<p>This study proposes a probabilistic framework for reliably estimating constitutive parameters under high strain-rate loading by integrating Monte Carlo sampling with the Differential Evolution (DE) optimization algorithm using Split Hopkinson Pressure Bar (SHPB) experimental data. Dynamic compression tests were conducted on Titanium Grade 1 specimens at three impact pressure conditions (0.5, 1.0, and 1.5 bar), yielding twelve stress–strain curves that exhibited noticeable variability, particularly under the low-pressure condition. To explicitly account for this experimental uncertainty, the extracted strain-rate, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\:K\)</EquationSource> </InlineEquation>, and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\:n\)</EquationSource> </InlineEquation>values were modeled using a multivariate normal distribution, from which 300,000 synthetic datasets were generated via Monte Carlo simulation. The DE algorithm was applied to each virtual dataset to estimate the strain-rate-dependent parameters <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\:D\)</EquationSource> </InlineEquation>and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\:E\)</EquationSource> </InlineEquation>of the Shin–Kim (S–K) constitutive model. Kernel Density Estimation (KDE) revealed a strong convergence of the estimated parameters toward distinct modal values, indicating the statistically most probable parameter combination that would emerge if the SHPB tests were repeated a large number of times. To validate the physical reliability of these probabilistically obtained parameters, finite element simulations of the SHPB setup were performed using LS-DYNA. The results demonstrated that the probabilistic parameters consistently achieved lower RMSE and higher <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\:{R}^{2}\)</EquationSource> </InlineEquation>values compared to the conventional average-based parameters across all pressure conditions, and the predicted strain-rate levels remained fully within the experimentally observed ranges. These findings confirm that the proposed probabilistic approach effectively captures inherent experimental variability and yields constitutive parameters that more accurately reproduce dynamic material behavior. The framework is broadly applicable to various constitutive models and materials, offering a robust pathway for uncertainty-aware parameter identification in high strain-rate material characterization.</p>

Probabilistic Estimation of Constitutive Parameters Using Monte Carlo Simulation and Differential Evolution in SHPB Experiments

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发表日期：

2026年5月12日

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Y.-B. Kim,
J. Kim

摘要

This study proposes a probabilistic framework for reliably estimating constitutive parameters under high strain-rate loading by integrating Monte Carlo sampling with the Differential Evolution (DE) optimization algorithm using Split Hopkinson Pressure Bar (SHPB) experimental data. Dynamic compression tests were conducted on Titanium Grade 1 specimens at three impact pressure conditions (0.5, 1.0, and 1.5 bar), yielding twelve stress–strain curves that exhibited noticeable variability, particularly under the low-pressure condition. To explicitly account for this experimental uncertainty, the extracted strain-rate, \(\:K\) , and \(\:n\) values were modeled using a multivariate normal distribution, from which 300,000 synthetic datasets were generated via Monte Carlo simulation. The DE algorithm was applied to each virtual dataset to estimate the strain-rate-dependent parameters \(\:D\) and \(\:E\) of the Shin–Kim (S–K) constitutive model. Kernel Density Estimation (KDE) revealed a strong convergence of the estimated parameters toward distinct modal values, indicating the statistically most probable parameter combination that would emerge if the SHPB tests were repeated a large number of times. To validate the physical reliability of these probabilistically obtained parameters, finite element simulations of the SHPB setup were performed using LS-DYNA. The results demonstrated that the probabilistic parameters consistently achieved lower RMSE and higher \(\:{R}^{2}\) values compared to the conventional average-based parameters across all pressure conditions, and the predicted strain-rate levels remained fully within the experimentally observed ranges. These findings confirm that the proposed probabilistic approach effectively captures inherent experimental variability and yields constitutive parameters that more accurately reproduce dynamic material behavior. The framework is broadly applicable to various constitutive models and materials, offering a robust pathway for uncertainty-aware parameter identification in high strain-rate material characterization.