Plastic strain distribution underneath a Vickers Indenter: Role of yield strength and work hardening exponent

Transcription

1 Acta Materialia 54 (2006) Plastic strain distribution underneath a Vickers Indenter: Role of yield strength and work hardening exponent G. Srikant a, N. Chollacoop b, U. Ramamurty a, * a Department of Metallurgy, Indian Institute of Science, Sir CV Raman Avenue, Bangalore , India b National Metals and Materials Technology Center (MTEC), Pathumthani 12120, Thailand Received 8 May 2006; accepted 20 June 2006 Available online 18 September 2006 Abstract The effects of yield strength, r y, and strain hardening exponent, n, on the plastic strain distribution underneath a Vickers indenter were explicitly examined by carrying out macro- and micro-indentation experiments on Al Zn Mg alloy that was aged for different times so as to obtain materials with different r y but with similar n, or the same r y but different n. Large Vickers indents were made (using a load of 700 N) that were subsequently sectioned along the median plane and the plastic strain distribution was determined by recourse to microindentation mapping. Comparison of the iso-strain contours shows that for similar values of n, higher r y leads to a smaller deformation field in both the indentation (z) and lateral (x) directions. Higher n, at the same r y, leads to a shallower deformation field. In all cases, it was found that strain fields are elliptical, with the long axis of the ellipse coinciding with the z-direction. The strain field ellipticity is sensitive to the work hardening behavior of the material, with higher n leading to lower ellipticity. While the lateral strain distribution is in agreement with the expanding cavity model, strain distribution directly ahead of the indenter tip appears to follow the Hutchison, Rice, and Rosengren fields ahead of a crack tip in an elastoplastic solid. Ó 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Hardness; Microindentation; Plastic deformation; Aluminum alloys; Aging 1. Introduction Since its discovery by Brinell in 1900 [1], the indentation technique has found widespread use in characterizing the mechanical response of materials through hardness measurements. With the advent of instrumented indentation wherein the load, P, and the depth of penetration, h, are measured continuously and analyzed to extract the elastoplastic properties of a material, it has become the method of choice for measuring properties at micro- and nanoscales of thin films, biological materials, and a variety of advanced materials. Development of instruments that can measure P h curves with a high degree of accuracy (with resolutions of micronewtons and nanometers) and a better understanding of contact mechanics problems through * Corresponding author. Tel.: ; Fax: address: ramu@met.iisc.ernet.in (U. Ramamurty). computational modeling are the two main factors responsible for this advancement. Among the various contact mechanics problems, understanding the plastic flow mechanisms and strain distribution in elastoplastic solids underneath the indenter has been an area of great interest as it helps the interpretation of the measured P h response or the hardness. On the basis of the slip-line field theory of plasticity, Hill et al. [2] proposed a cutting mechanism for the flow of material during wedge indentation of a rigid plastic material. Their analyses showed that the material was displaced sideways and upwards from the face of the edge with the plastic strain boundary passing through the tip of the indenter. Experimental observations on annealed brass (with a high work hardening exponent, n) employing both wedge and pyramidal indenters by Samuels and Mulhearn [3] showed that the plasticity was radial with hemispherical strain boundaries. They exploited the differential etching of brass with plastic /$30.00 Ó 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi: /j.actamat

2 5172 G. Srikant et al. / Acta Materialia 54 (2006) strain to obtain the strain contours and proposed a compression mechanism to explain the radial flow, which was analogous to the spherical cavity model derived by Hill [4]. Dugdale [5] contradicted this view and argued that the spherical cavity model could not be the basis of the compression mechanism, because the pressure required to expand a spherical cavity may not exactly match with the pressure required to create a shallow indentation. Marsh [6] carried out Vickers indentations on glass and observed that the indentation hardness theory adopted by Hill et al. [2] and Tabor [7] was not suitable for rationalizing the experimental results exactly. Marsh suggested that this may be due to the large elastic strains involved in glass indentations. On the basis of these observations, Marsh and later Hirst and Howse [8] working on wedge indentations proposed that the parameters like H/r y and E/r y (where r y is the yield strength, H the hardness, and E the elastic modulus) govern the transition from cutting to a compression type of mechanism. Furthermore, Atkins and Tabor [9] carried out conical indentations on split plasticine wedges and observed a change in the deformation mechanism from the cutting type to the compression type with increasing cone angles. They suggested that the angle of the indenter influenced the associated deformation mechanism. Subsequently, Johnson [10] considered both material parameters and the geometry of the indenter in the proposed expanding cavity model to explain the flow of material under the indenter. To date, this model has been the most popular and formed a cornerstone for understanding the indentation response of materials, especially metals. However, detailed plastic strain mapping underneath a Vickers indent in Cu by Chaudhri [11] shows that the plastic strain field underneath a sharp indenter is not always in accordance with the predictions of either the cutting or the compression models. More recently, finite element analysis (FEA) was utilized for understanding the strain distribution under sharp indenters [12,13]. Bhattacharya and Nix s [14] large-strain FEA of a cone indentation in Al and Si (which have relatively low and high r y, respectively) suggested that the plastic strain distribution is associated with a deformation mechanism intermediate to that of the cutting and the compression mechanisms depending on the E/r y ratio. Giannakopoulos et al. [13] used small-strain FEA to plot iso-strain contours for a Vickers indentation and observed that the shape of the contours was approximately hemispherical. Further, they observed that a transition between cutting and compression mechanisms occurs at a strain of 29%, which was in agreement with the experimental observation of Chaudhri [11] for maximum strain underneath the indenter. Mata et al. [12] carried out finite element simulations on elastic power-law plastic solids and observed that the shape of the strain contours was influenced by both r y and the strain hardening exponent, n. It was reported that a higher r y led to a radial flow mechanism because of the material s ability to sustain higher elastic strains. For lower values of n and r y, they observed that the strain contours broke towards the free surface leading to an uncontained deformation mode. Despite these recent advances in computational modeling of the strain distribution underneath an indenter, which is a complex function of both the material parameters and the geometry of the indenter (especially for a sharp indenter like Vickers), experimental work in understanding the deformation process underneath the indenter is limited. Such assessment is required not only for checking the veracity of FEA predictions but also to guide future developments in this area. However, performing clean parametric studies is difficult since the strain distribution is sensitive to the material properties E, r y, and n as well as the microstructure. Hence, conducting a parametric study (i.e., varying only one material parameter while keeping all the others constant) in real material systems is indeed a challenge. The age-hardenable Al alloys offer an opportunity for such conditions to be met. By selecting two different aging conditions, one under-aged and the other over-aged, it is possible to obtain two materials with the same r y but with different n values, because of the differences in the dislocation precipitate interaction mechanisms. While the dislocations shear the precipitates in the under-aged condition (leading to low values of n), they by-pass the incoherent precipitates through the Orowan looping mechanism in the over-aged condition (resulting in a higher n). Similarly, it is possible to obtain materials with different r y while keeping n constant. Note that E and microstructural parameters such as the grain size remain invariant for all the different aging conditions. Although the precipitate type and morphology change with aging time, the length scale of change (typically nanometer scale) is orders of magnitude smaller than the length scale of microindents. These features allow an investigation of the plastic strain distribution underneath a sharp indenter as a function explicitly of either r y or n, which is reported in this paper. 2. Materials and experiments A commercial 7075 Al alloy with a nominal composition (in wt.%) of Al 7.6Zn 4.6Mg 1.0Si was selected after a detailed survey of the available literature. Tensile specimens with 35 mm gage length and mm 2 cross section were machined from the as-received plates (thickness of 2 cm). The specimens were solutionized at a temperature of 480 C for 2 h and aged at a temperature of 150 C inan oil bath. As suggested in Ref. [15], this particular aging temperature gives the most symmetrical hardness, H, vs. aging time, t a, curve, which is desirable for the intended study. Three specimens were aged for each time interval. The Vickers hardness, H, of the aged samples was measured by indenting the grip section of the tensile specimens with an indentation load of 30 N. Subsequently, they were tested in uniaxial tension at a strain rate of 10 3 s 1 to measure r y (0.2% offset stress) and n. The variation of r y and H with t a is shown in Fig. 1, whereas Fig. 2 shows that of n with t a. On the basis of these plots, the alloys aged to 6,

3 G. Srikant et al. / Acta Materialia 54 (2006) Fig. 1. Variation of hardness, H, and yield strength, r y, with aging time, t a. Intersections of the horizontal line with the r y data give two different aging conditions with same r y but different n values. Fig. 2. Variation of the strain hardening exponent, n, with the aging time, t a. 11, 36, and 94 h were selected for the plastic strain mapping studies. The true stress, r, vs. true strain, e, curves of specimens subjected to these aging conditions are shown in Fig. 3, and the mechanical properties extracted from these curves are listed in Table 1. As seen from this table, the alloys aged to 6, 11, and 36 h have similar n but with increasing r y, whereas the 6 and 94 h aged alloys have similar r y but different n. For the strain mapping studies, blocks (3 cm long and 1 1cm 2 cross-section) were solutionized, aged to the prescribed period, polished to a surface finish of 1 lm using diamond paste, and then Vickers indented with a load of 700 N in order to create large indentation impressions. The diagonal lengths, D, of the impressions are listed in Table 1. Subsequently, the indented blocks were sectioned along the median plane of the indents using a 25 lm thick copper wire. Electrodischarge machining was employed so Fig. 3. Uniaxial true stress true strain curves for the selected aging conditions. that the influence of machining on the prior-deformed zone was minimal. The sectioned planes were carefully polished to 0.25 lm finish using diamond paste and microhardness measurements on the deformed region were carried out by employing a 0.5 N load. The diagonal length of these microindents was 22 lm. A distance of 80 lm (from center-to-center of the microindents) was maintained between the indents in either direction to minimize the strain field interference among these microindents. A typical hardness map is shown in Fig. 4. Note that the hardness variation along the x-axis (the coordinate adopted is shown in the inset of Fig. 4) was obtained by indenting along the x-axis on the x y plane (on blocks that were not sectioned). Prior to indenting, gentle polishing was done in order to remove the piled up material and hence obtain a flat plane suitable for microhardness measurements.

4 5174 G. Srikant et al. / Acta Materialia 54 (2006) Table 1 Relevant mechanical properties of the aged alloys Aging time, t a (h) Hardness, H (VHN) Yield stress, r y (MPa) Strain hardening exponent, n Diagonal length, D (lm) H 0 (Eq. (1)) m (Eq. (1)) Normalized pile-up, h p /h lm finish. An equation similar to the power-law hardening of the form H ¼ H 0 e m p ð1þ was found to describe the H vs. e p data well, implying that within the range of strains examined, all the aged samples obey power-law hardening. The hardness coefficient, H 0, and the power-law exponent, m, obtained for each aging condition are listed in Table Results Fig. 4. Typical microhardness map obtained from sectioned indent. The plastic strain distribution around the sectioned indents is visually represented by iso-strain contours shown in Figs. 6 and 7. Note that contour lines drawn in To convert the measured microhardness maps into equivalent plastic strain, e p, maps, first H vs. e p curves for each of the selected aging conditions (such as the typical one shown in Fig. 5) were generated by microhardness measurements (with an indentation load of 0.5 N) on cylindrical samples (15 mm in height and 10 mm in diameter) that were subjected to various levels of compressive plastic strains, e p, at a strain rate of 10 3 s 1. Measurements were carried out on axial sections of specimens polished to Fig. 6. Comparison of the plastic strain distribution underneath the Vickers indent of two differently aged alloys that have similar r y but different n (94 vs. 6 h). Fig. 5. Representative plot showing the variation of hardness, H, with the compressive uniaxial plastic strain, e p. These plots are used for converting the sub-surface H maps (an example is shown in the figure) into e p distribution maps. Fig. 7. Comparison of the plastic strain distribution underneath the Vickers indent of two differently aged alloys that have similar n but different r y (36 vs. 6 h).

5 G. Srikant et al. / Acta Materialia 54 (2006) these figures as well as those presented subsequently (unless otherwise specified) are merely to guide the eye. Here, all the distances were normalized by the diagonal length, D, of the respective macroindents (listed in Table 1) for comparison purposes. Fig. 6 compares the e p contours obtained on 6 and 94 h aged samples (similar r y but different n). The deformation field in the material with higher n is relatively shallow. This is illustrated in Fig. 8 where the intercepts of the strain contours with the z-axis (normalized with indent diagonal D), r z /D, and those with the x-axis, r x /D, are plotted against the lower bound of the plastic strain range of the contours A E shown in Fig. 6. Since higher n implies greater resistance offered by the material to plastic deformation, the spread of plasticity is limited in the alloy with higher n. Fig. 8 clearly shows that the normalized r x values at a given e p in both cases (filled and open square symbols) are similar, whereas a significant difference exists between the normalized r z values (filled and open circle symbols). This implies that the work hardening behavior of the material is relatively more influential in determining the plasticity spread in the z-direction (i.e., the loading direction) than in the x-direction. As discussed earlier, Table 1 shows that the alloys aged to 6, 11, and 36 h have similar n but with increasing r y. The 6 and 36 h aged cases are compared in Fig. 7, as these aged conditions show the largest difference in r y (with similar n values). It is seen that higher r y leads to a smaller deformation field in both x and z directions. This is in contrast to that seen in Fig. 6, where increasing n appears to affect only the depth of deformation field. Due to the similar results for both r x and r z data, only the normalized r z is plotted against the lower bound of the same plastic strain range in Fig. 8 for the 6, 11, and 36 h conditions (similar n but with increasing r y ) in Fig. 9, which clearly shows the systematically decreasing depth of plastic spread with increasing r y. z-axis intercept, r z /D Discussion 4.1. Strain field ellipticity 6 h 11 h 36 h plastic strain, ε p (%) 348 MPa 372 MPa 445 MPa Fig. 9. Variation of the normalized z-axis intercepts of the plastic strain contours obtained from aged alloys with similar n but different r y (6, 11, and 36 h), to illustrate the effect of yield strength on the plastic strain distribution. A common feature of the e p contours, observed in all the aged conditions, is that they are not hemispherical but are rather elongated in the z-direction. This observation is consistent with that of Chaudhri [11] for annealed Cu of a much higher work hardening exponent (n = 0.46) than the samples used here. To further illustrate the elongated nature of the deformation field, normalized r z is plotted against normalized r x in Fig. 10. Samuels and Mulhearn [3], who ingeniously exploited the ability of brass to etch differentially to reveal the plastic strain fields, found the Fig. 8. Variation of the normalized x and z axes intercepts of the plastic strain contours obtained from aged alloys with similar r y but different n (6 and 94 h). Note that the curve fits are only guides for the eye. Fig. 10. Normalized z-axis intercepts of the iso-strain contours plotted against the corresponding normalized x-axis intercepts, to illustrate the elliptic nature of the strain distribution underneath the Vickers indent.

6 5176 G. Srikant et al. / Acta Materialia 54 (2006) strain ellipticity, dr z /dr x h n = h n = h 36 h n = n = yield strength, σ y (MPa) Fig. 11. Strain ellipticity defined as the slope of the linear fits through the z- vs. x-axes intercept data (shown in Fig. 10), dr z /dr x, plotted against the yield strength. contours to be hemispherical and proposed the compressive radial flow mechanism that was further developed as the expanding cavity model. If the compressive radial flow mechanism proposed by Samuels and Mulhearn [3] was to be operative, the r z /r x ratio should be equal to one. However, Fig. 10 shows that it can be more than two in some cases, implying a different deformation mechanism. Chaudhri [16] concluded, on the basis of his experimental observations on spherical, conical, and pyramidal indentations of mild steel samples, that the deformation mechanism associated with the indentation process followed neither the cutting mechanism nor the compression mechanism. This conclusion appeared to be true even for the relatively low strain hardening Al alloys considered in the present work. In this context, it should be mentioned that relatively coarser resolution of the etching technique employed by Samuels and Mulhearn [3] may have contributed to the observed difference. The degree of deviation of the strain contours from the hemispherical shape or ellipticity is quantified by examining the slope of the straight-line fits through the r z vs. r x data shown in Fig. 10. The slopes, dr z /dr x, obtained for various aging conditions are plotted as a function of r y in Fig. 11. The data obtained on 6, 11, and 36 h (increasing r y but with similar n) indicate that the ellipticity is relatively independent of r y. On the other hand, comparison of the dr z /dr x values obtained from the 6 and 94 h (similar r y but higher n in the latter case) suggests that higher n leads to less ellipticity. This is another possible reason for the observation of hemispherical strain fields in brass, which is a known metal for high strain hardening [3]. It is also instructive to examine the strains close to the indenter tip and put them in perspective. The maximum strains observed under the indenter in this work are in the range By looking at the deformation mechanism underneath the conical indenter, Johnson [10] estimated the single parameter, which was interpreted as the ratio of the strain imposed by the indenter to the maximum strain before yielding, as C ¼ðE=r y Þ cotðbþ ð2þ where b is the cone semi-angle. On the basis of equivalent volume displaced by a Vickers indenter, b was found to be equal to 70.3 [17]. The (E/r y ) values for the differently aged Al alloys used in this work range between 157 and 203, which means that the C values range between 56.3 and 72 from Eq. (2). Johnson [10] himself noted that the expanding cavity model may not applicable for C > 50, due to a change in the mode of deformation (from expanding cavity to cutting-type mechanism that prevails under fully plastic conditions). It is interesting to note that in a metallic glass, which has low E and high r y (96 GPa and 1900 MPa, respectively) and is close to an elastic perfectly plastic solid, the plastic flow fields (decorated by shear bands) are hemispherical as C is 18, within the range of validity for the expanding cavity model [18]. Chaudhri [11] observed that the maximum strains to be in the range in well-annealed Cu samples, and proposed that the maximum strain was dependent on the strain hardening characteristics of the material. Although these maximum strains are similar to that observed in the current work, some difference does exist (the maximum strain in our case is 0.31 as against 0.36 in Cu). This difference may be due to the different material characteristics of annealed Cu, which has relatively high E (120 GPa) and n (0.463) with low r y (50 MPa). The C value of this annealed Cu was calculated from Eq. (2) to be 860 (compared to for the aged Al alloys used in this study), significantly larger than the upper limit for the validity of the expanding cavity model Strain decay and crack tip analogy In order to examine the nature of strain decay ahead of the indenter tip (z-axis) as well as in the lateral direction, e p µ 1/r q is fit into the data shown in Figs. 8 and 9, with q as the fitting parameter. The values of the exponent q that best describe the experimental trends are given in Table 2. It is seen that the strain decay in the x-direction is inversely proportional to r with q values very close to 1 (except in the case of the 94 h aged sample which has a slightly higher n). Kramer et al. [19] utilized the expanding cavity model of Table 2 Plastic strain decay exponents and constants obtained through power-law fitting through the e p vs. r (z and x axes intercepts) data Aging time (h) Exponent q obtained by fitting e p = K/r q r x r z K obtained by fitting e p = K/r x HRR exponent, 1/(1 + n) ± ± ± ±

7 G. Srikant et al. / Acta Materialia 54 (2006) Johnson [10] to propose a relation that determines the size of the plastic zone, d, as a function of the yield strength, r y, and indentation load, P, as follows: d ¼ 3P 0:5 ð3þ 2pr y The inverse relationship deduced from our lateral strain field is e p ¼ K ð4þ r where K is the constant of proportionality and r is the distance from the center of the indent. At the plastic zone boundary, e p = e 0.2% at r = d in Eq. (4) giving e 0:2% ¼ K ð5þ d Combining Eqs. (3) and (5), we get K ¼ 3P 0:5 ð6þ e 0:2% 2pr y A linear relation between the experimental K and (r y ) 0.5 data was observed (Fig. 12), confirming that the lateral flow of material around the indenter can be characterized with the expanding cavity model. In contrast to the x-direction, curve fits to the strain decay in the z-direction consistently yielded a q value that is less than 1, implying that the expanding cavity model is unable to capture the trends accurately. Interestingly, elastoplastic fracture mechanics solutions for the strain fields at the crack tip appear to agree with the trends. Note that a stress singularity, similar to that seen ahead of a crack tip, exists at the tip of a sharp indenter. In fact, it is the presence of a stress singularity at the tip of a sharp indenter that makes it possible to generate plasticity even in brittle materials. The stress, strain, and displacement fields ahead of a K h 11 h h [yield strengh, σ y (MPa)] h Fig. 12. Variation of the constant K from the lateral plastic strain distribution with the inversed square root of yield strength, to illustrate the crack tip analogy. crack tip in an elastoplastic solid are described by the HRR (Hutchison, Rice, and Rosengren [20,21]), fields and are widely used in fracture mechanics literature. For the small-scale yielding condition, the strain distribution directly ahead of the crack tip, e ij, has a 1/r 1/(1+n) functional dependence [21]. Therefore, q should scale with 1/(1 + n), if HRR-type fields were to govern the strain distribution ahead of a sharp indenter. From Table 2, it can be seen that, for the first approximation, the exponent q in our case is approximately equal to the HRR exponent 1/(1 + n), except for the case of the 94 h aged sample where a similar discrepancy was observed in the lateral strain distribution as discussed earlier. This similarity between crack tip and sharp indentation strain fields, while interesting, needs further investigation and critical examination. 5. Summary Experiments were conducted to understand the factors influencing the strain distribution around a Vickers indenter and hence to resolve the debate over the mechanism governing the deformation. An Al Zn Mg alloy was chosen as the material of study. From the aging curve, conditions were derived for which the alloy possessed similar yield strength but different strain hardening exponents and vice versa. This enabled comparison of the indentation response obtained from the under-aged and over-aged alloy for a constant yield strength condition and for a constant strain hardening condition. Hardness distributions were generated under the sectioned Vickers macroindents. These hardness plots were converted into equivalent plastic strain distributions that were described by strain contours. Comparison of the strain contours for constant yield stress and constant work hardening exponent condition reveals the dependence of the plastic strain distribution on r y and n values of a material. In all cases, it was found that strain fields are elliptical, with the long axis coinciding with the direction of indentation. The strain field ellipticity appears to be sensitive to the work hardening behavior of the material, with higher n leading to lower ellipticity. While the lateral strain distribution is in agreement with the expanding cavity model, strain distribution directly ahead of the indenter tip appears to follow the HRR fields of a crack tip in an elastoplastic solid. Acknowledgements We thank Mr. Niraj Nayan of the Vikram Sarabhai Space Centre, Indian Space Research Organization, Tiruvananthapuram, for providing the Al alloy used in this study and Mr. S. Sashidhara and Mr. Srinivasamurty for their experimental assistance. References [1] Brinell JA. In: 2ieme Conres. Internationale Méthodes d Essai, Paris, France; 1900.

8 5178 G. Srikant et al. / Acta Materialia 54 (2006) [2] Hill R, Lee EH, Tupper SJ. Proc R Soc A 1947;188:273. [3] Samuels LE, Mulhearn TO. J Mech Phys Solids 1957;5:125. [4] Hill R. The mathematical theory of plasticity. Oxford (UK): Clarendon Press; [5] Dugdale DS. J Mech Phys Solids 1957;6:85. [6] Marsh DM. Proc R Soc A 1963;279:420. [7] Tabor D. The hardness of metals. Oxford (UK): Clarendon Press; [8] Hirst W, Howse MGJW. Proc R Soc A 1969;311:429. [9] Atkins AG, Tabor D. J Mech Phys Solids 1965;13:149. [10] Johnson KL. J Mech Phys Solids 1970;18:115. [11] Chaudhri MM. Acta Mater 1998;46:3047. [12] Mata M, Anglada M, Alcala J. J Mater Res 2002;17:964. [13] Giannakopoulos AE, Larsson PL, Vestergaard R. Int J Solids Struct 1994;31:2679. [14] Bhattacharya AK, Nix WD. Int J Solids Struct 1991;27:1047. [15] Wood J, McCormick PG. Acta Metall 1987;35:247. [16] Chaudhri MM. Philos Mag Lett 1993;67:107. [17] Dao M, Chollacoop N, Van Vliet KJ, Venkatesh TA, Suresh S. Acta Mater 2001;49:3899. [18] Ramamurty U, Jana S, Kawamura Y, Chattopadhyay K. Acta Mater 2005;53:705. [19] Kramer D, Huang H, Kriese M, Robach J, Nelson J, Wright A, et al. Acta Metall 1998;47:333. [20] Hutchinson JW. J Mech Phys Solids 1968;16:13. [21] Rice JR, Rosengren GF. J Mech Phys Solids 1968;16:1.