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中文核心期刊

基于深度符号优化模型的材料弹塑性力学响应的压痕反演方法

An indentation inverse method for elasto-plastic mechanical response of materials based on deep symbolic optimization model

  • 摘要: 压痕测试因其小尺度、高分辨率和非破坏性等优势,已广泛应用于材料力学性能的快速表征与评估,特别是在微小尺度与复杂结构材料中展现出重要应用前景。然而,由于弹塑性材料本构行为高度非线性,传统反演方法通常依赖有限元模拟与试错型优化算法,不仅计算成本高,且难以兼顾预测精度与结果可解释性。为此,本文提出一种基于深度符号优化(Deep Symbolic Optimization, DSO)算法的压痕反演方法,通过学习压痕荷载–位移曲线特征与材料弹塑性参数之间的非线性映射关系,自动生成弹塑性参数之间的符号表达式。研究针对不同硬化指数与两类典型压头(锥形与球形)分别建立DSO反演模型,并将其与多层感知机、高斯过程回归和随机森林等三种主流机器学习方法进行系统对比,评估指标涵盖决定系数、均方根误差和平均绝对误差等。结果表明,DSO模型在大多数任务中均表现出更高的预测精度与更强的稳定性,尤其在屈服强度等强非线性参数拟合方面优势更为显著。此外,该方法生成的数学表达式具备良好的物理解释性,且在不同加载条件与材料类型下均展现出稳定的泛化能力,其预测值与真实材料参数之间的决定系数在0.8~0.96,为构建高效、可解释的材料性能反演模型提供了一种新路径

     

    Abstract: Instrumented indentation testing has been widely adopted for rapid and non-destructive evaluation of mechanical properties, especially in micro-scale and structurally complex materials, due to its small testing scale, high spatial resolution, and ease of implementation. However, because of the highly nonlinear and coupled nature of elasto-plastic constitutive behavior, conventional inversion approaches typically rely on finite element simulations and trial-and-error optimization procedures. These methods are often computationally expensive, lack physical interpretability, and are difficult to apply efficiently in practical engineering scenarios. To address these challenges, this study proposes an inverse analysis method based on the Deep Symbolic Optimization (DSO) algorithm. By learning the nonlinear mapping relationship between indentation load–displacement curve features and elasto-plastic material parameters, the DSO model generates symbolic expressions with explicit physical meaning. DSO-based inversion models are constructed under various hardening indices and two representative indenter types (conical and spherical). Furthermore, the proposed method is systematically compared with three commonly used machine learning approaches—multilayer perceptron, Gaussian process regression, and random forest—using evaluation metrics including the coefficient of determination (R2), root mean squared error, and mean absolute error. The results indicate that the DSO model achieves superior prediction accuracy and robustness in most cases, particularly in fitting highly nonlinear parameters such as yield strength. In addition, the generated symbolic expressions improve the physical interpretability compared to black-box models. The method also demonstrates strong generalization performance under different loading conditions and material types, with R2 values reaching 0.8–0.96. Overall, this work provides an efficient, accurate, and interpretable solution for inverse analysis in indentation-based characterization of elasto-plastic materials

     

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