International Journal of Solids and Structures

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1 International Journal of Solids and Structures 46 (9) 95 4 Contents lists aailale at ScienceDirect International Journal of Solids and Structures journal homepage: Comparison etween Berkoich, Vickers and conical indentation tests: A three-dimensional numerical simulation study N.A. Sakharoa a, J.V. Fernandes a, J.M. Antunes a,, *, M.C. Olieira a a CEMUC Department of Mechanical Engineering, Uniersity of Coimra, Rua Luís Reis Santos, Pinhal de Marrocos, Coimra, Portugal Escola Superior de Tecnologia de Arantes, Instituto Politécnico de Tomar, Rua 7 de Agosto de 88, Arantes, Portugal article info astract Article history: Receied Feruary 8 Receied in reised form Septemer 8 Aailale online Noemer 8 Keywords: Numerical simulation Indenter geometry Three-dimensional numerical simulations of Berkoich, Vickers and conical indenter hardness tests were carried out to inestigate the influence of indenter geometry on indentation test results of ulk and composite film/sustrate materials. The strain distriutions otained from the three indenters tested were studied, in order to clarify the differences in the load indentation depth cures and hardness alues of oth types of materials. For ulk materials, the differentiation etween the results otained with the three indenters is material sensitie. The indenter geometry shape factor,, for ealuating Young s modulus for each indenter, was also estimated. Depending on the indenter geometry, distinct mechanical ehaiours are osered for composite materials, which are related to the size of the indentation region in the film. The indentation depth at which the sustrate starts to deform plastically is sensitie to indenter geometry. Ó 9 Pulished y Elseier Ltd.. Introduction Depth-sensing indentation tests are used to determine the hardness and the Young s modulus of ulk materials and thin films. Usually, Berkoich and Vickers indenters are used. Thus, the importance of understanding the relationship etween the results of oth indenters is oious. In addition, the conical geometry is commonly used in i-dimensional numerical simulation studies as equialent to the Berkoich and Vickers indenters. Therefore, it is important to compare the results otained using the three indenters. To our knowledge, studies concerning the comparison of Berkoich, Vickers and conical indentation results are unusual. Only, a few experimental and numerical inestigations (Rother et al., 998; Min et al., 4), concerning the equialence of the results otained from specific ulk materials, hae een performed. Min et al. (4) studied the influence of the geometrical shape of Berkoich, Vickers, Knoop and conical indenters on load indentation depth cures and the strain field under the indentation for a copper specimen. Howeer, the comparison of the indentation ehaiour of ulk and composite materials with different indenter geometries still needs further inestigation. In the current study, three-dimensional numerical simulations of the indentation tests, in ulk and composite materials, were * Corresponding author. Address: CEMUC Department of Mechanical Engineering, Uniersity of Coimra, Rua Luís Reis Santos, Pinhal de Marrocos, Coimra, Portugal. Tel.: ; fax: address: jorge.antunes@dem.uc.pt (J.M. Antunes). performed using the Berkoich, Vickers and conical indenters. Regarding ulk materials, a systematic study is presented which has a ratio etween the residual indentation depth after unloading (h f ) and the indentation at the maximum load (h max ) in the range. < h f /h max <.98. The geometrical correction factor needed to determine the Young s modulus, was also studied for the three indenters, for oth ulk and composite materials. With regard to thin films, the study mainly focuses on the eginning of plastic deformation in the sustrate, which defines the critical penetration depth aoe which the composite hardness results depend on the sustrate s mechanical properties. The indentation test results, otained using the three indenter geometries, are examined y comparing the load indentation depth cures, the hardness alues and the strain distriutions in the indentation region.. Theoretical aspects As mentioned aoe, depth-sensing indentation measurements are used to determine the hardness and the Young s modulus. The hardness, H IT, is ealuated y (e.g., Olier and Pharr, 99) H IT ¼ P A ; where P is the maximum applied load and A is the contact area of the indentation, at the maximum load. The reduced Young s modulus, E r, is determined from (e.g., Sneddon, 965; Olier and Pharr, 99) ðþ -7683/$ - see front matter Ó 9 Pulished y Elseier Ltd. doi:.6/j.ijsolstr.8..3

2 96 N.A. Sakharoa et al. / International Journal of Solids and Structures 46 (9) 95 4 pffiffiffi p E r ¼ p ffiffiffi A C ; ðþ where is the geometrical correction factor for the indenter geometry and C is the compliance. The specimen s Young s modulus, E s,is otained using the definition: ¼ m s þ m i ; ð3þ E r E s E i where E and m are the Young s modulus and the Poisson s ratio, respectiely, of the specimen (s) and of the indenter (i). In this study, the indenter was considered rigid, and so ð m i Þ=E i ¼. The accuracy of the hardness and Young s modulus results, otained with Eqs. () (3), depends on the ealuation of contact area and compliance. In this study, the contact area, A, was ealuated using the contour of the indentation (see next section). Using this approach, contact area results are independent of the formation of pile-up and sink-in. The compliance C was ealuated y fitting the unloading part of the cure load indentation depth, (P h), using the power law (Antunes et al., 6) P ¼ P þ Tðh h Þ m ; where T and m are constants otained y fit and h is the indentation depth which corresponds to a load alue P, during unloading. In the fits, 7% of the unloading cure was used (Antunes et al., 6). Furthermore, another approach can e used for ealuating hardness and Young s modulus, allowing the Young s modulus to e otained when the hardness is known, and ice-ersa. This approach proposed y Joslin and Olier (99), uses the following equation, otained y comining Eqs. () and () P S ¼ p H IT : ð5þ 4 E r The ratio etween the maximum applied load (P) and the square of the stiffness (S =/C), P/S, is an experimentally measurale parameter that is independent of the contact area and so of the penetration depth (Joslin and Olier, 99). Moreoer, if the hardness and the Young s modulus are known, the determination of the correction factor is another useful application of Eq. (5). 3. Numerical simulation and materials The numerical simulations of the hardness tests were performed using the HAFILM in-house code, which was deeloped to simulate processes inoling large plastic deformations and rotations. This code considers the hardness tests a quasi-statistic process and makes use of a fully implicit algorithm of Newton Rapson type (Menezes and Teodosiu, ). Hardness tests simulations can e performed using any type of indenter and take into account the friction etween the indenter and the deformale ody. A detailed description of the HAFILM simulation code has preiously een gien (Antunes et al., 7). Numerical simulations of the hardness tests were performed using Berkoich, Vickers and conical indenters. These three geometries are modelled with parametric Bézier surfaces, which allow a fine description of the indenter tip, namely an imperfection such as the one which occurs in the real geometry (Antunes et al., ). For ideal Berkoich, Vickers and conical indenter geometries with half-angles of 65.7, 68 and 7.3, respectiely, the ratios etween the projected area and the square of the indentation depth are equal to 4.5, for all cases. In this study, the three indenters, shown in Fig., were modelled with tip imperfections, which consist in a plane normal to the indenters axis. The Berkoich, Vickers and conical indenter tips hae triangular, rectangular and circular shapes, respectiely, and an area of approximately.3 lm. ð4þ Fig.. Indenters geometry: (a) conical; () Berkoich; (c) Vickers. This alue corresponds to the imperfection usually osered in experimental Berkoich indenters (Antunes et al., 7). Due to the imperfection at the tip, the area function of the indenters differs from the ideal. Tale presents the area functions of the indenters used in the numerical simulations. As can easily e seen, the three area function equations in this tale represent equialent eolutions of the area ersus the indentation depth, in spite of their dissimilarity. The test sample used in numerical simulations of ulk materials has oth radius and thickness of 4 lm. It discretization was performed using three-linear eight-node isoparametric hexahedrons. The same sample was used in the simulation of the composite film/sustrate materials. In those cases, a coating with a thickness equal to.5 lm (nine layers of elements in the film) was added. Due to geometrical and material symmetries in the X = and Z = planes, only a quarter of the sample was used in the numerical simulation of the Vickers and conical hardness tests. For the Berkoich simulation, only a symmetry condition in the X = plane can e adopted. Thus, a half of the sample was used. In this context, the finite element meshes used in the numerical simulations with the Vickers and conical indenters were composed of 583 elements for the ulk materials and 97 for the thin films. In the case of the Berkoich simulations the numer of elements was 664 for the ulk materials and 8344 for the thin films. In all meshes the size of the finite elements in the indentation region was aout.55 lm. The mesh refinement was chosen in order to proide accurate alues of indentation contact area (Antunes et al., 6). Tale Area functions of the Vickers, p Berkoich and conical indenters. The ideal indentation depth for the area A is: h ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A=4:5. Indenter Area function, A (lm ) Berkoich 4.675h +.56h +.36 Vickers 4.56(h +.8) +.6(h +.8) Conical 4.5(h +.47)

3 N.A. Sakharoa et al. / International Journal of Solids and Structures 46 (9) At the maximum load, the contact area, A, was ealuated using geometrical considerations. For each contact node, the contact area is ealuated taking into account the contact status of its neighouring nodes, as suggested y Tanner (996), and also the normal distance etween these nodes and the indenter; this guarantees the accurate estimation of the contact area (Olieira, 6; Olieira et al., 8). The error alue for the contact area, and consequently for the hardness, was determined ased on the comparison etween the input and output alues of the Young s modulus (Eq. ()). The estimated hardness error is ±.6%. Contact friction was considered etween the indenter and the deformale ody, with a Coulom coefficient equal to.6 (Antunes et al., 6). Three-dimensional numerical simulations of the hardness tests for each indenter geometry were carried out on ulk materials and nine composites, up to the same maximum indentation depth, h max =.3 lm. The plastic ehaiour of the materials used in the numerical simulations was modelled considering that the stress, r and plastic strain, e, relationship was descried y the Swift law: r = k(e + e ) n, where k, e and n (work-hardening coefficient) are material constants (the material yield stress is: r y ¼ ke n ). The constant e was considered to e.5 for all simulations. In ulk material modelling, three different work-hardening coefficients (n =,.5,.5) and two Young s moduli (E = GPa and E = 6 GPa) were used. The Poisson s ratio, m, was.3 for ulk and composite materials. In case of composite materials, the same work-hardening alues of ulk materials were considered and the Young s modulus of the sustrate and the film was GPa, (E f / E s = ). The H f /H s ratio etween the hardness of the film (H f ) and sustrate (H s ) was always higher than. The mechanical properties of ulk and composite materials used are presented in Tales and 3, respectiely. 4. Results and discussion 4.. Bulk materials 4... Load indentation depth cures and hardness In this section, the load indentation depth cures and hardness alues otained in the simulations using the three indenter geometries are compared, for the ulk materials of Tale. In a general way, the load indentation depth cures are quite similar. Howeer, small differences etween cures can e easily osered when the h f /h max ratio decreases elow aout.65, whateer the work-hardening coefficient alue of the material. It must e noted that the h f /h max parameter does not depend on the indentation depth for a gien material (e.g., Bolshako and Pharr, 998) and is easily determined from the indentation cure. Moreoer, it is well known that this ratio is related to the material properties, particularly the H IT /E ratio etween the hardness H IT and the Young s modulus E. Fig. shows two examples of load (P) ersus indentation depth (h) cures otained with Berkoich, Vickers and conical indenters, where h corresponds to the ideal indentation depth determined as indicated in Tale. Fig. (a) and () correspond to materials with a work-hardening coefficient (n) of zero and a h f /h max ratio of to.74 (Fig. (a)) and.4 (Fig. ()). The two other examples of load indentation depth cures presented in Fig. correspond to materials with work-hardening coefficient alue of n =.5 and h f /h max equal to.88 (Fig. (c)) and.4 (Fig. (d)). For B3 and B4 materials (h f /h max =.74 and.88, respectiely), the cures otained are quite similar for all types of indenter tested. In the cases of B6 and B3 materials (h f / h max =.4 and.4), the cures otained can e easily separated. Neertheless, whateer the h f /h max alue, the highest leel of the cures corresponds to the one otained with the Berkoich indenter and the lowest leel to the one otained with the conical indenter. Moreoer, the difference etween the Berkoich and the Vickers cures is higher than etween the Vickers and the conical cures. Finally, it must e noted that similar results were otained for materials haing a work-hardening coefficient of n =.5. Fig. 3 shows hardness alues otained in the indentation tests with Vickers, H, and conical, H c, indenters normalized y the Berkoich indenter hardness, H, as a function of the ratio h f /h max (i.e., for ulk materials with different mechanical properties). As can e seen in Fig. 3, oth ratios H /H and H c /H are slightly lower than, whateer the alue of the ratio h f /h max, which means that the hardness alues for the Berkoich are always higher than those for the Vickers and conical indenters. Moreoer, the H /H and H c /H ratios decrease when h f /h max decreases. For alues of h f /h max close to, the H /H and H c /H ratios tend to approach, while for h f /h max alues close to., the H /H and H c /H ratios tend towards alues close to.96 and.94, respectiely. This is in agreement with the Tale Mechanical properties of the ulk materials used in the numerical simulation. The hardness results, H, were otained with the Berkoich indenter. Material Work-hardening coefficient, n r y (GPa) E (GPa) H (GPa) h f /h max B B B B4...6 B B B B B B B B B B B B B B B B B B

4 98 N.A. Sakharoa et al. / International Journal of Solids and Structures 46 (9) 95 4 Tale 3 Mechanical properties of the film, sustrate and composite materials used in the numerical simulations. The hardness results, for the film, H f, sustrate, H s and composite, H c, were otained with the Berkoich indenter. r f y ðr s y Þ and n f (n s ) are the yield stress and the work-hardening coefficient of the film (sustrate), respectiely. Composite r f y (GPa) n f r s y (GPa) n s H f (GPa) H s (GPa) H f =H s H c (GPa) C C C C C C C C C a c c 4 d c Fig.. Load indentation depth cures for the materials: (a) B3 (n = and h f /h max =.74); () B6 (n = and h f /h max =.4); (c) B4 (n =.5 and h f /h max =.88); (d) B3 (n =.5 and h f /h max =.4). Indenters:, Berkoich;, Vickers; c, conical. aoe discussed results concerning the load indentation depth cures, which show higher differentiation etween the cures otained with the three indenter geometries, for lower h f /h max alues than for higher ones. It must e mentioned that the most common materials show h f /h max alues higher than.7, and so the H /H and H c /H ratios are higher than.97 and.96, respectiely Indenter tip imperfection The tip imperfection of the indenter does not affect the hardness results, as discussed in this section. Actually, in a recent study, Antunes et al. (7) concluded that a correction of the geometry of the Vickers indenters with offset, using the respectie area function, is enough to otain accurate alues of the mechanical properties, namely the Young s modulus and the hardness. In this study, the modulation of the Vickers indenter was performed for fie different sizes of offset, and the load indentation depth cures ecome coincident after correction. Moreoer, when the eolution of k = P/h at each loading point (from Kick s law: P = kh, where P is the load and h is the indentation depth) was represented as function of the indentation depth for the different offset sizes, k = P/h ecomes constant and equal for all indenters, for high enough indentation alues (depending on the size of the offset). The geometrical imperfections of the indenters used in the present study were designed to make the numerical indenter as similar as possile to the experimental case. The size of the imperfection is equal to lowest tip imperfection size used in the preious study (an area of approximately.3 lm (Antunes et al., 7)). In order to ensure that the estimation of the material hardness is not affected y indenter tip imperfections, the eolution of k = P/h,at each point of the loading part of the load indentation depth cures (after correction with the area function), was represented as function of the indentation depth, for the three indenters used in current study. Fig. 4 shows two examples of this eolution. For indentation depths higher than. lm, the k = P/h alue ecomes constant for each indenter. This means that the loading cures are self similar after this indentation depth. Moreoer, for materials with high h f /h max alues, the constant leel is equal for the three indenters (see example of Fig. 4(a)), while for materials with low h f /h max alues, this leel can e easily separated, for the three indenters (Fig. 4()).

5 a Hc/H N.A. Sakharoa et al. / International Journal of Solids and Structures 46 (9) H /H.5.5 H /H =.38h f /h max h f /h max H c /H =.49h f /h max h f /h max Fig. 3. Hardness results otained for: (a) Vickers and () conical indenters. Both results are normalized y the hardness results otained for the Berkoich indenter Strain distriution Fig. 5 show the equialent plastic strain distriutions otained at maximum load for the materials presented in Fig. (a) and (). For B3 material, with h f /h max =.74 and n =, the maximum alue of equialent plastic strain is higher for the Berkoich indentation (.549) than for Vickers (.386) or conical (.366) (Fig. 5 (left hand side)). For Berkoich indentation, the maximum plastic strain region is located just at the surface in the edge regions of the indentation (Fig. 5(a)); for Vickers indentation, the maximum plastic strain region is sited eneath the indentation surface as well as at the surface, in the edge region of the indentation (Fig. 5()); in the case of the conical indentation the maximum plastic strain region is located just under the surface (Fig. 5(c)). So, the presence of edges in the indenter geometry can influence the plastic strain under the indentation. The case of low h f /h max alue (material B6) is shown in Fig. 5 (right hand side). This material, with h f /h max =.4 and n =, also presents the maximum alue of equialent plastic strain for the Berkoich indenter (.43), followed y the Vickers (.366) and conical (.36) ones (see the right hand side of Fig. 5(a c), respectiely). Howeer, for this material, differences are more attenuated and the region with maximum equialent plastic strain is located under the indentation surface, whateer the indentation geometry (for the Berkoich geometry only, high alues of plastic strain occur in regions in contact with the surface, near the edge of the indentation (Fig. 5(a)). These results are qualitatiely alid whateer the materials work-hardening coefficient (n = and also a 3 k, mn/μm...3 Berkoich Vickers conical indenter 45 k, mn/μm Berkoich Vickers conical indenter Fig. 4. Eolution of the alue of the constant k as function of the indentation depth, otained from the numerical simulation with Berkoich, Vickers and conical indenters: (a) B3 (n = and h f /h max =.74); () B6 (n = and h f /h max =.4).

6 N.A. Sakharoa et al. / International Journal of Solids and Structures 46 (9) 95 4 Fig. 5. Equialent plastic strain distriution otained at the maximum load for the materials B3 (left hand side) and B6 (right hand side) in numerical simulations with the indenters: (a) Berkoich; () Vickers; (c) conical. for n =.5 and.5), which suggests that the higher differentiation etween the load indentation depth cures, osered when the h f / h max alue decreases, is not related to differences in maximum equialent plastic strain alues. In fact, the differences etween the maximum equialent plastic strains osered using the three indenters are higher when the load indentation depth cures are closer, i.e., when the h f /h max alue approaches. Fig. 5 also shows that plastic strain distriutions are dependent on the indentation geometry. The plastic strain region is less spherical and slightly less deep for the Berkoich indenter than for the Vickers and conical ones. This is the case in oth materials ut mainly for the material with h f /h max =.4. These differences in the plastic strain region s geometry, otained with the three types of indenter, are proaly the main reason why the load indentation depth cures are not strictly identical Young s modulus The Young s modulus of the ulk materials was ealuated considering the results otained using the three types of indenter. Fig. 6 shows that the Young s modulus alues, E eal, are normalized y the alue used as input in the numerical simulation, E input,asa function of h f /h max (E eal was determined using =, in Eq. ()). The ratio E eal /E input is quite constant and always higher than, whateer the indenter used. The correction factor was estimated from the mean alue of E eal /E input, for each indenter. The alues otained were.8,.55 and.34 for the Berkoich (Fig. 6(a)), Vickers (Fig. 6()) and conical (Fig. 6(c)) indenters, respectiely. Eq. (5) was used in order to confirm the aoe correction factor alues. E r was determined from Eq. (3) using the input Young s modulus, E input, and the hardness alues, H IT, were determined using Eq. (), where the contact area is ealuated directly from numerical simulation results. Fig. 7 shows, for the three indenter geometries, the ratio P/S ersus H IT =E r otained for the ulk materials. All the straight lines in Fig. 7 pass through the origin of the axes as indicated y Eq. (5) (all cures match for H IT =E r ¼, i.e., for materials with rigid-plastic ehaiour, which corresponds to the ratio h f /h max = ). The factor is ealuated from the slope, x, of the straight lines, related with through x = p/4. Fig. 6 gies

7 N.A. Sakharoa et al. / International Journal of Solids and Structures 46 (9) 95 4 a Eeal /Einput Eeal /Einput c Eeal /Einput h f /h max h f /h max h f /h max Fig. 6. Young s modulus results, E eal, normalized y E input, otained for ulk the materials in numerical simulation with the indenters: (a) Berkoich; () Vickers; (c) conical. P/S P/S =.77H /E r P/S =.74H c /E r P/S =.678H /E r H IT /E r Berkoich, Vickers, conical indenter Fig. 7. Representation of the ratio P/S ersus H=E r of the ulk materials, for the three indenter geometry. factor alues of.74,.45 and.9 for Berkoich, Vickers and conical indenters, respectiely. These alues are quite similar to the ones preiously calculated from Fig. 6. In this context, it can e concluded that a factor greater than should e considered for the three indenters, when using Eq. () to determine the Young s modulus. Moreoer, the factor increases with deiation of the indenter s geometry from circular (conical indenter) to Vickers (four-sided pyramidal) and Berkoich (three-sided pyramidal). The same conclusion has een reached in seeral analytical and numerical studies (e.g., Antunes et al., 6; Bolshako and Pharr, 998; Cheng and Cheng, 999). Howeer, these preious studies proposed a wide range of alues, which generate some uncertainty aout the adequate alues. For example, ased on numerical simulation results, Dao et al. () propose alues for the correction factor of to.96,.7 and.6 for the Berkoich, Vickers and conical indenters, respectiely. In a recent threedimensional numerical simulation study, using seeral materials with different Young s moduli and work-hardening coefficients, a alue of.5 was found for the case of the Vickers indenter (Antunes et al., 6). Finally, is important to state that using the alues otained in the current study, using the data presented in Fig. 7, the maximum error in the ealuation of the Young s modulus, in each indiidual simulation, was at aout.4%,.8% and.7% for the Berkoich, Vickers and conical indenters, respectiely. 4.. Composite materials Numerical simulations of composite materials concern cases where the hardness of the film H f is higher than the hardness of the sustrate H s, as shown in Tale 3. Load indentation depth cures and strain distriutions were studied to improe understanding of the influence of the indenter geometry on the composite s ehaiour during indentation. The load indentation depth cures otained from Berkoich, Vickers and conical indentation tests on composite materials are not coincident, and show a non-negligile difference, for the H f / H s alues studied (where H f /H s > ). Examples of such load indentation depth cures are shown in Fig. 8, for three alues of the ratio etween the film (H f ) and the sustrate (H s ) hardness (H f /H s of aout., 4.33 and 8.67). Fig. 8 also shows the load indentation depth cures of the corresponding film and sustrate. It can e easily seen that the differences etween the Berkoich, Vickers and conical indentation cures are higher for the composite than for the respectie film and sustrate. The Berkoich indentation cure is higher than the Vickers, which in turn is higher than the conical, as for ulk materials. In addition, the numerical study of the composite materials hardness reealed that, for all the composites in Tale 3, the H c =H c ratio etween the Vickers and the Berkoich hardness and the H c c =H c ratio etween the conical and the Berkoich hardness are always lower than. This means that the hardness otained with the Berkoich indenter is always higher than with the Vickers and conical indenters, the same as for the ulk materials. Howeer, the osered differences are higher for composite than for ulk materials. In fact, for the composite materials studied, the ratio H c =H c is in the range.88 to.96, and decreases when the ratio H f /H s increases, as shown in Fig. 9, whateer the work-hardening coefficient alues of the film and the sustrate. In this figure, only the ratio H c =H c is shown, for simplification. In order to understand such differences etween the ehaiour of composite materials under Berkoich, Vickers and conical indentation tests, strain distriution at low indentation depths in the composite was studied. The C composite material is taken as an example. For a penetration depth of.5 lm, the sustrate is more deformed in the case of the Vickers and conical indenters than in the case of the Berkoich indenter, as shown in Fig.. The indentation depths at which the sustrate starts to deform plastically were also determined for all three indenters tested. The plastic response of sustrate material egins earlier for conical and Vickers indenters, at penetration depths of.7 and.9 lm, respectiely. For the Berkoich indenter the plastic deformation of the sustrate occurs for a penetration depth alue of.37 lm. The correspondent equialent plastic strain distriutions are represented in Fig. and confirm the delayed plastic response of the sustrate in the Berkoich case.

8 N.A. Sakharoa et al. / International Journal of Solids and Structures 46 (9) 95 4 a Berkoich Vickers Conical 4 3 Hc /Hc H f /H s Fig. 9. Eolution of the ratio H c =H c, etween the Vickers and the Berkoich hardness of the composite, of as function of the ratio H f =H s, etween the Berkoich hardness of the film and the sustrate. with preious results on ulk materials which show that the Vickers hardness alues are independent of the alue of the friction coefficient used in the simulations (Antunes et al., 6). 3 c Berkoich Vickers Conical Berkoich Vickers Conical 3 Fig. 8. Load indentation depth cures for the composites: (a) C ðh f =H s ¼ :Þ; () C4 ðh f =H s ¼ 4:33Þ; (c) C3 ððh f =H s ¼ 8:67ÞÞ. Materials:, film;, composite; 3, sustrate Friction coefficient Finally, in order to check the influence of the friction coefficient alue in the contact etween the indented materials and the different indenter geometries, numerical simulations with friction coefficients of.4 and.3 were also carried out for oth ulk and composite materials. Fig. shows examples of load indentation depth cures, corresponding to the B9 ulk material (Fig. (a)) and the C4 composite material (Fig. ()), otained with friction coefficients of.4,.6 and.3, for the cases of the Vickers and Berkoich indenter geometry. For all cases of ulk and composite materials studied, such as for the examples shown in Fig., no measurale differences were osered in the load indentation depth cures otained with each indenter s geometry, whateer the alue of the friction coefficient. This is in agreement Fig.. Equialent plastic strain distriution, at the penetration depth h.5 lm, otained in numerical simulation of the composite C ðh f =H s ¼ :Þ for the indenters: (a) Berkoich; () Vickers; (c) conical.

9 N.A. Sakharoa et al. / International Journal of Solids and Structures 46 (9) a Fig.. Load indentation depth cures otained with the friction coefficients equal to.3,.6 and.4 for: (a) the ulk material B9; () the composite C4. Indenters:, Berkoich;, Vickers. Fig.. Equialent plastic strain distriution, at the initial stages of sustrate plastic deformation, otained in numerical simulation of the composite C ðh f =H s ¼ :Þ for the indenters: (a) Berkoich (h.37 lm); () Vickers (h.9 lm); (c) conical (h.7 lm). It can e concluded that the ulk and composite load indentation depth responses to the hardness tests are independent of the friction coefficient (.4,.6 or.3), whateer the indenter used, i.e., the load indentation depth responses are not sensitie to the alue of the friction coefficient (at least in the studied range). Thus, een if the friction coefficient alue of.6, used in the simulations of preious sections, does not exactly correspond to the experimental one, the conclusions concerning the sensitiity of the load indentation depth response to the indenter type, are still alid. 5. Conclusions Three-dimensional numerical simulations of Berkoich, Vickers and conical indentation tests were performed in order to attain etter understanding of the influence of the geometry of equialent indenters on the materials ehaiour under indentation. The main following conclusions can e drawn: For ulk materials, coering a wide range of mechanical properties, the load indentation depth cures otained using the three indenters are difficult to distinguish, in cases of materials with high alues of the h f /h max ratio (typically for h f /h max >.65) ut can e separated for lower alues of h f /h max. Therefore, the hardness alues otained show the same kind of ehaiour. When comparing the results of the three indenters, the otained leels for load indentation depth cures and hardness are highest for the Berkoich and lowest for the conical. The results for the Vickers indenter lie etween the two others, eing closer to those for the conical indenter. Concerning the hardness, for the h f /h max = ratio (purely elastic ehaiour), the H /H and H c /H ratios tend towards the alues.947 and.93, respectiely; and when the ratio h f /h max = (rigid-plastic ehaiour), oth ratios H / H and H c /H tend towards a alue close to. This means that, when comparing Berkoich or Vickers experimental results or when replacing Berkoich or Vickers indenters with the conical one, in order to simplify numerical simulations, it is necessary to e cautious. For materials with low alues of the h f /h max ratio, the equialence etween conical and Vickers indenters is closer than etween conical and Berkoich indenters (or etween Vickers and Berkoich indenters). For materials with high alues of the h f /h max ratio, which corresponds in general to low to medium hardness, the equialence etween the three indenters can e considered acceptale. Some details of the equialent plastic strain distriutions are dependent on indenter geometry. In the case of ulk materials, the maximum alue of equialent plastic strain is higher for the Berkoich indenter than for the Vickers and conical ones, ut the differences are attenuated for low alues of h f /h max. This fact indicates that the increasing separation

10 4 N.A. Sakharoa et al. / International Journal of Solids and Structures 46 (9) 95 4 etween the load indentation depth cures with decreasing h f /h max alue is not related to differences in the maximum plastic strain alues. The main possile reason for such ehaiour is the differences in the geometry of the plastic strain regions otained with the three types of indenter. The equialent plastic strain distriution is less spherical and slightly less deep, for the Berkoich indenter than for the Vickers and conical ones for all materials, ut mainly for the materials with low alues of h f /h max. The results for composite materials where the ratio etween the film (H f ) and the sustrate (H s ) hardness s, H f /H s is higher than, show important distinctions in the function of the indenter geometry. The differences of the load indentation depth cures and of hardness under Berkoich, Vickers and conical indenters are greater than for ulk materials, and depend on the H f /H s ratio. Howeer, qualitatiely, the relatie position of the load indentation depth cures and hardness alues is similar to the case of ulk materials. It is shown that the sustrate s contriution to the composite plastic deformation starts later (i.e., for higher penetration depth) and is less important in the case of Berkoich indentation than for Vickers and conical indentation. This is certainly the most important reason for the amplification of the differences etween the responses under the three indenters for composite materials, when compared to ulk materials (knowing that for the studied composites the film is harder than the sustrate). So, in the case of composites, one must e more cautious when comparing the hardness results of the three indenters. For example, the ratio etween the Vickers and Berkoich hardnesses can attain.88, when the ratio etween the hardness of the film and the sustrate is 8.6. The ulk and composite load indentation depth responses to the hardness tests are independent of the friction coefficient alue (etween.4 and.3), whateer the indenter used. So, een if the friction coefficient used in the numerical simulations does not exactly correspond to the experimental one, the conclusions concerning the sensitiity of the load indentation depth response to the indenter type, are still alid. Acknowledgements through the Program POCTI (Portuguese Goernment and FEDER). One of the authors, N.A. Sakharoa, was supported y a grant for scientific research from the Portuguese Science and Technology Foundation. This support is gratefully acknowledged. References Antunes, J.M., Caaleiro, A., Menezes, L.F., Simões, M.I., Fernandes, J.V.,. Ultramicrohardness testing procedure with Vickers indenter. Surface and Coatings Technology 49, Antunes, J.M., Menezes, L.F., Fernandes, J.V., 6. Three-dimensional numerical simulation of Vickers indentation tests. 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Three-dimensional numerical simulation of the deep-drawing process using solid finite elements. Journal of Materials Processing Technology 97 ( 3), 6. Min, Li, Wei-min, C., Nai-gang, L., Ling-dong, W., 4. A numerical study of indentation using indenters of different geometry. Journal of Materials Research 9, Olieira, M.C., 6. Algorithms and Strategies for the Treatment of Large Deformation Frictional Contact Mechanics Application to sheet metal forming. Ph.D. Thesis, Uniersity of Coimra, Portugal. Olieira, M.C., Ales, J.L., Menezes, L.F., 8. Algorithms and strategies for treatment of large deformation frictional contact in the numerical simulation of deep drawing process. Archies of Computational Methods in Engineering 5, 3 6. Olier, W.C., Pharr, G.M., 99. An improed technique for determining hardness and elastic-modulus using load and displacement sensing indentation experiments. Journal of Materials Research 7 (6), Rother, B., Steiner, A., Dietrich, D.A., Jehn, H.A., Haupt, J., Giessler, W., 998. Depthsensing indentation measurements with Vickers and Berkoich indenters. Journal of Materials Research 3 (8), Sneddon, I.N., 965. The relation etween load and penetration in the axisymmetric Boussinesq prolem for a punch of aritrary profile. International Journal of Engineering Science 3, Tanner J.A., 996. Computational Methods for Frictional Contact with Applications to the Space Shuttle Oriter Nose-Gear Tire Deelopment of Frictional Contact Algorithm. NASA Technical Paper 3574, pp. 48. The authors are grateful to the Portuguese Foundation for Science and Technology (FCT) who financially supported this work,