structural Steels Application

Abstract

Constitutive modeling in plasticity is a critical topic in solid mechanics. However, modeling nonlinear plasticity remains a challenge due to the theoretical complexity in representing realistic material behavior. This work aims to develop a general material model based on a rational polynomial function for plasticity and to use Bayesian optimization to identify its parameters. As a data-driven approach, Bayesian optimization effectively estimates model parameters for a high-computational-cost model. In this work, automotive structural steel is selected as a representative example to benchmark the proposed approach. Our results demonstrate that the rational polynomial function effectively models the plasticity behavior for metallic alloy before the failure point, and Bayesian optimization successfully estimates the parameters. This novel framework also has the potential to be applied to other materials, whose constitutive models can be defined by stress–strain curves.

1. Introduction

In real situations, many materials show complex and nonlinear behavior in the plastic region that cannot be accurately represented by simple analytical models. Solving an inverse problem for inhomogeneous deformation is important to accurately identify constitutive model parameters or realistic stress–strain relationships. Most tensile coupon experiments exhibit non-uniform deformation when necking occurs. This non-uniformity prevents direct conversion of strain from displacement and stress from force using simple formulas or analytical solutions. Inverse analysis simulates the coupon test with precise geometries, boundary conditions, and loading scenarios. The stress–strain relation is determined through an iterative process by adjusting the simulation inputs until the outputs match the experimental data. The inverse finite element method can provide an accurate stress–strain relation beyond the necking point [1]. Therefore, identifying a constitutive model for a material using the inverse finite element method is necessary for most real scenarios, as deformation typically becomes non-uniform at higher strain levels in most material tests.

Conventionally, optimization methods, such as reduced domain random search [2], direct search [3,4], and gradient-based approaches [5,6], are used to identify material parameters from experiments. However, these methods often suffer from local minima and are highly sensitive to the initial parameter values. To address this challenge, the efficient global optimization (EGO) approach should be adopted.

The Markov Chain Monte Carlo (MCMC) method was applied to determine material model parameters. This method provides a global solution through probabilistic sampling. The Metropolis–Hastings algorithm was adopted to infer the neo-Hookean and Ogden material parameters of the sheep aorta [7]. The MultiNest algorithm by Feroz et al. [8] has been reported to calibrate isotropic hyperelastic models, including the Mooney–Rivlin and Ogden models [9,10]. The MCMC sampling method provides a posterior distribution of material parameters given the material model and experimental data to fit. However, it requires a large number of iterations to reach the tolerance, and the situation becomes more challenging when each iteration requires a high computational cost. Considering the high computational cost of most finite element simulations in real scenarios, this method is not suggested (or not feasible) in the inverse finite element method.

The surrogate model is a popular method that can reduce the computational cost [11]. Generally, it can be divided into parametric optimization and non-parametric optimization. Parametric optimization relies on predefined functional forms and known parameters to find the optimal solution, while non-parametric optimization uses flexible, data-driven approaches without assuming a specific model structure.

The response surface method [12,13] is a typical approach for parametric optimization. However, it is constrained by the number of material parameters and has difficulty achieving good performance in high-dimensional or highly nonlinear problems. Inspired by the anatomy of brain tissues, the neural network is a powerful algorithm in artificial intelligence applied to a wide variety of areas. Recently, neural networks have been investigated to build material models of the human brain cortex [14] as well as for ductile fracture and plasticity modeling [15]. Although the neural network has a strong nonlinear fit capacity for capturing nonlinear relationships, it heavily relies on the quality and diversity of the training dataset, which makes it not flexible enough to apply the trained network to other scenarios.

Bayesian optimization as a global optimization method is a popular non-parametric approach [16]. The main concept behind Bayesian optimization is that it uses a computationally cheap surrogate to mimic the behavior of the objective function. The general task of Bayesian optimization is to optimize a computationally expensive or black-box function. Bayesian optimization uses a Gaussian process as the surrogate model and an acquisition function to trade off exploitation and exploration for next-point estimation. As a sequential method [16], each iteration gradually builds up and enriches the surrogate model. It determines the next point by digesting all previous observations and then infers it using the acquisition function, which balances exploration and exploitation. Thus, it is a data-driven method. With higher-quality next-point sampling and faster convergence, it treats the objective function to be optimized as a “black box”. Thus, we only need to consider the inputs and the associated residual values; we do not need to compute the gradient, which is very computationally expensive. Because of its computationally inexpensive characteristics, this stochastic surrogate has been widely applied in inverse finite element methods to correlate simulations with experiments. Bayesian optimization has been widely applied to material parameter identification across various fields, including biomaterials, metals, etc., at both continuum and atomistic scales. It has been reported that the material parameter of the porcine meniscus was found using Bayesian optimization for the Yeoh model [17]. The shear stress and Armstrong–Frederick parameters in small-strain crystal plasticity were also determined using this approach [18]. The material growth parameters for ferromagnetic metals are evaluated using this efficient model [19]. In addition, the tensile strength, shear modulus, fracture toughness, etc., in the phase-field are corrected [20]. It also helps predict material properties at the atomistic level [21]. Nevertheless, to the best of my knowledge, no one has used Bayesian optimization to decide the material parameters for the strain-hardening phenomenon in plasticity.

Numerous studies [22,23,24] have investigated the plastic-hardening model for plastic deformation, as shown in

Table 1. Summary of stress vs. strain relations for strain hardening. ${\sigma }_Y$ is the yield stress as a function of initial yield strength ${\sigma }_0$, effective plastic strain ${\overline{\epsilon }}^p$, and other internal variables. K is the strength coefficient. Q is the saturation stress. b is the hardening rate parameter. n is the strain hardening exponent. E is the Young’s modulus. $ϵ$ is the strain. $n_1$ and $n_2$ are the two strain-hardening exponents. $K_1$ and $K_2$ are the two strength coefficients. C is the strain rate sensitivity factor. $\dot{\epsilon }$ is the strain rate. ${\dot{\epsilon }}_0$ is the reference strain rate. $T^{*}$ is the normalized temperature. m is the thermal softening coefficient. $p_1,p_2,\dots p_n$ and $q_1,q_2,\dots q_n$ are the rational polynomial coefficients.

Material Model	Stress vs. Strain Relation
Ludwik	${\sigma }_Y={\sigma }_0+K{\left({\overline{\epsilon }}^p\right)}^n$
Swift	${\sigma }_Y=K{\left({\overline{\epsilon }}^p+{\epsilon }_0\right)}^n$
Voce	${\sigma }_Y={\sigma }_0+Q\left(1-e^{-b{\overline{\epsilon }}^p}\right)$
Ramberg-Osgood	${\overline{\epsilon }}^p={\displaystyle \frac{{\sigma }_Y}{E}}+{\left({\displaystyle \frac{{\sigma }_Y}{K}}\right)}^n$
Normalized Ramberg-Osgood	${\sigma }_Y={\sigma }_0{\left({\displaystyle \frac{E{\overline{\epsilon }}^p}{{\sigma }_0}}\right)}^n$
Modified Ludwik Model	${\sigma }_Y=K_1+{\left({\overline{\epsilon }}^p\right)}^{n_1}+e^{K_2+n_2{\overline{\epsilon }}^p}$
Johnson–Cook	${\sigma }_Y=\left({\sigma }_0+K{\left({\overline{\epsilon }}^p\right)}^n\right)\left(1+Cln\left({\displaystyle \frac{\dot{\epsilon }}{{\dot{\epsilon }}_0}}\right)\right){\left(1-T^{*}\right)}^m$
Piecewise linear plasticity	Table includes ${\sigma }_Y\leftarrow {\overline{\epsilon }}^p$, i = 1, 2 ... N
Rational Polynomial (General)	${\sigma }_Y={\displaystyle \frac{p_1{\left({\overline{\epsilon }}^p\right)}^n+p_2{\left({\overline{\epsilon }}^p\right)}^{n-1}+\cdots +p_{n+1}}{{\left({\overline{\epsilon }}^p\right)}^m+q_1{\left({\overline{\epsilon }}^p\right)}^{m-1}+\cdots +q_m}}$
Rational Polynomial (Present)	${\sigma }_Y={\displaystyle \frac{p_1{\left({\overline{\epsilon }}^p\right)}^2+p_2\left({\overline{\epsilon }}^p\right)+p_3}{{\left({\overline{\epsilon }}^p\right)}^2+q_1\left({\overline{\epsilon }}^p\right)+q_2}}$

. However, these models impose significant constraints on the stress–strain behavior, which prevents them from accurately representing the experimental data of real materials. To overcome this issue, the piecewise linear plasticity model uses discrete stress vs. plastic strain data pairs, often obtained from experiments, to define the stress–strain relationship [25]. The grid search method combined with random search algorithms is designed to find the steel stress–strain curve [2]. The polynomial stress–strain curve for plasticity is developed for VT6 titanium alloy and steel 20HGR by fitting a stress–strain curve from the experiment directly [26]. However, those studies did not use advanced optimization or non-homogeneous deformation for more realistic boundary conditions, and none of the existing literature utilizes Bayesian optimization to develop curve-based material constitutive models.

This study aims to apply Bayesian learning, an efficient surrogate model, to determine a stress–strain curve-based plastic material model based on a tensile coupon experiment. The rational polynomial plasticity model is used to fit the stress–strain curve with a general representation of the arbitrary stress–strain curve. It can be seen that the rational polynomial plasticity model can fit almost all stress–strain curve cases for metallic material with fewer than five parameters. To demonstrate the framework, we used *MAT_024 (a piecewise linear plasticity model) in LS-DYNA [25] for the automotive structural steel as an example. This material model is also available in other commercial software, such as ABAQUS [27]. The automotive structural steel is selected because of its wide application in studying crash safety. The broad goal is to apply artificial intelligence to develop and formulate a novel framework for curve-based constitutive models.

This article is organized as follows. Section 2 outlines the experiment, finite element modeling, constitutive modeling, and optimization theory in detail. This is followed by the force vs. displacement correlation between the finite element and experimental data, along with the optimization results, in Section 3. Finally, Section 4 provides a discussion of the results along with the concluding remarks, while Section 5 presents the conclusion and future work.

2. Materials and Methods

In this section, the procedures for specimen preparation and the experimental setup are first described. The realistic dimensions for each tensile coupon specimen are measured and recorded. Then, the finite element model and novel constitutive model are introduced. Finally, the optimization methods used to fit force–displacement data are reviewed in detail. Figure 1 shows how those theories and methods above are related to each other, and the following subsections elaborate on the contents of each component.

2.1. Specimen Preparation

The specimens were cut from automobile parts rather than from raw materials from the market. The diagram dimensions of the ASTM E8 flat sub-size tensile specimen [28] are shown in Figure 2, and their actual dimensions are documented in

Table 2. Compact specimen dimensions. Specimen 1: Cut from the rear cross-member of a van’s ladder frame. Specimen 2: Cut from the front rail of a van’s ladder frame. Specimen 3: Cut from automotive chassis body-in-white structural steel. Note: the exact location information of Specimen 3 is no longer available due to a missing record. The length unit is in millimeters (mm).

Dimension	Specimen 1	Specimen 2	Specimen 3
Top Shoulder To Top Grip	8.69	11.16	3.97
Bottom Shoulder To Bottom Grip	5.60	9.00	6.45
Bottom Shoulder To Bottom Grip	8.93	11.99	7.44
Total Length	101.45	101.65	53.55
Reduced Length	31.06	29.77	29.49
Width	6.32	6.31	6.16
Thickness	2.02	2.02	1.23
Ext. Meter Len	25.40	25.40	25.40

.

2.2. Experiment Setup

The testing machine (MTS 810.12) was used to conduct the experiment. The real-time force was measured by an MTS high-accuracy servohydraulic system load cell (661.20 × 10⁻³) with a maximum capacity of 22 Kip. The 647 all-temperature hydraulic wedge grip was mounted to ensure no slipping of the specimen during loading. The MTS extensometer was used to record the strain value of the dog bone specimen. This setup is shown in Figure 3a. To investigate the elastic–plastic behavior, a quasi-static process for the specimen is required to minimize the strain-rate effect. The machine loading speed was programmed to achieve a strain rate of 0.01% (1/s).

2.3. Finite Element Model

The finite element model was built with the same boundary conditions, except for the loading speed, to simulate the experiment shown in Figure 4. Since viscosity was not considered, the simulation was a quasi-static process. In this explicit dynamic simulation, a higher loading speed was used to reduce the computational time. In this simulation, we adopted the smooth displacement to avoid the initial oscillation. Prescribed boundary conditions were applied to the node sets in the purple and yellow positions and the negative Z direction, as shown in Figure 4b. The fixed boundary conditions on that node set in both X and Y directions simulated the fixed grip. The reaction force was obtained from the cross-sectional force labeled by the red dashed line. The hexahedral element and single integration (one-point integration) were selected for the serious mesh deformation and higher computational efficiency. The finite element model, including the boundary condition, geometry, mesh, etc., is adopted from the literature [2]. We adopted a mesh size of 0.44 mm. For the mesh-size convergence study, a mesh size of 0.11 mm was considered the reference mesh size. The reaction force percentage $|F_c-F_r|/F_r$ = 0.115%, where $F_c$ and $F_r$ are the section forces for the 0.44 mm and 0.11 mm mesh sizes, respectively, at the last time step of the simulation. To balance computational efficiency during optimization iterations and accuracy, a 0.44 mm mesh size was selected. Each specimen was modeled according to the real geometry provided in Appendix A. The unit system for the material model is based on mm, ms, kg, kN, and GPa. A similar setup can be found in Wang et al. [2]. LS-DYNA R14.1.1 Double precision was used.

2.4. Constitutive Model in Continuum Mechanics

The elastic–plastic model is described under the continuum mechanics framework. To fit the force–displacement response from the experiment, the piecewise linear plastic model, *MAT_024 in LS-DYNA, is employed. In LS-DYNA, this material model is also called the tabulated input model, which means the material characteristic curves are tabulated in the model to describe the constitutive equations. *MAT_024 is selected because it can fully characterize the plastic behavior using a curve or multiple curves as input. The elastic range is based on a linear material model in *MAT_024. Young’s modulus of 210 GPa and Poisson’s ratio of 0.3 are used. The plastic mechanical behavior can be represented by an arbitrary yield stress versus effective plastic strain curve in the LCSS of *MAT_024. In this material model, a one-dimensional curve characterizes the uniaxial loading condition starting from the yield point, at which the yield stress (${\sigma }_Y$) value can be calculated by the effective stress formulation. The effective stress is a measure of the current applied stress state, while the yield stress represents the material’s resistance to plasticity. In LS-DYNA, the effective stress and effective plastic strain are mentioned together to emphasize their expressions as a thermodynamic conjugate pair, but physically, yield stress vs. effective plastic stress describes how the material’s mechanical behavior changes when plastic strain increases. Effective stress is determined from the external driving force, while yield stress is the internal material property. Their concepts are different, although their values are the same in plasticity. The specific strain rates and the curves of yield stress versus effective plastic strain are defined in a table to define the viscoplastic effect. The deviatoric stress tensor and effective plastic strain form a pair of thermodynamic conjugates that describe the irreversibility of the material and construct the dissipated energy [25].

In this study, we ignore the strain rate effect. The return-mapping algorithm built into the software achieves the deformation process in plasticity under applied forces. However, it is valuable to review the return-mapping algorithm embedded in this material model.

In the return-mapping algorithm [25,29,30,31,32], the effective plastic strain (${\overline{\epsilon }}^p$), the slope of the curve of yield stress vs. effective plastic strain (H), the von Mises strength (${\sigma }_{V0}$), the shifted stress tensor ($\mathsf{\xi }$), the true strain tensor ($\mathit{e}$), the Cauchy stress tensor ($\mathit{t}$), the deviatoric stress tensor ($\mathit{s}$), and the back stress tensor ($\mathit{\beta }$) are the variables in the algorithm. The von Mises strength is also called the initial von Mises stress or the initial yield stress (${\sigma }_0$). The effective stress is also called the von Mises stress or the equivalent stress.

The conventional form of the effective plastic strain is defined in Equation (1). The incremental form is shown in Equation (9).

$${\overline{\epsilon }}^p={\mathsf{\int }}_0^t\sqrt{{\displaystyle \frac{2}{3}}}∥\dot{{\mathbf{e}}^{\mathrm{p}}}\left(\tau \right)∥d\tau$$

(1)

The isotropic and kinematic rules can be found in Equation (2) with $c=1$ representing isotropic hardening and $c=0$ standing for kinematic hardening. When the yield surface expands uniformly, it is called isotropic hardening. As for kinematic hardening, the center of the yield surface moves to a new location. Values between 1 and 0 represent a combination of both. The trial value of the yield function ${\tilde{f}}_{n+1}$ at the $t_{n+1}$ time step is defined as

$${\tilde{f}}_{n+1}=∥{\tilde{\mathsf{\xi }}}_{n+1}∥-\sqrt{{\displaystyle \frac{2}{3}}}\left({\sigma }_{V0}+cH{\overline{\epsilon }}_n^p\right),$$

(2)

where

$${\tilde{\mathbf{t}}}_{n+1}=\lambda tr\left({\mathbf{e}}_{n+1}-{\mathbf{e}}_n^{\mathrm{p}}\right)\mathbf{I}+2\mu \left({\mathbf{e}}_{n+1}-{\mathbf{e}}_n^{\mathrm{p}}\right)$$

(3)

$${\tilde{\mathbf{s}}}_{n+1}={\tilde{\mathbf{t}}}_{n+1}-{\displaystyle \frac{1}{3}}tr\left({\tilde{\mathbf{t}}}_{n+1}\right)\mathbf{I}$$

(4)

$${\tilde{\mathsf{\xi }}}_{n+1}\equiv {\tilde{\mathbf{s}}}_{n+1}-{\mathit{\beta }}_n.$$

(5)

If ${\tilde{f}}_{n+1}>0$, we calculate the following updates as (If ${\tilde{f}}_{n+1}\le 0$, then no updates are needed).

$$\Delta \gamma ={\displaystyle \frac{{\tilde{f}}_{n+1}}{2\mu +\frac{2}{3}H}}$$

(6)

$${\mathbf{e}}_{n+1}^{\mathrm{p}}={\mathbf{e}}_n^{\mathrm{p}}+\Delta \gamma {\displaystyle \frac{{\tilde{\mathsf{\xi }}}_{n+1}}{∥{\tilde{\mathsf{\xi }}}_{n+1}∥}}$$

(7)

$${\mathit{\beta }}_{n+1}={\mathit{\beta }}_n+{\displaystyle \frac{2}{3}}(1-c)H\Delta \gamma {\displaystyle \frac{{\tilde{\mathsf{\xi }}}_{n+1}}{∥{\tilde{\mathsf{\xi }}}_{n+1}∥}}$$

(8)

$${\overline{\epsilon }}_{n+1}^p={\overline{\epsilon }}_n^p+\sqrt{{\displaystyle \frac{2}{3}}}\Delta \gamma$$

(9)

$${\mathbf{t}}_{n+1}=\lambda tr\left({\mathbf{e}}_{n+1}-{\mathbf{e}}_{n+1}^{\mathrm{p}}\right)\mathbf{I}+2\mu \left({\mathbf{e}}_{n+1}-{\mathbf{e}}_{n+1}^{\mathrm{p}}\right)$$

(10)

$${\mathbf{s}}_{n+1}={\mathbf{t}}_{n+1}-{\displaystyle \frac{1}{3}}tr\left({\mathbf{t}}_{n+1}\right)\mathbf{I}$$

(11)

With the above values, it is demonstrated that the value of the yield function indeed returns to zero, as shown in Appendix A.

$$f_{n+1}=∥{\mathbf{s}}_{n+1}-{\mathit{\beta }}_{n+1}∥-\sqrt{{\displaystyle \frac{2}{3}}}\left({\sigma }_{V0}+cH{\overline{\epsilon }}_{n+1}^p\right)=0$$

(12)

2.5. Rational Polynomial Function Applied in Constitutive Model

Physical phenomena can be naturally modeled by rational polynomial functions, which are important in engineering and science. One of the applications is the transfer function, which characterizes the relationship between output and input in control theory [33]. Rational polynomial functions can approximate a wide range of functions. Any continuous function on a closed interval can be approximated as closely as desired by a rational polynomial function. Its flexibility captures and recognizes complex patterns in nature. In machine learning, a polynomial kernel has been applied to classification tasks to solve nonlinear problems [34]. Rational polynomial regression is also applied to identify and predict trends [35].

Based on its high nonlinearity and simplicity, in this work, we design the yield stress (${\sigma }_Y$) versus effective plastic strain (${\overline{\epsilon }}^p$) curve using a rational polynomial, as shown in Equation (13). The rational polynomial function consists of a numerator and a denominator.

$${\sigma }_Y(p_1,p_2,\dots ,p_{n+1},q_1,q_2,\dots ,q_m,{\overline{\epsilon }}^p)={\displaystyle \frac{p_1{\left({\overline{\epsilon }}^p\right)}^n+p_2{\left({\overline{\epsilon }}^p\right)}^{n-1}+\cdots +p_{n+1}}{{\left({\overline{\epsilon }}^p\right)}^m+q_1{\left({\overline{\epsilon }}^p\right)}^{m-1}+\cdots +q_m}}$$

(13)

where $p_1,p_2,\dots ,p_n$ and $q_1,q_2,\dots ,q_n$ are the rational polynomial coefficients.

For our material, the three coefficients in the numerator and two coefficients in the denominator give the best fit.

$${\sigma }_Y={\displaystyle \frac{p_1{\left({\overline{\epsilon }}^p\right)}^2+p_2\left({\overline{\epsilon }}^p\right)+p_3}{{\left({\overline{\epsilon }}^p\right)}^2+q_1\left({\overline{\epsilon }}^p\right)+q_2}}.$$

(14)

The rational polynomial is converted to the piecewise plasticity model *MAT_024 and used in the LS-DYNA simulation for inverse analysis. Notice that other material may require different forms of the numerator and denominator in rational polynomials for accurate fitting of its plastic curve. Other choices include 3 in the numerator and 1 in the denominator, 3 in the numerator and 2 in the denominator, and 4 in the numerator and 2 in the denominator, etc.

2.6. Optimization

In this project, we first used Bayesian optimization to find the global solution. The result is then refined by a local optimization, Nelder–Mead simplex method [36] built in SciPy [37] using tolerance 1.0 × ${10}^{-4}$. In this work, the open resource Bayesian optimization [38] is applied. However, it is valuable to review the theory of Bayesian optimization. In this way, the package is no longer a black box. Bayesian optimization is used to find the global solution for $p_1$, $p_2$, $p_3$, $p_3$, $q_1$, and $q_2$ in Equation (14). In this optimization, $\mathit{\theta }$ is a vector of each parameter and it is written as $\mathit{\theta }=[p_1,p_2,p_3,p_3,q_1,q_2]$. It is known that the Gaussian process is defined as

$$f\left(\mathit{\theta }\right)\sim \mathcal{N}(m\left(\mathit{\theta }\right),k(\mathit{\theta },{\mathit{\theta }}^{\prime }\left)\right),$$

(15)

where $m(\mathit{\theta })$ is the prior mean. The function k calculates the covariance between the inputs $\mathit{\theta }$ and ${\mathit{\theta }}^{\prime }$. The document of this package uses Matern kernels as the kernel function k as follows:

$$k\left(\mathit{\theta },{\mathit{\theta }}^{\prime }\right)={\displaystyle \frac{1}{\Gamma \left(\nu \right)2^{\nu -1}}}{\left({\displaystyle \frac{\sqrt{2\nu }}{l}}d\left(\mathit{\theta },,{\mathit{\theta }}^{\prime }\right)\right)}^{\nu }{K}_{\nu }\left({\displaystyle \frac{\sqrt{2\nu }}{l}}d\left(\mathit{\theta },,{\mathit{\theta }}^{\prime }\right)\right),$$

(16)

where $d\left(\mathit{\theta },{\mathit{\theta }}^{\prime }\right)$, $K_{\nu }$, and $\Gamma$ denote the Euclidean distance, the modified Bessel function, and the gamma function, respectively. $\nu$ (smoothness parameter) and l (length scale parameter) are the hyperparameters [39]. In the open-source code, the hyperparameter $\nu$ is set to 2.5 together with the default l as 1.0 [40].

The Gaussian process has a useful property—the joint Gaussian distribution [41,42,43] of the observed values in history ${\mathbf{f}}_{1:t}$ given ${\mathit{\theta }}_{1:t}$ and any arbitrary value $f_{t+1}$ at any new point ${\mathit{\theta }}_{t+1}$ is given by

$$\left[\begin{array}{c}{\mathbf{f}}_{1:t}\\ f_{t+1}\end{array}\right]\sim \mathcal{N}\left(\left[\begin{array}{c}{\mathbf{m}}_{1:t}\\ m_{t+1}\end{array}\right],\left[\begin{array}{cc}\mathbf{K}& \mathbf{k}\\ {\mathbf{k}}^T& k({\mathit{\theta }}_{t+1},{\mathit{\theta }}_{t+1})\end{array}\right]\right),$$

(17)

with

$$\mathbf{K}=\left[\begin{array}{ccc}k({\mathit{\theta }}_1,{\mathit{\theta }}_1)& \dots & k({\mathit{\theta }}_1,{\mathit{\theta }}_t)\\ ⋮& \ddots & ⋮\\ k({\mathit{\theta }}_t,{\mathit{\theta }}_1)& \dots & k({\mathit{\theta }}_t,{\mathit{\theta }}_t)\end{array}\right],$$

(18)

$$\mathbf{k}={\begin{array}{cc}[k({\mathit{\theta }}_{t+1},{\mathit{\theta }}_1)& k({\mathit{\theta }}_{t+1},{\mathit{\theta }}_2)\dots k({\mathit{\theta }}_{t+1},{\mathit{\theta }}_t)]\end{array}}^T,$$

(19)

$${\mathbf{f}}_{1:t}={\begin{array}{cc}[f\left({\mathit{\theta }}_1\right)& f\left({\mathit{\theta }}_2\right)\dots f\left({\mathit{\theta }}_t\right)]\end{array}}^T,$$

(20)

$${\mathbf{m}}_{1:t}={\begin{array}{cc}[m\left({\mathit{\theta }}_1\right)& m\left({\mathit{\theta }}_2\right)\dots m\left({\mathit{\theta }}_t\right)]\end{array}}^T.$$

(21)

Following the Sherman–Woodbury–Morrison identity, this leads naturally to the distribution of the complete parameter space [39]. At any given value of ${\mathit{\theta }}_{t+1}$, the posterior distribution of the mean and variance is given by

$$P\left(f_{t+1}|{\mathbb{D}}_{1:t},{\mathit{\theta }}_{t+1}\right)\sim \mathcal{N}\left(\mu \left({\mathit{\theta }}_{t+1}\right),{\sigma }^2\left({\mathit{\theta }}_{t+1}\right)\right),$$

(22)

where

$$u\left({\mathit{\theta }}_{t+1}\right)=m_{t+1}\left({\mathit{\theta }}_{t+1}\right)+{\mathbf{k}}^T{\mathbf{K}}^{-1}({\mathbf{f}}_{1:t}-{\mathbf{m}}_{1:t}),$$

(23)

$${\sigma }^2\left({\mathit{\theta }}_{t+1}\right)=k({\mathit{\theta }}_{t+1},{\mathit{\theta }}_{t+1})-{\mathbf{k}}^T{\mathbf{K}}^{-1}\mathbf{k}$$

(24)

and ${\mathbb{D}}_{1:t}$ is the set of observations so far. In the open-source code, a noise identity matrix scaled by 1.0 × 10⁻¹⁰ is added to $\mathbf{K}$ in Equation (23) to make it easier to invert the matrix for good numerical stability [38,40], which is a typical procedure in numerical linear algebra.

The prior mean function has many forms, as summarized by Ath et al. [42]. However, without sufficient prior mean information, $\mathbf{m}$ is generally assumed to be zero [42,44,45,46]. In this open-source code [38], the GaussianProcessRegressor package from sklearn [40] is applied with a default prior mean of zero. Then Equation (23) is reduced to

$$u\left({\mathit{\theta }}_{t+1}\right)={\mathbf{k}}^T{\mathbf{K}}^{-1}{\mathbf{f}}_{1:t}.$$

(25)

The acquisition function evaluates the next point, considering the exploration (expectation) and exploitation (uncertainty). The lower confidence bound (LCB) is defined as

$${\alpha }_{LCB}(\mathit{\theta })=\mu \left(\mathit{\theta }\right)-\kappa \sigma \left(\mathit{\theta }\right),$$

(26)

while the upper confidence bound (UCB) is expressed as

$${\alpha }_{LCB}(\mathit{\theta })=\mu \left(\mathit{\theta }\right)+\kappa \sigma \left(\mathit{\theta }\right),$$

(27)

where $\kappa$ is a user-defined parameter that controls the trade-off between exploration and exploitation. The LCB is designed to minimize problems, while UCB is for maximization. This open-source code [38] uses UCB in Equation (27) with default $\kappa =2.576$.

Since the task in this work is to minimize the discrepancy between the simulation and the experiment, the sign of the residual value was flipped to convert the maximization code into a minimization program, specifically, minus one times the absolute value of the residual value. These functions are then minimized to obtain the next point ${\mathit{\theta }}_{t+1}$ for evaluating the cost function:

$${\mathit{\theta }}_{t+1}=\mathrm{argmin}{\alpha }_{LCB}\left(\mu \left(\mathit{\theta }\right),\sigma \left(\mathit{\theta }\right)\right).$$

(28)

This process is repeated until a user-defined stopping criterion is reached. In the open-source package, Equation (28) is minimized by random sampling (a low-cost approach) and L-BFGS-B, according to the documentation [38]. It can also be minimized using global optimization methods, such as genetic algorithms [36].

The root mean square error (RMSE) [47] is carried out by an optimization algorithm to minimize the objective value.

3. Results

3.1. Experimental Results

The force–displacement responses for each specimen are shown in Figure 5. It is a typical steel tensile coupon response. However, discrepancies are found among the specimens due to different dimensions and steel types. With the obvious plastic behaviors, the corresponding yield and failure points can be seen directly. In

Table 3. The corresponding yield point, initial cross-sectional area, and yield strength (${\sigma }_0$) for each specimen from Table 2.

	Specimen No.1	Specimen No.2	Specimen No.3
Yield point force (kN)	5.42	4.17	3.36
Initial cross-section area (${\mathrm{mm}}^2$)	12.77	12.75	7.58
Yield strength (GPa)	0.4244	0.3272	0.4435

, for each specimen, the yield point force is obtained from Figure 5 and the initial cross-section area is calculated by the width and thickness from Table 2. Thus, the yield strength can be calculated by the yield point force divided by the initial cross-section area. Specimen No.1 & Specimen No.2 share almost the same cross-sectional area. However, Specimen No.1’s yield point is significantly higher than that of Specimen No.2. This indicates that Specimen No.1’s nominal yield stress is higher than that of Specimen No.2. As for Specimen No.3, which has the lowest yield point and the smallest initial cross-sectional area, it shows a higher nominal stress, slightly exceeding that of Specimen No.1.

3.2. Force vs. Displacement Fit

The high accuracy of the material model we developed and the framework we proposed demonstrate good performance in the plastic deformation region, as shown in Figure 6a, Figure 7a, and Figure 8a. The force–displacement responses compare the experimental results (solid line) with the finite element results (dashed line). The finite element simulations fit the experimental data well before the failure point, which is defined as the point where the force begins to decrease. The force–displacement response is characterized by the yield stress–effective plastic strain curve in tension. This curve is used as the input for *MAT_024 in LS-DYNA, and its optimal solution is shown in Figure 6b, Figure 7b, and Figure 8b, which governs the different plastic behaviors of each specimen with different material compositions.

The Bayesian learning algorithm enables the novel plastic model in Equation (14) to finally find the parameters, which are shown in

Table 4. Best fit coefficients of rational polynomial in Equation (14) from Bayesian and Nelder–Mead simplex optimizations.

Specimen	${\mathit{p}}_1$	${\mathit{p}}_2$	${\mathit{p}}_3$	${\mathit{q}}_1$	${\mathit{q}}_2$
No.1	1436.04	70,626.59	1658.52	70,937.46	3793.77
No.2	214.21	320,496.07	16,346.98	533,919.11	51,904.17
No.3	1436.04	70,626.59	1658.52	70,937.46	3793.77

. Since this is a rational polynomial function, if the ratio of the order of $p_1$, $p_2$, $p_3$, and $q_1$, $q_2$ remains the same, the yield stress vs. effective plastic strain does not change. The rational polynomial is a ratio relation expression and, thus, has infinite unit selections for those parameters. For simplicity, the unit for each parameter is not written explicitly, and the curve generated by the parameters has the GPa unit in this work.

Significant improvements in the objective values are achieved. The Bayesian learning approach enables the system to refine and adapt the material parameters iteratively, based on the data provided from both experimental observations and all the previous FE simulations. The convergence history is summarized in Figure 9, which tracks the evolution of the optimization process over time. The blue curve records the best value for each iteration of Bayesian optimization. The green curve documents the Nelder–Mead simplex algorithm result. During the local optimization, the Nelder–Mead simplex algorithm used the final result of Bayesian optimization as the starting point to refine the value of the approximate global solution. The initial Bayesian optimization guesses, Bayesian optimization final values, and final improvements by the Nelder–Mead simplex algorithm are recorded in

Table 5. Best fit coefficients of rational polynomial based on Bayesian and Nelder–Mead simplex optimizations.

Specimen	BO Initial Value	BO Final Value	Nelder–Mead Final Value
No.1	1.204	0.479	0.380
No.2	1.144	0.474	0.422
No.3	1.487	0.237	0.204

. The details of the optimization histories are recorded in Figure 9.

4. Discussion

Our results demonstrate that the novel framework for plasticity we developed accurately fits the force–displacement data under investigation. In this section, we discuss the significance of our findings. First, we compare and analyze our novel plastic material model in detail within the context of existing literature. Then, we discuss the application of Bayesian optimization and the potential issues with the objective function.

The selection of material models is the first and most important step in the force–displacement correlation. In this work, a rational polynomial function of order two in both the numerator and denominator gives the best fit for determining the mechanical behavior of the automotive structural steel tensile coupon specimen under plastic deformation. Other combinations, in the order of numerator and denominator, have been tried, but they did not provide as good a fit as the one we selected. It is still worthwhile to discuss other types of elastic–plastic models for strain-hardening behavior as shown in Table 1. In the table, those frequently mentioned models include the Ludwik Power Law [48], Swift [48], Voce [48], Ramberg-Osgood [48], empirical representation [48], Ludwigson [49], etc. These models can be roughly classified into a power-law form with certain constraints on the shape of the stress–strain curve. Not all stress–strain curves can be fit into these models. On the other hand, the piecewise linear plasticity model [25,50] uses a number of stress–strain points as independent variables to define the plasticity. Therefore, all kinds of stress–strain relations for the engineering materials can be modeled. However, the large number of independent variables in the piecewise plasticity model makes it computationally costly for optimization. In this work, we introduced a rational polynomial plasticity model with only a few parameters. This model could accommodate almost all shapes of stress–strain curves used in the piecewise plasticity model. Based on these considerations, the rational polynomial, employed in the *MAT_024 model in LS-DYNA to describe the yield stress vs. effective plastic strain curve, is a better option. Although the rational polynomial function offers significant advantages, it naturally comes with challenges.

The isotropic material behavior is assumed up to the point of failure for the rational material model. However, materials often show strain-rate and temperature dependencies in plasticity and failure in the real world. According to the Johnson–Cook model [51], the material model is classically used. However, this material model is based on the Ludwigson model [49] scaled by the rate and temperature effect, making it not fit well for the stress–strain relation of many engineering materials. For better accuracy, the tabulated Johnson–Cook model, as implemented (e.g., *MAT_224) in LS-DYNA, is used instead. In this model, a table consisting of many stress–strain curves (each representing a case under a certain strain rate and temperature) is used to define the material plasticity. The more advanced version (e.g., *MAT_264 in LS-DYNA) also includes the material anisotropic effect. Notice that all these tabulated material models are based on a piecewise linear plasticity model (*MAT_024 in LS-DYNA), with a single stress–strain curve. The purpose of developing the rational material model is to replace the single-curve input (e.g., *MAT_024 in LS-DYNA) with a more compact mathematical formulation, which retains the model generality and reduces the optimization cost. Eventually, this approach will be a tool for defining curves in more complex tabulated material models.

The optimization scheme we selected provides a good fit with high efficiency in practice. This framework performs well, as expected, to evaluate the model parameters in the computationally expensive model. Since our finite element model only requires around twenty-five seconds, more iterations are set up in Bayesian optimization to obtain a better global solution. Then, the local optimization receives a better starting point and requires fewer iterations to meet the tolerance. The open-source code of Bayesian optimization [38] was employed, since it offers a convenient and clean approach to study the “Block-Box” function. Thus, it provides a convenient and robust procedure for other real engineering applications. However, the acquisition function is designed to be optimized by L-BFGS-B, which is a local optimization. This L-BFGS-B implementation can be modified into a global algorithm, such as a genetic algorithm [36], since it is open-source. However, this modification would increase the computational cost by trading off exploitation and exploration in the acquisition function for the next point. While Bayesian optimization can be challenged by high-dimensional problems [52], the five-parameter model used in this study remains well within its effective range and yields an excellent fit.

In practical engineering tasks, it is necessary to consider the cost of time. This work aims to propose a very practical method to solve a real engineering problem. The 500-iteration number for Bayesian optimization was used based on the observation of the final fit quality between simulation and experiment. It provides a practical balance between exploration of the parameter space and total computational cost, together with local optimization refinement as the second step. Alternatively, the minimum relative error or tolerance for the best-fit values in the residual history can be used as the stopping criterion to compare with the current approach for evaluation. This offers different strategies for terminating the optimization process.

The construction of the objective function is critical, and Bayesian calibration is a way to consider the experiment’s uncertainty by including the standard deviation and mean value of the experiment. As the uncertainty in experimental noise is unavoidable, it is beneficial to include it in the material calibration process. However, since only one dataset is available for each material type, the RMSE is used to evaluate the difference between the simulation and experimental results of the force–displacement responses.

5. Conclusions and Further Work

This study proposes a novel framework to identify the stress–strain relationship in plasticity and characterize strain hardening. It is the first work to use optimization to directly predict a stress–strain curve, which has a significant impact on the constitutive model. There are two important contributions. First, the isotropic plastic behavior can be approximated by a rational polynomial function. Second, Bayesian optimization demonstrates its capacity in the plasticity to characterize the proposed strain-hardening material model. Finally, this novel framework demonstrates a strong performance in the plastic deformation region to correlate the experimental force vs. displacement data, which provides a significant potential to characterize other material models.

Further work should include additional experiments to account for the standard deviation in optimization for Bayesian calibration. More trial simulations are suggested to test the potential of the material model and identify its limitations in capturing behavior around or after the failure point. Other types of coupon tests, such as compression, torsion, and shear, are also needed to fully understand the material model by considering more deformation types.