Identification of model parameters from elastic/elasto-plastic spherical indentation

Thomas Niederkofler a, Andreas Jäger a, Roman Lackner b a Institute for Mechanics of Materials and Structures (IMWS), Department of Civil Engineering, Vienna University of Technology, Vienna, Austria b Material-Technology Innsbruck (MTI), Department of Civil Engineering, University of Innsbruck, Innsbruck, Austria Identification of model parameters from elastic/elasto-plastic spherical indentation Spherical indentation is commonly used to study the elastic and visco-elastic response of materials at different length scales. However, depending on the strength properties of the investigated material and the geometric properties, such as sphere radius and penetration depth, plastic material response might occur, making parameter identification from load penetration curves a difficult task. In this paper, the influence of the aforementioned parameters is investigated by means of numerical simulations, considering elastoplastic material response. The numerical results obtained are presented in dimensionless format, finally providing (i) general rules for the specification of test parameters in the case of spherical indentation, on the one hand, and in the case plastic deformation takes place (ii) relations for the determination of plastic model parameters (such as the deviatoric strength) from the loading phase of the indentation test. Keywords: Nanoindentation; Spherical indentation; Elastoplasticity; Parameter identification 1. Motivation Instrumented indentation is a well-known tool for the characterization of materials at various length scales, from the nanometer to the meter range. During indentation tests, a tip with defined shape (see Fig. 1a) penetrates the specimen surface with the applied load P(N) and the penetration h(m) recorded as a function of time. Commonly, each indent consists of a loading, holding, and unloading phase. In the most general case, the material response is characterized by elastic, plastic, and viscous behavior. In general, elastic material parameters are determined from the unloading branch of the load penetration history under the assumption of purely elastic unloading [1]. Parameter identification of materials exhibiting time-dependent behavior (e. g., polymers, bitumen, etc.) requires back calculation of model parameters from the holding phase of the measured penetration history [2 4]. For identification of both elastic and viscous parameters, analytical solutions describing the penetration history are available [5 13] and employed for parameter identification. As regards identification of plastic parameters, on the other hand, no analytical solutions for the penetration history exist. Hence, numerical methods need to be applied for the identification of model parameters from load penetration curves. Such a numerical approach has recently been applied for the special case of conical indenter shapes [14, 15], extracting so-called scaling relations for conical viscoelastic-cohesive indentation from numerical results. In this line, the scaling-relation approach given in [14, 15] is extended towards the case of spherical indenter shapes in this paper. For this purpose, the elastic as well as elasto-plastic material response is studied by means of numerical simulations and compared (for the elastic case) to existing analytical solutions. These analytical solutions for spherical indentation into an elastic halfspace are presented in the following section, including the presentation of the dimensionless format of the parameters used throughout this paper. In Section 3, the employed finite element (FE) model and the chosen simulation parameters (tip radii and material parameters) are given. The results from elastic and elasto-plastic simulations are presented in Section 4. In Section 5, the identification of plastic material param- 926 Int. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 7

BBasic Fig. 1. Contact between rigid axisymmetric indenter of shape f(q) and an infinite halfspace (P is the applied load, h is the penetration, and a is the contact radius) and load penetration curve and increase in contact radius for indentation of spherical tip into elastic halfspace given by the Hertz and the Sneddon solution (R =1m,E = 1 Pa, m = 0.3). eters from instrumented indentation tests on the basis of the numerical results obtained is discussed. Furthermore, the onset of plastic deformation during elasto-plastic spherical indentation is identified. 2. Analytical solutions for spherical penetration According to the literature, there exist two analytical models to describe the contact between a sphere and an elastic halfspace: 1. In [16], two elastic spheres are considered, assuming that (i) the elastic limit is not passed, (ii) the contact area is significantly smaller than the radii R 1 (m) and R 2 (m) of the spheres, and (iii) only normal stress acts at the contact surface. The relation between the contact force P(N) and the contact deformation h(m) is given by P ¼ KR 0:5 h 1:5 ð1þ where the contact curvature 1=R(m 1 ) and the contact modulus K(Pa) are defined as: 1 R ¼ 1 þ 1 and K ¼ 4 1 m 2 1 þ 1 1 m2 2 ð2þ R 1 R 2 3 E 1 E 2 with R 1 (m), m 1 ( ), E 1 (Pa) and R 2 (m), m 2 ( ), E 2 (Pa) as the radius, Poisson s ratio, and Young s modulus of the two spheres, respectively. For the case of a rigid sphere (R 1 ¼ R, E 1 ¼1) penetrating an elastic halfspace (R 2 ¼1, E 2 ¼ E), the contact modulus K reduces to K ¼ 4 E 3 ð1 m 2 Þ ¼ 4 3 M ð3þ where M is the so-called indentation modulus. Inserting Eq. (3) into Eq. (2) gives the applied load P as a function of penetration h for the Hertz solution as: P ¼ 4 E 3 ð1 m 2 Þ R0:5 h 1:5 ð4þ 2. The so-called Sneddon solution [5] describes the contact between an axisymmetric tip and an infinite elastic halfspace. Here, the tip shape is described by a smooth function f(q) (see Fig. 1a), where q is the radius of the axisymmetric tip. According to [5], the penetration h(m) and the applied load P(N) are given by: Z a f 0 ðqþ dq E q 2 f 0 ðqþ dq h ¼ a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and P ¼ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 q 2 1 m 2 a 2 q 2 q¼0 q¼0 (5) where a(m) is the radius of the projected contact area A c (m 2 ), f 0 ¼ df =dq, and E(Pa) and m( ) are Young s modulus and Poisson s ratio, respectively. In the case of a spherical tip with tip radius R(m), the parameters C 0 ( ) and C 1 (m) of the tip-shape function f ðqþ ¼ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C1 2 2C þ 4C 0q 2 p C 1 ð6þ 0 are C 0 = p( ) and C 1 =2Rp(m), yielding p f ðqþ ¼R ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 2 q 2 ð7þ Specializing Eq. (5) for the case of a spherical tip shape defined by the tip-shape function given in Eq. (7), one gets hðaþ ¼0:5a log R þ a ð8þ R a and E PðaÞ ¼ 2ð1 m 2 a 2 þ R 2 R þ a log 2aR ð9þ Þ R a According to Eq. (8), the penetration h depends only on geometrical properties, such as the contact radius a and the tip radius R. The applied load P, on the other hand, is a function of a, R, and the mechanical properties of the elastic halfspace, represented by Young s modulus E and Poisson s ratio m. Figure 1b shows a comparison of the load penetration curves obtained from the Hertz and the Sneddon solutions, respectively, for the same radius R and elastic material properties. For small penetrations, both solutions are in good agreement. For increasing penetration, however, the Hertz solution which is only valid for small penetrations leads to an underestimation of the penetration. Z a Int. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 7 927

BBasic Fig. 2. Sneddon solution for different Poisson s ratios. In case of the spherical elastic indentation, the underlying problem is described by five physical quantities: the applied load P(N), Young s modulus E(N mm 2 ), the radius of the spherical indenter R(mm), the penetration h(mm), and Poisson s ratio m. Considering the Hertz solution (see Eq. (4)) and applying dimensional analysis, the dimensionless variable P is introduced as P p ¼ Eh 1:5 R 0:5 ¼ p m; h ¼ h ð10þ R depending only on two dimensionless parameters, i. e., m and h ¼ h=r. Considering the Hertz solution (Eq. (4)) in Eq. (10), one gets H P ¼ 4 31 ð m 2 ¼ const: ð11þ Þ with H P being independent of h. As regards the Sneddon solution, S P decreases continuously with increasing h (see Fig. 2). Figure 2 shows the influence of Poisson s ratio m on S P ð h Þ, with lower values for S P for decreasing m. When plastic deformation is taken into account, an additional dimensionless variable is introduced, p reading for the von-mises yield criterion f ðrþ ¼ ffiffiffiffiffiffiffi 2J 2 c 0, with J 2 (Pa 2 ) as the second invariant of the stress deviator and c (Pa) as the deviatoric strength: p c ¼ c=e (c is related to the uniaxial strength r Y by r Y ¼ ffiffiffiffiffiffiffi 3=2 c). Accordingly, P depends on three dimensionless parameters, reading P p ¼ Eh 1:5 R 0:5 ¼ p m; h ¼ h R ; c ¼ c ð12þ E 3. Finite element model and considered model parameters Fig. 3. Numerical model and finite element mesh (L =50lm). For the simulation of the indentation process, an axisymmetric model is employed (see Fig. 3), indicating the refinement of the FE mesh in the vicinity of the indenter tip. Numerical simulations are performed considering elastic as well as elasto-plastic material response. The elastic material behavior is described by Young s modulus E(Pa) and Poisson s ratio m( ). In the case of plastic material behavior, the deviatoric strength c(pa) defines the elastic limit (von-mises yield criterion, ideally-plastic behavior). For all simulations, three different radii of the spherical indenter are considered, with R = 1, 2.5 and 5 lm. 4. Presentation of results 4.1. Results from elastic simulations Elastic FE simulations are performed for different Poisson s ratios, Young s moduli, and sphere radii (R = 1, 2.5 and 5 lm). Figure 4a shows the numerical results obtained for m = 0.3. According to Fig. 4a, the numerical results for P are independent of Young s modulus and tip radius, depending only on Poisson s ratio as also observed for the Hertz solution H P. The oscillating shape of the curves results from the discrete nature of the numerical model, with increasing stiffness as a new node of the FE mesh comes into contact with the spherical tip. Comparing the numerical results with the analytical solution, P is underestimated by the Sneddon solution, which is explained by the underlying simplification of zero radial displacement at the contact surface, resulting in a too stiff material sample response (see also [15]). In order to account for the observed deviation between the numerical results and the analytical Sneddon solution, the Sneddon solution is modified by a two-step procedure. In the first step, the numerical results are approximated by: ~ P ¼ k log 10 ð h Þþd ð13þ with k as the slope in the logarithmic scale. Figure 4b shows the approximation function ~ P for m = 0.3 obtained from a best-fit analysis performed within 0 < h < 0:1. The soobtained values for k are given in Table 1 for all considered Poisson s ratios. 928 Int. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 7

BBasic Table 1. Parameter k for approximation of numerical results and parameter b for determination of the modified Sneddon solution for different Poisson s ratios m. m k b Fig. 4. Spherical indentation (R = 1, 2.5 and 5 lm) into elastic medium (m = 0.3): Numerical results versus Sneddon solution and approximation of numerical results. In the second step, the Sneddon solution is shifted towards the obtained approximation functions, introducing the parameter b, defined by 1 b ¼ ~ P ð h ¼ 0:01Þ S P ð h ¼ 0:01Þ ð14þ The computed values of b for the considered Poisson s ratios are given in Table 1. Finally, b and k are employed for modifying the Sneddon solution, now reading mods P 1 ð h ; mþ ¼ S P ð h; mþþ½bðmþ 1Š S P ð h ¼ 0:01; mþþkðmþ log h 0:01 ð15þ P h = 1%used for determination of the parameter b refers to a rather moderate value for the penetration/radius ratio H/R, corresponding well to the underlying assumption of elastic material response. 0.15 0.033 1.030 0.2 0.028 1.028 0.25 0.023 1.026 0.3 0.019 1.024 0.35 0.014 1.023 0.45 0.000 1.027 Fig. 5. Spherical indentation (R = 1, 2.5 and 5 lm) into elastic medium (m = 0.3): Modified Sneddon solution compared with numerical results. Figures 5 shows the modified Sneddon solution mods P with the results from the numerical simulations for m = 0.3. 4.2. Results from elasto-plastic simulations In the case of spherical indentation into an elasto-plastic material, the amount of plastic deformation depends on (i) the resistance of the material to plastic deformation, i. e., the deviatoric strength c for materials described by the von-mises yield criterion, (ii) the radius of the tip, influencing the stress state under the tip, and (iii) the maximum applied load. Figure 6a shows the numerical results for m = 0.3, three tip radii (R = 5, 2.5 and 1 lm), and different c/e ratios, indicating that: (i) P does not depend on the selected tip radius for a constant c/e ratio and (ii) P does not depend on the values of c and E as long as the respective c/e ratio is the same. In the case plastic deformation occurs, P starts to decrease below the respective value of P of the elastic solution. The oscillating shape of the numerical results, stemming from the discretized nature of the FE discretization, is eliminated by smoothing, giving the numerical results as shown in Fig. 6b. Int. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 7 929

BBasic Fig. 6. Spherical indentation into elastic-plastic medium (m = 0.3, c/e = 0.0005, 0.001, 0.002, 0.005, 0.01, 0.015, 0.025, 0.05, 0.075, 0.25, 0.5): Dimensionless plot of numerical simulation results and smoothed numerical results. 5. Discussion and concluding remarks Fig. 7. Identification of plastic model parameters: Experimental results according to [17] and comparison of numerical results (m = 0.3) and experimental results for St35. The identification of elastic material parameters from indentation tests of elasto-plastic materials is commonly performed by using the so-called Oliver and Pharr method [1]. According to Oliver and Pharr, Young s modulus of a material can be determined from the initial slope of the unloading curve of the load penetration history. The exact shape of the indenter (obtained from the calibration procedure) and Poisson s ratio are also required. As regards identification of plastic material parameters, commonly the hardness of the material is employed for characterizing the inelastic behavior, defined as the ratio of the maximum load and the contact area between tip and sample surface. For identification of plastic model parameters (such as, e. g., strength properties of yield criteria) from spherical indentation tests, the numerical results presented in the previous section are used. With known tip shape (tip radius), Poisson s ratio, and Young s modulus, the deviatoric strength of a material can be determined, employing the following procedure: 1. Perform indentation test and record load and penetration continuously during loading and unloading; 2. Determine Young s modulus using the Oliver and Pharr method; 3. Compute P and plot loading path of indentation test in dimensionless form as presented in the previous section; 4. Compare experimental result with results from numerical simulations presented in the previous section, giving access to the c/e ratio of the considered material sample. Finally, the deviatoric strength c is computed, using the c/e ratio and the known Young s modulus. This procedure is illustrated for identification of strength properties of steel (St35). The results from spherical penetration tests on St35 were taken from [17] (see Fig. 7a). Young s modulus E and Poisson s ratio m of this material are 210 000 MPa and 0.3, respectively. According to [17], the tests are performed using a spherical indenter with a tip radius of 1.25 mm. Figure 7b shows the dimensionless plot of the experimental results together with the results from numerical simulations for m = 0.3. Comparing the numerical results 930 Int. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 7

BBasic Fig. 8. Spherical indentation: Boundary between elastic and elasto-plastic material response for m = 0.3 and m = 0.0, 0.15, 0.3, 0.45. Fig. 9. Smoothed numerical results for spherical indentation for different c/e ratios and different Poisson s ratios. Int. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 7 931

BBasic with the experiment result, the c/e-ratio is found as c/e & 0.0013, laying within 0.0011 < c/e < 0.0017, giving a uniaxial strength of p r Y ¼ ffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 3=2 c ¼ E 3=2 c=e pffiffiffiffiffiffiffi ¼ 210 000 3=2 0:0013 ¼ 334 MPa ð16þ This value lies within the range between the yield and ultimate strength of St35, given by r Y ¼ 235 MPa and r u ¼ 360 440 MPa. The higher value for the identified strength property of 334 MPa is explained by the assumption of ideally-plastic material behavior in the numerical simulations. Thus, the identified strength property represents both the yield and ultimate strength of St35. The extension of the relations for identification of strength properties presented in this paper towards hardening is a topic of ongoing research. In the first part of the loading phase, spherical indentation is characterized by a pure elastic material response. However, with increasing load, plastic deformation develops. The onset of plastic deformation depends on the tip radius as well as the material behavior in terms of Poisson s ratio, Young s modulus, and deviatoric strength. The numerical results presented in the previous section provide the value of h corresponding to the onset of plastic deformation (see Fig. 8a for m = 0.3). Figure 8b gives h corresponding to the onset of plastic deformation as a function of c for different Poisson s ratios. Based on the numerical results and the findings presented in this paper, the following conclusions can be drawn: 1. Depending on the tip shape, the material properties, and the maximum load, spherical indentation is characterized by either pure elastic or elasto-plastic material response. The onset of plastic deformation can be determined using the given relations between the dimensionless parameters h and c defining the boundary line between elastic and elasto-plastic indentation. 2. In case of indentation tests performed in the elastic regime, the presented modified Sneddon solution can be employed for identification of elastic material parameters. This modified analytical solution is based on the numerical simulations taking into account effects neglected by the Sneddon solution, such as radial displacements of points at the contact surface. 3. If the elastic limit is exceeded, the indentation process is characterized by elasto-plastic material behavior. In this case, no analytical solutions, required for identification of plastic model parameters, are available. However, plastic model parameters can be determined using the dimensionless representation of the numerical results presented in this paper. This approach was illustrated for steel St35. The respective functions required for parameter identification are given in Fig. 9 for all considered Poisson s ratios. References [1] W. Oliver, G. Pharr: J. Mater. Res. 7 (1992) 1564. [2] L. Cheng, X. Xia, L. Scriven, W. Gerberich: Mech. Mater. 37 (2005) 213. [3] M. Vandamme, F.-J. Ulm: Int. J. Sol. Struct. 43 (2006) 3142. [4] A. Jäger, R. Lackner, J. Eberhardsteiner: Mecanica 42 (2007) 293. [5] I. Sneddon: Int. J. Eng. Sci. 3 (1965) 47. [6] E. Lee: Q. Appl. Math. 13 (1955) 183. [7] J. Radok: Q. Appl. Math. 15 (1957) 189. [8] E. Lee, J. Radok: J. Appl. Mech. 27 (1960) 438. [9] S. Hunter: J. Mech. Phys. Sol. 8 (1960) 219. [10] G. Graham: Int. J. Eng. Sci. 3 (1965) 27. [11] G. Graham: Int. J. Eng. Sci. 5 (1967) 495. [12] T. Ting: J. Appl. Mech. 33 (1966) 845. [13] T. Ting: J. Appl. Mech. 35 (1968) 248. [14] Ch. Pichler: Multiscale characterization and modeling of creep and autogeneous shrinkage of early-age cement-based materials. Ph.D. thesis, Vienna University of Technology, Vienna (2007). [15] C. Pichler, R. Lackner, F.-J. Ulm: Int. J. Mater. Res. 99 (2008) 836. [16] H. Hertz: Journal für die reine und angewandte Mathematik 92 (1882) 156. [17] S. Kucharski, Z. Mróz: J. Eng. Mater. Techn. 123 (2001) 245. (Received November 4, 2008; accepted February 4, 2009) Bibliography DOI 10.3139/146.110133 Int. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 7; page 926 932 # Carl Hanser Verlag GmbH & Co. KG ISSN 1862-5282 Correspondence address Prof. Roman Lackner Material-Technology Innsbruck (MTI) Department of Civil Engineering, University of Innsbruck Technikerstraße 13, A-6020 Innsbruck, Austria Tel.: +43 512 507 6600 Fax: +43 512 507 96518 E-mail: Roman.Lackner@uibk.ac.at You will find the article and additional material by entering the document number MK110133 on our website at www.ijmr.de 932 Int. J. Mat. Res. (formerly Z. Metallkd.) 100 (2009) 7