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Journal ArticleDOI

A study on the determination of mechanical properties of a power law material by its indentation force–depth curve

23 Aug 2006-Philosophical Magazine (Taylor & Francis)-Vol. 86, Iss: 19, pp 2881-2905
TL;DR: In this paper, a novel optimization approach is proposed to extract mechanical properties of a power law material whose stress-strain relationship may be expressed as a power-law from its given experimental indentation P-h curve.
Abstract: In this study a novel optimization approach is proposed to extract mechanical properties of a power law material whose stress–strain relationship may be expressed as a power law from its given experimental indentation P–h curve. A set of equations have been established to relate the P–h curve to mechanical properties E, σ y and n of the material. For the loading part of a P–h curve this approach is based on the assumption that the indentation response of an elastic–plastic material is a linear combination of the corresponding elastic and elastic–perfect plastic materials. For the unloading part of the P–h curve it is based on the assumption that the unloading response of the elastic–plastic material is a linear combination of the full contact straight line and the purely elastic curve. Using the proposed optimization approach it was found that the mechanical properties of an elastic–plastic material usually cannot be decided uniquely by using only a single indentation P–h curve of the material. This is be...

Summary (3 min read)

1 Introduction

  • Compressive forces are normally used to form micro-components through micro-forming.
  • Sharp indenters widely used for micro- and nano- indentation tests are of Berkovich or Vickers types.
  • A lot of efforts have been made in the recent few years to derive approaches to extract these mechanical properties from a single or multi-set of P-h curves, and will be briefl reviewed in the following sections.
  • It has been shown by many numerical simulations [8, 9, 11, 14, 18-22] that Eq.(1) is a good approximation for elastic, elastic–perfect plastic and elastic–plastic materials.

1.2 Unloading curve and Young’s modulus

  • The Young’s modulus E can be estimated from the unloading curve which is assumed to be purely elastic.
  • In order to resolve this issue, Oliver and Pharr [6,25] introduced another method, taking into account the large elastic recovery during the unloading process of hard materials.
  • After the parameters B and m are determined, the unloading slope at the maximum load can be evaluated to be, 1( )m m r S mB h h −= − (12) Another major difference between Doerner and Nix’s and Oliver and Pharr’s methods is the approaches to determine the contact area at the maximum load.
  • Since hc obtained by Doerner and Nix’s, Eq.(10), and Oliver and Pharr’s methods, Eq.(14), is always smaller than hm, they can only explain the sink-in expressions.

1.3 Dimensional analysis and universal functions

  • Cheng et al [9, 8] and Tunvisut et al. [27] have used dimensional analysis to propose a number of dimensionless universal functions, with the aid of computational data points calculated via the FE method.
  • For this approach, one of the most complete studies has been published recently by Dao et al. [1].
  • Using these functions, the relationships between characteristic parameters, C, S and hr/hm, of a P-h curve and the mechanical properties of a material, E, σy and n, have been set up and then were used to extract mechanical properties of power-law materials by co-equation solving.
  • On the other hand, their analysis is less precise for determining the strain hardening coefficient n.
  • Even if the mean value gives a good estimate of the expected value, the errors are high and at least six experimental curves on the same material were required.

2.1 Numerical model

  • Axisymmetric FE models were constructed to simulate the indentation response of elastic-plastic solids using the commercial FE code ABAQUS.
  • Fig. 3(a) shows the FE model for axisymmetric calculations.
  • The semi-infinite substrate of the indented solid was modelled using 9600 four-noded, bilinear axisymmetric quadrilateral elements, where a fine mesh near the contact region and a gradually coarser mesh further from the contact region were designed to ensure numerical accuracy.
  • At the maximum load, the minimum number of contact elements in the contact zone was no less than 35 in each FEM computation.
  • An example of the self- adaptive mesh and the Mises stress contour at the maximum load is shown in Fig. 3(b).

2.2.1 The loading curve

  • 2.1.1 Loading curves for elastic and elastic-perfect plastic materials.
  • It was found that this weighting scheme worked very well.

2.2.2 The unloading curve

  • Zeng and Chiu’s weighting scheme as described in section 1.4 is firstly used in an attempt to describe the indentation-unloading curve (in fact, the upper 50% of the unloading curve) of an elastic-plastic material by weighting the unloading curves of the corresponding elastic and elastic-perfect plastic materials.
  • As shown in Fig. 9, if the contact area does not change during unloading, the unloading curve would be a straight line, Pfc, with a slope at the maximum load (note this is different from the straight line for the corresponding elastic-perfect plastic material as proposed in Zeng and Chiu’s method).
  • The unloading response of the elastic-plastic material should be a combination of the full contact straight line and the purely elastic curve.
  • It was found that excellent fitting can be obtained for at least the upper 50% of the unloading curve by the proposed weighting scheme.

2.2.3 The residual depth

  • An important characteristic parameter of the P-h curve is the residual depth, hr, after complete unloading (also shown in Table 1 for each simulation case).
  • Following the same least square fitting approach used in previous sections.
  • With the loading and unloading parts of a P-h curve being related to the mechanical properties of a material by Eq.(21) and (26), respectively, the mechanical properties of the matereial are determined by minimising the errors between the FE and predicted (using Eq.(21) and (26)) P-h curves, as shown in Fig. 14.
  • The first term in the equation defines the errors in the loading phase and the second in the unloading.
  • The Bates and Watts’ optimisation method improves the efficiency of optimisation by normalising optimised parameters in situations where several parameters, which have different magnitude but are confined within their own individual boundaries, need to be optimised.

3 Case studies-effectiveness of the proposed optimisation approach

  • To use The Bates and Watts’ optimisation method [29], the lower and upper boundaries of the parameters to be optimised have to be defined.
  • Here the optimisation parameters are the mechanical properties of the material of concern.
  • In most cases, some background information of the material of concern should be given.
  • The number of the matched sets of material properties whose residuals are not more than 10% of the global minimum is given in Table 2.
  • Using the Young’s modulus predicted by Oliver and Pharr’s method in the two parameter optimisation seems not to be a good approach, since Oliver and Pharr’s method often overestimates the Young’s modulus.

4 Discussion

  • This study proposes a novel optimisation approach to extract mechanical properties of a power law material from its given experimental (or FE simulation) indentation P-h curve.
  • The present study shows that the non-uniqueness problem is a commonplace for nearly all P-h curves and the matched materials can be found in a deterministic way through optimisation algorithm.
  • For three parameter (i.e. E, σy and n) optimisation, although the best matched material is identified in this study (Table 2), it could be difficult to identify it in reality since the material properties are not exactly known, particularly when there is large number of matched materials to choose from.
  • Further investigation is carried out in this direction.
  • The difficulty in relating the unloading P-h curve to mechanical properties is the main factor that affects the prediction accuracy of the proposed optimisation scheme.

5 Conclusions

  • A novel optimisation approach to extract mechanical properties of a power law material from its given experimental (or FE simulation) indentation P-h curve is proposed.
  • It was found that the prediction accuracy of material properties can be improved by this approach since the entire P-h curve data, except the lower 50% of the unloading curve, are used.
  • Using the proposed optimisation approach, it was found that mechanical properties of an elastic-plastic material usually cannot be determined uniquely by using only a single indentation P-h curve of the material.
  • This is because in general a few matched set of mechanical properties were found to produce a given P-h curve.
  • If the best matched material is identified, the predictions of mechanical properties are quite accurate.

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A study on the determination of mechanical properties
of a power law material by its indentation force-depth
curve
Jiaming Luo, Jianguo Lin, Trevor Dean
To cite this version:
Jiaming Luo, Jianguo Lin, Trevor Dean. A study on the determination of mechanical properties of a
power law material by its indentation force-depth curve. Philosophical Magazine, Taylor & Francis,
2006, 86 (19), pp.2881-2905. �10.1080/14786430600640528�. �hal-00513678�

For Peer Review Only
A study on the determination of mechanical properties of a
power law material by its indentation force-depth curve
Journal:
Philosophical Magazine & Philosophical Magazine Letters
Manuscript ID:
TPHM-05-Nov-0520
Journal Selection:
Philosophical Magazine
Date Submitted by the
Author:
22-Nov-2005
Complete List of Authors:
Luo, Jiaming; University of Birmingham, Mechanical Engineering
Lin, Jianguo; University of Birmingham, Mechanical and
Manufacturing Engineering
Dean, Trevor; University of Birmingham, Mechanical and
manufacturing Engineering
Keywords:
indentation, finite-element modelling, mechanical properties,
numerical simulation
Keywords (user supplied):
Strain hardening, Optimisation
http://mc.manuscriptcentral.com/pm-pml
Philosophical Magazine & Philosophical Magazine Letters

For Peer Review Only
1
A study on the determination of mechanical properties of a power law material by its indentation
force-depth curve
J LUO, J LIN* and T A DEAN
Department of Manufacturing and Mechanical Engineering, School of Engineering, University of
Birmingham, Edgbaston, Birmingham, B15 2TT, UK
Abstract: In this study a novel optimisation approach is proposed to extract mechanical properties of a
power law material whose stress-strain relationship may be expressed as a power law from its given
experimental indentation P-h curve. A set of equations have been established to relate the P-h curve to
mechanical properties, E,
σ
y
and n, of the material. For the loading part of a P-h curve, this approach is based
on the assumption that the indentation response of an elastic-plastic material is a linear combination of the
corresponding elastic and elastic-perfect plastic materials. For the unloading part of the P-h curve, it is based
on the assumption that the unloading response of the elastic-plastic material is a linear combination of the full
contact straight line and the purely elastic curve. Using the proposed optimisation approach, it was found that
the mechanical properties of an elastic-plastic material usually cannot be decided uniquely by using only a
single indentation P-h curve of the material. This is because in general a few matched sets of mechanical
properties were found to produce a given P-h curve. It is however possible to identify the best matched set of
mechanical properties by knowing some background information of the material. If the best matched material
is identified, the predictions of mechanical properties are quite accurate.
Keywords: Instrumented indentation; Mechanical properties; Finite element simulation;
Optimisation; Metals; Strain hardening
1 Introduction
Compressive forces are normally used to form micro-components through micro-forming. However, it is
difficult to determine material properties via compressive tests of micro-scale materials. Thus micro- or
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2
nano-indentation tests are often used for this purpose. Recent technological advances have led to the general
availability of depth-sensing instrumented indentation experiments in micro- and nano- scales [1-9]. These
tests provide accurate measurements of the continuous variation of indentation force, P, down to µN, as a
function of the indentation depth, h, down to nm. In addition to the determination of traditional hardness,
experimental investigations and theoretical/computational studies of indentation have been conducted on
many material systems in order to systematically extract material properties from P versus h curves (P-h
curves) obtained from instrumented indentation [5, 6, 8, 9, 10-14]. Instrumented indentation has been
recently used to study the deformation mechanisms at dislocation-level [15, 16].
Sharp indenters widely used for micro- and nano- indentation tests are of Berkovich or Vickers types.
In the simulation community, they are often simplified to a conical sharp indenter with an included half
angle, θ, of 70.3° [1]. This angle is chosen since the projected area/depth ratio of the two-dimensional
(axisymmetrical) cone is the same as that for the Berkovich or Vickers indenters and a general finding is that
computational P-h responses of the conical, Berkovich and Vickers indentations are virtually identical if
they have a same projected area/depth ratio. The contact geometry of a conical indenter with a material is
shown in Fig. 1, where θ is the included half angle of the conical indenter and a
c
and h
c
are the contact
radius and depth, respectively. Fig. 1 shows the pile-up impression of the indentation where h
c
>h
m
, and h
m
is
the depth at the maximum load. Dependent on the mechanical properties, there are cases of sink-ins where
h
c
<h
m
.
A schematic illustration of a typical P-h curve is shown in Fig. 1. During loading, the response
generally follows the relation described by Kick’s Law,
2
P Ch
= (1)
where C is a constant depending on the geometry of the indenter tip and workpiece material properties.
The average contact pressure, P
av
=P
m
/A
m
(A
m
is the true projected contact area measured at the maximum
load
P
), is defined to be the hardness of the indented material. The maximum indentation depth
m
h
occurs
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3
at
P
, and the initial unloading slope is defined as
ul
h
m
h
dP
S
d
= , where
ul
P
is the unloading force. The
residual indentation depth after complete unloading is
r
h
. As discussed by Giannakopoulos and Suresh [13],
C, S and h
r
/h
m
are three independent characteristic parameters that can be directly obtained from a single P-h
curve. These characteristic parameters have been used to determine elastic-plastic properties of the indented
materials and will be discussed in more detail later.
For many pure and alloyed engineering metals, their plastic behaviour can be closely approximated by a
power law description [17], as shown schematically in Fig. 2. A simple elastic-plastic, true stress–true strain
behaviour is assumed to be
y
n
y
E for
R for
ε σ σ
σ
ε σ σ
=
(2)
where E is the Young’s modulus, R a strength coefficient, n the strain hardening exponent,
σ
y
the initial
yield stress. For
y
σ σ
>
, one can approximately express:
y p
ε ε ε
= +
(3)
where
ε
y
is strain at yield stress and
ε
P
is the plastic strain, then
(1 )
n
y p
y
E
σ σ ε
σ
= + (4)
Therefore, to describe the mechanical properties of a power law material, three quantities, E,
σ
y
and n
are needed. A lot of efforts have been made in the recent few years to derive approaches to extract these
mechanical properties from a single or multi-set of P-h curves, and will be briefly reviewed in the following
sections.
1.1 Loading part of a P-h curve
It has been shown by many numerical simulations [8, 9, 11, 14, 18-22] that Eq.(1) is a good approximation
for elastic, elastic–perfect plastic and elastic–plastic materials. The results obtained by the finite element
(FE) analysis [18] is cited below, which considered a sharp Berkovich indentation on an elastic material,
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Citations
More filters
Journal ArticleDOI
TL;DR: In this article, Luo et al. proposed an approach based on the assumption that the instrumented indentation force-depth response of an elastic-plastic material is a linear combination of the corresponding elastic and perfect plastic materials.

80 citations

Journal ArticleDOI
TL;DR: In this article, non-linear optimisation algorithms are developed and integrated with finite element (FE) analysis to determine and improve the accuracy of the elastic-plastic mechanical properties of a power-law material without the concept of dimensionless analysis and a representative strain.

71 citations

Journal ArticleDOI
Minh-Quy Le1
TL;DR: In this paper, a set of dimensionless functions are constructed for cone indenters of half included angles of 60° and 70.3°, based on extensive finite element analysis, and an inverse analysis procedure is suggested to estimate material properties.

47 citations


Cites background from "A study on the determination of mec..."

  • ...Due to the self-similarity (Cheng and Cheng, 2004), there can be multiple combinations of mechanical properties ðE; Y; and nÞ, that give rise to almost indistinguishable indentation load–depth curves of a single conical/pyramid indenter (Cheng and Cheng, 1999; Capehart and Cheng, 2003; Alkorta et al., 2005; Tho et al., 2005; Luo et al., 2006)....

    [...]

  • ...1 for notations), see the recent review by Oliver and Pharr (2004): S ¼ b 2ffiffiffi p p E ffiffiffiffiffiffi Am p ð1Þ where b is a correction factor and E is determined by the elastic modulus E and Ei, and Poisson’s ratios m and mi of the indented material and the indenter, respectively, as below: E ¼ 1 m 2 E þ 1 m 2 i Ei 1 ð2Þ Due to the self-similarity (Cheng and Cheng, 2004), there can be multiple combinations of mechanical properties ðE; Y; and nÞ, that give rise to almost indistinguishable indentation load–depth curves of a single conical/pyramid indenter (Cheng and Cheng, 1999; Capehart and Cheng, 2003; Alkorta et al., 2005; Tho et al., 2005; Luo et al., 2006)....

    [...]

  • ...…there can be multiple combinations of mechanical properties ðE; Y; and nÞ, that give rise to almost indistinguishable indentation load–depth curves of a single conical/pyramid indenter (Cheng and Cheng, 1999; Capehart and Cheng, 2003; Alkorta et al., 2005; Tho et al., 2005; Luo et al., 2006)....

    [...]

Journal ArticleDOI
TL;DR: In this paper, a discussion of issues associated with the measurement of strength and strain, particularly with the latter assessed using non-contact optical techniques based on Digital Image Correlation, is presented.
Abstract: Testing in the micro regime is relevant to many technological issues, including, for example, evaluating the performance of established plant or new components through removal of small scale samples; assessing point to point variation in welded structures; characterising expensive advanced alloys that typically are only available in small quantities; and testing of sub-sized components. The use of miniature testpieces requires attention to the scale of the underlying microstructure relative to the overall testpiece dimensions, and presents challenges to the measurement of stress and strain in small volumes. The subject is extensive and the emphasis of the current paper is on a discussion of issues associated with the measurement of strength and strain, particularly with the latter assessed using non-contact optical techniques based on Digital Image Correlation. The current article thus concentrates on the testing situation at the intermediate scale (in the range 0.5-5 mm) where testpieces are small compared with the size of engineering components and in the range where representative mechanical behaviour of the underlying microstructure must be considered. Metrology issues are addressed through specific case studies concerning; micro uniaxial tests; small punch tests; sub-sized ceramic strength tests and the microindentation technique. Particular attention is paid to the uncertainties in measurement arising from the use of microscale tests to provide quantifiable levels of confidence in the ability of such tests to discriminate material behaviour.

35 citations


Cites background from "A study on the determination of mec..."

  • ...However, Luo et al.(124,125) have highlighted the difficulties of calculating the coefficients in these expressions from indents of a single geometry and noted that the use of more than one indenter geometry may be necessary to obtain a unique solution to the fitting of a suitable constitutive expression for uniaxial deformation behaviour....

    [...]

  • ...The trend towards the calculation of uniaxial constitutive mechanical response (either stress–strain or creep) from depth sensing data was noted.4,123–128 However, Luo et al.124,125 have highlighted the difficulties of calculating the coefficients in these expressions from indents of a single geometry and noted that the use of more than one indenter geometry may be necessary to obtain a unique solution to the fitting of a suitable constitutive expression for uniaxial deformation behaviour....

    [...]

Journal ArticleDOI
Minh-Quy Le1
TL;DR: In this article, the authors investigated the indentation response of elasto-plastic solids for conical indenters of half included angles of 60° and 70.3°.

35 citations

References
More filters
Journal ArticleDOI
TL;DR: In this paper, the authors used a Berkovich indenter to determine hardness and elastic modulus from indentation load-displacement data, and showed that the curve of the curve is not linear, even in the initial stages of the unloading process.
Abstract: The indentation load-displacement behavior of six materials tested with a Berkovich indenter has been carefully documented to establish an improved method for determining hardness and elastic modulus from indentation load-displacement data. The materials included fused silica, soda–lime glass, and single crystals of aluminum, tungsten, quartz, and sapphire. It is shown that the load–displacement curves during unloading in these materials are not linear, even in the initial stages, thereby suggesting that the flat punch approximation used so often in the analysis of unloading data is not entirely adequate. An analysis technique is presented that accounts for the curvature in the unloading data and provides a physically justifiable procedure for determining the depth which should be used in conjunction with the indenter shape function to establish the contact area at peak load. The hardnesses and elastic moduli of the six materials are computed using the analysis procedure and compared with values determined by independent means to assess the accuracy of the method. The results show that with good technique, moduli can be measured to within 5%.

22,557 citations

Journal ArticleDOI
TL;DR: In this article, a solution of the axisymmetric Boussinesq problem is derived from which are deduced simple formulae for the depth of penetration of the tip of a punch of arbitrary profile and for the total load which must be applied to the punch to achieve this penetration.

3,959 citations

Book
01 Aug 1988
TL;DR: This book offers a balanced presentation of the theoretical, practical, and computational aspects of nonlinear regression and provides background material on linear regression, including the geometrical development for linear and nonlinear least squares.
Abstract: Wiley-Interscience Paperback Series The Wiley-Interscience Paperback Series consists of selected books that have been made more accessible to consumers in an effort to increase global appeal and general circulation. With these new unabridged softcover volumes, Wiley hopes to extend the lives of these works by making them available to future generations of statisticians, mathematicians, and scientists. "The authors have put together an extraordinary presentation of concepts and methods concerning the use and analysis of nonlinear regression models ...highly recommend[ed] ...for anyone needing to use and/or understand issues concerning the analysis of nonlinear regression models." -Technometrics "[This book] provides a good balance of relevant theory and application with many examples ...[and it] provides the most balanced approach to theory and application appropriate for a first course in nonlinear regression modeling for graduate statistics students." -Mathematical Reviews "[This book] joins a distinguished list of publications with a reputation for balancing technical rigor with readability, and theory with application. [It] upholds tradition ...[and is] a worthwhile reference for the marketing researcher with a serious interest in linear models. " -Journal of Marketing Research This book offers a balanced presentation of the theoretical, practical, and computational aspects of nonlinear regression and provides background material on linear regression, including the geometrical development for linear and nonlinear least squares. The authors employ real data sets throughout, and their extensive use of geometric constructs and continuing examples makes the progression of ideas appear very natural. The book also includes pseudocode for computing algorithms.

3,202 citations


"A study on the determination of mec..." refers methods in this paper

  • ...To use the Bates and Watts’ optimization method [29], the lower and upper boundaries of the parameters to be optimized have to be defined....

    [...]

  • ...For a particular set of P–h data (obtained experimentally or by FE simulation as is the case in this study), the Bates and Watts’ optimization method [29], is used to determine the mechanical properties of the material by minimizing the residual defined in equation (28)....

    [...]

Journal ArticleDOI
TL;DR: In this paper, a method for obtaining hardness and Young's modulus from the data obtained from these types of instruments is described, where the elastic displacements are determined from data obtained during unloading of the indentation.
Abstract: Depth-sensing indentation instruments provide a means for studying the elastic and plastic properties of thin films. A method for obtaining hardness and Young’s modulus from the data obtained from these types of instruments is described. Elastic displacements are determined from the data obtained during unloading of the indentation. Young’s modulus can be calculated from these measurements. In addition, the elastic contribution to the total displacement can be removed in order to calculate hardness. Determination of the exact shape of the indenter at the tip is critical to the measurement of both hardness and elastic modulus for indentation depths less than a micron. Hardness is shown to depend on strain rate, especially when the hardness values are calculated from the data along the loading curves.

2,653 citations

Journal ArticleDOI
TL;DR: In this paper, Sneddon's analysis for the elastic contact between a rigid, axisymmetric punch and an elastic half space is used to show that a simple relationship exists between the contact stiffness, the contact area, and the elastic modulus that is not dependent on the geometry of the punch.
Abstract: Results of Sneddon's analysis for the elastic contact between a rigid, axisymmetric punch and an elastic half space are used to show that a simple relationship exists between the contact stiffness, the contact area, and the elastic modulus that is not dependent on the geometry of the punch. The generality of the relationship has important implications for the measurement of mechanical properties using load and depth sensing indentation techniques and in the measurement of small contact areas such as those encountered in atomic force microscopy.

1,363 citations

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In this study a novel optimisation approach is proposed to extract mechanical properties of a power law material whose stress-strain relationship may be expressed as a power law from its given experimental indentation P-h curve.