Hardness Prediction Based On P-h Curves And Inverse Material Parameters Estimation.
Applied Mechanics and Materials Vol 776 (2015) pp 233-238
© (2015) Trans Tech Publications, Switzerland
doi:10.4028/www.scientific.net/AMM.776.233
Submitted: 2015-02-13
Accepted: 2015-04-10
Hardness Prediction Based On P-h Curves
And Inverse Material Parameters Estimation
I Nyoman Budiarsa1, a
1
Mechanical Engineering. University of Udayana.Bali. Indonesia
a
Email: nyoman.budiarsa@me.unud.ac.id
Keywords: Finite Element Model, Indentation, P-h Curve, HV, HRB
Abstract. The Finite Element model of Vickers indentation has been developed. The model was
validated against published testing data. An approach to predict the P-h curves from constitutive
material properties has been developed and evaluated based the relationship between the curvature
and material properties and representative stress. The equation and procedure established was then
successfully used in predict the full Vickers indentation P-h curve. FE Spherical indentation models
of different radius have been developed and replay file model was developed that is able to produce
data of different materials properties. Two new approaches to characterise the P-h curves of
spherical indentation have been developed and evaluated. One is the full curve fitting approach
while the other is depth based approach. Both approaches were proven to be adequate and effective
in predicting indentation P-h curves. The concept and methodology developed is successfully used
to predict hardness values (HV and HRB) of materials through direct analysis and validated with
experimental data on selected sample of steels. The approaches (i.e. predict hardness from P-h
curves) established was successfully used to produce hardness values of a wide range of material
properties, which is then used to establish the relationship between the hardness values (HV and/or
HRB) with representative stress. This provided a useful tool to evaluate the feasibility of using
hardness values in predicting the constitutive material parameters with reference to accuracy and
uniqueness by mapping through all potential materials ranges
Introduction
Many works has been explored in searching a way to inversely predict material properties from
indentation tests [1]. Most of the research has been focusing on using full P-h curves [2] while the
links established between the hardness and constitutive materials properties are mostly based on
empirical data. For example, for elasto-plastic metals, most of the property-hardness data available
had been mainly using strength (yield strength and ultimate tensile strength) [3] as it is difficult to
quantify the contribution of the work hardening coefficient. This is not ideal, as increasingly, the
work hardening coefficient is required for situation where a detailed FE model is required. An
established link between materials properties, indentation curves and hardness values would also
provide a useful tool to explore the feasibility of predicting both yield strength and work hardening
coefficients based on hardness values and establish a full understanding with confidence about the
issue such as uniqueness (with only one set of material properties it the testing results) or nonuniqueness (more than one sets of material properties fit the testing results), which has been a major
problem of inverse material properties identification. The ability of identification of all possible
candidate material property sets that match the testing results will also pave the way for future
improvement of inverse program by using additional measurable data.
Materials and Experimental
Material used in this research were carbon steel with various compositions of carbon content
(0.10 % C, 0.54 % C and 0.85 % C). The tensile tests were performed using a material testing
machine with extensometer. Sample steel used is solid rod-shaped elliptical of 5 mm in diameter
and 90 mm long has a holder on the edges. The samples were sectioned, mounted in resin before
All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans
Tech Publications, www.ttp.net. (ID: 36.85.186.83-09/05/15,13:02:18)
234
Recent Decisions in Technologies for Sustainable Development
being polished with diamond paste. The Rockwell hardness test was performed using: Wilson
Rockwell hardness tester (ACCO Wilson instrument division, USA). The indentation using
spherical indenter B scale, with R= 0.79 mm (Diameter steel ball =1/16 in). For Vickers hardness
tester using a direct load method. The indenter is in the form of right pyramid with a square base
and angle of 136o between opposite face. The load range available is from 98.07mN to 19.61 N. On
the Vickers hardness tester, the hardness is measured as the diagonal diamond impression load on
the specimen displayed directly on the LCD touch panel display. The accuracy of the measurement
of the diagonal length has been checked with an optical microscope. One of the samples (0.54%
carbon) was processed through different heat treatments (annealing, quenching and tempering). The
sample was normalised at 8400 C, then annealed, quenched and tempered at T=2000 C, T= 4000 C
and T= 6000 C. This will effectively generate material samples with different hardness to modulus
(H/E) ratios as the Young’s modulus of steel doesn’t significantly with the heat treatments. These
hardness data (HV and/or HRB) be used to validate the representative stress based hardness
evaluation and property prediction program to be developed.
Numerical Model and Results
Finite element model (FE model) of the Vickers indentation designed using commercial code
ABAQUS. The Vickers indenter has the form of the right pyramid with a square base and an angle
of 136 between opposite face. Only a quarter of the indenter and material column was simulated
as a result of plane symmetric geometry. The sample size is more than 10 times the maximum
indentation depth, which is sufficiently large to avoid any sample size effect or boundary effect[4]
The bottom face of the material volume was fixed in all degrees of freedom (DOF) and two side
faces (A and B) Fig. 1(a). Were symmetrically fixed in y and x direction. The element type used is
C3D8R (reduced integration element used in stress/displacement analysis). Fig.1(b). Contact was
defined at indenter-specimen interface with a friction coefficients of 0.2. The FE model of Vickers
indentation test was verified by comparing the numerical results from this work with some
published modelling and experimental results. Typical result is shown in Figure 1(c). The materials
properties in the FE model were adopted from the data used by [1], and then the predicted P-h
curves were compared to the published numerical and experimental data. As shown in the curves,
for both materials, the predicted results agree well with the experimental data. This suggests that the
model is accurate and valid.
2169
P(N)
100
50
0
0.00 0.01 0.02 0.03 0.04 0.05
h(mm)
9066
120
12
100
10
80
8
P (N)
B
P(N)
A
4139
60
6
40
4
20
2
0
0
0.02
h(mm)
0.04
0.06
Material 1, This work numerical results
Material 1, Published numerical results
Material 1, Published experimental data
Material 2, This work numerical results
Material 2, Published numerical results
0
0.000
0.005h (mm)0.010
0.015
(a)
(b)
(c)
Figure 1(a). FE model of the Vickers indentation and P-h curve (b). Typical Effect of frictional
conditions on modelling. (c). Comparison of numerical results with some published experimental
Figure 2(a). shows the FE model of Spherical indentation. A 2-D axial symmetric model was used
due to the symmetry of the spherical indenter. The indenter was assumed to be rigid body as it is
much harder than the indented material. The type element of the material is standard axial
symmetric element: CAX4R and CAX3 (4-node bilinear asymmetric quadrilateral and 3-node linear
asymmetric triangle element). The movement of the indenter was simulated by displacing a rigid
Applied Mechanics and Materials Vol. 776
235
arc (rigid body) along the Z axis. In the model, the sample size can be changed to ensure that the
sample is much larger than the indenter radius/contact area during the indentation to avoid potential
sample size and boundary effects [4]. The thickness and width of the model used is 3mm in both
side. The bottom line of the model was fixed in all degree of freedoms (DOF) and the central line
was symmetrically constrained. Figure 2(b). shows typical P-h curve (Force vs. Indentation depth)
during loading and unloading phase of a typical elastic-plastic materials with different indenter
sizes. The loading curve represents the resistance of material to indenter penetration, while
difference between the loading and unloading curve represents the energy loss [5] Figure 2(c)
shows the comparison between the FE force-displacement data and corresponding result using a
known analytical solution for indentation of linear elastic materials As shown in the figures, the FE
results show a good agreement with the analytical solution.
500
100
400
75
Force, N
P, N
Published Experimental data
Numerical results proposed
Spherical R= 0.79 mm
125
50
25
300
200
100
0
0.000
0.010
0.020
h,mm
0.030
0
0
0.02
0.04
0.06
h, mm
0.08
(a)
(b)
(c)
Figure 2(a).FE Model of the spherical indentation and graph of plastic zone expansion during
spherical indentation (b). Typical force indentation depth (P-h) curves for the spherical indentation
(c).Comparison of numerical results with published experimental data of indentation with a
spherical indenter (R=1.25 mm)
Inverse Material Properties Prediction Based On Vickers and HRB Hardness Values
The work presented showed that hardness can be related to the stress of the indented material,
σr, corresponding to a representative strain, εr, which represents the mean plastic strain after
yielding and the hardness values can be predicted based on the indentation P-h curves [2], Base on
the concept prediction developed, it is implied that the hardness values could be potentially being
directly linked to the representative stresses over the material ranges studied. This is investigated by
firstly determine the HV or the HRB values of materials over a wide range of properties, then a
direct Hardness- representative stresses (σr) relation is explored using data fitting process. Figure
3(a) plots the HRB data against the constitutive material properties (yield stress(σy) and the work
hardening efficient(n)). Figure 3(b). Shows the surface plot of HV vs material properties (σy and n).
In Nonlinear Regression relationship between hardness - constitutive material properties can be
formulated as:
HRB=111.67 e
.
HV =26.62 E+06 e
.
.
.
.
.
.
.
(1)
.
.
(2)
If εr is a particular plastic strain point, the stress at the point representative stress, σr At this plastic
strain point, the stress can also be expressed as:
236
Recent Decisions in Technologies for Sustainable Development
σr = σy 1 +
(3)
εr
The curve could be simplified as a linear line with an equation of :
HRB = 0.0748 ln (
.
HV = 0.3115 σr + 11.186
) - 0.2945
(4)
(5)
These relationships (Eq.4 and 5) established allow direct hardness prediction from material
properties. This is assessed using the two steel materials as example, the predicted HV and HRB
showed a similar level of agreement with the experimental data. In the case of the 0.1 C Steel, the
hardness value is 98.368% of the measured value; the HRB values is within 107% of the measured
value; In the case of Mild steel, the predicted Hv value is within 98.611% of the measured value,
HRB is within 102% of the measured value. Similar agreement has been found in other materials
(within 5% error range). This suggests that these can be used to predict the hardness values with
sufficient accuracy with the measurement error ranges.
(a)
(b)
Figure 3(a) Surface plot of the HRB (b). HV data over a wide range of material properties (σy, n).
Furthermore the work carried out to investigate the feasibility of using the hardness values to
predict the Yield stress and the work hardening coefficients with a particular focus on uniqueness
issue making use of the methodologies and equations established in this work. The simulation space
included a group of hardness data covering a wide range of material properties. In this process, a
simulation space can be constructed conveniently using the representative stress approach (Eq.4 and
5). In this work, the yield stress was varied from 100 to 900 MPa with an increment of 10MPa. The
strain hardening coefficients used were from 0.01 to 0.3 with an increment of 0.01. This covers over
2400 hardness data for Hv and HRB respectively. The results were then recorded and stored into a
database to form a simulation space. In each case, the optimum material parameters, which
produces the P-h curves match or close to the experimental results (in this case, hardness values),
were determined by mapping the objective function (Eq. 6) [6]
G=
(
(
)
(
)
)
(6)
G is the objective function that needs to be minimised. This simple format allows easier
interpretation of the relative error of the results. The materials with lower objective functions are
potentially the target material parameter sets. Figure 4(a) Show the surface plot of objective
Applied Mechanics and Materials Vol. 776
237
function vs. material parameters of hardness value HV and Fig.4(b) hardness value HRB. The
contour band represents different objective function values. The figure shows that many sets of
material properties have very close objective function values at the valley, which suggests that there
are a large group of materials that can produce hardness value matching the experimental data.
whereas in the surface plot of objective functions vs. material parameters for the dual indenter
approach. As shown in the Figures 4(c), for both materials, there are still many sets of material
properties with very close objective function values at the valley. It clear shows that there are a
large group of material which give similarly Hv and HRB values in both cases.
Lower.Obj.Function
Lower.Obj.Function
Lower.Obj.Function
(a)
(b)
(c)
Figure 4(a) Show the surface plot of objective function vs. material parameters of HV for the
0.1%C Steel. (b) HRB for the 0.1% C Steel. (c) surface plot of objective functions vs. material
parameters for the dual indenter approach (HV and HRB) for the 0.1% C Steel.
Figure 5(a) plots the materials sets taken at the valley of Figure 4(a) and 4(b). These data represent
the material sets that have similar HV values and HRB values close to the experimental data of mild
steel (within 1%). The data for the Vickers hardness is much scattered while there are many
material data with low objective function for HRB. It is interesting to see that, even though, the
shape of the indenters and the way the hardness has been calculated is different between HV and
HRB, but the data with low objective function falls on a similar straight line. This explains why
there is no improvement with the dual indenter approach. Figure 5(b) shows the stress strain curves
of the selected material property sets (σy, n) from Figure 5(a). It is clearly shown that all these
materials sets go through the same point with similar representative stress. Figure 6(a) plots the
loading and unloading curves of Vickers indentation from FE indentation model. It is clearly shown
that both the loading and the unloading cures are close to identical. Figure 6 (b) shows the loading
curves of the Rockwell hardness; again, the P-h curves are almost identical. These suggest that
there are multiple materials that have the same HV and HRB values.
1500
600
Mild Steel (Hv)
500
Mild Steel (HRB)
Stress, σy (Mpa)
Yield Stress (MPa)
700
400
300
200
100
0
0
0.1
0.2
n
0.3
0.4
1000
500
0
0.00
σy=600 MPa,n=0.03
σy=500 MPa,n=0.10
σy=340 MPa,n=0.22
0.20
Strain, Є
0.40
(a)
(b)
Figure 5(a) Materials sets with an objective function within 1%. (b) Comparison between the
stress-strain curves of selected material sets with similar objective function
Recent Decisions in Technologies for Sustainable Development
7
P(N)
6
σy=600 MPa,n=0.03
σy=500 MPa,n=0.10
σy=340 MPa,n=0.22
(a)
5000
5
4000
4
3000
P(N)
238
3
2
(b)
2000
1000
1
0
0.000
σy=600 MPa,n=0.03
σy=500 MPa,n=0.10
σy=340 MPa,n=0.22
0.005
h(mm)
0.010
0.015
0
0.00
0.05
0.10
h(mm)
0.15
Figure 6(a) Comparison of P-h curves with material sets of similar low objective function of the
Vickers hardness and (b) of the Spherical indentations
The work clearly highlighted that uncertainty of uniqueness is a major challenge for the application
of inverse modelling, which has been the main concern of inverse modelling based on the
indentation approach [1,7] If the physical process is non unique, in a search based method. the
results may converge to a point (material sets) at a local minimal rather than globe minimal point,
thus identify the wrong properties. For elasto-plastic materials, the method used in this work could
effectively to identify any possible material property sets with the same hardness by using the
representative stress method rather than repeating limited number of FE modelling.
Summary
A new method to predict the P-h curves from constitutive material properties has been developed
and evaluated for both Vickers and spherical indentation. the models developed for P-h curve
prediction are further applied to determine HV and Rockwell hardness values. The predicted
hardness values are compared with the experimental data. The program is then used to produce
hardness data over a wide range of material properties providing a simulation space to establish the
relationship between hardness (HV and/or HRB) and representative stress. This is used to evaluate
the feasibility of using hardness values in predicting the constitutive material parameters with
particular focus on the uniqueness of the results by mapping through a large number of potential
materials. The identification of all potential materials set will effectively narrow the material
searching range and provide the possibility to identify the true material based pre-knowledge or
other measurable data
References
[1] Dao M., Chollacoop N., Van Vliet K. J., Venkatesh T. A. and Suresh S., Computational
modelling of the forward and reverse problems in instrumented sharp indentation, Acta
Materialia, Vol. 49 (2001) pp. 3899–3918
[2] Budiarsa I N., Jamal M., P-h Curves and Hardness Value Prediction for Spherical Indentation
Based on the Representative Stress Approach, App. Mech.and Mat. Vol. 493 (2014) pp 628-633
[3] Busby J. T., Hash M. C., Was G. S., The relationship between hardness and yield stress in
irradiated austenitic and ferritic steels, Journal of Nuclear Materials 336 (2005) 267-278
[4] Johnson K. L., Contact Mechanics, Cambridge: Cambridge University Press, UK, 1985
[5] Swaddiwudhipong S., Tho K. K. , Liu Z. S. and Zeng K., Material characterisation based on
dual indenters, International Journal of Solids and Structures, Vol. 42 (2005) pp. 69-83
[6] Bolzon G., Maier G., Panico M.,Material model calibration by indentation, imprint mapping and
inverse analysis, Inter. Journal of Solids and Structures, Vol. 41 (2004) pp. 2957-2975.
[7] Chen X., Ogasawara N., Zhao M. and Chiba N., 2007, On the uniqueness of measuring
elastoplastic properties from indentation: The indistinguishable mystical materials, Journal of
the Mechanics and Physics of Solids 55, (2007) pp. 1618–1660
Hardness Prediction Based On
P-h Curves And Inverse Material
Parameters Estimation
by I Nyoman Budiarsa
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Mode Analysis of Coherently Coupled Array
Based on Proton Implantation", Applied
Mechanics and Materials, 2015.
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Ibrahim, Alaa M., E.M. Hassen, M. A. AbdelRahman, and Emad A. Badawi. "Estimating
the Activation Enthalpy for Defect Formation
3%
2%
1%
1%
1%
1%
in 5754 Alloys by Using Nuclear, Electrical
and Mechanical Methods", Defect and
Diffusion Forum, 2012.
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X. X. Su. "Rapid Identification of Nonlinear
Material Parameters of Foams Based on
Neural Network", IFMBE Proceedings, 2010
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Sudarshan, O.R. Nandagopan, Ajith Kumar,
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"Mechanical properties of vinylester/glass
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© (2015) Trans Tech Publications, Switzerland
doi:10.4028/www.scientific.net/AMM.776.233
Submitted: 2015-02-13
Accepted: 2015-04-10
Hardness Prediction Based On P-h Curves
And Inverse Material Parameters Estimation
I Nyoman Budiarsa1, a
1
Mechanical Engineering. University of Udayana.Bali. Indonesia
a
Email: nyoman.budiarsa@me.unud.ac.id
Keywords: Finite Element Model, Indentation, P-h Curve, HV, HRB
Abstract. The Finite Element model of Vickers indentation has been developed. The model was
validated against published testing data. An approach to predict the P-h curves from constitutive
material properties has been developed and evaluated based the relationship between the curvature
and material properties and representative stress. The equation and procedure established was then
successfully used in predict the full Vickers indentation P-h curve. FE Spherical indentation models
of different radius have been developed and replay file model was developed that is able to produce
data of different materials properties. Two new approaches to characterise the P-h curves of
spherical indentation have been developed and evaluated. One is the full curve fitting approach
while the other is depth based approach. Both approaches were proven to be adequate and effective
in predicting indentation P-h curves. The concept and methodology developed is successfully used
to predict hardness values (HV and HRB) of materials through direct analysis and validated with
experimental data on selected sample of steels. The approaches (i.e. predict hardness from P-h
curves) established was successfully used to produce hardness values of a wide range of material
properties, which is then used to establish the relationship between the hardness values (HV and/or
HRB) with representative stress. This provided a useful tool to evaluate the feasibility of using
hardness values in predicting the constitutive material parameters with reference to accuracy and
uniqueness by mapping through all potential materials ranges
Introduction
Many works has been explored in searching a way to inversely predict material properties from
indentation tests [1]. Most of the research has been focusing on using full P-h curves [2] while the
links established between the hardness and constitutive materials properties are mostly based on
empirical data. For example, for elasto-plastic metals, most of the property-hardness data available
had been mainly using strength (yield strength and ultimate tensile strength) [3] as it is difficult to
quantify the contribution of the work hardening coefficient. This is not ideal, as increasingly, the
work hardening coefficient is required for situation where a detailed FE model is required. An
established link between materials properties, indentation curves and hardness values would also
provide a useful tool to explore the feasibility of predicting both yield strength and work hardening
coefficients based on hardness values and establish a full understanding with confidence about the
issue such as uniqueness (with only one set of material properties it the testing results) or nonuniqueness (more than one sets of material properties fit the testing results), which has been a major
problem of inverse material properties identification. The ability of identification of all possible
candidate material property sets that match the testing results will also pave the way for future
improvement of inverse program by using additional measurable data.
Materials and Experimental
Material used in this research were carbon steel with various compositions of carbon content
(0.10 % C, 0.54 % C and 0.85 % C). The tensile tests were performed using a material testing
machine with extensometer. Sample steel used is solid rod-shaped elliptical of 5 mm in diameter
and 90 mm long has a holder on the edges. The samples were sectioned, mounted in resin before
All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans
Tech Publications, www.ttp.net. (ID: 36.85.186.83-09/05/15,13:02:18)
234
Recent Decisions in Technologies for Sustainable Development
being polished with diamond paste. The Rockwell hardness test was performed using: Wilson
Rockwell hardness tester (ACCO Wilson instrument division, USA). The indentation using
spherical indenter B scale, with R= 0.79 mm (Diameter steel ball =1/16 in). For Vickers hardness
tester using a direct load method. The indenter is in the form of right pyramid with a square base
and angle of 136o between opposite face. The load range available is from 98.07mN to 19.61 N. On
the Vickers hardness tester, the hardness is measured as the diagonal diamond impression load on
the specimen displayed directly on the LCD touch panel display. The accuracy of the measurement
of the diagonal length has been checked with an optical microscope. One of the samples (0.54%
carbon) was processed through different heat treatments (annealing, quenching and tempering). The
sample was normalised at 8400 C, then annealed, quenched and tempered at T=2000 C, T= 4000 C
and T= 6000 C. This will effectively generate material samples with different hardness to modulus
(H/E) ratios as the Young’s modulus of steel doesn’t significantly with the heat treatments. These
hardness data (HV and/or HRB) be used to validate the representative stress based hardness
evaluation and property prediction program to be developed.
Numerical Model and Results
Finite element model (FE model) of the Vickers indentation designed using commercial code
ABAQUS. The Vickers indenter has the form of the right pyramid with a square base and an angle
of 136 between opposite face. Only a quarter of the indenter and material column was simulated
as a result of plane symmetric geometry. The sample size is more than 10 times the maximum
indentation depth, which is sufficiently large to avoid any sample size effect or boundary effect[4]
The bottom face of the material volume was fixed in all degrees of freedom (DOF) and two side
faces (A and B) Fig. 1(a). Were symmetrically fixed in y and x direction. The element type used is
C3D8R (reduced integration element used in stress/displacement analysis). Fig.1(b). Contact was
defined at indenter-specimen interface with a friction coefficients of 0.2. The FE model of Vickers
indentation test was verified by comparing the numerical results from this work with some
published modelling and experimental results. Typical result is shown in Figure 1(c). The materials
properties in the FE model were adopted from the data used by [1], and then the predicted P-h
curves were compared to the published numerical and experimental data. As shown in the curves,
for both materials, the predicted results agree well with the experimental data. This suggests that the
model is accurate and valid.
2169
P(N)
100
50
0
0.00 0.01 0.02 0.03 0.04 0.05
h(mm)
9066
120
12
100
10
80
8
P (N)
B
P(N)
A
4139
60
6
40
4
20
2
0
0
0.02
h(mm)
0.04
0.06
Material 1, This work numerical results
Material 1, Published numerical results
Material 1, Published experimental data
Material 2, This work numerical results
Material 2, Published numerical results
0
0.000
0.005h (mm)0.010
0.015
(a)
(b)
(c)
Figure 1(a). FE model of the Vickers indentation and P-h curve (b). Typical Effect of frictional
conditions on modelling. (c). Comparison of numerical results with some published experimental
Figure 2(a). shows the FE model of Spherical indentation. A 2-D axial symmetric model was used
due to the symmetry of the spherical indenter. The indenter was assumed to be rigid body as it is
much harder than the indented material. The type element of the material is standard axial
symmetric element: CAX4R and CAX3 (4-node bilinear asymmetric quadrilateral and 3-node linear
asymmetric triangle element). The movement of the indenter was simulated by displacing a rigid
Applied Mechanics and Materials Vol. 776
235
arc (rigid body) along the Z axis. In the model, the sample size can be changed to ensure that the
sample is much larger than the indenter radius/contact area during the indentation to avoid potential
sample size and boundary effects [4]. The thickness and width of the model used is 3mm in both
side. The bottom line of the model was fixed in all degree of freedoms (DOF) and the central line
was symmetrically constrained. Figure 2(b). shows typical P-h curve (Force vs. Indentation depth)
during loading and unloading phase of a typical elastic-plastic materials with different indenter
sizes. The loading curve represents the resistance of material to indenter penetration, while
difference between the loading and unloading curve represents the energy loss [5] Figure 2(c)
shows the comparison between the FE force-displacement data and corresponding result using a
known analytical solution for indentation of linear elastic materials As shown in the figures, the FE
results show a good agreement with the analytical solution.
500
100
400
75
Force, N
P, N
Published Experimental data
Numerical results proposed
Spherical R= 0.79 mm
125
50
25
300
200
100
0
0.000
0.010
0.020
h,mm
0.030
0
0
0.02
0.04
0.06
h, mm
0.08
(a)
(b)
(c)
Figure 2(a).FE Model of the spherical indentation and graph of plastic zone expansion during
spherical indentation (b). Typical force indentation depth (P-h) curves for the spherical indentation
(c).Comparison of numerical results with published experimental data of indentation with a
spherical indenter (R=1.25 mm)
Inverse Material Properties Prediction Based On Vickers and HRB Hardness Values
The work presented showed that hardness can be related to the stress of the indented material,
σr, corresponding to a representative strain, εr, which represents the mean plastic strain after
yielding and the hardness values can be predicted based on the indentation P-h curves [2], Base on
the concept prediction developed, it is implied that the hardness values could be potentially being
directly linked to the representative stresses over the material ranges studied. This is investigated by
firstly determine the HV or the HRB values of materials over a wide range of properties, then a
direct Hardness- representative stresses (σr) relation is explored using data fitting process. Figure
3(a) plots the HRB data against the constitutive material properties (yield stress(σy) and the work
hardening efficient(n)). Figure 3(b). Shows the surface plot of HV vs material properties (σy and n).
In Nonlinear Regression relationship between hardness - constitutive material properties can be
formulated as:
HRB=111.67 e
.
HV =26.62 E+06 e
.
.
.
.
.
.
.
(1)
.
.
(2)
If εr is a particular plastic strain point, the stress at the point representative stress, σr At this plastic
strain point, the stress can also be expressed as:
236
Recent Decisions in Technologies for Sustainable Development
σr = σy 1 +
(3)
εr
The curve could be simplified as a linear line with an equation of :
HRB = 0.0748 ln (
.
HV = 0.3115 σr + 11.186
) - 0.2945
(4)
(5)
These relationships (Eq.4 and 5) established allow direct hardness prediction from material
properties. This is assessed using the two steel materials as example, the predicted HV and HRB
showed a similar level of agreement with the experimental data. In the case of the 0.1 C Steel, the
hardness value is 98.368% of the measured value; the HRB values is within 107% of the measured
value; In the case of Mild steel, the predicted Hv value is within 98.611% of the measured value,
HRB is within 102% of the measured value. Similar agreement has been found in other materials
(within 5% error range). This suggests that these can be used to predict the hardness values with
sufficient accuracy with the measurement error ranges.
(a)
(b)
Figure 3(a) Surface plot of the HRB (b). HV data over a wide range of material properties (σy, n).
Furthermore the work carried out to investigate the feasibility of using the hardness values to
predict the Yield stress and the work hardening coefficients with a particular focus on uniqueness
issue making use of the methodologies and equations established in this work. The simulation space
included a group of hardness data covering a wide range of material properties. In this process, a
simulation space can be constructed conveniently using the representative stress approach (Eq.4 and
5). In this work, the yield stress was varied from 100 to 900 MPa with an increment of 10MPa. The
strain hardening coefficients used were from 0.01 to 0.3 with an increment of 0.01. This covers over
2400 hardness data for Hv and HRB respectively. The results were then recorded and stored into a
database to form a simulation space. In each case, the optimum material parameters, which
produces the P-h curves match or close to the experimental results (in this case, hardness values),
were determined by mapping the objective function (Eq. 6) [6]
G=
(
(
)
(
)
)
(6)
G is the objective function that needs to be minimised. This simple format allows easier
interpretation of the relative error of the results. The materials with lower objective functions are
potentially the target material parameter sets. Figure 4(a) Show the surface plot of objective
Applied Mechanics and Materials Vol. 776
237
function vs. material parameters of hardness value HV and Fig.4(b) hardness value HRB. The
contour band represents different objective function values. The figure shows that many sets of
material properties have very close objective function values at the valley, which suggests that there
are a large group of materials that can produce hardness value matching the experimental data.
whereas in the surface plot of objective functions vs. material parameters for the dual indenter
approach. As shown in the Figures 4(c), for both materials, there are still many sets of material
properties with very close objective function values at the valley. It clear shows that there are a
large group of material which give similarly Hv and HRB values in both cases.
Lower.Obj.Function
Lower.Obj.Function
Lower.Obj.Function
(a)
(b)
(c)
Figure 4(a) Show the surface plot of objective function vs. material parameters of HV for the
0.1%C Steel. (b) HRB for the 0.1% C Steel. (c) surface plot of objective functions vs. material
parameters for the dual indenter approach (HV and HRB) for the 0.1% C Steel.
Figure 5(a) plots the materials sets taken at the valley of Figure 4(a) and 4(b). These data represent
the material sets that have similar HV values and HRB values close to the experimental data of mild
steel (within 1%). The data for the Vickers hardness is much scattered while there are many
material data with low objective function for HRB. It is interesting to see that, even though, the
shape of the indenters and the way the hardness has been calculated is different between HV and
HRB, but the data with low objective function falls on a similar straight line. This explains why
there is no improvement with the dual indenter approach. Figure 5(b) shows the stress strain curves
of the selected material property sets (σy, n) from Figure 5(a). It is clearly shown that all these
materials sets go through the same point with similar representative stress. Figure 6(a) plots the
loading and unloading curves of Vickers indentation from FE indentation model. It is clearly shown
that both the loading and the unloading cures are close to identical. Figure 6 (b) shows the loading
curves of the Rockwell hardness; again, the P-h curves are almost identical. These suggest that
there are multiple materials that have the same HV and HRB values.
1500
600
Mild Steel (Hv)
500
Mild Steel (HRB)
Stress, σy (Mpa)
Yield Stress (MPa)
700
400
300
200
100
0
0
0.1
0.2
n
0.3
0.4
1000
500
0
0.00
σy=600 MPa,n=0.03
σy=500 MPa,n=0.10
σy=340 MPa,n=0.22
0.20
Strain, Є
0.40
(a)
(b)
Figure 5(a) Materials sets with an objective function within 1%. (b) Comparison between the
stress-strain curves of selected material sets with similar objective function
Recent Decisions in Technologies for Sustainable Development
7
P(N)
6
σy=600 MPa,n=0.03
σy=500 MPa,n=0.10
σy=340 MPa,n=0.22
(a)
5000
5
4000
4
3000
P(N)
238
3
2
(b)
2000
1000
1
0
0.000
σy=600 MPa,n=0.03
σy=500 MPa,n=0.10
σy=340 MPa,n=0.22
0.005
h(mm)
0.010
0.015
0
0.00
0.05
0.10
h(mm)
0.15
Figure 6(a) Comparison of P-h curves with material sets of similar low objective function of the
Vickers hardness and (b) of the Spherical indentations
The work clearly highlighted that uncertainty of uniqueness is a major challenge for the application
of inverse modelling, which has been the main concern of inverse modelling based on the
indentation approach [1,7] If the physical process is non unique, in a search based method. the
results may converge to a point (material sets) at a local minimal rather than globe minimal point,
thus identify the wrong properties. For elasto-plastic materials, the method used in this work could
effectively to identify any possible material property sets with the same hardness by using the
representative stress method rather than repeating limited number of FE modelling.
Summary
A new method to predict the P-h curves from constitutive material properties has been developed
and evaluated for both Vickers and spherical indentation. the models developed for P-h curve
prediction are further applied to determine HV and Rockwell hardness values. The predicted
hardness values are compared with the experimental data. The program is then used to produce
hardness data over a wide range of material properties providing a simulation space to establish the
relationship between hardness (HV and/or HRB) and representative stress. This is used to evaluate
the feasibility of using hardness values in predicting the constitutive material parameters with
particular focus on the uniqueness of the results by mapping through a large number of potential
materials. The identification of all potential materials set will effectively narrow the material
searching range and provide the possibility to identify the true material based pre-knowledge or
other measurable data
References
[1] Dao M., Chollacoop N., Van Vliet K. J., Venkatesh T. A. and Suresh S., Computational
modelling of the forward and reverse problems in instrumented sharp indentation, Acta
Materialia, Vol. 49 (2001) pp. 3899–3918
[2] Budiarsa I N., Jamal M., P-h Curves and Hardness Value Prediction for Spherical Indentation
Based on the Representative Stress Approach, App. Mech.and Mat. Vol. 493 (2014) pp 628-633
[3] Busby J. T., Hash M. C., Was G. S., The relationship between hardness and yield stress in
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[4] Johnson K. L., Contact Mechanics, Cambridge: Cambridge University Press, UK, 1985
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elastoplastic properties from indentation: The indistinguishable mystical materials, Journal of
the Mechanics and Physics of Solids 55, (2007) pp. 1618–1660
Hardness Prediction Based On
P-h Curves And Inverse Material
Parameters Estimation
by I Nyoman Budiarsa
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