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New procedure to determine steel mechanical parameters from the spherical indentation technique
Mechanics of Materials 34 (2002) 243–254 www.elsevier.com/locate/mechmat
New procedure to determine steel mechanical parameters from the spherical indentation technique A. Nayebi, R. El Abdi *, O. Bartier, G. Mauvoisin Laboratoire de Recherche en M ecanique Appliqu ee de l’Universit e de Rennes, I.U.T. de Rennes. 3, rue du Clos Courtel – B.P. 90422, 35704 Rennes cedex 7, France Received 2 April 2001; received in revised form 2 January 2002
Abstract A new numerical and experimental approach for determining mechanical properties of steels is presented. This method uses the instrumented spherical-indentation technique. A relationship between applied load, indenter displacement, flow stress and strain hardening exponent of steels, is given. This method is based on the minimisation of error between the experimental curve (applied load–indenter displacement curve) and the theoretical curve which is a function of the mechanical properties of studied materials. Comparison between the results obtained by the proposed method and experimental tensile tests, confirms the interest of the proposed method. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Indentation method; Finite-element modelling; Hardness; Strain hardening exponent; Yield stress
1. Introduction Indentation experiments have been performed for nearly one hundred years for measuring the hardness of materials. Long time ago, as early as the 19th century, the work of Hertz on rubbers appeared. Indentation of rubber can be interpreted in terms of elastic parameters using Hertz’s theory (1882) for a spherical indenter, or using Sneddon’s equations for a conical indenter (Sneddon, 1965). For metals in a plastic state, Tabor (1951) was the first to point out the relation between Vickers
*
Corresponding author. Fax: +33-223-234101. E-mail address: [email protected] (R. El Abdi).
(HV) or Brinell (HB) hardness and yield stress ry , namely HV 3ry and HB 2:8ry :
ð1Þ
This experimental observation was backed up by theoretical analyses of plastic flow, based on the Slip Line Fields method for spherical indenters (Ishlinsky, 1944), for a cone (Lockett, 1963) and the generalisation to the Vickers pyramid by Haddow and Johnson (1961). These studies confirmed the value of HV 3ry for blunt indenters, and also derived solutions for sharper indenters. Purely plastic analyses proved insufficient for hard metals where elasticity cannot be neglected. Following observations by Marsh (1964), an analogy with the classical problem of the growth
0167-6636/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 6 3 6 ( 0 2 ) 0 0 1 1 3 - 8
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Nomenclature D a d dr dmax r ry m E
diameter of ball contact radius indenter displacement residual depth of penetration maximum penetration depth stress yield stress Poisson ratio Young’s modulus
of a cavity in an infinite elastoplastic medium was used to determine the relationship between the hardness number (or average indentation pressure) and parameters mixing geometry with elastic and plastic properties of the indented material (Johnson, 1970; Edlinger et al., 1993; Giannakopoulos and Suresh, 1999; Venkatesh et al., 2000). The traditional method of determining the yield stress and the strain hardening exponent of a steel is the tensile test (or compression test) which requires the machining of samples which are, most of the time, cylindrical. On the other hand, the spherical-indentation method can be used for determining material stress–strain behaviour in a non-destructive and localised fashion. Moreover, it does not need precise sample machining (Field and Swain, 1995). The standard indentation measurement consists of pushing an indenter into the sample under precise load control. The indenter is inserted to some specified depth and then extracted, giving the loading force over the whole depth range. Recent years have seen increasing interest in indentation because of the significant improvement in the indentation equipment and the need for measuring the mechanical properties of materials on small scales. From the load–displacement curve of an indenter during indentation, the hardness and Young’s modulus (for example) may be obtained from the peak load and the initial slope of the unloading curves. The initial rate of change following the decrease in force (i.e., the slope) is defined as the sample stiffness.
n epl eel Pav F Fmax S HV q; h; z
strain hardening exponent plastic strain elastic strain average contact pressure applied load maximum applied load projected contact area Vickers Hardness cylindrical co-ordinates
Mechanical analysis requires formulation of the sharp indentation problem which is inevitably three-dimensional and at first glance seems quite impossible to solve, mainly because of material and geometric non-linearities, various dissipative mechanisms such as plasticity, phase transformation, micro-cracking and the presence of residual stresses. The properties of the indenter, such as its sharpness, are also very important to the analysis. The combination of all these effects makes the problem difficult to address analytically, and for this reason it was left unsolved in the past. Today’s computational capacity is, however, sufficient for a detailed investigation of the important features of sharp indentation tests. Concerning the problems for which the indenter has an axisymmetric form (cones, spheres, etc.), a model using two dimensional finite elements (2D) often remains sufficient. The problem of stress–strain curve determination from indentation tests has been studied in the past. The use of the stress–strain curve must lead to information about mechanical characteristic parameters. Tabor (1951) has shown a direct relation between indentation diameter and average applied strain, average pressure and corresponding stress. Field and Swain (1995) have used micro-spherical indentation for determining a small volume of material mechanical properties. They have used a stepwise indentation with a partial unloading technique for separating elastic and plastic components of indentation at each step. Later, Jayaraman et al. (1998), using different angle conical
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indenters, gave a method for determining the nonlinear part of stress–strain curve. Taljat et al. (1998) used spherical indentation for determining in another way the stress–strain curve. They have determined the relations for maximum and minimum applied strain and relating stresses, function of contact diameter, indenter diameter and hardening exponent, but they have not given an exact method for determining the hardening exponent. Huber and Tsakmakis (1999a,b) have used a neural network and cyclic indentation for determining the stress–strain curve. Bouzakis and Vidakis (1999) have given a new procedure which uses finite element simulation of successive ball indentation with increasing load. These methods are usually difficult and time consuming, for example Huber’s method needs cyclic loading and Bouzakis’ method needs a finite element simulation and measurement of imprint profile for every stress–strain curve point. Giannakopoulos and Suresh (1999) have used two parts of the loading and unloading Vickers indentation curve for yield stress and hardening exponent determination. They determine yield stress and stress at 0.29 plastic strain so the hardening exponent can be determined from these values. But in another paper (Venkatesh et al., 2000), the authors specified that their approach cannot distinguish two different materials which have the same yield stress ry and the same r0:29 stress obtained for a plastic strain equal to 0.29. The use of a Vickers indenter leads to the load–displacement F ¼ ad2 relation which is based on the determination of only one parameter a and cannot lead to the uniqueness. This was shown by Cheng and Cheng (1999). The aim of this work is to present a new procedure which leads to determining the yield stress and the strain hardening exponent of steels using an instrumented indentation technique and a theoretical curve. The experimental indentation test leads to obtaining the load–displacement curve. The theoretical curve is a relationship between applied load, indenter displacement, yield stress ry and the strain hardening exponent n of steels, which was determined by numerical simulations. The procedure is based on the error minimisation between the experimental (applied load–ind-
245
enter displacement curve) and the theoretical curves. This optimisation leads to having ry and n. Several cylindrical samples from different steel types, are submitted to a tensile test and the experimental values were compared with those numerically obtained from the proposed method. Finally, we give a method which leads to obtaining the Vickers hardness from knowing ry and n. This hardness value is compared to the one obtained from the classical Vickers indentation test.
2. Experimental procedure 2.1. Tensile test procedure The tensile test has been used to have experimental comparison values. Tensile tests have been carried out in a Lloyd LR30K machine. The advance speed pertinent to the tensile tests is 2 mm/ min. Cylindrical specimens 6mm in median diameter, having an effective length of 24 mm, were used as tensile specimens (Fig. 1). Because of large strains, an extensometer was used in the middle of the sample. A traditional system of acquisition makes it possible to record the tensile curve. Fig. 1 gives the 100C6 steel stress–strain curve. This tensile curve is fitted to Eq. (2), and the best values of ry and n so obtained were listed in Table 1. Eq. (2) represents the strain–stress curve of a metal: n
r ¼ Ken ¼ en ry =ðry =EÞ ;
ð2Þ
where e is the total strain (e ¼ epl þ eel ), and n is the strain hardening exponent. n ¼ 0 indicates a perfectly plastic material.
Fig. 1. Tensile curve for the 100C6 steel.
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Table 1 Comparison between proposed method and tensile test results Steels
ry , obtained from tensile test
ry , obtained from indentation proposed method
n, obtained from tensile test
n, obtained from indentation proposed method
Measured hardness HV30
Calculated hardness HV30
16NC6 100C6 35CD4 35NC15 Z38CDV5 Z15CN17-03 Z2CN18-10 42CD4 XC18 XC38 XC48 XC65 XC80
345 430 473 790 390 804 200 515 350 576 419 345 303
423 422 457 748 387 811 276 479 325 492 429 384 308
0.17 0.14 0.09 0.08 0.17 0.10 0.21 0.10 0.10 0.11 0.16 0.24 0.17
0.15 0.15 0.10 0.09 0.19 0.11 0.18 0.13 0.12 0.13 0.15 0.22 0.17
206 206 194 287 223 304 155 214 149 232 205 232 167
202 216 195 290 223 321 155 215 155 241 227 287 178
The tested materials used in the present investigation cover a wide range with respect to ry (200–800 MPa) and n (0.07–0.25). 2.2. Indentation procedure The relationship between applied load F and the characteristic contact displacement d represents an important material characteristic in indentation with a sphere of radius R. Spherical indenters are widely studied experimentally, because of their relative simple geometry, and theoretically, because of their facility to provide essential information on both elastic and plastic deformation properties of the tested material. A brief description of the experimental procedures is given in this section. The schematic set-up of the test apparatus is shown in Fig. 2. This indentation device is used to measure the F–d curve
during indentation. It was obtained with a load cell, displacement sensor and a spherical indenter (with a standardised sphere diameter D ¼ 1:587 mm). Indentation depth d was monitored using a displacement sensor mounted in the base which sensed the displacement of a steel arm mounted in a very stiff load train immediately above the indentation tip. The potential of the indentation system to allow accurate measurement is illustrated by the following parameters: displacement resolution 1 lm and force resolution 1 N. Load and displacement are simultaneously recorded for a loading curve. After three indentations, the curves have been averaged after zero point determination. As a numerical simulation procedure, maximum indenter displacement was held about 100 lm. The samples have been cut into cylindrical pieces of 10 mm depth from the same bars of traction samples. 3. Proposed model The aim of this section is to determine the flowstress and the strain hardening exponent starting from an instrumented test of indentation. For a homogeneous material, we seek to express the displacement d of the indenter according to applied load F. The selected function can be written as follows:
Fig. 2. Schematic set-up of the indentation test.
d ¼ Aðry ; nÞF Bðry ;nÞ :
ð3Þ
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Functions A and B depend on the yield stress ry and the strain hardening exponent n, and are deduced from the analysis of the results obtained by finite elements simulations on a series of materials. Simulation of the force–displacement relation for the indentation of a half-space using a spherical indenter with a high elastic modulus, was performed using the axisymmetric and large strain elastoplastic feature of the Castem 2000 finite element code (Millard, 2000), with uniaxial stress– strain data as input. Materials with various yield stresses and hardening components were studied. The quasi-static nature of the process allows using the static analysis performed by the programme. Underlying the approach in this code is the discretization of the continuum involved. Also, an important feature of this programme involves the ability to model contact between the indenter and the sample as a sliding interface. Initially, there is contact at only one point between the indenter and the sample. Progressively as the indenter progresses, the programme automatically establishes the junction between the contact surface nodes of the sample surface and those of the indenter. An elastic sphere indenting an elastoplastic steel for an axisymmetric half-space under normal contact was considered and modelled using a finite-element model in which the boundary conditions are given in Fig. 3; the boundary conditions may be expressed as follows: rz ðq; 0Þ ¼ 0
for jqj > a;
ð4Þ
sqh ðq; 0Þ ¼ 0
for jqj > a;
ð5Þ
where a is the contact radius. Far from the indentation zone, the displacements are cancelled: u ¼ v ¼ 0 as q; z ! 1:
ð6Þ
A finite element mesh of the half-space is given in Fig. 4. The programme was used with 6316 elements and 5986 nodes. A total of at least 30 elements were allowed to come in contact with the substrate in order to provide sufficient resolution in the computation of the field around the indenter. Other smoothness meshes were tested. The conclusion was that the results were identical.
247
Fig. 3. Geometry and boundary conditions.
Table 2 gives several values of ry and n which represent various studied materials. For reducing numbers of simulations and parameters, we have used an indenter with a constant diameter as D ¼ 1:5875 mm and Young’s modulus as E ¼ 650 GPa. All the other steel parameters (Young’s modulus E ¼ 210 GPa, Poisson ratio m ¼ 0:3) remain constant. The friction coefficient at the indenter–material contact was varied from 0.0 to 1.0. Various friction coefficients were used, and their influence on the F–d curve is negligible. Thus, for the studied materials, it will be considered constant and equal to 0.2 (Taljat et al., 1998). For all homogeneous steels cited in Table 1, functions A and B have been determined as B ¼ 1= ð 0:151ry þ 0:609Þn þ 0:09ry þ 0:975 ; h i B 2:9nr 0:323 y : A ¼ ð 3294 þ 22170r0:8 y Þe ð7Þ Yield stress ry is in GPa. For 35CD4 steel, Fig. 5 gives the load–displacement curve obtained from experimental testing and the proposed theoretical curve (3). Using
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Fig. 4. Finite-element meshing of half-space and zoom of the surface contact.
Table 2 Series of tested materials ry (MPa)
n
n
n
n
n
400 600 800 1000 1200
0.0 0.0 0.0 0.0 0.0
0.1 0.1 0.1 0.1 0.1
0.2 0.2 0.2 0.2 0.2
0.3 0.3 0.3 0.3 0.3
0.4 0.4 0.4 0.4 0.4
n ¼ 0:10 whereas the values obtained from the experimental tensile test are ry ¼ 473 MPa and n ¼ 0:09. For several steels, ry and n values obtained from the proposed method and tensile test are listed in Table 1. The two methods lead to close results.
4. Hardness–flow stress relation From ry and n values, we can have the hardness values. A method for the determination of the hardness value is proposed. 4.1. Average contact pressure Pav
Fig. 5. Load–displacement curve for the 35CD4 steel.
an optimisation procedure, we minimise the error between the proposed theoretical curve and the experimental indentation curve (F–d). This optimisation yields parameters ry and n which make error minimisation. For 35CD4 steel, the error minimisation leads to ry ¼ 457 MPa and to
The spherical cavity expansion model expresses the hardness value (average contact pressure) to elastic modulus ratio (Pav =E) as a function of the flow stress to elastic modulus ratio ðry =EÞ for very hard or elastic materials. Thus, the linear hardness–flow stress correlation obtained from the Slip-Line Field theory can be rewritten in terms of Pav =E and ry =E. Due to the 1=E dependence in these relations, the elastic modulus was used as the normalising parameter. As a first step in developing the hardness–flow stress correlation using the elastic modulus as the normalising parameter, the dependence of equivalent plastic strain epl on
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the equivalent flow stress ry was given by the simple Ramberg–Osgood equation: N
epl ¼ Lðry =EÞ :
ð8Þ
Any definition of the characteristic plastic strain should be such that it is (i) dependent on the geometry of the self-similar indenter, and (ii) independent of the depth of penetration (Jayaraman et al., 1998). For a given material, an experimentally determinable parameter that can satisfy both these conditions is average pressure Pav . If the characteristic plastic strain epl is related to average pressure Pav by a function similar to Eq. (8), we have N0
epl ¼ L0 ðPav =EÞ :
ð9Þ
Such a definition identifies the material being indented in terms of the new parameter L0 and N 0 , while satisfying conditions (i) and (ii) mentioned above. Thus, for a given indenter geometry, Eqs. (8) and (9) give a hardness–flow stress relation of the form: G
Pav =E ¼ Cðry =EÞ ;
ð10Þ
where 0
C ¼ ðL=L0 Þ1=N and G ¼ N =N 0 :
ð11Þ
This theory connects Pav and ry through two constants C and G which depend on material properties and which are difficult to determine. This relation gives only the average contact pressure Pav and not the Vickers hardness. The following paragraph presents a method for obtaining Vickers hardness from ry and n. 4.2. Vickers hardness determination from mechanical properties Fig. 6 is a schematic of the load–penetration depth (F–d) curve for a sharp indenter. During loading, the curve generally follows the power relation F ¼ W dm , where W and m depend on the mechanical material parameters and on the indenter geometry. The stress–strain curve of a metal can be characterised by a Hollomon-type Eq. (2).
Fig. 6. A schematic representation of load versus indenter displacement data for an indentation experiment.
The average contact pressure Pav is defined as the ratio of the applied load F divided by the projected contact area S. Finite-element simulation of elastoplastic indentation along with experiments (e.g., Giannakopoulos et al., 1994) provides the following results: Pav 0:27 E tanð22°Þ þ r 1 þ ln ¼ r ; 0:29 y E 3ry E ð12Þ Fmax E ¼ 7:143 r 1 þ ln þ r ; 0:29 y 2 ry dmax
ð13Þ
where 22° is the actual indentation angle of the Vickers indenter, Fmax is the maximum value of the applied load F and dmax is the maximum penetration depth. E the effective modulus, is defined by
1 1 m2 1 m2ind E ¼ ; ð14Þ þ E Eind where E and m are Young’s modulus and Poisson ratio, respectively, for the specimen and Eind and mind are the same parameters for the indenter. In Eq. (12), r0:29 is the stress corresponding to the characteristic plastic strain of 0.29 for the indented material, deduced from Eq. (2):
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n r0:29 ¼ K epl ð0:29Þ þ eel :
ð15Þ
By accounting for the effects of strain hardening on pile-up and sinking down, and on the true contact area, Giannakopoulos and Suresh (1999) proposes the following relation between the maximum contact area Smax and the maximum penetration depth dmax : Smax 2 ¼ 9:96 12:64ð1 T Þ þ 105:42ð1 T Þ 2 dmax 229:57ð1 T Þ3 þ 157:67ð1 T Þ4 Pav with T ¼ : E
ð16Þ
Usually T is the experimentally measured stiffness of the upper portion of the unloading data. On the other hand, the effective elastic modulus E of the indenter-specimen is defined as (Giannakopoulos and Suresh, 1999) 1 dF E ¼ pffiffiffiffiffiffiffiffiffi : ð17Þ c Smax dd The constant c ¼ 1:142 for the Vickers pyramid indenter, 1.167 for Berkovich indenter and 1.128 for the circular conical indenter of any included apex angle. dF =dd is the slope of the F–d curve of the initial stages of unloading from Fmax (Fig. 6). Thus, the use of Eq. (17) allows us to extract the value of dF =dd. Fig. 7 shows a cross section of an indentation and identifies the parameters used. At any time during loading, the total indenter displacement d is written as d ¼ dc þ d s ;
where dc is the vertical distance along which contact is made and ds is the displacement of the surface at the perimeter of contact. At peak load, the load and the displacement are Fmax : and dmax , respectively. Upon unloading, the elastic displacements are recovered, and when the indenter is fully withdrawn, the final depth of the residual hardness impression is dr . The Sneddon’s force–displacement relationship for conical indenter yields (Oliver and Pharr, 1992): dr ¼ dmax 2
Fmax : ð dF =ddÞ
ð19Þ
For a pyramidal indenter with a half-angle of 68° and for a material under a load F, we can determine the Vickers hardness HV as HV ¼ 1:8544
F : 49d2r
ð20Þ
In our case, the maximum applied load Fmax is 294 N (HV30 ). From Eq. (13), we can deduce the value of the maximum penetration depth dmax and calculating T ¼ Pav =E from Eqs. (12) and (16) leads to obtaining Smax . Knowing Smax and E , Eq. (17) gives dF =dd, so from Eqs. (19) and (20) the Vickers hardness can be calculated. Fig. 8 gives the hardness evolution in function of n and ry . Overall, hardness evolution is an increasing function of ry , and for n ¼ 0 (perfectly plastic material) hardness is a linear function. Hardness is also an
ð18Þ
Fig. 7. A schematic representation of a section through an indentation.
Fig. 8. Hardness evolution versus the flow stress and the strain hardening exponent.
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between the experimental measured hardness values and the calculated Vickers hardness obtained from the presented method, in which n and ry obtained from tensile tests are introduced. Fig. 9 shows that the obtained results from the two methods are similar.
5. Discussion Fig. 9. Comparison between calculated and measured hardness HV30 .
increasing function of n, but this growth varies with the ry value. We have carried out the Vickers indentation test on many steels and Table 1 gives the comparison
Several authors who use the indentation test to determine the parameters of a material are satisfied with an error of 20% (Venkatesh et al., 2000), moreover for brittle materials, the tensile test does not allow going beyond a few percent deformation. However the indentation test makes it possible to go beyond this limit. Steels presented in Fig. 10 failed around 10% deformation under the
(a)
(b)
(c)
(d) Fig. 10. Tensile and indentation curves (r ¼ Ken ) for different steels.
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tensile test. But the presented curves are extrapolated for a best comparison to the indentation results. Using the behaviour law (2), we can have the stress evolution according to the strain. For four steels (XC65, XC80, 100C6, 35CN15 steels), Figs. 10(a)–(d) show that the results deduced by the two methods lead to similar curves. Most of the steels which we used obey the Hollomon relation; since the experimental results are close to the values obtained from the suggested method (see Table 1). On the other hand, Z2CN18-
10 steel presents a behaviour law different from that of Hollomon. Therefore, for this material, it will be necessary to introduce its true behaviour law obtained from an experimental test. This true behaviour law should be introduced in the proposed model. Otherwise, Table 1 shows that the XC65 steel, which contains less carbon than XC80 steel, has a hardness (HV30 ¼ 232) higher than that of XC80 steel (HV30 ¼ 167). Which goes against common sense. A metallurgical analysis (Fig. 11) indicates
Fig. 11. Metallurgical analysis for different steels: (a) XC65 steel; (b) XC80 steel; (c) 100C6 steel; (d) 35CN15 steel.
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that the XC65 steel used (Fig. 11(a)), has a lamellar pearlitic structure and XC85 steel has a globulized structure (Fig. 11(b)), thus less hard. 100C6 steel is also globulized but has a higher carbon rate (1%) (Fig. 11(c)). Therefore it will be harder than XC80 steel (HV30 (100C6) ¼ 206). The 35CN15 steel sample presents a bainitic structure which appears fuzzy (Fig. 11(d)). We can hardly distinguish carbides aligned in needle packages. It is a self-quenching steel, thus very hard and its hardness is equal to 287. In general, the hardness values obtained from the Vickers indentation test and those obtained from the proposed method are similar (Table 1).
6. Conclusion This paper presents a new procedure for determining mechanical parameters of steel using the indentation test. Unlike the tensile test, the indentation test does not require the machining of samples. An experimental indentation test coupled with a proposed method leads to the strain hardening exponent and the yield stress for steels. On the other hand, a theoretical approach is given (Eqs. (12)–(20)) and leads to the Vickers hardness from the strain hardening exponent and the yield stress. The proposed model gives the indenter displacement evolution according to the loading d ¼ Aðry ; nÞF Bðry ;nÞ . This model is based on a minimisation procedure of error between the proposed model and the finite element simulation results. From the indentation displacement–load curve, the model yields the steel mechanical parameters, ry and n. The proposed method involves the following steps: 1. Experimentally determine the (F–d) curve during loading by a spherical indenter with radius of a 1.5875 mm. 2. Use minimisation procedure between the (F–d) relation (3) and the experimental curve, we can determine ry and n.
253
3. With the above theoretical framework (Eqs. (12)–(20)), the Vickers hardness can be calculated.
References Bouzakis, K.D., Vidakis, N., 1999. Superficial plastic response determination of hard isotropic materials using ball indentations and a FEM optimisation technique. Mater. Charact. 42 (1), 1–12. Cheng, Y.T., Cheng, C.T., 1999. Can stress–strain relationships be obtained from indentation curves using conical and pyramidal indenters? J. Mater. Res. 14, 3493– 3496. Edlinger, M.L., Gratacos, P., Montmitonnet, P., Felder, E., 1993. Finite element analysis of elastoplastic indentation with a deformable indenter. Eur. J. Mech. A/Solids 12 (5), 679–698. Field, J.S., Swain, M.V., 1995. Determining the mechanical properties of small volumes of material from submicrometer spherical indentations. J. Mater. Res. 10 (1), 101– 112. Giannakopoulos, A.E., Suresh, S., 1999. Determination of elastoplastic properties by instrumented sharp indentation. Scr. Mater. 40 (10), 1191–1198. Giannakopoulos, A.E., Larsson, P.L., Vestergaard, R., 1994. Analysis of Vickers indentation. Int. J. Solid Struct. 31 (19), 2679–2708. Haddow, J.B., Johnson, W., 1961. Indentation with pyramids I: theory. Int. J. Mech. Sci. 3, 229–238. € ber die Ber€ Hertz, H., 1882. U uhrung festischer K€ orper. Z. Reine und Angewande Math. 92, 156–171. Huber, N., Tsakmakis, C., 1999a. Determination of constitutive properties from spherical indentation data using neural networks Part I: the case of pure kinematic hardening in plasticity laws. J. Mech. Phys. Solids 47 (7), 1569– 1588. Huber, N., Tsakmakis, C., 1999b. Determination of constitutive properties from spherical indentation data using neural networks Part II: plasticity with nonlinear isotropic and kinematic hardening. J. Mech. Phys. Solids 47 (7), 1589–1607. Ishlinsky, A.J., 1944. Axisymmetric plastic flow in the Brinell test. Prik. Math. Mekh. 8, 201–224 (in Russian). Jayaraman, S., Hahn, G.T., Oliver, W.C., Rubin, C.A., Bastias, P.C., 1998. Determination of monotonic stress–strain curve of hard materials from ultra-low-load indentation tests. Int. J. Solid Struct. 35 (5-6), 365–381. Johnson, K.L., 1970. The correlation of indentation experiments. J. Mech. Phys. Solids 18, 115–126. Lockett, F.J., 1963. Indentation of a rigid plastic material by a conical indenter. J. Mech. Phys. Solids 11, 345–355. Marsh, D.M., 1964. Plastic flow in glass. Proc. R. Soc. A 279, 420–435.
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Millard, A., 1992. Castem 2000. Guide du development. Rapport D.E.M.T./92 300, C.E.A. Oliver, W.O., Pharr, G.M., 1992. An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. J. Mater. Res. 7 (6), 1564–1583. Sneddon, I.N., 1965. The relation between load and penetration in the axisymmetric Boussineq problem for a punch of arbitrary profile. Int. J. Eng. Sci. 3, 47–57.
Tabor, D., 1951. The Hardness of Metal. Clarendon Press, Oxford. Taljat, B., Zacharia, T., Kosel, F., 1998. New analytical procedure to determine stress–strain curve from spherical indentation data. Int. J. Solid Struct. 35 (33), 4411–4426. Venkatesh, T.A., Vliet, K.J., Giannakopoulos, A.E., Suresh, S., 2000. Determination of elasto-plastic properties by instrumented indentation: Guidelines for property extraction. Scr. Mater. 42 (9), 833–836.
New procedure to determine steel mechanical parameters from the spherical indentation technique A. Nayebi, R. El Abdi *, O. Bartier, G. Mauvoisin Laboratoire de Recherche en M ecanique Appliqu ee de l’Universit e de Rennes, I.U.T. de Rennes. 3, rue du Clos Courtel – B.P. 90422, 35704 Rennes cedex 7, France Received 2 April 2001; received in revised form 2 January 2002
Abstract A new numerical and experimental approach for determining mechanical properties of steels is presented. This method uses the instrumented spherical-indentation technique. A relationship between applied load, indenter displacement, flow stress and strain hardening exponent of steels, is given. This method is based on the minimisation of error between the experimental curve (applied load–indenter displacement curve) and the theoretical curve which is a function of the mechanical properties of studied materials. Comparison between the results obtained by the proposed method and experimental tensile tests, confirms the interest of the proposed method. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Indentation method; Finite-element modelling; Hardness; Strain hardening exponent; Yield stress
1. Introduction Indentation experiments have been performed for nearly one hundred years for measuring the hardness of materials. Long time ago, as early as the 19th century, the work of Hertz on rubbers appeared. Indentation of rubber can be interpreted in terms of elastic parameters using Hertz’s theory (1882) for a spherical indenter, or using Sneddon’s equations for a conical indenter (Sneddon, 1965). For metals in a plastic state, Tabor (1951) was the first to point out the relation between Vickers
*
Corresponding author. Fax: +33-223-234101. E-mail address: [email protected] (R. El Abdi).
(HV) or Brinell (HB) hardness and yield stress ry , namely HV 3ry and HB 2:8ry :
ð1Þ
This experimental observation was backed up by theoretical analyses of plastic flow, based on the Slip Line Fields method for spherical indenters (Ishlinsky, 1944), for a cone (Lockett, 1963) and the generalisation to the Vickers pyramid by Haddow and Johnson (1961). These studies confirmed the value of HV 3ry for blunt indenters, and also derived solutions for sharper indenters. Purely plastic analyses proved insufficient for hard metals where elasticity cannot be neglected. Following observations by Marsh (1964), an analogy with the classical problem of the growth
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Nomenclature D a d dr dmax r ry m E
diameter of ball contact radius indenter displacement residual depth of penetration maximum penetration depth stress yield stress Poisson ratio Young’s modulus
of a cavity in an infinite elastoplastic medium was used to determine the relationship between the hardness number (or average indentation pressure) and parameters mixing geometry with elastic and plastic properties of the indented material (Johnson, 1970; Edlinger et al., 1993; Giannakopoulos and Suresh, 1999; Venkatesh et al., 2000). The traditional method of determining the yield stress and the strain hardening exponent of a steel is the tensile test (or compression test) which requires the machining of samples which are, most of the time, cylindrical. On the other hand, the spherical-indentation method can be used for determining material stress–strain behaviour in a non-destructive and localised fashion. Moreover, it does not need precise sample machining (Field and Swain, 1995). The standard indentation measurement consists of pushing an indenter into the sample under precise load control. The indenter is inserted to some specified depth and then extracted, giving the loading force over the whole depth range. Recent years have seen increasing interest in indentation because of the significant improvement in the indentation equipment and the need for measuring the mechanical properties of materials on small scales. From the load–displacement curve of an indenter during indentation, the hardness and Young’s modulus (for example) may be obtained from the peak load and the initial slope of the unloading curves. The initial rate of change following the decrease in force (i.e., the slope) is defined as the sample stiffness.
n epl eel Pav F Fmax S HV q; h; z
strain hardening exponent plastic strain elastic strain average contact pressure applied load maximum applied load projected contact area Vickers Hardness cylindrical co-ordinates
Mechanical analysis requires formulation of the sharp indentation problem which is inevitably three-dimensional and at first glance seems quite impossible to solve, mainly because of material and geometric non-linearities, various dissipative mechanisms such as plasticity, phase transformation, micro-cracking and the presence of residual stresses. The properties of the indenter, such as its sharpness, are also very important to the analysis. The combination of all these effects makes the problem difficult to address analytically, and for this reason it was left unsolved in the past. Today’s computational capacity is, however, sufficient for a detailed investigation of the important features of sharp indentation tests. Concerning the problems for which the indenter has an axisymmetric form (cones, spheres, etc.), a model using two dimensional finite elements (2D) often remains sufficient. The problem of stress–strain curve determination from indentation tests has been studied in the past. The use of the stress–strain curve must lead to information about mechanical characteristic parameters. Tabor (1951) has shown a direct relation between indentation diameter and average applied strain, average pressure and corresponding stress. Field and Swain (1995) have used micro-spherical indentation for determining a small volume of material mechanical properties. They have used a stepwise indentation with a partial unloading technique for separating elastic and plastic components of indentation at each step. Later, Jayaraman et al. (1998), using different angle conical
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indenters, gave a method for determining the nonlinear part of stress–strain curve. Taljat et al. (1998) used spherical indentation for determining in another way the stress–strain curve. They have determined the relations for maximum and minimum applied strain and relating stresses, function of contact diameter, indenter diameter and hardening exponent, but they have not given an exact method for determining the hardening exponent. Huber and Tsakmakis (1999a,b) have used a neural network and cyclic indentation for determining the stress–strain curve. Bouzakis and Vidakis (1999) have given a new procedure which uses finite element simulation of successive ball indentation with increasing load. These methods are usually difficult and time consuming, for example Huber’s method needs cyclic loading and Bouzakis’ method needs a finite element simulation and measurement of imprint profile for every stress–strain curve point. Giannakopoulos and Suresh (1999) have used two parts of the loading and unloading Vickers indentation curve for yield stress and hardening exponent determination. They determine yield stress and stress at 0.29 plastic strain so the hardening exponent can be determined from these values. But in another paper (Venkatesh et al., 2000), the authors specified that their approach cannot distinguish two different materials which have the same yield stress ry and the same r0:29 stress obtained for a plastic strain equal to 0.29. The use of a Vickers indenter leads to the load–displacement F ¼ ad2 relation which is based on the determination of only one parameter a and cannot lead to the uniqueness. This was shown by Cheng and Cheng (1999). The aim of this work is to present a new procedure which leads to determining the yield stress and the strain hardening exponent of steels using an instrumented indentation technique and a theoretical curve. The experimental indentation test leads to obtaining the load–displacement curve. The theoretical curve is a relationship between applied load, indenter displacement, yield stress ry and the strain hardening exponent n of steels, which was determined by numerical simulations. The procedure is based on the error minimisation between the experimental (applied load–ind-
245
enter displacement curve) and the theoretical curves. This optimisation leads to having ry and n. Several cylindrical samples from different steel types, are submitted to a tensile test and the experimental values were compared with those numerically obtained from the proposed method. Finally, we give a method which leads to obtaining the Vickers hardness from knowing ry and n. This hardness value is compared to the one obtained from the classical Vickers indentation test.
2. Experimental procedure 2.1. Tensile test procedure The tensile test has been used to have experimental comparison values. Tensile tests have been carried out in a Lloyd LR30K machine. The advance speed pertinent to the tensile tests is 2 mm/ min. Cylindrical specimens 6mm in median diameter, having an effective length of 24 mm, were used as tensile specimens (Fig. 1). Because of large strains, an extensometer was used in the middle of the sample. A traditional system of acquisition makes it possible to record the tensile curve. Fig. 1 gives the 100C6 steel stress–strain curve. This tensile curve is fitted to Eq. (2), and the best values of ry and n so obtained were listed in Table 1. Eq. (2) represents the strain–stress curve of a metal: n
r ¼ Ken ¼ en ry =ðry =EÞ ;
ð2Þ
where e is the total strain (e ¼ epl þ eel ), and n is the strain hardening exponent. n ¼ 0 indicates a perfectly plastic material.
Fig. 1. Tensile curve for the 100C6 steel.
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Table 1 Comparison between proposed method and tensile test results Steels
ry , obtained from tensile test
ry , obtained from indentation proposed method
n, obtained from tensile test
n, obtained from indentation proposed method
Measured hardness HV30
Calculated hardness HV30
16NC6 100C6 35CD4 35NC15 Z38CDV5 Z15CN17-03 Z2CN18-10 42CD4 XC18 XC38 XC48 XC65 XC80
345 430 473 790 390 804 200 515 350 576 419 345 303
423 422 457 748 387 811 276 479 325 492 429 384 308
0.17 0.14 0.09 0.08 0.17 0.10 0.21 0.10 0.10 0.11 0.16 0.24 0.17
0.15 0.15 0.10 0.09 0.19 0.11 0.18 0.13 0.12 0.13 0.15 0.22 0.17
206 206 194 287 223 304 155 214 149 232 205 232 167
202 216 195 290 223 321 155 215 155 241 227 287 178
The tested materials used in the present investigation cover a wide range with respect to ry (200–800 MPa) and n (0.07–0.25). 2.2. Indentation procedure The relationship between applied load F and the characteristic contact displacement d represents an important material characteristic in indentation with a sphere of radius R. Spherical indenters are widely studied experimentally, because of their relative simple geometry, and theoretically, because of their facility to provide essential information on both elastic and plastic deformation properties of the tested material. A brief description of the experimental procedures is given in this section. The schematic set-up of the test apparatus is shown in Fig. 2. This indentation device is used to measure the F–d curve
during indentation. It was obtained with a load cell, displacement sensor and a spherical indenter (with a standardised sphere diameter D ¼ 1:587 mm). Indentation depth d was monitored using a displacement sensor mounted in the base which sensed the displacement of a steel arm mounted in a very stiff load train immediately above the indentation tip. The potential of the indentation system to allow accurate measurement is illustrated by the following parameters: displacement resolution 1 lm and force resolution 1 N. Load and displacement are simultaneously recorded for a loading curve. After three indentations, the curves have been averaged after zero point determination. As a numerical simulation procedure, maximum indenter displacement was held about 100 lm. The samples have been cut into cylindrical pieces of 10 mm depth from the same bars of traction samples. 3. Proposed model The aim of this section is to determine the flowstress and the strain hardening exponent starting from an instrumented test of indentation. For a homogeneous material, we seek to express the displacement d of the indenter according to applied load F. The selected function can be written as follows:
Fig. 2. Schematic set-up of the indentation test.
d ¼ Aðry ; nÞF Bðry ;nÞ :
ð3Þ
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Functions A and B depend on the yield stress ry and the strain hardening exponent n, and are deduced from the analysis of the results obtained by finite elements simulations on a series of materials. Simulation of the force–displacement relation for the indentation of a half-space using a spherical indenter with a high elastic modulus, was performed using the axisymmetric and large strain elastoplastic feature of the Castem 2000 finite element code (Millard, 2000), with uniaxial stress– strain data as input. Materials with various yield stresses and hardening components were studied. The quasi-static nature of the process allows using the static analysis performed by the programme. Underlying the approach in this code is the discretization of the continuum involved. Also, an important feature of this programme involves the ability to model contact between the indenter and the sample as a sliding interface. Initially, there is contact at only one point between the indenter and the sample. Progressively as the indenter progresses, the programme automatically establishes the junction between the contact surface nodes of the sample surface and those of the indenter. An elastic sphere indenting an elastoplastic steel for an axisymmetric half-space under normal contact was considered and modelled using a finite-element model in which the boundary conditions are given in Fig. 3; the boundary conditions may be expressed as follows: rz ðq; 0Þ ¼ 0
for jqj > a;
ð4Þ
sqh ðq; 0Þ ¼ 0
for jqj > a;
ð5Þ
where a is the contact radius. Far from the indentation zone, the displacements are cancelled: u ¼ v ¼ 0 as q; z ! 1:
ð6Þ
A finite element mesh of the half-space is given in Fig. 4. The programme was used with 6316 elements and 5986 nodes. A total of at least 30 elements were allowed to come in contact with the substrate in order to provide sufficient resolution in the computation of the field around the indenter. Other smoothness meshes were tested. The conclusion was that the results were identical.
247
Fig. 3. Geometry and boundary conditions.
Table 2 gives several values of ry and n which represent various studied materials. For reducing numbers of simulations and parameters, we have used an indenter with a constant diameter as D ¼ 1:5875 mm and Young’s modulus as E ¼ 650 GPa. All the other steel parameters (Young’s modulus E ¼ 210 GPa, Poisson ratio m ¼ 0:3) remain constant. The friction coefficient at the indenter–material contact was varied from 0.0 to 1.0. Various friction coefficients were used, and their influence on the F–d curve is negligible. Thus, for the studied materials, it will be considered constant and equal to 0.2 (Taljat et al., 1998). For all homogeneous steels cited in Table 1, functions A and B have been determined as B ¼ 1= ð 0:151ry þ 0:609Þn þ 0:09ry þ 0:975 ; h i B 2:9nr 0:323 y : A ¼ ð 3294 þ 22170r0:8 y Þe ð7Þ Yield stress ry is in GPa. For 35CD4 steel, Fig. 5 gives the load–displacement curve obtained from experimental testing and the proposed theoretical curve (3). Using
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Fig. 4. Finite-element meshing of half-space and zoom of the surface contact.
Table 2 Series of tested materials ry (MPa)
n
n
n
n
n
400 600 800 1000 1200
0.0 0.0 0.0 0.0 0.0
0.1 0.1 0.1 0.1 0.1
0.2 0.2 0.2 0.2 0.2
0.3 0.3 0.3 0.3 0.3
0.4 0.4 0.4 0.4 0.4
n ¼ 0:10 whereas the values obtained from the experimental tensile test are ry ¼ 473 MPa and n ¼ 0:09. For several steels, ry and n values obtained from the proposed method and tensile test are listed in Table 1. The two methods lead to close results.
4. Hardness–flow stress relation From ry and n values, we can have the hardness values. A method for the determination of the hardness value is proposed. 4.1. Average contact pressure Pav
Fig. 5. Load–displacement curve for the 35CD4 steel.
an optimisation procedure, we minimise the error between the proposed theoretical curve and the experimental indentation curve (F–d). This optimisation yields parameters ry and n which make error minimisation. For 35CD4 steel, the error minimisation leads to ry ¼ 457 MPa and to
The spherical cavity expansion model expresses the hardness value (average contact pressure) to elastic modulus ratio (Pav =E) as a function of the flow stress to elastic modulus ratio ðry =EÞ for very hard or elastic materials. Thus, the linear hardness–flow stress correlation obtained from the Slip-Line Field theory can be rewritten in terms of Pav =E and ry =E. Due to the 1=E dependence in these relations, the elastic modulus was used as the normalising parameter. As a first step in developing the hardness–flow stress correlation using the elastic modulus as the normalising parameter, the dependence of equivalent plastic strain epl on
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the equivalent flow stress ry was given by the simple Ramberg–Osgood equation: N
epl ¼ Lðry =EÞ :
ð8Þ
Any definition of the characteristic plastic strain should be such that it is (i) dependent on the geometry of the self-similar indenter, and (ii) independent of the depth of penetration (Jayaraman et al., 1998). For a given material, an experimentally determinable parameter that can satisfy both these conditions is average pressure Pav . If the characteristic plastic strain epl is related to average pressure Pav by a function similar to Eq. (8), we have N0
epl ¼ L0 ðPav =EÞ :
ð9Þ
Such a definition identifies the material being indented in terms of the new parameter L0 and N 0 , while satisfying conditions (i) and (ii) mentioned above. Thus, for a given indenter geometry, Eqs. (8) and (9) give a hardness–flow stress relation of the form: G
Pav =E ¼ Cðry =EÞ ;
ð10Þ
where 0
C ¼ ðL=L0 Þ1=N and G ¼ N =N 0 :
ð11Þ
This theory connects Pav and ry through two constants C and G which depend on material properties and which are difficult to determine. This relation gives only the average contact pressure Pav and not the Vickers hardness. The following paragraph presents a method for obtaining Vickers hardness from ry and n. 4.2. Vickers hardness determination from mechanical properties Fig. 6 is a schematic of the load–penetration depth (F–d) curve for a sharp indenter. During loading, the curve generally follows the power relation F ¼ W dm , where W and m depend on the mechanical material parameters and on the indenter geometry. The stress–strain curve of a metal can be characterised by a Hollomon-type Eq. (2).
Fig. 6. A schematic representation of load versus indenter displacement data for an indentation experiment.
The average contact pressure Pav is defined as the ratio of the applied load F divided by the projected contact area S. Finite-element simulation of elastoplastic indentation along with experiments (e.g., Giannakopoulos et al., 1994) provides the following results: Pav 0:27 E tanð22°Þ þ r 1 þ ln ¼ r ; 0:29 y E 3ry E ð12Þ Fmax E ¼ 7:143 r 1 þ ln þ r ; 0:29 y 2 ry dmax
ð13Þ
where 22° is the actual indentation angle of the Vickers indenter, Fmax is the maximum value of the applied load F and dmax is the maximum penetration depth. E the effective modulus, is defined by
1 1 m2 1 m2ind E ¼ ; ð14Þ þ E Eind where E and m are Young’s modulus and Poisson ratio, respectively, for the specimen and Eind and mind are the same parameters for the indenter. In Eq. (12), r0:29 is the stress corresponding to the characteristic plastic strain of 0.29 for the indented material, deduced from Eq. (2):
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n r0:29 ¼ K epl ð0:29Þ þ eel :
ð15Þ
By accounting for the effects of strain hardening on pile-up and sinking down, and on the true contact area, Giannakopoulos and Suresh (1999) proposes the following relation between the maximum contact area Smax and the maximum penetration depth dmax : Smax 2 ¼ 9:96 12:64ð1 T Þ þ 105:42ð1 T Þ 2 dmax 229:57ð1 T Þ3 þ 157:67ð1 T Þ4 Pav with T ¼ : E
ð16Þ
Usually T is the experimentally measured stiffness of the upper portion of the unloading data. On the other hand, the effective elastic modulus E of the indenter-specimen is defined as (Giannakopoulos and Suresh, 1999) 1 dF E ¼ pffiffiffiffiffiffiffiffiffi : ð17Þ c Smax dd The constant c ¼ 1:142 for the Vickers pyramid indenter, 1.167 for Berkovich indenter and 1.128 for the circular conical indenter of any included apex angle. dF =dd is the slope of the F–d curve of the initial stages of unloading from Fmax (Fig. 6). Thus, the use of Eq. (17) allows us to extract the value of dF =dd. Fig. 7 shows a cross section of an indentation and identifies the parameters used. At any time during loading, the total indenter displacement d is written as d ¼ dc þ d s ;
where dc is the vertical distance along which contact is made and ds is the displacement of the surface at the perimeter of contact. At peak load, the load and the displacement are Fmax : and dmax , respectively. Upon unloading, the elastic displacements are recovered, and when the indenter is fully withdrawn, the final depth of the residual hardness impression is dr . The Sneddon’s force–displacement relationship for conical indenter yields (Oliver and Pharr, 1992): dr ¼ dmax 2
Fmax : ð dF =ddÞ
ð19Þ
For a pyramidal indenter with a half-angle of 68° and for a material under a load F, we can determine the Vickers hardness HV as HV ¼ 1:8544
F : 49d2r
ð20Þ
In our case, the maximum applied load Fmax is 294 N (HV30 ). From Eq. (13), we can deduce the value of the maximum penetration depth dmax and calculating T ¼ Pav =E from Eqs. (12) and (16) leads to obtaining Smax . Knowing Smax and E , Eq. (17) gives dF =dd, so from Eqs. (19) and (20) the Vickers hardness can be calculated. Fig. 8 gives the hardness evolution in function of n and ry . Overall, hardness evolution is an increasing function of ry , and for n ¼ 0 (perfectly plastic material) hardness is a linear function. Hardness is also an
ð18Þ
Fig. 7. A schematic representation of a section through an indentation.
Fig. 8. Hardness evolution versus the flow stress and the strain hardening exponent.
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between the experimental measured hardness values and the calculated Vickers hardness obtained from the presented method, in which n and ry obtained from tensile tests are introduced. Fig. 9 shows that the obtained results from the two methods are similar.
5. Discussion Fig. 9. Comparison between calculated and measured hardness HV30 .
increasing function of n, but this growth varies with the ry value. We have carried out the Vickers indentation test on many steels and Table 1 gives the comparison
Several authors who use the indentation test to determine the parameters of a material are satisfied with an error of 20% (Venkatesh et al., 2000), moreover for brittle materials, the tensile test does not allow going beyond a few percent deformation. However the indentation test makes it possible to go beyond this limit. Steels presented in Fig. 10 failed around 10% deformation under the
(a)
(b)
(c)
(d) Fig. 10. Tensile and indentation curves (r ¼ Ken ) for different steels.
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tensile test. But the presented curves are extrapolated for a best comparison to the indentation results. Using the behaviour law (2), we can have the stress evolution according to the strain. For four steels (XC65, XC80, 100C6, 35CN15 steels), Figs. 10(a)–(d) show that the results deduced by the two methods lead to similar curves. Most of the steels which we used obey the Hollomon relation; since the experimental results are close to the values obtained from the suggested method (see Table 1). On the other hand, Z2CN18-
10 steel presents a behaviour law different from that of Hollomon. Therefore, for this material, it will be necessary to introduce its true behaviour law obtained from an experimental test. This true behaviour law should be introduced in the proposed model. Otherwise, Table 1 shows that the XC65 steel, which contains less carbon than XC80 steel, has a hardness (HV30 ¼ 232) higher than that of XC80 steel (HV30 ¼ 167). Which goes against common sense. A metallurgical analysis (Fig. 11) indicates
Fig. 11. Metallurgical analysis for different steels: (a) XC65 steel; (b) XC80 steel; (c) 100C6 steel; (d) 35CN15 steel.
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that the XC65 steel used (Fig. 11(a)), has a lamellar pearlitic structure and XC85 steel has a globulized structure (Fig. 11(b)), thus less hard. 100C6 steel is also globulized but has a higher carbon rate (1%) (Fig. 11(c)). Therefore it will be harder than XC80 steel (HV30 (100C6) ¼ 206). The 35CN15 steel sample presents a bainitic structure which appears fuzzy (Fig. 11(d)). We can hardly distinguish carbides aligned in needle packages. It is a self-quenching steel, thus very hard and its hardness is equal to 287. In general, the hardness values obtained from the Vickers indentation test and those obtained from the proposed method are similar (Table 1).
6. Conclusion This paper presents a new procedure for determining mechanical parameters of steel using the indentation test. Unlike the tensile test, the indentation test does not require the machining of samples. An experimental indentation test coupled with a proposed method leads to the strain hardening exponent and the yield stress for steels. On the other hand, a theoretical approach is given (Eqs. (12)–(20)) and leads to the Vickers hardness from the strain hardening exponent and the yield stress. The proposed model gives the indenter displacement evolution according to the loading d ¼ Aðry ; nÞF Bðry ;nÞ . This model is based on a minimisation procedure of error between the proposed model and the finite element simulation results. From the indentation displacement–load curve, the model yields the steel mechanical parameters, ry and n. The proposed method involves the following steps: 1. Experimentally determine the (F–d) curve during loading by a spherical indenter with radius of a 1.5875 mm. 2. Use minimisation procedure between the (F–d) relation (3) and the experimental curve, we can determine ry and n.
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3. With the above theoretical framework (Eqs. (12)–(20)), the Vickers hardness can be calculated.
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