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Procedía Engineering 130 (2015) 1612 - 1621

Procedía Engineering

www.elsevier.com/loeate/procedia

14th International Conference on Pressure Vessel Technology

Determination of Area Reduction Rate by Spherical Indentation

B. Zoua, K.S. Guana>* , S. B.Wua

aThe Key Laboratory of Safety Science of Pressurized System, Ministry of Education, School of Mechanical and Power Engineering, East China University of Science and Technology, 130 Meilong Road, Shanghai,200237,China

Abstract

Rate of area reduction is an important mechanical characterization to appraise the plasticity of metals, which is always obtained from the uniaxial tensile test. A methodology is proposed to determine the area reduction rate by continuous ball indentation test technique. The continuum damage accumulation theory has been adopted in this work to identify the failure point in the indentation. According to this theory, the damage is accumulated in the material during the test with voids nucleating and growing, which can be divided into two stages due to the variation in the degree of constraint. The turning point between two stages can be identified as the failure point where the material beneath the indenter collapses. The corresponding indentation depth of this point can be obtained and used to estimate the area reduction rate. The local strain limit criterion proposed in the ASME VIII-2 2007 alternative rules is also adopted in this research to convert the multiaxial strain of indentation test to uniaxial strain of tensile test. This method can be useful in engineering practice to evaluate the material degradation under severe working condition due to the non-destructive nature of ball indentation test. In order to validate the method, spherical indentation test is performed on ferritic steel 16MnR and B16, then the results are compared with that got from the traditional uniaxial tensile test.

© 2015 The Authors.PublishedbyElsevierLtd. Thisis an open access article under the CC BY-NC-ND license

(http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the organizing committee of ICPVT-14

Keywords: Ball indentation test; Area reduction rate; Local strain limit criterion; Continuum damage accumulation theory

1. Introduction

Area reduction rate is a fundamental parameter to evaluate the plasticity of metals, which is usually got from the tensile test. This parameter indicates the plastic flow ability of ductile materials. Unfortunately, the traditional

* Corresponding author. Tel.: +0086-021-64253055; fax: +0086-021-64253055. E-mail address: guankaishu@ecust.edu.cn

1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the organizing committee of ICPVT-14

doi: 10.1016/j.proeng.2015.12.340

uniaxial tensile test is destructive and cannot be used to evaluate the mechanical property of equipment in-service. In order to monitor the plastic flow property of materials under working condition, other methods must be introduced. As an alternative way to evaluate the equipment mechanical property, ball indentation technique was used to get many material property parameters such as strain-hardening exponent [1,2], elastic modulus [3-5] and fracture toughness [6-10]. The technique can be applied to the equipment in-service due to its non-destructive nature. In the current study, the ball indentation test technique is used to estimate the area reduction rate of metals. Fig. 1 shows the load-displacement curve obtained through ball indentation test. The curve can be divided into two parts, loading and unloading part. In the unloading process, the deformation of material elastically recover which shows the elastic property so it can be used to calculate the elastic parameters like elastic modulus. As showed in Fig. 1, there is residual plastic deformation hr after unloading, which means the deformation of material contains plastic deformation that cannot recover. During the ball indentation test, there isn't any significant phenomenon like rupture to indicate the failure point where the material collapses. In order to get the area reduction rate of metals, the failure point must be identified at first.

A continuum damage mechanism is adopted in the present work to identify the failure point with the assumption that the damage is accumulated when the material experiences the loading-unloading-reloading process in the continuous ball indentation test. During loading process, the metal beneath the indenter experiences the localized plastic deformation, at the same time, voids nucleate and grow under the effect of localized shear stress. Then the load will be partially removed and the deformation will recover to certain extent. In the unloading process, the material will undergo a temporary tensile stress, which can show the tensile property of the material. The elastic modulus calculated from the unloading curves is dropping sharply during the continuous test process. Many factors may induce the decrease of elastic modulus, i.e. localized plastic strain, voids nucleation and growth. All of these factors eventually lead to the deterioration of the material property. In other words, the elastic modulus can be used to characterize the damage level, since the elastic modulus decreases with the damage accumulation.

In order to validate the model proposed in this work, continuous ball indentation test was performed on the ferritic steel 16MnR and B16, and at the same time, the uniaxial tensile test was also performed on the materials. Then area reduction rate was calculated from the load-depth curve by using the method proposed in this work. The indentation area reduction rate was compared with the area reduction rate obtained from the traditional uniaxial tensile test. There is deviation between the area reduction rates obtained by two different methods. The sources of error are also discussed in this paper.

DISPLACEMENT, h

Fig. 1. Load-Displacement curve obtained from ball indentation test.

2. Methodology

In the traditional tensile test, the area reduction rate can be obtained after the rupture of the sample. However, in the ball indentation test, there isn't any significant phenomenon to indicate the failure of metals. As a result, a failure point must be found at first and then the area reduction rate can be calculated based on the indentation depth at that point. In present study, the damage in the material can be seemed as isotropic. Kachanov [12] use Damage variable D to describe the level of damage in the material, which can be defined as:

where s is the cross-sectional area of load region and Sd is the reduced area due to the appearance and growth of micro-defects. The damage variable D can be converted to the void volume fraction f:

where V is the volume of material and Vd is the reduced volume due to the appearance and growth of micro-defects. In the Eq (1) and Eq (2), s and v are, respectively, the area of a circle and the volume of the sphere which has the same radius with the circle. The relationship between f and D can be expressed as:

D = ■

According to the Lemaitre's strain-equivalence principle [13],

Ed = E (1 - D)

where Ed and E are, respectively, elastic modulus of damaged material and elastic modulus of undamaged material. Elastic modulus can be obtained from the unloading curve through Oliver-Pharr method [3].

Ed =■

i4A i-i

where E; and Vi are, respectively, the elastic modulus and Poisson ratio of the indenter, S is the slope of unloading curve, Ed and v are, respectively, the elastic modulus and Poisson ratio of the material under the indenter, A is the contact area, which can be calculated as follows:

where d and dc are, respectively, the diameter of indenter and the contact area. The diameter of contact area, d c, can be calculated from the indentation depth h,

= sj0.252 -(0.25 - h)2

From Eq (1) ~ Eq (7), the relationship between void volume fraction and h can be obtained. There is a specific damage level or void volume fraction corresponding to every given indentation depth, which means the damage variable at the specific indentation depth can be identified according to the indentation curve. With the indentation depth increasing, voids nucleate and grow, and when the void volume fraction increase to a certain value, the failure happens. The critical depth h can be calculated from Eq (1) ~Eq (7) if the total void volume fraction at total failure is known, which will be discussed later. With the damage accumulation theory, the failure point in ball indentation test can be identified. Then the area reduction rate can be estimated on the basis of the deformation degree at the failure point. The strain of indentation depth can be defined as [14]:

e = 0.2-^ d

The strain calculated through Tabor method is a multiaxial strain which is different from the strain in the uniaxial tensile test. According to ASME VIII-2 2007 alternative rules [15], the multiaxial strain limit can be converted to uniaxial strain limit through the relation:

£L = £LU * eXP

V 1 + m2

where sL and sLa respectively represent the strain limit under the multiaxial strain state and the uniaxial strain state, TF is the stress triaxiality factor of the material beneath the indenter which can be calculated from finite element method. TF can be expressed as:

TF °m _+a2 +03 )/3_

TF --- ' == (10)

" ^ ) + (^1 ) + (a2 - )]/2

where m2 and asl is the material parameters. m2 is defined on the basis of the ratio between yield and tensile strength, m2 = 0.6 -(l -cs/<rb) for ferritic steels and m2 = 0.75 -(l — as/ab) for stainless steels. Many researchers

[16-19] have studied the method of determining the hardening law of the materials through spherical indentation test, where the yield and tensile strength can be obtained. Thus the process of obtaining yield and tensile strength is not discussed in this paper. asl is only related to the type of steels, for ferritic steels asl - 2.2 and asl - 0.6 for stainless steels. The area reduction rate can be calculated through the relation [20]:

¥ = {\-e'£Lu )xl00% (11)

where Y is the area reduction rate and eLu is the strain limit under uniaxial strain state. eLu can be seemed as the

fracture strain in uniaxial tension test. Therefore, through the method mentioned above, one may calculate the area reduction rate if the total void volume fraction is determined.

3. Experimental details

In order to validate the theoretical model, the continuous indentation test was carried out on the pressure vessel steel 16MnR and B16 using a DDL 20 machine (Changchun Inc., Jilin, China) whose load and displacement resolution is 0.1N and l^m. The chemical composition of 16MnR and B16 is showed in table 1. The size of the sample is 30x30x15mm, which is big enough compared with the indenter whose diameter is only 0.5mm. The size effect of samples can be neglected. The maximum indentation depth is 0.1mm and the loading-unloading circle was applied at 0.01mm intervals with the loading and unloading rate is fixed at 0.1mm/min. Lemaitre and Chaboche [21] and Bonora [22] et al estimate the elastic modulus of material from unloading curve other than reloading curve to avoid the effect of back-stress relaxation. [23] The slope of unloading curve is essential to calculate the elastic modulus, which can be obtained as showed in Fig. 1. The experiment is controlled by displacement and the unloading ratio is around 0.5 since the beginning of the unloading curve is adequate to provide all parameters for the calculation of elastic modulus. The curve obtained from experiment is showed in Fig. 2. The indenter used in this research is a tungsten carbide ball with E; = 600GPa and Vi = 0.03.

The uniaxial tensile test is also carried out on the same material to verify the result of the continuous ball indentation test. The uniaxial tensile test uses Instron-8032 testing machine with loading rate fixing at 1.5mm/min.

Table 1. The chemical composition of 16MnR and B16 steel (wt-%)

Elements (%) C Mn P S Si Cr Mo Al Ni V Nb Ti Cu

16MnR 0.17 1.48 0.022 0.004 0.26 0.004 0.005 0.022 0.02 - 0.004 0.013 0.03

B16 0.36 0.45 0.035 0.04 0.15 0.8 0.65 0.015 _ 0.3 _ _ _

Displacement, h(mm)

Fig. 2. Experimental load vs displacement curve.

4. Results and Discussion

4.1. The change in elastic modulus

The decrease of elastic modulus in the continuous indentation test can indicate the accumulation of damage in the material, which can be divided into two stages as showed in the Fig. 3.

At the first stage, the elastic modulus drops sharply mainly because a highly concentrated deformation field is generated beneath the indenter due to effect of highly constraint from the surrounding material, as showed in Fig. 4(a), which means the damage in the material is accumulated quickly.

Fig. 3. The elastic modulus with increasing indentation depth.

Fig. 4. Schematic ofindentation fist stage (a) and second stage (b).

Then the deformation comes into the second stage in which the elastic modulus drops slowly because the deformation spreads to the surface of the sample and the constraint is released to a certain extent, which results the stress-strain concentration cannot be as high as the first stage. The drop of elastic modulus in the indentation test represents the decay of metallic property, which is not uniform since constrain state is changed.

4.2. The failure point in the indentation test

Since the accumulation of damage in the sample is not uniform during the indentation test and there is a turning area between the first stage and the second stage. The turning point can be identified as the failure point of the material beneath the indenter. The corresponding relationship between void volume fraction and indentation depth can be obtained from Eq (1) ~Eq (7), as the showed in the Fig. 5.

16MnR B16

0.04 0.06

Displacement,h(mm)

Fig. 5. The void volume fraction with increasing indentation depth.

0.00 0.01 0.02 0.03 0.04 0.05 0 06 0 07 0-03 0.09 0.10

Displacement,h(tnm)

Fig. 6. The void volume fraction versus indentation depth (16MnR).

Area reduction rate(Tensile),HJ(%)

Fig.7. The comparison of area reduction rate obtained from indentation and tensile test.

Zener [24] and Dyskin et al [25] proposed dislocation pile-up model and angled crack growth mechanism respectively, which can be used to explain the void nucleation and growth under the compressive force. The formation of voids results from the effect of localized shear stress beneath the indenter. The void volume fraction increases dramatically with the elastic modulus dropping sharply because the high constraint induce highly concentrated shear stress. Under the effect of the highly concentrated shear stress, the property of material is quickly decaying. Then the deformation steps into the second stage, as showed in Fig. 4 (b), the constraint is released with the deformation expands to the surface of metal causing the growth of void volume fraction slowing down. In order to get the turning point, the linear fit is carried out on two stages of indentation test, as showed in Fig. 6, the intersection of two straight lines is defined as the failure point. The indentation depth of the failure point can be obtained from this point, and then the area reduction rate can be calculated through Eq (1) ~ Eq (11). As showed in Fig. 6, the failure indentation depth of 16MnR is 0.015mm and the corresponding void volume fraction is 0.31 which is consistent with the results of other researchers. Andersson [26] proposed the void volume fraction of ductile material is 0.25 when the crack begin to grow through numerical analysis method. Junk-Suk Lee et al also adopted that critical void volume fraction is 0.25 and validated it through interrupted tensile test. During the indentation test, voids nucleate in the ductile material under the effect of localized shear stress. When the void volume fraction arrives to a critical value, critical void volume fraction, voids begin to grow, which means the material reaches the fracture initiation point. With the indentation depth increasing further, voids begin to coalesce and the void volume fraction increases to failure point which means the material will collapse soon. The void volume fraction at total failure point is 0.31, which is a litter higher than the critical void volume fraction used by Andersson and Junk-Suk Lee et al.

4.3. Calculation of area reduction rate

According to the method mentioned above, the failure indentation depth of 16MnR is 0.015mm, the corresponding multiaxial strain is 0.068 which is calculated through Eq (7) and Eq (8). The stress triaxiality is also needed, which can be calculated from Eq (10) with finite element method. A GTN model was built in the finite element program ABAQUS to identify the absolute value of TF which is around 1.8. The material parameter m2 is defined on the basis of yield and tensile strength which also can be obtained from spherical indentation test. Many people have discussed the method of obtaining tensile property through ball indentation test, which will not be discussed here. In this research the value of yield and tensile strength is directly obtained from traditional tensile test.

As a kind of ferritic steels, the parameter asl of 16MnR is 2.2. Through the Eq (9), the value of uniaxial strain is 1.

Then the area reduction rate can be calculated from Eq (11), the result is showed in table 2. The comparison of area reduction rate calculated by indentation and tensile test is also showed in Fig. 7.

Table 2. Area reduction rate obtained from ball indentation and traditional tensile test.

Area reduction rate (%) Indentation test Tensile test

16MnR 63.2 48.5

B16 73.9 55.6

As showed in Fig. 7, the area reduction rate got from indentation test is significantly higher than the results obtained from tensile test. Several reasons may induce this result. Since there is no fracture happening in the indentation test, the deformation can be more adequate. In the tensile test, fracture happens before the deformation completing, which causes the deformation degree in the tensile test is lower than that in the indentation test.

The local strain limit criterion mentioned in the ASME VIII-2 2007 alternative rules is a kind of approximate method used in engineering practice, which adopt some conservative material constants to ensure the safety of vessels. These material parameters may bring some unexpected influence to the result, which need to be discussed in further study. Though two kinds of ferritic steels are covered in this investigation, other types of steels are also needed to be taken into account.

5. Conclusion

(1) The damage is accumulated in the material during the ball indentation test with voids nucleating and growing. The reduction of elastic modulus can indicate the damage evolution which can be divided into two stages. In the first stage, the elastic modulus drops sharply with the damage quickly accumulating in the material due to the constraint from the surrounding material of deformed area. Then the deformation steps into the second stage that the dropping of elastic modulus slows down because of the release of constraint. In this stage, the growing rate of void volume fraction is also approaching to a constant value since the degree of constraint is decreasing.

(2) The turning point between two stages can be identified as the failure point of material beneath the indenter. In order to obtain the failure point, linear fit of two stages is carried out and the intersection is defined as the failure point. The indentation depth at total failure can be identified and the corresponding void volume fraction is obtained, which is consistent with other scholars.

(3) The local strain limit criterion is adopted in this research to transform the multiaxial strain in the indentation test into the uniaxial strain in the tensile test. The area reduction rate can be calculated through the uniaxial strain. Two metallic materials, 16MnR and B16, are adopted in this study. The area reduction rate obtained through spherical indentation technique is a little higher than that got from uniaxial tensile test.

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