A unified model for obtaining stress-strain relationship under spherical indenter loading and test application

• Published in: Applied mathematics and mechanics Vol. 46; no. 8; pp. 1591 - 1608
• Main Authors: Wang, Haomin, Cai, Lixun, Xiao, Huairong
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2025; Springer Nature B.V
• Edition: English ed.
• Subjects: Applications of Mathematics; Classical Mechanics; Constitutive relationships; Equivalence principle; Fluid- and Aerodynamics; Goodness of fit; Hardness tests; Indenters; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Partial Differential Equations; Stress-strain curves; Stress-strain relationships; Tensile strength; Tensile tests; test method; steel material; O346; TG115.5; spherical indentation; Hollomon law; 74-05; energy density equivalence
• ISSN: 0253-4827; 1573-2754
• DOI: 10.1007/s10483-025-3283-6

A dimensionless load-displacement model based on the energy-density equivalence principle is proposed to obtain the stress-strain relationships of metallic materials under monotonic indentations with various diameters of spherical indenters. Finite element simulations are carried out to verify the constitutive relations from the new model, involving indentations made with various spherical indenters. For each indenter, some quasi-static spherical indentation tests are conducted on the materials with 40 preset constitutive relationships. The results indicate that the stress-strain curves predicted by the model align with the preset curves under 200 loading conditions. Moreover, the goodness-of-fit between the predicted stress-strain curves and the preset curves exceeds 0.96 for all indenters and materials. In the end, the indentation tests are conducted by the spherical indenters with the diameters of 1.587 mm for fifteen metallic materials and 1 mm for eight metallic materials. The results show that the stress-strain curves obtained by the spherical indentation based on the new model closely match those obtained from the uniaxial tensile tests. The relative errors for both the proof strength at 0.2% plastic extension and the tensile strength are below 5%.