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

doi:10.1007/s10483-025-3283-6

Abstract

Abstract:

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%.

Key words: energy density equivalence, spherical indentation, test method, Hollomon law, steel material

2010 MSC Number:

O346

Haomin WANG, Lixun CAI, Huairong XIAO. A unified model for obtaining stress-strain relationship under spherical indenter loading and test application. Applied Mathematics and Mechanics (English Edition), 2025, 46(8): 1591-1608.

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Figures/Tables 20

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Table 1

Comparison of loading indices for different materials"

Material	$m D = 1 mm$	$m D = 1.587 mm$	Relative error/%
Cr17Ni4Cu4Nb	1.121	1.090	2.70
Cr13	1.101	1.081	1.90
Cr12MoV	1.106	1.142	3.30
Cr2MoVA	1.104	1.116	1.00
CrMoA	1.112	1.118	0.50
Cr2Ni4MoV	1.112	1.125	1.10
Super pure-30Cr2Ni4MoV	1.113	1.119	0.53
Cr13Mo	1.092	1.099	0.60

Table 1

Fig. 8

Table 2

Uniaxial tensile mechanical properties of the tested material"

Tested material	$E$ /GPa	$R p0 .2$ /MPa	$R m$ /MPa
Cr17Ni4Cu4Nb	202.2	963.3	1 013.0
Cr13	218.8	739.6	883.3
Cr12MoV	216.9	772.2	929.0
Cr2MoVA	211.7	931.4	1 046.0
CrMoA	212.4	699.4	862.5
Cr2Ni4MoV	203.2	1 076.0	1 150.0
Super pure-30Cr2Ni4MoV	204.0	1 012.0	1 103.0
Cr13Mo	215.3	609.8	757.7

Table 2

Fig. 9

Table 3

Test verification of the elastic modulus E prediction (hmax=60 μm)"

$D$ /mm	Material	Uniaxial tensile test $E$ /GPa	DUM-SPHI $E$ /GPa	Relative error/%
1.587	05Cr17Ni4Cu4Nb	202.2	196.5	2.90
	2Cr13	218.8	219.9	0.48
	2Cr12MoV	216.9	219.8	1.30
	25Cr2MoVA	211.7	217.1	2.60
	35CrMoA	212.4	216.7	2.00
	30Cr2Ni4MoV	203.2	210.4	3.50
	Super pure-30Cr2Ni4MoV	204.0	212.7	4.30
	1Cr13Mo	215.3	213.7	0.73
1	05Cr17Ni4Cu4Nb	202.2	213.9	5.80
	2Cr13	218.8	224.5	2.60
	2Cr12MoV	216.9	221.1	1.90
	25Cr2MoVA	211.7	219.1	3.50
	35CrMoA	212.4	215.4	1.40
	30Cr2Ni4MoV	203.2	213.5	5.10
	Super pure-30Cr2Ni4MoV	204.0	207.1	1.50
	1Cr13Mo	215.3	223.6	3.80

Table 3

Fig. 10

Fig. 11

Table 4

Comparison of Rp0.2 and Rm (D=1.587 mm)"

Material	Uniaxial tensile test		DUM-SPHI		Relative error
Material	$R p0 .2$ /MPa	$R m$ /MPa	$R p0 .2$ /MPa	$R m$ /MPa	$R p0 .2$	$R m$
Cr17Ni4Cu4Nb	963.3	1 013.0	1 008.0	1 040.0	4.60/%	2.70/%
Cr13	739.6	883.3	747.4	889.7	1.10/%	0.73/%
Cr12MoV	772.2	929.0	747.8	912.1	3.20/%	1.80/%
Cr2MoVA	931.4	1 045.8	921.1	1 043.6	1.10/%	0.21/%
CrMoA	699.4	862.5	666.7	822.8	4.70/%	4.60/%
Cr2Ni4MoV	1 076.0	1 150.0	1 024.0	1 111.0	4.90/%	3.40/%
Super pure-30Cr2Ni4MoV	1 012.0	1 103.0	963.9	1 067.0	4.80/%	3.10/%
Cr13Mo	609.8	757.7	639.3	788.2	4.80/%	4.00/%
P92	511.4	718.5	504.1	724.7	1.40/%	0.86/%
A508-III	426.8	682.1	411.6	683.3	3.60/%	0.18/%
SA302B	463.3	591.5	467.3	602.1	0.87/%	1.80/%
Mn	219.2	536.8	226.6	546.2	3.40/%	1.80/%
Cr	764.3	889.5	743.0	871.5	2.80/%	2.00/%
MnR	359.1	553.7	349.3	555.7	2.70/%	0.36/%
T91	656.8	823.1	624.1	807.0	4.90/%	1.90/%

Table 4

Fig. 12

Table 5

Comparison of Rp0.2 and Rm (D=1 mm)"

Material	Uniaxial tensile test		DUM-SPHI		Relative error
Material	$R p0 .2$	$R m$	$R p0 .2$	$R m$	$R p0 .2$	$R m$
Cr17Ni4Cu4Nb	963.3	1 013.0	942.3	1 057.0	2.20/%	4.40/%
Cr13	739.6	883.3	706.9	863.2	4.40/%	2.30/%
Cr12MoV	772.2	929.0	758.3	909.7	1.80/%	2.10/%
Cr2MoVA	931.4	1 046.0	907.7	1 030.0	2.60/%	1.50/%
CrMoA	699.4	862.5	684.7	837.8	2.10/%	2.90/%
Cr2Ni4MoV	1 076.0	1 150.0	1 069.0	1 135.0	0.71/%	1.30/%
Super pure-30Cr2Ni4MoV	1 012.0	1 103.0	999.0	1 077.0	1.30/%	2.40/%
Cr13Mo	609.8	757.7	610.5	764.0	0.11/%	0.83/%

Table 5

Table 6

Summary of the goodness-of-fit"

$D$ /mm	05Cr17Ni4Cu4Nb	2Cr13	2Cr12MoV	25Cr2MoVA	35CrMoA
1.587	0.96	0.99	0.98	0.99	0.95
1	0.95	0.97	0.98	0.99	0.97
$D$ /mm	30Cr2Ni4MoV	Super pure-30Cr2Ni4MoV	1Cr13Mo	P92	SA302B
1.587	0.96	0.97	0.95	0.99	0.99
1	0.99	0.99	0.98	–	–
$D$ /mm	16Mn	A508-III	30Cr	16MnR	T91
1.587	0.98	0.99	0.99	0.98	0.98
1	–	–	–	–	–

Table 6

Table 7

Repeatability verification table"

Test point	Uniaxial tensile test		DUM-SPHI		Relative error		$G$
Test point	$R p0 .2$ /MPa	$R m$ /MPa	$R p0 .2$ /MPa	$R m$ /MPa	$R p0 .2$	$R m$	$G$
P1	963.3	1 012	928.8	968.9	3.6/%	4.3/%	0.95
P2	963.3	1 012	915.9	976.8	4.9/%	3.5/%	0.97
P3	963.3	1 012	925.9	968.8	3.9/%	4.3/%	0.96
P4	963.3	1 012	935.8	970.7	2.9/%	4.1/%	0.95
P5	963.3	1 012	939.4	981.3	2.5/%	3.1/%	0.97

Table 7

Fig. 13

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