In situ nanoindentation method for characterizing tensile properties of AISI 1045 steel based on mesomechanical analysis

Abstract

A novel method for characterizing the tensile properties of AISI 1045 steel is proposed by combining the method of in situ nanoindentation test and the theory of mesomechanical analysis. First, the load–depth curves of exact location of ferrite, pearlite and grain boundary on the surface of AISI 1045 steel are obtained by 30 groups of in situ nanoindentation tests. The constitutive equation (stress–strain function) of the real-time metallographic structure is obtained by nanoindentation analysis of the above curves. Then, based on the principle of mesomechanical analysis, the computational representative volume element models are reconstructed according to the three metallographic images of AISI 1045 steel surface collected by the test equipment. Finally, taking the constitutive equation of the real-time metallographic structure as the input condition, the finite element analysis of the above representative volume element models are carried out. The data resulted from finite element analysis are taken as the tensile mechanical properties of AISI 1045 steel. The elastic modulus of AISI 1045 steel calculated is as the same as that by the traditional nanoindentation method. And, the error is less than 6% compared with the tensile test, which is within the range of the elastic modulus of the material. The error between the yield strength calculated and tensile test results is 3.4%. Due to the influence of surface cracks on the plastic deformation ability of AISI 1045 steel during tension, the error between the strain hardening index calculated and tensile test results is 7.4%. The results show that it is a more accurate nondestructive testing method in the point of material damage mechanism. On the premise of using more accurate representative volume element modelling way, this method is suitable for testing more materials.

Keywords

In situ nanoindentation mesomechanical analysis tensile properties representative volume element finite element analysis

Introduction

The relationship between nanoindentation curves and tensile properties of materials has attracted wide attention. The conventional nanoindentation methods mainly focus on characterizing the mechanical properties of homogeneous materials with single structure^1–3 or local areas on the surface of materials.^4–6 And, the conventional inverse problem is adopted in using either the curve or the residual imprint, or the combination of both.^7,8 The stress–strain functions of materials have also be obtained by calculating the metallographic content of materials.^9,10 However, the conventional mesomechanical analysis often fails to introduce in situ mechanical properties of local structures. In addition, the influence of metallographic distribution on the mechanical properties of materials cannot be ignored, especially the grain boundary size and texture.^11,12

In this article, a novel method for characterizing the tensile properties of AISI 1045 steel is proposed based on mesomechanical analysis using self-made in situ nanoindentation test equipment. It is a more accurate nondestructive testing method in the point of material damage mechanism. What’s more, it avoids the limitation of conventional nanoindentation on the location of sampling points. At the same time, the requirement to input material’s real mechanical properties’ parameters of metallographic structure is realized in mesomechanical analysis. First, a matrix nanoindentation test of 5 × 6 points on a specimen is carried out. The stress–strain functions at the nanoindentation points on the surface of the specimens are calculated using nanoindentation data. Second, metallographic structures (ferrite, pearlite and grain boundary) at the nanoindentation points are confirmed by in situ follow-up images. Third, three computational representative volume element (RVE) models are reconstructed from three metallographic images of the specimen. Finally, the RVE models are used as the simulation object, and the stress–strain functions of ferrite, pearlite and grain boundary are taken as the input conditions. The mechanical properties of the RVE models under uniaxial tensile stress are simulated and analysed, and the obtained constitutive curves are used as macroscopic tensile stress–strain functions of AISI 1045 steel. The in situ tensile test results of the nanoindentated specimen show that the mechanical properties calculated by this method are consistent with the damage mechanism of the material.

In this article, theoretical analysis and experimental study on normalized AISI 1045 steel with uniform metallographic structure are carried out. According to the principle adopted in this research, it is suitable for metal materials with uniform metallographic structure, or metal materials that can be created RVE models.

Experiment and methods

Experimental equipment and materials

In this study, a self-made in situ nanoindentation-tensile testing device is used, as shown in Figure 1(a). The device has been calibrated by commercial products. Figure 1(a) also shows the loading curve and control program interface. A Vickers carbide head is selected as the indenter, which is driven by a piezoceramic, as shown in Figure 1(b). The elastic modulus of the cemented carbide indenter is 710 GPa. The indenter has no wear and the indentation area is calculated according to the ideal state. The horizontal distance between the centerline of the carbide head and the centre of the microscope lens of the device is fixed. After acquiring data to generate the nanoindentation curve of the specimen, the device moves the microscope based on the aforementioned distance, and metallographic images of the indentation position are then obtained.

Figure 1.

Experimental device and specimen: (a) testing device, (b) indenter selected and (c) shape and size of specimen.

AISI 1045 steel is a widely used carbon steel. To facilitate the reconstruction of computational RVE models, normalized AISI 1045 steel with a uniform metallographic structure is selected as the research object. The specific shape and size of specimen are shown in Figure 1(c). To conveniently observe the metallographic structure, the central 2mm area of the specimen is polished and corroded with 4% nitric acid hydrochloride. To avoid problems associated with tracking in the nanoindentation area, the other parts of the specimen are blackened.

Experimental procedure

The experimental procedure carried out in this study is divided into three parts, as shown in Figure 2: data acquisition for the generation of the nanoindentation curve and the calculation of the mechanical properties of the metallographic structure represent the first step; reconstructing the computational RVE models is the second step; and the finite element analysis of the computational RVE models is the third step.

Step 1. In this study, we first collect the nanoindentation load–displacement curves of 5 × 6 matrix points; the distance between the points is 10 µm, as shown in Figure 3(a). Due to the limitation of the microscopic magnification, the images of the residual indentation area are not particularly clear, as shown in Figure 3(a). Therefore, the nanoindentation points are marked in the figure. As shown in Figure 3(a), the area is composed of ferrite, pearlite and grain boundary.

Figure 2.

Flowchart of the experimental process.

Figure 3.

Distribution of nanoindentation points and P–h curves: (a) distribution of nanoindentation points and (b) load–depth curves.

To avoid the size effect, the nanoindentation displacement cannot exceed the thickness of a metallographic structure.^13,14 To avoid the influence of roughness effect, the depth of the compression cannot be too small.^15,16 Therefore, 20 mN is selected as the nanoindentation force, under which the metallographic structure performance should be fully reflected. The nanoindentation displacement is no greater than 600 nm.

The mechanical properties of different metallographic structures can be calculated by combining the corresponding nanoindentation curves as shown in Figure 3(b) and in situ images of the nanoindentation points. The nanoindentation curves of ferrite, pearlite and grain boundary in AISI 1045 steel are different.^2,17–19 The loading curve of nanoindentation determines the plasticity, hardening index and strength of the material. The unloading curve determines the elastic behaviour and elastic modulus of the material. The formula for the stress–strain relationship for plastic power-law hardening models commonly used to study metal materials is as follows⁴

\begin{matrix} σ = f (ε) = \\ {\begin{matrix} E ε & (σ \leq σ_{y}) \\ R ε^{n} = σ_{y} {(1 + \frac{E}{σ_{y}} ε_{p})}^{n} & (σ > σ_{y}) \end{matrix} \end{matrix}

(1)

In the formula, σ is the macroscopic stress of the material; ε is the macroscopic strain of the material; E is the elastic modulus of the material; R is the strength coefficient of the material; n is the strain hardening index; σ_y and ε_y are the yield stress and the yield strain of the material, respectively; and ε_p is the effective strain, which is larger than ε_y in the total strain, that is, the plastic strain. Therefore, the stress–strain function can be solved by solving for the parameters E,σ_y and n.

According to the dimensional function based on 76 types of metal materials established by Dao,⁴ reverse analysis of nanoindentation,²¹ and determination of residual indentation depth,²² the metallographic structure mechanical properties are computed as shown in Figure 4. H is the hardness of the material; h_max is the maximum pressure depth; h_c is the contact depth between the indenter and the material; P_max represents the maximum indentation load; α is the shape correlation coefficient of the indenter,²²α = 0.75; β is the shape correlation coefficient of the indenter, β = 1.05; A_c represents the residual area after unloading; E_r represents the reduced modulus; S =dP/dh|_h = hmax represents the elastic recovery slope after unloading, which can be obtained by the nanoindentation P–h curves;²³E_i represents the elastic modulus of the Vickers carbide head, E_i = 710 GPa; E represents the elastic modulus of the specimen; v represents Poisson’s ratio of the measured material, v = 0.25; v_i represents Poisson’s ratio of the Vickers carbide head, v_i = 0.21; and σ_r and ε_r are the characteristic stress and characteristic strain of the material, respectively, ε_r = 0.033.⁴

Figure 4.

Computational process of metallographic structure mechanical properties.

The formulas Π₁ and Π₂ mentioned in Figure 4 are as follows

\begin{matrix} Π_{1} = & - 1.131 {[\ln (\frac{E_{r}}{σ_{0.033}})]}^{3} + 13.635 {[\ln (\frac{E_{r}}{σ_{0.033}})]}^{2} \\ - 30.594 [\ln (\frac{E_{r}}{σ_{0.033}})] + 29.267 \end{matrix}

(2)

\begin{matrix} Π_{2} = (- 1.40557 n^{3} + 0.77526 n^{2} + 0.15830 n - 0.06831) {[\ln (\frac{E_{r}}{σ_{0.033}})]}^{3} \\ + (17.93006 n^{3} - 9.22091 n^{2} - 2.37733 n + 0.86259) {[\ln (\frac{E_{r}}{σ_{0.033}})]}^{2} \\ + (- 79.99715 n^{3} + 40.55620 n^{2} + 9.00157 n - 2.54543) [\ln (\frac{E_{r}}{σ_{0.033}})] \\ + (122.65069 n^{3} - 63.88418 n^{2} - 9.58936 n + 6.20045) \end{matrix}

(3)

The stress–strain function and mechanical parameters at the nanoindentation positions can be calculated according to formulas (1)–(3) and the flowchart in Figure 4. Combining the abovementioned formulas with the distribution of nanoindentation points and P–h curves in Figure 3, the mechanical parameters at each nanoindentation point can be calculated as shown in Table 1.

Table 1.

Mechanical parameters calculated.

Numbers of indentation point	Maximum depth, h_max (μm)	Error (%)	Hardness, H (GPa)	Error (%)	Hardening index, n	Error (%)	Elastic modulus, E (MPa)	Error (%)	Yield stress, σ_y (MPa)	Error (%)	Metallographic structure
5	0.5380	−0.0557	3.0774	−1.4001	0.3545	−2.7168	285.9534	14.3660	312.6968	11.1477	Ferrite
6^a	0.5619	4.3842	2.8586	−8.4105	0.3232	−11.3063	212.6841	−14.9378	423.322	50.4693
11	0.5554	3.1767	2.8769	−7.8242	0.3611	−0.9056	276.5531	10.6064	272.0033	−3.3167
17	0.5257	−2.3407	3.3185	6.3247	0.3735	2.4973	212.1545	−15.1496	265.1057	−5.7685
18	0.522	−3.0281	3.3853	8.4650	0.3768	3.4029	204.2282	−18.3197	260.294	−7.4788
21	0.5477	1.7462	3.0537	−2.1595	0.3628	−0.4391	193.3699	−22.6624	281.3961	0.0219
25	0.541	0.5016	3.0147	−3.4091	0.3575	−1.8935	327.942	31.1592	296.5106	5.3944
Mean value	0.5383		3.1211		0.3644		250.0335		281.3344
Standard deviation	0.0128		0.1930		0.0089		54.4187		20.0353
Coefficient of variation	0.0237		0.0618		0.0244		0.2176		0.0712
1^a	0.5643	12.0250	2.7384	−24.1314	0.3826	26.8568	366.6951	44.4669	198.5797	−69.9015	Grain boundary
2	0.5319	5.5966	3.2621	−9.6221	0.2848	−5.5703	193.5684	−23.7398	685.6030	3.9159
3	0.5157	2.3824	3.4845	−3.4604	0.2601	−13.7600	201.6416	−20.5592	886.3201	34.3384
4	0.5119	1.6280	3.5198	−2.4824	0.2695	−10.6432	214.8649	−15.3496	835.5959	26.6502
7	0.5075	0.7544	3.4934	−3.2138	0.3413	13.1631	293.0755	15.4630	406.3917	−38.4037
10	0.5087	0.9927	3.5811	−0.7841	0.3315	9.9138	208.6466	−17.7995	473.613	−28.2151
16	0.5036	−0.01993	3.5278	−2.2609	0.2922	−3.117	323.3964	27.4085	709.9631	7.6081
19	0.4991	−0.9132	3.6084	−0.0277	0.2936	−2.6525	310.9755	22.5151	708.0241	7.3143
22	0.4952	−1.6875	3.7314	3.3801	0.3335	10.5769	255.1254	0.5118	474.1475	−28.1341
23	0.4952	−1.6875	3.7768	4.6379	0.3196	5.9682	225.4902	−11.1636	562.0496	−14.8109
24	0.4886	−2.9978	3.7941	5.1172	0.3432	13.7931	297.8392	17.3398	425.7284	−35.4729
28	0.4833	−4.050	3.9235	8.7023	0.2488	−17.5066	267.4653	5.3734	1090.000	65.2099
Mean value	0.5037		3.6094		0.3016		253.8263		659.7669
Standard deviation	0.0136		0.1846		0.0339		47.4213		216.9074
Coefficient of variation	0.0270		0.0511		0.1125		0.1868		0.3288
8	0.5282	−1.1047	3.2841	3.6353	0.3005	−11.1473	211.7331	−17.1826	599.7298	47.5367	Pearlite
9	0.5276	−1.2170	3.2371	2.1522	0.3435	1.5671	257.6238	0.7671	370.3240	−8.8984
12	0.5404	1.1796	3.0664	−3.2346	0.3454	2.1289	261.0119	2.0923	348.3750	−14.2979
13	0.5257	−1.5727	3.3658	6.21356	0.3169	−6.2980	185.5652	−27.4180	523.1556	28.6990
14	0.5239	−1.9098	3.2724	3.2662	0.3262	−3.5482	274.4154	7.3349	461.0485	13.4203
15	0.5207	−2.5089	3.4349	8.3941	0.3095	−8.4861	188.0041	−26.4640	572.0411	40.7251
20	0.5343	0.0374	3.1567	−0.3850	0.3122	−7.6878	247.6146	−3.1479	520.4076	28.0230
26	0.5404	1.1796	3.0376	−4.1434	0.3649	7.8947	300.4013	17.4991	269.8773	−33.6088
27	0.5343	0.0374	3.1786	0.3061	0.3398	0.4731	227.7708	−10.9096	383.8443	−5.5723
29	0.5525	3.4450	2.8877	−8.8737	0.3873	14.5180	315.5142	23.4103	190.6677	−53.0947
30	0.5471	2.4340	2.9362	−7.3432	0.3745	10.7333	342.6349	34.0183	231.9781	−42.9322
Mean value	0.5341		3.1689		0.3382		255.6627		406.4954
Standard deviation	0.0101		0.1730		0.0284		50.7897		139.7170
Coefficient of variation	0.0189		0.0546		0.0839		0.1987		0.3437

All errors are computed as (data–mean value)/mean value × 100%, where data represents a variable: h_max, H, n, E and σ_y.

Since h_max is a direct measurement, two sets of data – groups 1 and 6 – with large h_max error are excluded in calculating the mean value.

Table 1 shows that the hardness of different metallographic structures vary greatly, while the elastic modulus of different metallographic structures vary slightly. The hardness of pearlite is higher than that of ferrite, and the hardness of the grain boundary is higher than that of pearlite. The specific reasons are as follow: the hardness of ferrite mainly depends on its alloy element content and crystal defects.²⁴ Pearlite is mainly composed of lamellar cementite and ferrite. The hardness of pearlite depends on the proportion of ferrite and cementite and on the extent of lamellar spacing. The elastic modulus is an inherent property of a material; therefore, the elastic modulus of materials with the same chemical composition and undergoing the same process are essentially the same.²⁵

h_max is the data collected directly. Because the errors of h_max in group 1 and group 6 are larger, the two sets of data are excluded. According to the statistical Q-Q chart of Parameter Distribution (Figure 5) and Shapiro–Wilk (SW) test chart of normal distribution (Table 2), the obtained data of ferrite, pearlite and grain boundary conform to normal distribution. Therefore, the data can effectively characterize the properties of the corresponding metallographic structure.

Figure 5.

Statistical Q-Q chart of parameter distribution.

Table 2.

SW test chart of normal distribution.

Shapiro–Wilk		Statistic data	Sample size	Significance
Ferrite	h_max	0.959	6	0.812
	H	0.922	6	0.518
	n	0.911	6	0.440
	E	0.896	6	0.348
	σ_y	0.936	6	0.626
Grain boundary	h_max	0.973	11	0.917
	H	0.963	11	0.813
	n	0.918	11	0.306
	E	0.912	11	0.261
	σ_y	0.93	11	0.406
Pearlite	h_max	0.949	11	0.628
	H	0.973	11	0.915
	n	0.949	11	0.630
	E	0.966	11	0.848
	σ_y	0.947	11	0.611

SW: Shapiro–Wilk.

At the same time, the experimental results show that the mechanical parameters of the same metallographic structure are different. This finding demonstrates that under the same material conditions, if the grain shape varies, due to the different paths of force transfer, the material can still have different macroscopic mechanical properties. Therefore, the macroscopic properties are related not only to the properties of the microstructure but also to the size and orientation of the microphase. Previous studies have shown that the cross-sectional shape and hardness of grains are different, even when cross-sectional areas of grains are equal and the depths of nanoindentation are the same.²⁶

The mechanical parameters of grain boundary are partially similar to those of ferrite and pearlite because grain boundary are composed of adjacent metallographic structures. In this article, the average values of the mechanical parameters obtained at the indentation points on grain boundary are selected as the mechanical properties of grain boundary. The average values of the mechanical parameters of ferrite, pearlite and grain boundary listed in Table 3 are used as the material properties for simulation analysis. The average value of the last row in the table is the average value of the mechanical parameters corresponding to the three metallographic structures.

Step 2. The macroscopic mechanical properties of materials are determined by the materials’ microstructure, and the constitutive relations of the metallographic structures and grain boundary are different. Thus, previous scholars have calculated the macroscopic mechanical properties of materials by calculating the volume fraction of the metallographic composition. In Table 1, the grain boundary of the same metallographic structure has little difference from the mechanical properties of the metallographic structure. So, this method does not consider the effect of grain boundary distribution on the mechanical properties of materials. Therefore, it is necessary to transform the metallographic images into a file format that the simulation software can recognize (reconstructed RVE models) and calculate the tensile mechanical properties of the RVE models. When the RVE size reaches 24 µm, the required accuracy of the model can be achieved.^27,28 In this article, square metallographic sections measuring 60, 100 and 200 μm are selected.

Table 3.

Mean values of mechanical properties of metallographic structures.

Metallographic structure	Mesomechanical parameters		Macro-mechanical parameters
	Maximum depth, h_max (μm)	Hardness, H (GPa)	Hardening index, n	Elastic modulus, E (MPa)	Yield stress, σ_y (MPa)
Ferrite	0.5419	3.0836	0.3585	244.4979	301.6184
Pearlite	0.5341	3.1689	0.3382	255.6627	406.4954
Grain boundary	0.4977	3.5368	0.3084	263.2320	621.3347
Average	0.5246	3.2631	0.3350	254.4642	443.1495

RVE reconstruction requires image binarization to ensure that metallographic images are in black and white. To improve the smoothness of the edges between ferrite and pearlite, the edge eclosion of the metallographic structures is needed. The eclosion areas are regarded as the grain boundary region of the metallographic structure. The binarization images are extracted and saved as DXF format files so that they can be opened in Creo software. The accuracy of the DXF files extracted after eclosion is not able to meet the requirements of the ABAQUS software. The DXF file must be remodelled in the Creo software. The images of the metallographic structure before and after the reconfiguration are shown in Figure 6. According to the nanoindentation data in Table 1, the grain boundary mechanical properties of the same metallographic structure have little change. So, the RVE models of the reconstruction ignore the grain boundary of the same metallographic structure.

Step 3. In this study, the ABAQUS software is used for the tensile simulation of the RVE models.

Figure 6.

Metallographic structure and reconstructed RVE models: (a) 60 µm × 100 µm; (b) 100 µm × 100 µm; and (c) 200 µm × 200 µm.

Results and discussion

The simulation stress cloud in Figure 7 shows that the nanoindentation hardness of the grain boundary is higher than that of the grain interior. The significant difference in the mesomechanical properties between the grain boundary and grain interior is the main reason that the macro-mechanical properties of the samples are affected by grain size.²⁹

Figure 7.

Von Mises stress cloud.

When the RVE model is loaded with displacement, the ferrite structure, which exhibits a lower yield strength, enters the plastic stage first, and the toughness of pearlite prevents the rapid plastic deformation of ferrite. The distribution of grain boundary increases the bearing capacity of ferrite grains. Deformation mainly occurs in ferrite. Therefore, the macro-mechanical properties of materials are related not only to the mechanical properties of their microstructures but also to the grain orientation of microphases and the distribution of grain boundary.

The logarithm of the plastic part in formula (1) is as follows^14,30

\ln σ = \ln R + n \ln ε

(4)

Therefore, the hardening index, n can be calculated by formula (5)

n = \frac{\ln σ_{2} - \ln σ_{1}}{\ln ε_{2} - \ln ε_{1}}

(5)

Due to the extension of the material length direction and the shrinkage of the material in the width direction according to the principle of volume invariance during tension, the stress in the middle area of the stress nephogram of the simulation results is clearly larger than that in the surrounding area.

In the same displacement region, it is clear that the stress at the grain boundary is larger, which affects the macroscopic strength of the material. This finding confirms the strengthening effect of fine grains. For metal materials, there is the same trend in hardness and strength. That is to say, in the process of tensile test, when the same displacement is stretched, the more stress the harder structure bears. The simulated true stress–strain curve is shown in Figure 8. Because the specimens are homogeneous metallographic materials, the results of the three RVE models are essentially the same. The mean values of the mechanical parameters are E = 253.244 GPa, n = 0.3507, and σ_Rp0.2 = 406 MPa.

Figure 8.

True stress–strain curves of FEA.

The in situ image of the plastic stage in Figure 8 shows that the grain orientations on the two sides of the grain boundary are different, and the slip direction and the slip surface are not consistent with each other. It is just the small cracks in different directions that reduce the plasticity of materials. Macroscopically, the strength and hardness of the material are decreased.

The elastic modulus E = 235 GPa, hardening index n = 0.327, and yield strength σ_y = 385 MPa are calculated from the tensile data. The ferrite composition shows good slip deformation ability and deformation compatibility under a tensile load. Therefore, the deformation of ferrite plays a leading role in the macro-deformation of the materials. Moreover, the hardening index n of the material is close to that of ferrite. The strain hardening index n is the embodiment of the properties of the materials in the plastic stage, which is mainly affected by the order in which each metallographic structure enters the plastic stage. Because dislocations that destroy ferrite do not easily cross grain boundary, grain boundary increases the strength of the whole material. The results of the tension experiment are consistent with those of the simulation. The reason why value of n calculated by FEA is a little larger than that of the experiment is caused by the yield curve of the material, whereas the hardening index decreases due to the defect of the specimen and the dislocation motion between the slip surfaces during the test as shown in Figure 9.

Figure 9.

In situ images and true stress–strain curves of the tensile test.

Due to the different heat treatment methods, the mechanical properties of different batches of materials are different. Therefore, the in situ tensile data of the nanoindented specimens are used as a reference. The result of tensile test is n = 0.3270, E = 235.9211 GPa, and σ_y = 385.3821 MPa. The differences between simulation results and tensile test data of each group are shown in Table 4.

Table 4.

Differences of mechanical properties calculated results.

Group	Macro-mechanical parameters
	Hardening exponent, n	Differences (%)	Elastic modulus, E (GPa)	Differences (%)	Yield stress, σ_y (MPa)	Differences (%)
RVE model 1	0.3483	6.5138	251.2913	6.5150	410.6282	6.5510
RVE model 2	0.3524	7.7676	254.8184	8.0100	420.3476	9.0730
RVE model 3	0.3514	7.4618	253.6232	7.5034	398.6293	3.4374
Means of 28 sampling points (conventional nanoindentation calculation method)	0.3295	0.7645	253.7455	7.5553	479.1751	24.3377

RVE: representative volume element.

The reasons for the significant difference between simulation results and experimental results are as follows: one is the non-uniqueness of indentation test identification results^31–33 and the other is the effect of small cracks on material surface during tensile test. The results of elastic modulus are the same as those of the traditional nanoindentation method, because the elastic modulus of the material is determined by the chemical composition of the materials and has nothing to do with the metallographic structure. The deviation between the elastic modulus and the tensile test results is within the allowable deviation of the elastic modulus. However, the average data of yield stress or strain hardening index of nanoindentation point cannot be used as macroscopic mechanical properties.³⁴ The yield stress is determined by the structure of the material,³⁵ and the maximum RVE model is the closest to the tensile test results. Because of the small cracks appearing in the tensile process, the material’s resistance to plastic deformation decreases and the strain hardening index decreases, so the strain hardening index differs greatly from the conclusion of the tensile test.

Conclusion

In this study, in situ nanoindentation and mesomechanical analysis are used to establish the relationship between macro-mechanical properties and mesomechanical properties. The study provides a new testing and simulation analysis method based on micromechanics for the tensile properties of materials in service.

The mesomechanical properties of grain boundary have a great influence on the hardness and strength of specimens, but have little influence on the elastic modulus. In situ tensile images can prove the strengthening effect of grain boundary on the plastic deformation of materials and confirm that it is necessary to establish the grain boundary mechanical function as an input condition for simulation.

In this article, an experimental study of materials with a homogeneous structure is conducted. For highly heterogeneous materials, the mechanical properties can also be analysed by this method as long as artificial RVE generation approaches.^36–39

Macroscopic properties are related not only to the properties of the microstructure but also to the size and orientation of the microphase.

Footnotes

Handling Editor: Rubén Lostado Lorza

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by Special Projects for the National Natural Science Funds for Excellent Young Scholar (grant no. 51422503), the National Natural Science Foundation of China (grant no. 51275198), Fund Guiding on Strategic Adjustment of Jilin Provincial Economic Structure Project (grant no. 2014Z045), Major Project of Jilin Province Science and Technology Development Plan (20150203014GX), Jilin Provincial Industrial Innovation Special Fund Project (2016C030), and Jilin Provincial Middle and Young Scientific and Technological Innovation Talent and Team Project (20170519001JH).

ORCID iDs

Cong Li

Hongwei Zhao

References

London

Baker

et al . A robust inverse analysis method to estimate the local tensile properties of heterogeneous materials from nano-indentation data. Int J Mech Sci 2017; 123: 162–176.

Choi

Wung

Kwon

Analysis of mechanical property distribution in multiphase ultra-fine-grained steels by nanoindentation. Scripta Mater 2001; 45: 1401–1406.

Gautam

Mousumi

Sinha

et al . Characterization of cast stainless steel weld pools by using ball indentation technique. Mat Sci Eng A-Struct 2009; 513–514: 389–393.

Dao

Chollacoop

Van Vliet

et al . Computational modeling of the forward and reverse problems in instrumented sharp indentation. Acta Mater 2001; 49: 3899–3918.

Antunes

Fernandes

Menezes

et al . A new approach for reverse analyses in depth-sensing indentation using numerical simulation. Acta Mater 2007; 55: 69–81.

Cheng

Zhang

Ruimi

et al . Quantifying the effects of tempering on individual phase properties of DP980 steel with nanoindentation. Mat Sci Eng A-Struct 2016; 667: 240–249.

Massimiliano

Vladimir

An inverse analysis approach based on a POD direct model for the mechanical characterization of metallic materials. Comput Mater Sci 2014; 95: 302–308.

Meng

Piotr

Balaji

Identification of material properties using indentation test and shape manifold learning approach. Comput Method Appl M 2015; 297: 239–257.

Sudharshan

PO.

A direct comparison of high temperature nanoindentation creep and uniaxial creep measurements for commercial purity aluminum. Acta Mater 2016; 111: 31–38.

10.

Ishikawa

Kyono

Hitoshi

et al . Microscopic deformation and strain hardening analysis of ferrite-bainite dual-phase steels using micro-grid method. Acta Mater 2015; 97: 257–268.

11.

Ehsan

Alberto

David

et al . Grain size and texture dependence on mechanical properties, asymmetric behavior and low temperature superplasticity of Zk60mg alloy. Mater Character 2015; 107: 70–78.

12.

Yeol

Choo

Kwon

Analysis of mechanical property distribution in multiphase ultra-fine-grained steels by nanoindentation. Scripta Mater 2001; 45: 1401–1406.

13.

Diez-Pascual

Gomez-Fatou

Ania

Nanoindentation in polymer nanocomposites. Prog Mater Sci 2015; 67: 1–94.

14.

Silva

Pinto

Kuznetsov

Precipitation and grain size effects on the tensile strain-hardening exponents of an API X80 steel pipe after high-frequency hot-induction bending. Metals 2018; 8: 168.

15.

Chen

Ahadi

Zhou

Quantitative study of roughness effect in nanoindentation on AISI316L based on simulation and experiment. Proc IMechE, Part C: J Mechanical Engineering Science 2017; 231: 4067–4075.

16.

Tianhan

Yaorong

Nano-crystalline Ni-Ti alloy thin films fabricated using magnetron co-sputtering from elemental targets: effect of substrate conditions. Mat Sci Eng A-Struct 2016; 616: 733–745.

17.

Xiong

Liu

et al . Role of melt pool boundary condition in determining the mechanical properties of selective laser melting Alsi10mg alloy. Mat Sci Eng A-Struct 2019; 740: 148–156.

18.

et al . Effect of grain boundary characteristic on intergranular corrosion and mechanical properties of severely sheared Al-Zn-Mg-Cu alloy. Mat Sci Eng A-Struct 2018; 732: 53–62.

19.

Argandona

Palacio

Berlanga

et al . Effect of the temperature in the mechanical properties of austenite, ferrite and sigma phases of duplex stainless steels using hardness, microhardness and nanoindentation techniques. Metals 2017; 7: 219.

20.

Lee

Kim

Reverse analysis of nano-indentation using different representative strains and residual indentation profiles. Mater Design 2009; 30: 3395–3404.

21.

Huang

Hongwei

Determination of residual indentation depth h(f) in incomplete or irregular unloading curves. Meas Sci Technol 2014; 25: 087003.

22.

Sneddon

IN.

The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile. Int J Eng Sci 1965; 3: 47–57.

23.

Akatsu

Numata

Shinoda

Effect of the elastic deformation of a point sharp indenter on nanoindentation behavior. Materials 2017; 10: 270.

24.

Pandey

Mahapatra

Kumar

et al . Dissimilar joining of CSEF steels using autogenous tungsten-inert gas welding and gas tungsten arc welding and their effect on delta-ferrite evolution and mechanical properties. J Manuf Proces 2017; 31: 247–259.

25.

Savrai

Makarov

Schastlivtsev

VM.

Elastic modulus of pearlitic steel and its change during cyclic loading. Deform Razrush Mater 2010; 7: 15–19.

26.

Manika

Maniks

Size effects in micro-and nanoscale indentation. Acta Mater 2006; 54: 2049–2056.

27.

Zabihyan

Mergheim

Javili

Aspects of computational homogenization in magneto-mechanics: boundary conditions, RVE size and microstructure composition. Int J Solid Struct 2018; 130: 105–121.

28.

Ramazani

Mukherjee

Prahl

Modelling the effect of microstructural banding on the flow curve behaviour of dual-phase (DP) steels. Comput Mater Sci 2011; 52: 46–54.

29.

Heritier

Guillot

Dillmann

Microstructural characterization and mechanical properties of iron reinforcements in buildings from the medieval and modern periods in France. Int J Archit Herit 2019; 13: 507–513.

30.

Cao

SZ.

Semiempirical formulae for mechanical properties controlled by strength and ductility of power-law hardening metallic materials. J Mater Eng Perform 2018; 27: 6149–6154.

31.

Alkorta

Martínez

José

et al . On the elastic effects in power-law indentation creep with sharp conical indenters. J Mater Res 2005; 23: 182–188.

32.

Chen

Ogasawara

Zhao

On the uniqueness of measuring elastoplastic properties from indentation: the indistinguishable mystical materials. J Mech Phys Solid 2007; 55: 1618–1660.

33.

Meng

Breitkopf

QG.

An insight into the identifiability of material properties by instrumented indentation test using manifold approach based on P-h curve and imprint shape. Int J Solid Struct 2017; 106: 13–26.

34.

Tanaka

Takaki

Tsuchiyama

Effect of grain size on the yield stress of cold worked iron. ISIJ Int 2018; 58: 1927–1933.

35.

Geng

Liu

Tong

et al . Experimental report on the nano-indentation testing of textured stainless steel 904L and 316L. Proc Eng 2015; 99: 1268–1274.

36.

Fillafer

Krempaszky

Werner

On strain partitioning and micro-damage behavior of dual-phase steels. Mat Sci Eng A-Struct 2014; 614: 180–192.

37.

Hou

Cai

Sapanathan

et al . Micromechanical modeling of the effect of phase distribution topology on the plastic behavior of dual-phase steels. Comput Mater Sci 2019; 158: 243–254.

38.

Hou

Sapanathan

Dumon

et al . A novel artificial dual-phase microstructure generator based on topology optimization. Comput Mater Sci 2016; 123: 188–200.

39.

Madej

Adam

Mateusz

Experimental and numerical two- and three-dimensional investigation of porosity morphology of the sintered metallic material. Arch Civ Mech Eng 2018; 18: 1520–1534.