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Nondestructive evaluation of material strength using depth-sensing indentation
Engineering Structures 30 (2008) 2206–2210 www.elsevier.com/locate/engstruct
Nondestructive evaluation of material strength using depth-sensing indentation Jale Tezcan ∗ , Kent J. Hsiao Department of Civil and Environmental Engineering, Southern Illinois University, 1230 Lincoln Dr., 62901, Carbondale, IL, United States Available online 15 August 2007
Abstract Successful application of any structural repair or strengthening method requires knowledge of strength and stiffness properties of the materials used in the construction. Depth-sensing indentation is a relatively new material testing method that significantly expands the capabilities of conventional hardness tests. Due to the mathematical complexity of the indentation problem, an analytical solution of the load–penetration depth (P–h) behavior is not available for most indenters. This study derives the P–h equations for a truncated cone indenter with arbitrary tip radius and included angle, which covers a wide range of geometries including the cylinder and the cone. A step by step procedure for material strength evaluation is introduced and validated using recorded indentation data. The analytical approach described in this paper can be used in evaluating the modulus of elasticity, stiffness and hardness of common construction materials without resorting to empirical equations. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Nondestructive testing; Depth-sensing indentation; Elastic modulus; Stiffness; Hardness
1. Introduction A variety of hardness tests are performed in industry to predict strength and durability characteristics of a myriad of materials at different scales. However, these tests have several drawbacks that limit their range of applicability and effectiveness. One of the problems in traditional hardness testing is the difficulty in interpreting the test results. Since hardness is not a fundamental material property, results from different hardness tests are not directly comparable. Empirical equations or conversion tables are required to evaluate the material properties of interest. The fact that most hardness tests are destructive is another factor that limits their in-situ application. Moreover, penetration resistance tests require that the residual imprint be measured optically, which introduces potential errors due to operator judgment. Application of the conventional hardness tests to assess material strength for historical structures presents additional challenges. To protect the cultural heritage, the testing has to be carried out in a minimally invasive way. ∗ Corresponding author. Tel.: +1 618 453 6125; fax: +1 618 453 3044.
E-mail address: [email protected] (J. Tezcan). c 2007 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2007.06.011
Depth-sensing indentation is a relatively new approach that offers a virtually nondestructive means of material characterization. Since only small volumes of materials are required, spatial variation of surface mechanical properties can be evaluated by this method. The hardness and the modulus of elasticity are the properties that can most readily be obtained from an indentation test [1,2]. In the past two decades, numerous researchers have developed various procedures to evaluate additional mechanical properties such as yield stress, bond strength, strain hardening, strain rate sensitivity and material damping from the indentation data. Recently, there have been efforts to incorporate indentation testing in structural materials. Zhu et al. [3] studied the bond and interfacial properties around steel reinforcement in selfcompacting concrete using a depth-sensing nanoindentation technique. Trtik [4] used the microindentation test to analyze the variation of modulus of elasticity and creep of an ordinary Portland cement matrix as a function of the distance from the closest aggregate. Hossain [5] investigated the durability of pumice concrete at high temperatures using a microindentation method. Indentation testing has also been successfully applied in measuring strength properties of structural steel [6] and wood [7].
J. Tezcan, K.J. Hsiao / Engineering Structures 30 (2008) 2206–2210
The analysis of load–indentation depth (P–h) curves is often based on Sneddon’s elastic contact model [8]. Oliver and Pharr [9] used Sneddon’s analysis to develop a procedure to extract the modulus of elasticity, hardness and stiffness of the indented material. Due to the mathematical complexity of contact analysis, analytical formulas describing the indentation behavior are available for only a few simple geometries; equations for the rest are usually determined empirically through a curve fitting procedure. Briscoe [10] investigated the penetration behavior of indenters with tip defects and concluded that the curve fitting approach should be used with caution since the load–depth behavior is highly sensitive to the geometry of the indenter tip. Careful selection of the indenter tip is very important in maximizing the reliability of the test results. The indenter tip geometry should be selected considering the expected behavior of the sample material and the objective of the test. Sharp indenters such as the pyramid and the cone are preferred in fracture toughness estimations, since they produce high stress and strain concentrations in the vicinity of the contact area, thereby facilitating crack formation. However, sharp indenters might initiate cracking at very low loads, making it difficult to determine the elastic properties. When testing brittle materials for stiffness and modulus of elasticity, high stress concentrations should be avoided. This can be achieved by lowering the applied load, or using an indenter with a larger tip area. It is also important to ensure full contact between the sample and the indenter tip, since the test results are analyzed assuming uniform stress distribution at the contact area. In addition, the geometry of the indenter should reveal the variation of the contact area as the indenter is withdrawn. A cylindrical punch, due to its flat tip, will help achieve uniform stress distribution; however, it has a constant cross sectional area. On the other hand, a cone has a well defined depth–area relationship; however, it will create high stress concentrations. A truncated cone with appropriate tip radius and included angle can overcome the limitations of both the cone and the cylinder. The purpose of this paper is to formulate a procedure for evaluating material properties to be used in conjunction with indentation testing. Specifically, we derive the P–h equations for a truncated cone indenter with arbitrary tip radius and included angle. By varying these parameters, P–h behavior for a range of axisymmetric geometries can be obtained. This paper starts with a general introduction to material characterization using depth-sensing indentation. In Section 3, the P–h equations for a truncated cone are derived. Section 4 presents verification of the derived equations for two special cases with known closed-form solutions. A step by step procedure for material strength evaluation is introduced in Section 5. Finally, a practical application of the procedure is demonstrated in Section 6. 2. General procedure of material characterization using indentation testing During an indentation experiment, the penetration depth (h) is measured as the indenter is driven into and withdrawn
2207
Fig. 1. Schematic representation of indentation with a truncated cone.
Fig. 2. Typical load–penetration depth data from indentation tests.
from the material under controlled loading. Fig. 1 shows the geometric model of a truncated cone indenting a sample material. As the indenter is driven into the material, both elastic and plastic deformations occur. When the indenter is removed, the elastic portion of the deformation is recovered. Typical outcome of an indentation test is a load–penetration depth curve as shown in Fig. 2. Sneddon [8] derived the P–h equations for axisymmetric indenters and showed that the penetration depth (h), is related to the shape function of the indenter f (x), through the equation Z 1 f 0 (x) h= dx, (1) √ 1 − x2 0 where x is a dimensionless parameter, and the load required to produce this penetration is Z 1 2 0 x f (x) P = 2Er rc dx. (2) √ 1 − x2 0 The reduced modulus of elasticity (Er ) is a property that accounts for bidirectional displacements in both the indenter and the sample, and is given by [11] 1 − νi2 1 − νs2 1 = + , Er Ei Es
(3)
where E i , νi and E s , νs are the moduli of elasticity and the Poisson’s ratios of the indenter and the sample, respectively. The hardness (H ) of a material is a measure of resistance to deformation. One common way of quantifying the hardness
2208
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Dividing the radial distance by the contact radius (rc ), and applying the change of variables x = rrc , the shape function can be rewritten in terms of the dimensionless parameter x. The resulting shape function and its derivative are given in Eqs. (7) and (8), respectively ri 0 for 0 < x ≤ r c f (x) = (7) ri
consists of evaluating the mean contact pressure at maximum load through the formula Pmax , (4) A where Pmax and A are the maximum applied load and the projected area of contact at that load, respectively. Determining A from optical measurements is a challenge even with modern imaging instruments. In indentation testing, the availability of continuous load–penetration data makes it possible to calculate A empirically, by using a function which relates the contact area to the distance from the indenter tip. The stiffness (S) of the tested material can be determined from the unloading part of the indentation data, since unloading represents elastic recovery of the deformation [12,13]. In most cases, S can be determined with sufficient accuracy by fitting a curve to the upper 25% of the unloading data. Once A and S are known, the reduced modulus of elasticity (Er ) can be solved using equation [14] H=
2 √ S=√ AEr . π
(5)
Finally, the modulus of elasticity of the material (E s ) is calculated using Eq. (3). It is important to note that the indentation testing will evaluate the material properties only at the locations that come into contact with the indenter. Strength and stiffness properties of composite materials can be determined by testing the individual components and using the appropriate rule of mixtures. 3. Analytical derivation of P–h relationship for a truncated cone Since the truncated cone has an axisymmetric geometry, Sneddon’s method [8] can be used to derive the P–h relationship. For the indenter shown in Fig. 1, the tip radius and the included half angle are denoted by ri and ψ, respectively. The indenter shape function can be expressed as a function of the radial distance (r ), as 0 if r < ri f (r ) = (6) (r − ri ) cot(ψ) if r > ri .
To simplify the foregoing analysis, we introduce the ri −1 dimensionless contact parameter, φ = cos rc . It is obvious from the geometry of penetration that the possible values of φ range from 0 to π2 radians. The penetration depth corresponding to the parameter φ is found by substituting Eq. (8) in Eq. (1) as h = ri cot(ψ)
φ . cos(φ)
(9)
The load required to produce this penetration is calculated from Eq. (2) as φ P = ri2 cot(ψ)Er tan(φ) + . (10) cos2 (φ) Eqs. (9) and (10) define the elastoplastic P–h behavior for the truncated cone. The half included angle (ψ) can be eliminated between Eqs. (10) and (9) to yield sin(φ) 1 P = E r ri h + , (11) φ cos(φ) where P is the applied load, h is the indenter displacement, Er is the reduced modulus of elasticity, ri is the indenter tip radius and φ is the dimensionless contact parameter. 4. Verification of the derived P–h equations for two special cases To examine the validity of the equations derived in Section 3, the P–h behavior for varying indenter shape parameters, i.e. tip radius and half angle, is depicted in Fig. 3. All three axes have been put into non-dimensional form. The axes x, y and z represent the dimensionless contact parameter φ, the depth parameter rhi , and the load parameter P 2 , respectively. E r ri
The boundaries defined by φ = 0 and φ = π2 are of particular interest since they correspond to the cylinder and the cone, respectively. Closed form P–h expressions for the two shapes are Pcylinder = 2Er ri h
(12)
2 Er tan(ψ)h 2 . (13) π These two geometries will be analyzed as special cases of the derived solution. Pcone =
J. Tezcan, K.J. Hsiao / Engineering Structures 30 (2008) 2206–2210
2209
Fig. 5. Variation of contact radius with load.
Fig. 4. Load–penetration data showing indentation with a truncated cone.
Case I: The value φ = 0 corresponds to the cylinder, since the contact radius has to be equal to the indenter tip radius, i.e., −1 ri = 0 ↔ r c = ri . (14) φ = cos rc
be expressed in terms of S by eliminating A between Eqs. (4) and (5) to yield the formula
The limit of Eq. (10) as φ → 0 is sin(φ) 1 P = Er ri h lim + = 2Er ri h, φ→0 φ cos(φ)
The procedure for evaluating the strength of material tested under a truncated cone indenter can be summarized as follows: (15)
which is the same as the closed form P–h equation for a cylinder as given in Eq. (12). Case II: The value φ = π2 corresponds to the cone, since the tip radius has to be zero, i.e., ri π φ = cos−1 = ↔ ri = 0. (16) rc 2 The P–h equation is obtained using Eq. (9) in Eq. (11) and taking the limit as ri → 0, sin(φ) h tan(ψ) 2 + P = lim Er ri h = Er tan(ψ)h 2 , (17) ri →0 φ ri φ π which is the same as the closed form P–h equation for a cone as given in Eq. (13). 5. Procedure of material characterization using a truncated cone The equations derived in Section 3 allow evaluation of material strength without resorting to empirical equations. The modulus of elasticity (E s ) can be directly calculated from the indentation data, as opposed to the traditional method described in Section 2, where errors associated with empirical determination of A are propagated into the calculation of E s . This is an obvious advantage over the traditional method, since accuracy of the E s is critical in evaluating structural safety. Direct calculation of the modulus of elasticity also allows determining the hardness (H ) without explicitly calculating the projected area of contact (A). Since the stiffness (S) is readily determined from the upper portion of the unloading data, H can
H=
Pmax π
2Er S
2
.
(18)
(a) Determine Pmax and h max values from the indentation data. (b) Determine S from the slope of the initial unloading curve. (c) Calculate φ, Er and E s from Eqs. (9), (11) and (3), respectively. (d) Calculate H from Eq. (18). 6. A practical example To demonstrate how the procedure described in Section 5 can be used in evaluating material strength properties, a single indentation testing was performed on a carbon–carbon composite sample (K196A). This material was previously tested by other researchers under a Berkovich indenter [15]. This particular material was chosen to allow direct comparison between the indentation test results under two different indenters. R The experiment was conducted using the Nano Indenter XP system. A diamond tip indenter with radius ri = 2.5 µm and included half angle of ψ = 30◦ was used. For the diamond indenter, the modulus of elasticity and the Poisson’s ratio are E i = 1141 GPa and υi = 0.07, respectively. The recorded indentation data and the variation of contact radius with loading are shown in Figs. 4 and 5, respectively. The maximum penetration depth h max = 1425 nm and the maximum load Pmax = 118.3 mN can be read from Fig. 4. The stiffness of the material is found from the initial slope of the unloading curve. Fitting a line to the upper 25% portion of the unloading data, the stiffness is found to be S = 192 kN/m. The dimensionless contact parameter is calculated using Eq. (9) as φ = 0.33456. The reduced elastic modulus is found from Eq. (11) as Er = 16.2 GPa. To determine the elastic modulus of the material from Eq. (3), prior knowledge of Poisson’s ratio
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is needed. However, a rough estimate of the Poisson’s ratio will not introduce a large error since most engineering materials, as well as carbon–carbon composites, have a Poisson’s ratio between 0.20 and 0.35. Using Eq. (3), the moduli of elasticity corresponding to υs = 0.2 and υs = 0.35 are found to be E s = 14.63 GPa and E s = 13.37 GPa, respectively. Finally, the hardness is determined from Eq. (18), as H = 1.08 GPa. The reported values of the modulus of elasticity, the stiffness and the hardness of the tested material are E s = 15.8 GPa, S = 171 kN/m and H = 1.43 GPa, respectively [15]. The modulus of elasticity and the stiffness values calculated using the method described in this paper are in good agreement with the reported values. The discrepancy in the hardness values should not raise much concern since hardness is not a fundamental material property but a relative measure of deformation resistance of materials. 7. Conclusion In this paper, the load–penetration depth (P–h) equations for a truncated cone indenter with arbitrary tip radius and included angle were derived. The validity of the equations was demonstrated by considering two indenters, the cone and the cylinder, for which closed form equations are available in the literature. A step by step procedure for material strength evaluation was introduced and validated using recorded test data. It was concluded that the procedure described in this paper offers a reliable method for evaluating the modulus of elasticity, stiffness and hardness of materials, without resorting to empirical data. References [1] Newey AD, Wilkins AMA, Pollock AHM. An ultra-low-load penetration hardness tester. Journal of Physics E: Scientific Instruments 1982;15: 119–22.
[2] Pethica JB, Hutchings R, Oliver WC. Hardness measurements at penetration depths as small as 20 nm. Philosophical Magazine A 1983; 48(4):593–606. [3] Zhu W, Sonebi M, Bartos P. Bond and interfacial properties of reinforcement in self-compacting concrete. Materials and Structures 2004;37:442–8. [4] Trtik P, Bartos PJM. Micromechanical properties of cementitious composites. Materials and Structures 1999;32:388–93. [5] Hossain KMA. Macro and microstructural investigations on strength and durability of pumice concrete at high temperature. Journal of Materials in Civil Engineering 2006;18(4):527–36. [6] Sung-Hoon K, Eun-chae J, Dongil K. Determining Brinell hardness from analysis of indentation load–depth curve without optical measurement. Journal of Engineering Materials and Technology 2005;127(1): 154–158. [7] Zickler GA, Schoberl T, Paris O. Mechanical properties of pyrolysed wood: A nanoindentation study. Philosophical Magazine A 2006;86: 1373–86. [8] Sneddon IN. The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile. International Journal of Engineering Science 1965;3:47–57. [9] Oliver WC, Pharr GM. An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. Journal of Materials Research 1992;7:1564–83. [10] Briscoe BJ, Sebastian KS, Adams MJ. The effect of indenter geometry on the elastic response to indentation. Journal of Physics D: Applied Physics 1994;27:1156–62. [11] Stilwell NA, Tabor D. Elastic recovery of conical indentations. Proceedings of the Physical Society 1961;78:169–79. [12] Doerner MF, Nix WD. A method for interpreting the data from depthsensing indentation instruments. Journal of Materials Research 1986;1: 601–9. [13] Oliver WC, Pharr GM. Measurement of hardness and elastic modulus by instrumented indentation: Advances in understanding and refinements to methodology. Journal of Materials Research 2004;19(1):3–20. [14] Pharr GM, Oliver WC, Brotzen FR. On the generality of the relationship among contact stiffness, contact area and elastic modulus during indentation. Journal of Materials Research 1992;7(3):613–7. [15] Marx DT, Riester L. Mechanical properties of carbon–carbon composite components determined from nanoindentation. Carbon 1999;37: 1679–1684.
Nondestructive evaluation of material strength using depth-sensing indentation Jale Tezcan ∗ , Kent J. Hsiao Department of Civil and Environmental Engineering, Southern Illinois University, 1230 Lincoln Dr., 62901, Carbondale, IL, United States Available online 15 August 2007
Abstract Successful application of any structural repair or strengthening method requires knowledge of strength and stiffness properties of the materials used in the construction. Depth-sensing indentation is a relatively new material testing method that significantly expands the capabilities of conventional hardness tests. Due to the mathematical complexity of the indentation problem, an analytical solution of the load–penetration depth (P–h) behavior is not available for most indenters. This study derives the P–h equations for a truncated cone indenter with arbitrary tip radius and included angle, which covers a wide range of geometries including the cylinder and the cone. A step by step procedure for material strength evaluation is introduced and validated using recorded indentation data. The analytical approach described in this paper can be used in evaluating the modulus of elasticity, stiffness and hardness of common construction materials without resorting to empirical equations. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Nondestructive testing; Depth-sensing indentation; Elastic modulus; Stiffness; Hardness
1. Introduction A variety of hardness tests are performed in industry to predict strength and durability characteristics of a myriad of materials at different scales. However, these tests have several drawbacks that limit their range of applicability and effectiveness. One of the problems in traditional hardness testing is the difficulty in interpreting the test results. Since hardness is not a fundamental material property, results from different hardness tests are not directly comparable. Empirical equations or conversion tables are required to evaluate the material properties of interest. The fact that most hardness tests are destructive is another factor that limits their in-situ application. Moreover, penetration resistance tests require that the residual imprint be measured optically, which introduces potential errors due to operator judgment. Application of the conventional hardness tests to assess material strength for historical structures presents additional challenges. To protect the cultural heritage, the testing has to be carried out in a minimally invasive way. ∗ Corresponding author. Tel.: +1 618 453 6125; fax: +1 618 453 3044.
E-mail address: [email protected] (J. Tezcan). c 2007 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2007.06.011
Depth-sensing indentation is a relatively new approach that offers a virtually nondestructive means of material characterization. Since only small volumes of materials are required, spatial variation of surface mechanical properties can be evaluated by this method. The hardness and the modulus of elasticity are the properties that can most readily be obtained from an indentation test [1,2]. In the past two decades, numerous researchers have developed various procedures to evaluate additional mechanical properties such as yield stress, bond strength, strain hardening, strain rate sensitivity and material damping from the indentation data. Recently, there have been efforts to incorporate indentation testing in structural materials. Zhu et al. [3] studied the bond and interfacial properties around steel reinforcement in selfcompacting concrete using a depth-sensing nanoindentation technique. Trtik [4] used the microindentation test to analyze the variation of modulus of elasticity and creep of an ordinary Portland cement matrix as a function of the distance from the closest aggregate. Hossain [5] investigated the durability of pumice concrete at high temperatures using a microindentation method. Indentation testing has also been successfully applied in measuring strength properties of structural steel [6] and wood [7].
J. Tezcan, K.J. Hsiao / Engineering Structures 30 (2008) 2206–2210
The analysis of load–indentation depth (P–h) curves is often based on Sneddon’s elastic contact model [8]. Oliver and Pharr [9] used Sneddon’s analysis to develop a procedure to extract the modulus of elasticity, hardness and stiffness of the indented material. Due to the mathematical complexity of contact analysis, analytical formulas describing the indentation behavior are available for only a few simple geometries; equations for the rest are usually determined empirically through a curve fitting procedure. Briscoe [10] investigated the penetration behavior of indenters with tip defects and concluded that the curve fitting approach should be used with caution since the load–depth behavior is highly sensitive to the geometry of the indenter tip. Careful selection of the indenter tip is very important in maximizing the reliability of the test results. The indenter tip geometry should be selected considering the expected behavior of the sample material and the objective of the test. Sharp indenters such as the pyramid and the cone are preferred in fracture toughness estimations, since they produce high stress and strain concentrations in the vicinity of the contact area, thereby facilitating crack formation. However, sharp indenters might initiate cracking at very low loads, making it difficult to determine the elastic properties. When testing brittle materials for stiffness and modulus of elasticity, high stress concentrations should be avoided. This can be achieved by lowering the applied load, or using an indenter with a larger tip area. It is also important to ensure full contact between the sample and the indenter tip, since the test results are analyzed assuming uniform stress distribution at the contact area. In addition, the geometry of the indenter should reveal the variation of the contact area as the indenter is withdrawn. A cylindrical punch, due to its flat tip, will help achieve uniform stress distribution; however, it has a constant cross sectional area. On the other hand, a cone has a well defined depth–area relationship; however, it will create high stress concentrations. A truncated cone with appropriate tip radius and included angle can overcome the limitations of both the cone and the cylinder. The purpose of this paper is to formulate a procedure for evaluating material properties to be used in conjunction with indentation testing. Specifically, we derive the P–h equations for a truncated cone indenter with arbitrary tip radius and included angle. By varying these parameters, P–h behavior for a range of axisymmetric geometries can be obtained. This paper starts with a general introduction to material characterization using depth-sensing indentation. In Section 3, the P–h equations for a truncated cone are derived. Section 4 presents verification of the derived equations for two special cases with known closed-form solutions. A step by step procedure for material strength evaluation is introduced in Section 5. Finally, a practical application of the procedure is demonstrated in Section 6. 2. General procedure of material characterization using indentation testing During an indentation experiment, the penetration depth (h) is measured as the indenter is driven into and withdrawn
2207
Fig. 1. Schematic representation of indentation with a truncated cone.
Fig. 2. Typical load–penetration depth data from indentation tests.
from the material under controlled loading. Fig. 1 shows the geometric model of a truncated cone indenting a sample material. As the indenter is driven into the material, both elastic and plastic deformations occur. When the indenter is removed, the elastic portion of the deformation is recovered. Typical outcome of an indentation test is a load–penetration depth curve as shown in Fig. 2. Sneddon [8] derived the P–h equations for axisymmetric indenters and showed that the penetration depth (h), is related to the shape function of the indenter f (x), through the equation Z 1 f 0 (x) h= dx, (1) √ 1 − x2 0 where x is a dimensionless parameter, and the load required to produce this penetration is Z 1 2 0 x f (x) P = 2Er rc dx. (2) √ 1 − x2 0 The reduced modulus of elasticity (Er ) is a property that accounts for bidirectional displacements in both the indenter and the sample, and is given by [11] 1 − νi2 1 − νs2 1 = + , Er Ei Es
(3)
where E i , νi and E s , νs are the moduli of elasticity and the Poisson’s ratios of the indenter and the sample, respectively. The hardness (H ) of a material is a measure of resistance to deformation. One common way of quantifying the hardness
2208
J. Tezcan, K.J. Hsiao / Engineering Structures 30 (2008) 2206–2210
Dividing the radial distance by the contact radius (rc ), and applying the change of variables x = rrc , the shape function can be rewritten in terms of the dimensionless parameter x. The resulting shape function and its derivative are given in Eqs. (7) and (8), respectively ri 0 for 0 < x ≤ r c f (x) = (7) ri
consists of evaluating the mean contact pressure at maximum load through the formula Pmax , (4) A where Pmax and A are the maximum applied load and the projected area of contact at that load, respectively. Determining A from optical measurements is a challenge even with modern imaging instruments. In indentation testing, the availability of continuous load–penetration data makes it possible to calculate A empirically, by using a function which relates the contact area to the distance from the indenter tip. The stiffness (S) of the tested material can be determined from the unloading part of the indentation data, since unloading represents elastic recovery of the deformation [12,13]. In most cases, S can be determined with sufficient accuracy by fitting a curve to the upper 25% of the unloading data. Once A and S are known, the reduced modulus of elasticity (Er ) can be solved using equation [14] H=
2 √ S=√ AEr . π
(5)
Finally, the modulus of elasticity of the material (E s ) is calculated using Eq. (3). It is important to note that the indentation testing will evaluate the material properties only at the locations that come into contact with the indenter. Strength and stiffness properties of composite materials can be determined by testing the individual components and using the appropriate rule of mixtures. 3. Analytical derivation of P–h relationship for a truncated cone Since the truncated cone has an axisymmetric geometry, Sneddon’s method [8] can be used to derive the P–h relationship. For the indenter shown in Fig. 1, the tip radius and the included half angle are denoted by ri and ψ, respectively. The indenter shape function can be expressed as a function of the radial distance (r ), as 0 if r < ri f (r ) = (6) (r − ri ) cot(ψ) if r > ri .
To simplify the foregoing analysis, we introduce the ri −1 dimensionless contact parameter, φ = cos rc . It is obvious from the geometry of penetration that the possible values of φ range from 0 to π2 radians. The penetration depth corresponding to the parameter φ is found by substituting Eq. (8) in Eq. (1) as h = ri cot(ψ)
φ . cos(φ)
(9)
The load required to produce this penetration is calculated from Eq. (2) as φ P = ri2 cot(ψ)Er tan(φ) + . (10) cos2 (φ) Eqs. (9) and (10) define the elastoplastic P–h behavior for the truncated cone. The half included angle (ψ) can be eliminated between Eqs. (10) and (9) to yield sin(φ) 1 P = E r ri h + , (11) φ cos(φ) where P is the applied load, h is the indenter displacement, Er is the reduced modulus of elasticity, ri is the indenter tip radius and φ is the dimensionless contact parameter. 4. Verification of the derived P–h equations for two special cases To examine the validity of the equations derived in Section 3, the P–h behavior for varying indenter shape parameters, i.e. tip radius and half angle, is depicted in Fig. 3. All three axes have been put into non-dimensional form. The axes x, y and z represent the dimensionless contact parameter φ, the depth parameter rhi , and the load parameter P 2 , respectively. E r ri
The boundaries defined by φ = 0 and φ = π2 are of particular interest since they correspond to the cylinder and the cone, respectively. Closed form P–h expressions for the two shapes are Pcylinder = 2Er ri h
(12)
2 Er tan(ψ)h 2 . (13) π These two geometries will be analyzed as special cases of the derived solution. Pcone =
J. Tezcan, K.J. Hsiao / Engineering Structures 30 (2008) 2206–2210
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Fig. 5. Variation of contact radius with load.
Fig. 4. Load–penetration data showing indentation with a truncated cone.
Case I: The value φ = 0 corresponds to the cylinder, since the contact radius has to be equal to the indenter tip radius, i.e., −1 ri = 0 ↔ r c = ri . (14) φ = cos rc
be expressed in terms of S by eliminating A between Eqs. (4) and (5) to yield the formula
The limit of Eq. (10) as φ → 0 is sin(φ) 1 P = Er ri h lim + = 2Er ri h, φ→0 φ cos(φ)
The procedure for evaluating the strength of material tested under a truncated cone indenter can be summarized as follows: (15)
which is the same as the closed form P–h equation for a cylinder as given in Eq. (12). Case II: The value φ = π2 corresponds to the cone, since the tip radius has to be zero, i.e., ri π φ = cos−1 = ↔ ri = 0. (16) rc 2 The P–h equation is obtained using Eq. (9) in Eq. (11) and taking the limit as ri → 0, sin(φ) h tan(ψ) 2 + P = lim Er ri h = Er tan(ψ)h 2 , (17) ri →0 φ ri φ π which is the same as the closed form P–h equation for a cone as given in Eq. (13). 5. Procedure of material characterization using a truncated cone The equations derived in Section 3 allow evaluation of material strength without resorting to empirical equations. The modulus of elasticity (E s ) can be directly calculated from the indentation data, as opposed to the traditional method described in Section 2, where errors associated with empirical determination of A are propagated into the calculation of E s . This is an obvious advantage over the traditional method, since accuracy of the E s is critical in evaluating structural safety. Direct calculation of the modulus of elasticity also allows determining the hardness (H ) without explicitly calculating the projected area of contact (A). Since the stiffness (S) is readily determined from the upper portion of the unloading data, H can
H=
Pmax π
2Er S
2
.
(18)
(a) Determine Pmax and h max values from the indentation data. (b) Determine S from the slope of the initial unloading curve. (c) Calculate φ, Er and E s from Eqs. (9), (11) and (3), respectively. (d) Calculate H from Eq. (18). 6. A practical example To demonstrate how the procedure described in Section 5 can be used in evaluating material strength properties, a single indentation testing was performed on a carbon–carbon composite sample (K196A). This material was previously tested by other researchers under a Berkovich indenter [15]. This particular material was chosen to allow direct comparison between the indentation test results under two different indenters. R The experiment was conducted using the Nano Indenter XP system. A diamond tip indenter with radius ri = 2.5 µm and included half angle of ψ = 30◦ was used. For the diamond indenter, the modulus of elasticity and the Poisson’s ratio are E i = 1141 GPa and υi = 0.07, respectively. The recorded indentation data and the variation of contact radius with loading are shown in Figs. 4 and 5, respectively. The maximum penetration depth h max = 1425 nm and the maximum load Pmax = 118.3 mN can be read from Fig. 4. The stiffness of the material is found from the initial slope of the unloading curve. Fitting a line to the upper 25% portion of the unloading data, the stiffness is found to be S = 192 kN/m. The dimensionless contact parameter is calculated using Eq. (9) as φ = 0.33456. The reduced elastic modulus is found from Eq. (11) as Er = 16.2 GPa. To determine the elastic modulus of the material from Eq. (3), prior knowledge of Poisson’s ratio
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is needed. However, a rough estimate of the Poisson’s ratio will not introduce a large error since most engineering materials, as well as carbon–carbon composites, have a Poisson’s ratio between 0.20 and 0.35. Using Eq. (3), the moduli of elasticity corresponding to υs = 0.2 and υs = 0.35 are found to be E s = 14.63 GPa and E s = 13.37 GPa, respectively. Finally, the hardness is determined from Eq. (18), as H = 1.08 GPa. The reported values of the modulus of elasticity, the stiffness and the hardness of the tested material are E s = 15.8 GPa, S = 171 kN/m and H = 1.43 GPa, respectively [15]. The modulus of elasticity and the stiffness values calculated using the method described in this paper are in good agreement with the reported values. The discrepancy in the hardness values should not raise much concern since hardness is not a fundamental material property but a relative measure of deformation resistance of materials. 7. Conclusion In this paper, the load–penetration depth (P–h) equations for a truncated cone indenter with arbitrary tip radius and included angle were derived. The validity of the equations was demonstrated by considering two indenters, the cone and the cylinder, for which closed form equations are available in the literature. A step by step procedure for material strength evaluation was introduced and validated using recorded test data. It was concluded that the procedure described in this paper offers a reliable method for evaluating the modulus of elasticity, stiffness and hardness of materials, without resorting to empirical data. References [1] Newey AD, Wilkins AMA, Pollock AHM. An ultra-low-load penetration hardness tester. Journal of Physics E: Scientific Instruments 1982;15: 119–22.
[2] Pethica JB, Hutchings R, Oliver WC. Hardness measurements at penetration depths as small as 20 nm. Philosophical Magazine A 1983; 48(4):593–606. [3] Zhu W, Sonebi M, Bartos P. Bond and interfacial properties of reinforcement in self-compacting concrete. Materials and Structures 2004;37:442–8. [4] Trtik P, Bartos PJM. Micromechanical properties of cementitious composites. Materials and Structures 1999;32:388–93. [5] Hossain KMA. Macro and microstructural investigations on strength and durability of pumice concrete at high temperature. Journal of Materials in Civil Engineering 2006;18(4):527–36. [6] Sung-Hoon K, Eun-chae J, Dongil K. Determining Brinell hardness from analysis of indentation load–depth curve without optical measurement. Journal of Engineering Materials and Technology 2005;127(1): 154–158. [7] Zickler GA, Schoberl T, Paris O. Mechanical properties of pyrolysed wood: A nanoindentation study. Philosophical Magazine A 2006;86: 1373–86. [8] Sneddon IN. The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile. International Journal of Engineering Science 1965;3:47–57. [9] Oliver WC, Pharr GM. An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. Journal of Materials Research 1992;7:1564–83. [10] Briscoe BJ, Sebastian KS, Adams MJ. The effect of indenter geometry on the elastic response to indentation. Journal of Physics D: Applied Physics 1994;27:1156–62. [11] Stilwell NA, Tabor D. Elastic recovery of conical indentations. Proceedings of the Physical Society 1961;78:169–79. [12] Doerner MF, Nix WD. A method for interpreting the data from depthsensing indentation instruments. Journal of Materials Research 1986;1: 601–9. [13] Oliver WC, Pharr GM. Measurement of hardness and elastic modulus by instrumented indentation: Advances in understanding and refinements to methodology. Journal of Materials Research 2004;19(1):3–20. [14] Pharr GM, Oliver WC, Brotzen FR. On the generality of the relationship among contact stiffness, contact area and elastic modulus during indentation. Journal of Materials Research 1992;7(3):613–7. [15] Marx DT, Riester L. Mechanical properties of carbon–carbon composite components determined from nanoindentation. Carbon 1999;37: 1679–1684.