International Journal of Solids and Structures

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1 International Journal of Solids and Structures 48 (2011) Contents lists available at ScienceDirect International Journal of Solids and Structures journal homepage: Numerical and experimental indentation tests considering size effects E. Harsono a, S. Swaddiwudhipong a,, Z.S. Liu b,, L. Shen c a Department of Civil and Environmental Engineering, National University of Singapore, E1A-07-03, 1 Engineering Drive 2, Singapore , Singapore b Institute of High Performance Computing, A STAR, 1 Fusionopolis Way, #16-16 Connexis, Singapore , Singapore c Institute of Materials Research and Engineering, A STAR, 3 Research Link, Singapore , Singapore article info abstract Article history: Received 14 July 2009 Received in revised form 10 November 2010 Available online 13 December 2010 Keywords: Conventional mechanism based strain gradient plasticity Finite element method Indentation size effect Indentation test Simulated indentation A series of nanoindentation experiments with maximum depths varying from 1200 to 2500 nm were conducted to study indentation size effects on copper, aluminium alloy and nickel. As expected, results from classical plasticity simulation deviate significantly from experimental data for indentation at micron and submicron levels. C 0 continuity finite element analysis incorporating the conventional theory of mechanism-based strain-gradient (CMSG) plasticity has been carried out to simulate spherical and Berkovich indentation tests at micron level where size effect is observed. The results from both numerical and actual spherical and Berkovich indentation tests demonstrate that materials are significantly strengthened for deformation at this level and the proposed approach is able to provide reasonably accurate results. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction Material characterization using instrumented indentation tests has been extended to new applications as a result of technological advances in microelectronics and nano-technology. Not only hardness of material but also other mechanical properties such as Young s modulus, yield strength and strain hardening exponent can be deduced from load displacement indentation curves. Most of indentation tests have been intensively conducted at indentation depths from micron down to submicron levels to accommodate the needs of material properties of small volumes in the fields of MEMS and NEMS. In many of these applications, material properties are shown to be inconsistent with those provided by classical plasticity approach, exhibiting a strong size effect. Gains in strength at such small deformation comparable to the material length scales have been reported for many tests on metallic materials. Numerous experiments (micro- and nanoindentation tests (see e.g. Atkinson, 1995; Ma and Clarke, 1995; Nix, 1989; Stelmashenko et al., 1993); twisting of copper wires of micron diameters by Fleck et al. (1994) micro-bend tests by Haque and Saif (2003)) have shown significant size-dependent effects when the material and deformation length scales are of the same order at micron and submicron levels. Finite element simulations employing classical plasticity theories are unable to capture these size-dependent effects. The size effects cannot be simulated via Corresponding authors. Tel.: / ; fax: addresses: cvesomsa@nus.edu.sg (S. Swaddiwudhipong), liuzs@ihpc. a-star.edu.sg (Z.S. Liu). classical plasticity theories as no material length scale is introduced. Fleck et al. (1994) proposed the theory of strain gradient plasticity requiring additional higher-order stress and consequently leading to significantly greater formulation and computational efforts. Gao et al. (1999) and Huang et al. (2000) proposed the mechanism-based strain gradient (MSG) plasticity guided by the Taylor dislocation concept to model the indentation size effect. Huang et al. (2004) further developed the conventional mechanism-based strain gradient (CMSG) plasticity theory confining the presence of the strain gradient plasticity in the material constitutive equation without involving the higher-order stress components. Adopting this approach, Swaddiwudhipong et al. (2005, 2006) formulated C 0 continuity solid, plane and axisymmetric finite elements incorporating strain gradient plasticity to simulate various indentation tests and other physical problems involving deformation at micron and submicron levels. Alternatively, the strain gradient plasticity may also be determined via the differences in numerical values of the plastic at various locations. The formulation was derived based on the classical continuum plasticity framework taking into consideration Taylor dislocation model. Higher order variables and consequently higher-order continuity conditions are not required and the direct application of conventional plasticity algorithms in finite element modelling is applicable. Indentation size effect (ISE) has been studied extensively for both sharp and spherical indentation tests. The measured hardness of metallic materials increases with decreasing indentation depth for conical and Berkovich tips (McElhaney et al., 1998; Nix and Gao, 1998; Oliver and Pharr, 1992; Stelmashenko et al., 1993; /$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi: /j.ijsolstr

2 E. Harsono et al. / International Journal of Solids and Structures 48 (2011) Tho et al., 2006) and decreasing indenter radius for spherical indenters (Lim and Chaudhri, 1999; Spary et al., 2006; Swadener et al., 2002). Tho et al. (2006) performed experimental and numerical studies on copper and aluminium alloy Al7075 to investigate the size effect of Berkovich indentation tests. Their findings showed that the strength of indented materials increased when indentation depth reduced. Another ISE study was conducted by Zong et al. (2006) on fcc single crystals (Ni, Au and Ag). They presented nano- and micro-indentation test results and theoretical study of indentation size effects for those crystalline materials. In their study, a three-sided pyramidal Berkovich tip was used as the indenter for nano-indentation tests while a Vicker diamond tip used for micro-indentation tests. They employed MSG theories proposed by Gao et al. (1999), Huang et al. (2000), and Nix and Gao (1998) to study the size dependence of the crystals at submicron levels. Strong size effects in the hardness were observed in all specimens. The ISE has also been studied using spherical indenters (Swadener et al., 2002; Qu et al., 2004, 2005; Spary et al., 2006; Hou et al., 2008). Lim and his co-workers (1999) have reported that the size effect increases with decreasing indenter radius as observed in polycrystalline and single crystal oxygen free copper. Swadener et al. (2002) have proposed that the size effects observed in conical indentation can be related to those of spherical indentation using the contact radius. They found that the size effect is a function of the indentation depth for sharp indenter tips (e.g. conical and Berkovich) and the indenter tip radius for a spherical indenter depending on the expression of the average geometrically necessary dislocation density. Qu et al. (2004) implemented CMSG in order to study the ISE when indentation depths approaching the nanometer scale. Qu et al. (2005) reported the size effect in the spherical indentation of iridium. They proposed an analytical spherical indentation model to predict the indentation hardness of indented materials. In the present study, spherical indentation tests were conducted on copper and aluminium alloy Al7075. Indentation tests were designed for various maximum indentation depths of 1200, 1800 and 2500 nm. The ISE for spherical indentation test on copper and the aluminium alloy Al7075 reported here was done in the same framework adopted as for the Berkovich indentation size effect reported earlier by Tho et al. (2006). Another series of experimental study of size effects were conducted on nickel by using a three-sided pyramidal Berkovich tip for various depths of indentation ranging from 350 to 2500 nm. The objective of the study is to verify that the CMSG model incorporating the strain gradient effect be able to simulate indentation size effects observed in the experimental results of pure metals and metallic alloys, especially in copper, Al7075 and nickel. 2. Numerical model Two-dimensional axisymmetric finite elements were adopted to model the target materials for simulated spherical indentation tests and three-dimensional elements for Berkovich indentation tests in the present study. The far field effect and convergence study were carried out. The former study showed that a domain size of 100 micron by 100 micron is sufficiently large to simulate indentation tests using a spherical indenter tip with radius of 5 - microns. The domain size of micron by 200 micron by 150 micron for length AH, HI and AJ respectively indicated in Fig. 1 is required to safely avoid the boundary effect near the indenter tip. Based on the convergence study, a total of 8328 CAX8 elements were used in the formulation of the spherical indentation of 5-micron radius tip. On the other hand, to simulate the indentation by Berkovich indenter possessing a threefold symmetry, only one-sixth of the target materials had to be considered in the 3D model. The finite element mesh for the target material comprising 5338 second-order solid elements (C3D20) was adopted in the latter. In this study, the indenter was modelled as a rigid body while the target as a deformable body. The penalty approach was employed to model the contact problem between the indenter and the target. A constant value Poisson s ratio of 0.3 and a friction coefficient of 0.15 between the contact surfaces were adopted for both simulated spherical and Berkovich indentation tests. A finer mesh was used near the contact region where high stress gradient was expected and the element size was gradually coarser elsewhere. 3. Size effect via conventional mechanism-based strain gradient plasticity The dislocation density can be related to the shear strength by the Taylor (1938) dislocation model: p s ¼ alb q ffiffiffiffiffi T ð1þ wheres is the shear strength, l the shear modulus, a an empirical constant, b the Burgers vector and q T the total dislocation density. The total dislocation density q T comprises the statistically stored dislocation density (SSD), q S, and the geometrically necessary dislocation (GND) density, q G. CMSG plasticity was formulated by Huang et al. (2000, 2004) guided by the Taylor dislocation model. The flow stress of Taylor dislocation model can be expressed as follows qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r f ¼ f 2 ðe p Þþlg p ð2þ l ¼ rb Mal 2 ð3þ where is the reference stress, f(e p ) represents the stress and plastic strain relationship in uniaxial tension, l the material length scale and g p the effective plastic strain gradient. For facecentered-cubic materials for a random polycrystal, M = 3.06 (Bishop and Hill, 1951) and for r ¼ 1:90 (Arsenlis and Parks, 1999; Huang et al., 2000), the material length scale can be rewritten as 2 l ¼ 18b ð4þ al Fig. 1. Typical Berkovich indentation model.

3 974 E. Harsono et al. / International Journal of Solids and Structures 48 (2011) The variation of the value of M for other materials structures can be incorporated into the values of other parameters such as a. The material length scale in (4) represents three combined factors i.e. elasticity denoted through the shear modulus l, plasticity by the presence of the yield strength r Y and dislocation as expressed via the magnitude of Burger vector b, to capture the essential components affecting the behaviour of material during its transition from elastic condition to plastic state at submicron level. The plastic strain rate, _e p, can be expressed in the power law visco-plastic model as presented by Hutchinson (1976) and Kok et al. (2002). _e p ¼ _e r m e ð5þ r f rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 _e ¼ e 3 _ 0 ij _e 0 ij ð6þ _e 0 ij ¼ _e ij 1 e 3 _ kk d ij ð7þ where r e is the von Mises effective stress, r Y the initial yield stress, _e effective strain rate, m = 20 in this study, is the rate-sensitivity exponent and _e 0 ij the deviatoric strain rate. The stress-plastic strain relation for a power law hardening solids is expressed as N r Y f ðe p Þ¼r Y 1 þ ep ð8þ e Y where N denotes the plastic work hardening exponent, the values of which vary from 0.0 to 0.6. After substituting the uniaxial flow stress by the flow stress and incorporating the strain gradient effects, the plastic strain rate can be rewritten as _e p ¼ _e r m e ¼ _e r f r Y r e 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 5 f 2 ðe p Þþln p The deviatoric strain rate can be derived from J 2 -flow theory of plasticity (Huang et al., 2000).! _e 0 ij ¼ _e ij 1 e 3 _ kk d ij ¼ 1 2l _r0 ij þ 3 m _e r e pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2r 0 e f 2 ðe p Þþlg p ij ð10þ m While the stress rate can be expressed as "! _r ij ¼ K _e kk d ij þ 2l _e 0 ij 3 m # _e r e pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2r 0 e f 2 ðe p Þþlg p ij ð9þ ð11þ More details of effective strain gradient derivations for the axisymmetric elements and solid elements were reported earlier by Swaddiwudhipong et al. (2005, 2006). Swadener et al. (2002) derived the average geometrically necessary dislocation density by assuming that the dislocations are distributed approximately in a hemispherical volume, V =2pa 3 /3. Therefore, the average geometrically necessary dislocation density as a function of indenter geometry can be written as q G ¼ 3nAð2=nÞ bðn þ 1Þ hð1 2=nÞ ð12þ where A is a constant, b the magnitude of the Burger s vector, n the indenter geometry and h the indentation depth. The values of n describe the indenter geometries. The value of n = 1 implies conical and pyramidal indenters whereas n = 2 signifies the spherical tips and when n = 1 represent the flat punches. 4. Experimental details We conducted the nanoindentation tests using Nano Indenter Ò XP system available at the Institute of Materials Research and Engineering (IMRE), Singapore. The experiments were designed carefully to study the indentation size effects of spherical indenter tips on copper and the aluminium alloy Al7075 while the tests with a standard Berkovich tip were performed to study the size effect of nickel. The preparation of the specimens involved the casting of the sample materials in the epoxy resin in the form of cylinders of about 3 cm in diameter and 1 cm in depth. In order to prevent the inconsistency of the test results via minimizing the effect of surface roughness and the formation of the surface oxide, the samples were subjected to wet mechanical grinding using #500, #2400 and #4000 silicon carbide papers and subsequently mechanically polished using 6, 3 and 1 lm diamond suspensions followed by the polishing using 0.1 lm silica solution. A diamond spherical indenter with a tip radius of 5 lm was used to indent the copper and Al7075 samples to various maximum depths varying from 1200 to 2500 nm. The nickel samples, on the other hand, were indented with a standard Berkovich tip at maximum depths ranging from 350 to 2500 nm. Experimental sequence comprises the following: (1) loading at a constant loading rate of 1 mn/s to maximum load (or indentation depth); (2) unloading at the same rate to 10% of the maximum load; (3) holding the load for 100sec to monitor the drift of the system setup (the thermal drift correction will be applied to all data points after testing); and (4) withdrawing of the indenter tip from the sample. Displacements and loads were measured with a resolution of nm and 0.05 ln, respectively. Each series of tests at a specified maximum depth comprises ten repeated indentation tests. Except for one case for indentation on nickel at maximum depth of 1500 nm, there are at least seven sets of consistent test results for each series. The interval of 100 lm was maintained between two adjacent indentations throughout the experiments. 5. Results and discussions Numerical analyses were conducted using C 0 continuity axisymmetric finite element incorporating strain gradient plasticity to simulate the response of copper and Al7075 under spherical indentation tests at various maximum depths. While C 0 continuity 3D solid elements for Berkovich indentation model adopting the same strain gradient plasticity theory were employed to study size effects of nickel. Material properties for copper are E = GPa, r Y = 78.4 MPa, N = 0.3 (Dao et al., 2001) and E = 70.1 GPa, r Y = 500 MPa, N = for Al7075 (Qiu et al., 2003). Material properties for nickel are found to be E = 207 GPa, r Y = 80 MPa, N = 0.24 based on its uniaxial stress strain curve. The values of the magnitude of Burger vector of 0.25 nm for these materials as reported by Qiu et al. (2003) and Huang et al. (2006) were adopted in this study. The values of a in the intrinsic material length formula in (4) are required but are not usually available in the literature as the values are influenced by several parameters including the types and structures of materials and the actions applied onto the materials. In this study, the values of material length scales were iteratively established via experimental results. They were identified to be respectively lm and lm for copper and Al7075 under spherical indentation and lm for nickel subjected to Berkovich indentation. By adopting the same value of material length scales for the same target materials under the indentation tests using the same indenter tips, the CMSG model is able to demonstrate reasonably accurately the size effects on the indentation tests at various indentation depths conducted and reported in the present study. Figs. 2 and 3 depict the comparison of numerical simulation employing classical and CMSG plasticity theory and experimental results obtained from spherical indentation tests for copper and Al7075. While the comparisons of results for

4 E. Harsono et al. / International Journal of Solids and Structures 48 (2011) Fig. 2. Numerical and test results using spherical indenter tip for copper with a maximum depth of (a) 1200 nm (b) 1800 nm (c) 2500 nm. Berkovich indentation tests on nickel are illustrated in Fig. 4. It can be shown that classical plasticity solutions deviate significantly from experimental data. On the contrary, results from numerical model incorporating CMSG plasticity theory are able to predict rather accurately the experimental values. Small deviations at the initial part on the loading curve of actual spherical indentation tests are observed in Figs. 2 and 3. These deviations are most likely due to the presence of a thin oxide layer formed on the surface of the specimens. As copper oxides are easier to form than those of the aluminium alloy Al7075 and hence the oxidation layer of the former is thicker than the latter resulting Fig. 3. Numerical and test results using spherical indenter tip for Al7075 with a maximum depth of (a) 1200 nm (b) 1800 nm (c) 2500 nm. in more loads needed to indent the copper specimen at the beginning of indentation test as compared to that of Al7075, as shown in Figs. 2 and 3. This observation is usually less apparent when sharp indenter tips are employed as the oxide layer is mostly cut through in the latter case. Smooth loading curves and no apparent deviations were noted in all Berkovich indentation tests on nickel, as depicted in Fig. 4. Strong size effects were observed for both spherical and Berkovich indentation tests on copper, aluminium alloy Al7075 and nickel, as illustrated in Figs It is interesting to note that the deviations of spherical indentation curves predicted by the classical plasticity theory and experimental results for different

5 976 E. Harsono et al. / International Journal of Solids and Structures 48 (2011) Fig. 4. Numerical and test results using standard Berkovich indenter tip for nickel with a maximum depth of (a) 350 nm (b) 1000 nm (c) 1500 nm (d) 2000 nm e) 2500 nm. depths are almost the same. This implies that size effects in spherical indentation tests are independent of the indentation depth. The conclusions support Swadener et al. (2002) s stipulation that the hardness of material indented by a spherical indenter tip is not affected by the indentation depth, but observed to rise with the decrease of the radius of indenter tip. The phenomenon can be explained through the presence of the average geometrically necessary dislocation density q G, which is a function of the indenter geometry as expressed in (12) (Swadener et al., 2002). A spherical indenter can be approximated by a parabolic geometry with n = 2. By substituting n = 2 to. 12, q G is no longer a function of h, and therefore, the size effects demonstrated in Figs. 2, 3 are

6 E. Harsono et al. / International Journal of Solids and Structures 48 (2011) independent of the indentation depth. However, for pyramidal, Berkovich and conical tips (n = 1), the indentation size effect varies with the indentation depth, h. The deviation of load indentation curves predicted by the classical plasticity theory from experimental results becomes larger when the indentation depth reduces from 2500 to 350 nm for the latter case. Fig. 4 confirms that the size effect for pyramidal and conical tips depends on the indentation depth and further supports the validity of (12). 6. Conclusions In this study, spherical indentation tests were conducted on copper and the aluminium alloy Al7075 and Berkovich indentation tests for nickel to investigate the size effect phenomenon. These materials are observed to gain in strength and hardness values when the tests were conducted at a few hundreds to thousands of nanometers. Numerical models employing material length scale derived based on the CMSG theory and those of classical models were adopted in the analyses and the results compared with experimental values. Classical plasticity simulations produce results with significantly lower indentation response while those obtained from CMSG model incorporating the strain gradient plasticity demonstrate reasonably good agreement with experimental results for both spherical and Berkovich indentation tests. The study indicates that incorporating the effects of strain gradient plasticity in the formulation of governing equation is essential for accurate simulation when the material and characteristic length scales of non-uniform plastic deformation are of the same order at micron or sub-micron level. Acknowledgements The authors gratefully acknowledge the contribution of Dr. Tho Kee Kiat and the financial supports from the Singapore Ministry of Education s ACRF Tier 1 Funds through Grant R References Arsenlis, A., Parks, D.M., Crystallographic aspects of geometricallynecessary and statistically-stored dislocation density. Acta Materialia 47, Atkinson, M., Further analysis of the size effect in indentation hardness tests of some metals. Journal of Materials Research 10, Bishop, J.F.W., Hill, R., A theory of plastic distortion of a polycrystalline aggregate under combined stresses. Philosophical Magazine 42, Dao, M., Chollacoop, N., Van Vliet, K.J., Venkatesh, T.A., Suresh, S., Computational modeling of the forward and reverse problems in instrumented sharp indentation. Acta Materialia 49, Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W., Strain gradient plasticity: theory and experiment. 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International Journal of Solids and Structures 43, Swadener, J.G., George, E.P., Pharr, G.M., The correlation of the indentation size effect measured with indenters of various shapes. Journal of the Mechanics and Physics of Solids 50, Taylor, G.I., Plastic strain in metals. Journal of the Institute of Metals 13, Tho, K.K., Swaddiwudhipong, S., Hua, J., Liu, Z.S., Numerical simulation of indentation with size effect. Materials Science and Engineering A 421, Zong, Z., Lou, J., Adewoye, O.O., Elmustafa, A.A., Hammad, F., Soboyejo, W.O., Indentation size effects in the nano- and micro-hardness of fcc single crystal metals. Materials Science and Engineering A 434, Edy Harsono graduated from Institut Teknologi Bandung (ITB) in 2004 and thereafter he worked as a structural engineer in Laboratory of Structures and Materials, ITB. In 2005, he continued his postgraduate study at National University of Singapore. Edy received his PhD degree in civil engineering from the National University of Singapore in His research interests include material characterization, neural network and applied mechanics. Somsak Swaddiwudhipong is currently a Professor in the Department of Civil and Environmental Engineering at the National University of Singapore. His research work is mainly on Computational Mechanics, Material Characterization and Offshore Structures. He currently serves as a member of the editorial boards of the ASEAN Engineering Journal, International Journal of Applied Mechanics, and Structural Engineering Mechanics, an International Journal.

7 978 E. Harsono et al. / International Journal of Solids and Structures 48 (2011) Zishun Liu holds Bachelor of Eng. and Master of Eng. Degrees in Applied Mechanics and Solid Mechanics from Xi an Jiaotong University, and Master of Eng. and PhD Degrees in Structural Mechanics from the National University of Singapore. He is currently a Research Scientist in Institute of High Performance Computing, A STAR, Singapore. His research work is mainly on the Solid Mechanics, Computational Mechanics and Material Characterization. He is also an Editor-In-Chief of International Journal of Applied mechanics. Lu Shen is currently a senior research officer in the Materials Science and Characterisation Laboratory at Institute of Materials Research and Engineering, A STAR, Singapore. Her research work is on deformation behaviour of polymer nanocomposites (PNCs) and mechanical characterization of thin film and multi-layer structures.