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An analytical model of the surface roughness of an aluminum alloy treated with a surface nanocrystallization and hardening process
Scripta Materialia 52 (2005) 259–263 www.actamat-journals.com
An analytical model of the surface roughness of an aluminum alloy treated with a surface nanocrystallization and hardening process K. Dai, J. Villegas, L. Shaw
*
Department of Materials Science and Engineering, Institute of Materials Science, University of Connecticut, U-3136 Storrs, CT 06269, USA Received 2 September 2004; received in revised form 7 October 2004; accepted 18 October 2004 Available online 5 November 2004
Abstract An analytical model is developed to predict the surface roughness of Al-5052 plates treated with the surface nanocrystallization and hardening process. The peak-to-valley distances calculated using the analytical model compare reasonably well with the experimental data as well as those predicted using the finite element analysis. Ó 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Severe plastic deformation; Aluminum alloys; Surface structure; Micromechanical modeling
1. Introduction Severe plastic deformation (SPD) has been known to be one of the effective methods for making nanocrystalline (NC) materials [1,2]. Recently such material processing strategy has been extended to make bulk materials with a NC-surface and coarse-grained interior [3–10]. Ultrasonic shot peening [3,4], high-energy shot peening [5], surface mechanical attrition or surface mechanical attrition treatment [6–8], and surface nanocrystallization and hardening (SNH) [9,10] are examples of these new processes. The surface nanocrystallization in these processes relies on severe plastic deformation at the surface layer of the metallic component induced by repeated impacts of high-energy balls under a controlled atmosphere [3–10]. Strengthening introduced by these surface SPD-based processes has been demonstrated in several metallic materials [5,10,11].
*
Corresponding author. Tel.: +1 860 486 2592; fax: +1 860 486 4745. E-mail address: [email protected] (L. Shaw).
A potential downside of the SNH process is the rough surface caused by repeated impacts of balls which could be several millimeters in diameter. The rough surface created may take away the beneficial effects of a surface nanocrystalline microstructure. This roughness may induce stress concentration at specific points, and thus facilitate crack initiation, especially under fatigue loading conditions [12,13]. On the other hand, if surface roughness is controlled properly, strengthening due to the SNH treatment can result in improvement in fatigue resistance [11]. In order to provide guidelines in controlling the surface roughness during the SNH process, a finite element model (FEM) has been developed recently [14]. This previous study established that the surface roughness of the material subject to the SNH treatment is mainly dictated by the indentation process of the impacting balls [14]. Furthermore, the peak-to-valley (PV) distance predicted from the FEM matches the maximum PV value measured experimentally very well [14]. In spite of the predictability of the FEM, it is desirable to have an analytical model that can be used to quantitatively predict the surface roughness and evaluate the effects
1359-6462/$ - see front matter Ó 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2004.10.021
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K. Dai et al. / Scripta Materialia 52 (2005) 259–263
of various SNH process parameters on the surface roughness. Such an analytical model is especially useful for those who do not have access to finite element codes. In this study, an analytical model has been developed to predict the PV value of the surface roughness of Al5052 plates treated with the SNH process. The analytical results are compared with the experimental data obtained from 3D non-contact optical profilometry roughness measurements and those calculated from the previous FEM [14]. This study enables the prediction of the effects of major SNH process parameters on surface roughness of the specimen and, in the future, the extension of those predictions to the performance of the component tested under fatigue conditions.
2. Experimental procedure Al-5052 plates with a nominal thickness of 4.7 mm was purchased from McMaster-Carr, New Brunswick, NJ at the condition of Temper H32 (i.e. strain-hardened and then stabilized using a low temperature thermal treatment). Al-5052 is a FCC single-phase material and has the nominal chemical composition of 2.2– 2.8% Mg, 0.15–0.35% Cr, <0.45% Si, <0.45% Fe, <0.1% Cu, <0.1% Mn, <0.1% Zn, and Al balance (wt.%). Discs 49 mm in diameter were cut out of the Al-5052 plates and were cleaned with acetone and then ethanol. After cleaning, the discs with an initial peakto-valley (PV) value of 1.6 lm were used directly for the SNH treatment without any grinding and polishing. The Al-5052 disc was loaded at the one end of a cylindrical container with internal dimensions of 38.0 mm in diameter and 57.0 mm deep made of hardened steel. The disc was held in place via mechanical locking by pushing the disc against the rigid cover of the container. As a result, one side of the disc would be subjected to impacts of high velocity balls, while the other side of the disc was held rigidly against the thick cover of the container. Five tungsten carbide/cobalt (WC/Co) balls with a diameter of either 7.94 mm or 4.95 mm were loaded into the container to provide the desired impact on the surface of the Al-5052 disc. Loading of the disc and balls was conducted in a glove box and the container was filled with argon before the SNH treatment. The high velocity of the balls in the SNH process (5 m/s) was achieved by shaking the container three dimensionally using a Spex 8000 Mill. Such 3D shaking provided kinetic energy to the balls and generated the complex pattern of motion of the balls inside the container [15]. Various processing times, ranging from 5 min to 180 min, were applied. Each Al-5052 disc was processed on one side only and no temperature control was employed during the SNH process. The roughness evolution of the sample surface as a function of the SNH processing time was monitored using a three-
dimensional non-contact optical profilometer (Zygo NewView 5000, Zygo Corp., Middlefield, CT).
3. Description of the analytical model An analytical model is proposed to estimate the peakto-valley value on the sample surface impacted by a rigid high-velocity ball. The PV value is defined as the distance measured from the highest peak to the lowest valley on the surface of the sample, as shown in Fig. 1 [16]. In the model the metal plate being SNH-processed is treated as a semi-infinite body and is the only body having the deformation during the ball impact (Fig. 2). The gravity of the ball is neglected since it is very small when compared with the impacting force. According to NewtonÕs second law, the equation of motion during contact is m
dv ¼ P m A dt
ð1Þ
where m and v represent the mass and velocity of the ball, respectively; t is time; Pm is the mean normal pressure on the plate; and A is the projected contact area. The negative sign in Eq. (1) is to account for the reduction of the ball velocity. Additionally, one has the following relations dv dZ dv dv ¼ ¼v dt dt dZ dZ
ð2Þ
PV
Fig. 1. The peak-to-valley (PV) value as an indicator of the maximum surface roughness.
v
P(r) r Z
2a Z Fig. 2. The illustration of deformation on a semi-infinite body being impacted by a rigid ball under fully plastic contact. The area within the dashed line represents the plastic zone in the deformed body.
K. Dai et al. / Scripta Materialia 52 (2005) 259–263
and 4 m ¼ pR3 qb 3
Z pv ¼ c2 R
A ¼ pa2 2pRZ
ð4Þ
where a is the radius of the projected contact area. Substituting Eqs. (2)–(4), into Eq. (1) and integrating each side over the indent depth from Z1 to Z2 during the impacting process, one obtains Z Z2 1 P m Z dZ ¼ qb R2 ðv21 v22 Þ ð5Þ 3 Z1 where v1 and v2 are the velocities of the ball corresponding to the indent depths Z1 and Z2, respectively. It is further assumed that the contact is fully plastic, i.e. plastic flow occurs on the entire contact surface [17– 19]. This assumption is reasonable for the SNH treatment of Al alloys because Al-5052 alloy is soft and the impacting energy is high. Thus, for a fully plastic contact, Pm can be approximately expressed by a mean contact pressure under static load by a spherical indenter [20] n 6k 8a ð6Þ Pm ¼ n þ 2 9pR where n and k are the strain hardening exponent and strength coefficient, respectively, in the following power-law strain hardening relation of the plate r ¼ ken
ð3Þ
where Z is the indent depth (Fig. 2), and qb and R are the density and radius of the ball, respectively. Based on the geometry (Fig. 2) and neglecting Z2 term (because Z R), one can also obtain
ð7Þ
where r and e are the true stress and strain, respectively. Note that using Eqs. (6) and (7) implies a low strain rate insensitivity of the Al-5052 plate. This is reasonable since previous studies [21] have indicated that many aluminum alloys are strain-rate insensitive. To integrate Eq. (5), let Z1 = 0, then v1 is the velocity of the impacting ball right before contacting the surface of the sample. Furthermore, choose Z2 as the total indent depth and thus v2 = 0. Substituting Eq. (6) and these boundary conditions into Eq. (5), one has n 2 9p ðn þ 2Þðn þ 4Þqb v21 ðnþ4Þ pffiffiffi Z2 ¼ R ð8Þ 36k 8 2 Note that Z2 is not only the total indent depth, but also the PV value, Zpv, when the piling-up or sinking-in is small enough to be neglected. When the effect of piling-up or sinking-in around the spherical indent is considered, the PV value is corrected as follows. For the material with n smaller than 0.213, piling-up occurs around the spherical indenter and the PV value is [22]
261
2 n 9p ðn þ 2Þðn þ 4Þqb v21 ðnþ4Þ pffiffiffi ðn < 0:213Þ 36kcnþ2 8 2 ð9aÞ
where c2 1 = 0.276 1.748n + 2.451n2 1.469n3 [22]. For the material with n larger than 0.213 (which leads to c2 1 < 0), sinking-in occurs around the spherical indent and the PV value is [22] n 2 9p ðn þ 2Þðn þ 4Þqb v21 ðnþ4Þ pffiffiffi Z pv ¼ R ðn > 0:213Þ 36kcnþ2 8 2 ð9bÞ
4. Results and discussion The parameters used in the experiment (Section 2) are listed as follows: for Al-5052 plates, n = 0.12 and k = 332 MPa [9]; for WC/Co balls, qb = 14.5 g/cm3 and v1 = 5 m/s. Substituting these parameters into Eq. (9a) (the piling-up effect is considered here since n is smaller than 0.213), one obtains Zpv = 50.7 lm when R = 2.5 mm, and Zpv = 79.8 lm when R = 3.95 mm. These data along with the maximum PV values measured experimentally and the predicted PV value from the previous finite element model [14] are listed in Table 1 for comparison. It can be seen that the finite element predictions are very close to the experimental maximum PV values with discrepancy less than 11%. In contrast, the discrepancy between the predictions from the analytical model and the measured maximum PV values is relatively large with 23% difference for the Al-plates treated with 5-mm balls and 27% difference for the Alplates treated with 7.9-mm balls. The large discrepancy between the experimental data and the analytical model predictions is due to the expression of the mean contact pressure, Pm, in Eq. (6), which is an empirical equation under static load by a spherical indenter [20]. As a result of the approximation and the assumed relationship between the mean contact pressure and the uniaxial yield strength [20], Pm is overestimated. Thus, it can be seen from Eq. (5) that the PV value is then underestimated. In spite of the relatively large discrepancy between the experimental data and the analytical model predictions, the analytical model is valuable in evaluating the dependency of surface roughness on the SNH parameters.
Table 1 The PV values measured experimentally and predicted from the models
5-mm Balls 7.9-mm Balls *
Ref. [14].
Experiment* (lm)
Finite element model* (lm)
Analytical model (lm)
65 110
72 105
50 80
262
K. Dai et al. / Scripta Materialia 52 (2005) 259–263
Based on Eqs. (9a) and (9b), one can show that the surface roughness of the specimen treated with the SNH process has the following dependencies on the radius of the impacting ball, the impacting velocity, the ball density, and the strength coefficient of the specimen. 4
Z pv / R;
Z pv / v1nþ4 ;
2
2
Z pv / k nþ4
Z pv / qbnþ4 ;
ð10Þ
Eq. (10) shows that the surface roughness increases linearly with the radius of the ball, whereas the effects of the ball velocity and density on surface roughness are described by power laws. Substituting n = 0.12 (for Al5052) into Eq. (10), one has Z pv / v0:97 1
and
Z pv / q0:48 b
ð11Þ
Thus, for Al-5052 alloys the effect of reducing the radius of the impacting ball on reducing the surface roughness of Al-5052 is similar to that of reducing the impacting velocity, whereas reducing the ball density has less influence on the surface roughness than the ball radius and velocity. Eq. (10) also indicates that the surface roughness is inversely proportional to the strength coefficient of the material being treated. Again, assuming the material being treated is an Al-5052 alloy, one has Z pv / k 0:48
ð12Þ
which indicates that increasing the strength coefficient of the material being treated will reduce the surface roughness. In order to verify the predictability of Eq. (10), numerical simulations based on a finite element model established in the previous study [14] have been conducted. Fig. 3 shows the effect of the strength coefficient of the plate on the PV value. Clearly, the finite element analysis shows that Zpv is indeed varied with k defined by a power law with a power of 0.52 which is very close to the analytical prediction of 0.48 shown in Eq. (12). Effects of other parameters, R, v1 and qb, have also been checked using finite element modeling (not
shown here), and all are found to be consistent with the predictions of Eq. (10). It should be pointed out that the effect of some SNH parameters may not be able to be expressed explicitly, as those shown in Eq. (10). Under such conditions, curves showing the relationship between the surface roughness and the SNH parameter would be good choices. Shown in Fig. 4 are the curves showing the PV value as a function of the strain hardening exponent, n, of the material being treated. There are two curves in this figure, one corresponding to the strength coefficient, k, as a constant and the other to the yield strength, ry, as a constant. This is necessary because of the power-law strain hardening relationship we used, Eq. (7). This relationship dictates that the yield strength at e = 0.002 and k cannot be kept constant simultaneously if n is changing. Thus, the two curves have been evaluated. The physical meanings of the two curves are as follows. For the curve with ry = constant, increasing n increases the work hardening rate and the flow stress for a given strain. Thus, the PV value decreases as n increases because of the increased resistance to plastic deformation. For the curve with k = constant, increasing n decreases the yield strength of the material. As a result, although the material has a higher work hardening rate (larger n), the flow stress at a given strain (when e < 1) is lower for the material with higher n than that for the material with lower n. Consequently, the PV value increases as n increases because of the decreased resistance to plastic deformation. It is also noted from Fig. 4 that the PV value change is very small for the curve with ry = constant when sinking-in occurs (i.e. in the region where n > 0.213). This phenomenon is due to the fact that when sinking-in occurs, the contact area decreases as the flow stress increases. As a result, the PV value does not change much because the increased flow stress of the material (due to the increased n) is compensated by the decreased contact area.
180.00
5
160.00
y = -0.5156x + 7.6462
4.8
PV (µm)
ln (Zpv) (µm)
4.9
4.7 4.6
120.00
Sinking-in regime
k = constant
σy = constant
100.00 80.00
4.5 4.4 5.2
Piling-up regime
140.00
60.00 5.4
5.6
5.8
6
6.2
ln(k) (MPa)
Fig. 3. The PV value predicted from the finite element model as a function of the strength coefficient of the plate, k, predicting a relationship of Zpv / k0.52. Other parameters for the simulation are: (i) single impact; (ii) a 7.9-mm WC/Co ball; (iii) n = 0.12; and (iv) v1 = 5 m/s.
0
0.1
0.2
0.3
0.4
0.5
0.6
Strain hardening exponent, n
Fig. 4. The PV value predicted from the analytical model as a function of the strain hardening exponent, n, when k is constant (k = 332 MPa) or the yield strength is constant (194 MPa). Other parameters are: (i) single impact; (ii) a 7.9-mm WC/Co ball; and (iii) v1 = 5 m/s. The piling-up and sinking-in regimes are shown.
K. Dai et al. / Scripta Materialia 52 (2005) 259–263
Finally, it should be noted that in many cases the surface roughness is preferably characterized using the root-mean-square average (RMS) or arithmetic-mean value (AMV) rather than the PV value. In a previous study [14], it was shown that RMS and AMV exhibit the same trend as the PV value and all of them reach the maximum value at the same processing time. Thus, although the analytical model proposed here does not provide estimation of the root-mean-square average and arithmetic-mean value, it does offer the guideline on how to minimize the surface roughness since the PV value and RMS and AMV have the same dependence on experimental conditions.
5. Concluding remarks An analytical model based on the fully plastic contact theory has been developed to predict the surface roughness of Al-5052 plates treated with the SNH process. The predicted surface roughness compares reasonably well with the experimental measurement. The analytical model developed can be utilized to evaluate the dependence of the surface roughness on various SNH processing and material parameters. These dependences provide the fundamental knowledge for optimization of the SNH process to minimize the surface roughness.
Acknowledgment The authors acknowledge the financial support by the National Science Foundation through Grant No. DMR-0207729.
References [1] Valiev RZ, Islamgaliev RK, Alexandrov IV. Bulk nanostructured materials from severe plastic deformation. Prog Mat Sci 2000;45:103–89. [2] Vinogradov A, Hashimoto S, Kopylov VI. Enhanced strength and fatigue life of ultra-fine grain Fe–36Ni invar alloy. Mater Sci Eng A 2003;355:277–85. [3] Tao NR, Sui ML, Lu J, Lu K. Surface nanocrystallization of iron induced by ultrasonic shot peening. Nanostruct Mater 1999;11(4):433–40. [4] Wu X, Tao N, Hong Y, Xu B, Lu J, Lu K. Microstructure and evolution of mechanically induced ultrafine grain in surface layer of Al-alloy subjected to USSP. Acta Mater 2002;50:2075–84.
263
[5] Liu G, Wang SC, Lou XF, Lu J, Lu K. Low carbon steel with nanostructured surface layer induced by high-energy shot peening. Scripta Mater 2001;44:1791–5. [6] Tao NR, Wang ZB, Tong WP, Sui ML, Lu J, Lu K. An investigation of surface nanocrystallization mechanism in Fe induced by surface mechanical attrition treatment. Acta Mater 2002;50:4603–16. [7] Zhang HW, Hei ZK, Liu G, Lu J, Lu K. Formation of nanostructured surface layer on AISI 304 stainless steel by means of surface mechanical attrition treatment. Acta Mater 2003;51:1871–81. [8] Zhu KY, Vassel A, Brisset F, Lu K, Lu J. Nanostructure formation mechanism of a-titanium using SMAT. Acta Mater 2004;52:4101–10. [9] Villegas J, Dai K, Shaw L. Surface roughness evolution in the surface nanocrystallization and hardening (SNH) process. In: Srivatsan T, Varin R, editors. Processing and fabrication of advanced mateirials: XII. Materials Park, OH: ASM International; 2003. p. 358–72. [10] Villegas J, Dai K, Shaw L, Liaw P. Experiments and modeling of the surface nanocrystallization and hardening (SNH) process. In: Shaw L, Suryanarayana C, Mishra R, editors. Processing and properties of structural nanomaterials. Warrendale, PA: TMS; 2003. p. 61–8. [11] Yuan W, Villegas J, Tian JW, McDaniels R, Shaw L, Liaw PK, et al. Fatigue behavior of a Hastelloy C-2000 superalloy processed by the surface nanocrystallization and hardening process. Acta Mater. Submitted for publication. [12] Itoga H, Tokaji K, Nakajima M, Ko H-N. Effect of surface roughness on step-wise S–N characteristics in high strength steel. Int J Fatigue 2003;25:379–85. [13] Rodopoulos CA, Curtis SA, de los Rios ER, SolisRomero J. Optimization of the fatigue resistance of 2024-T351 aluminum alloys by controlled shot peening—methodology, results and analysis. Int J Fatigue 2004;26:849–56. [14] Dai K, Villegas J, Stone Z, Shaw L. Finite element modeling of the surface roughness of 5052 Al alloy subjected to a surface severe plastic deformation process. Acta Mater 2004;52(20): 5771–82. [15] Wang W. Available from: http://www.geocities.com/www67/ MA.html#figure4. [16] MetroPro Reference Guide OMP-0347D, Zygo Corp., Middlefield, CT, p. 3.51–3.52. [17] Al-Obaid YF. Shot peening mechanics: experimental and theoretical analysis. Mech Mat 1995;19:251–60. [18] Shaw MC, DeSalvo G. On the plastic flow beneath a blunt axisymmetric indentor. Trans ASME J Eng Ind 1970;92:480–94. [19] Ahn JH, Kwon D. Derivation of plastic stress–strain relationship from ball indentation: examination of strain definition and pileup effect. J Mat Res 2001;16(11):3170–8. [20] Matthews JR. Indentation hardness and hot pressing. Acta Metall 1980;28(3):311–8. [21] Pageau G, Vaziri R, Poursartip A. An evaluation of metal matrix composites as ballistic protection materials. In: 23rd International SAMPE Technical Conference. 1991. pp. 639–50. [22] Alcala´ J, Barone AC, Anglada M. The influence of plastic hardening on surface deformation modes around vickers and spherical indents. Acta Mater 2000;48(13):3451–64.
An analytical model of the surface roughness of an aluminum alloy treated with a surface nanocrystallization and hardening process K. Dai, J. Villegas, L. Shaw
*
Department of Materials Science and Engineering, Institute of Materials Science, University of Connecticut, U-3136 Storrs, CT 06269, USA Received 2 September 2004; received in revised form 7 October 2004; accepted 18 October 2004 Available online 5 November 2004
Abstract An analytical model is developed to predict the surface roughness of Al-5052 plates treated with the surface nanocrystallization and hardening process. The peak-to-valley distances calculated using the analytical model compare reasonably well with the experimental data as well as those predicted using the finite element analysis. Ó 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Severe plastic deformation; Aluminum alloys; Surface structure; Micromechanical modeling
1. Introduction Severe plastic deformation (SPD) has been known to be one of the effective methods for making nanocrystalline (NC) materials [1,2]. Recently such material processing strategy has been extended to make bulk materials with a NC-surface and coarse-grained interior [3–10]. Ultrasonic shot peening [3,4], high-energy shot peening [5], surface mechanical attrition or surface mechanical attrition treatment [6–8], and surface nanocrystallization and hardening (SNH) [9,10] are examples of these new processes. The surface nanocrystallization in these processes relies on severe plastic deformation at the surface layer of the metallic component induced by repeated impacts of high-energy balls under a controlled atmosphere [3–10]. Strengthening introduced by these surface SPD-based processes has been demonstrated in several metallic materials [5,10,11].
*
Corresponding author. Tel.: +1 860 486 2592; fax: +1 860 486 4745. E-mail address: [email protected] (L. Shaw).
A potential downside of the SNH process is the rough surface caused by repeated impacts of balls which could be several millimeters in diameter. The rough surface created may take away the beneficial effects of a surface nanocrystalline microstructure. This roughness may induce stress concentration at specific points, and thus facilitate crack initiation, especially under fatigue loading conditions [12,13]. On the other hand, if surface roughness is controlled properly, strengthening due to the SNH treatment can result in improvement in fatigue resistance [11]. In order to provide guidelines in controlling the surface roughness during the SNH process, a finite element model (FEM) has been developed recently [14]. This previous study established that the surface roughness of the material subject to the SNH treatment is mainly dictated by the indentation process of the impacting balls [14]. Furthermore, the peak-to-valley (PV) distance predicted from the FEM matches the maximum PV value measured experimentally very well [14]. In spite of the predictability of the FEM, it is desirable to have an analytical model that can be used to quantitatively predict the surface roughness and evaluate the effects
1359-6462/$ - see front matter Ó 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2004.10.021
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K. Dai et al. / Scripta Materialia 52 (2005) 259–263
of various SNH process parameters on the surface roughness. Such an analytical model is especially useful for those who do not have access to finite element codes. In this study, an analytical model has been developed to predict the PV value of the surface roughness of Al5052 plates treated with the SNH process. The analytical results are compared with the experimental data obtained from 3D non-contact optical profilometry roughness measurements and those calculated from the previous FEM [14]. This study enables the prediction of the effects of major SNH process parameters on surface roughness of the specimen and, in the future, the extension of those predictions to the performance of the component tested under fatigue conditions.
2. Experimental procedure Al-5052 plates with a nominal thickness of 4.7 mm was purchased from McMaster-Carr, New Brunswick, NJ at the condition of Temper H32 (i.e. strain-hardened and then stabilized using a low temperature thermal treatment). Al-5052 is a FCC single-phase material and has the nominal chemical composition of 2.2– 2.8% Mg, 0.15–0.35% Cr, <0.45% Si, <0.45% Fe, <0.1% Cu, <0.1% Mn, <0.1% Zn, and Al balance (wt.%). Discs 49 mm in diameter were cut out of the Al-5052 plates and were cleaned with acetone and then ethanol. After cleaning, the discs with an initial peakto-valley (PV) value of 1.6 lm were used directly for the SNH treatment without any grinding and polishing. The Al-5052 disc was loaded at the one end of a cylindrical container with internal dimensions of 38.0 mm in diameter and 57.0 mm deep made of hardened steel. The disc was held in place via mechanical locking by pushing the disc against the rigid cover of the container. As a result, one side of the disc would be subjected to impacts of high velocity balls, while the other side of the disc was held rigidly against the thick cover of the container. Five tungsten carbide/cobalt (WC/Co) balls with a diameter of either 7.94 mm or 4.95 mm were loaded into the container to provide the desired impact on the surface of the Al-5052 disc. Loading of the disc and balls was conducted in a glove box and the container was filled with argon before the SNH treatment. The high velocity of the balls in the SNH process (5 m/s) was achieved by shaking the container three dimensionally using a Spex 8000 Mill. Such 3D shaking provided kinetic energy to the balls and generated the complex pattern of motion of the balls inside the container [15]. Various processing times, ranging from 5 min to 180 min, were applied. Each Al-5052 disc was processed on one side only and no temperature control was employed during the SNH process. The roughness evolution of the sample surface as a function of the SNH processing time was monitored using a three-
dimensional non-contact optical profilometer (Zygo NewView 5000, Zygo Corp., Middlefield, CT).
3. Description of the analytical model An analytical model is proposed to estimate the peakto-valley value on the sample surface impacted by a rigid high-velocity ball. The PV value is defined as the distance measured from the highest peak to the lowest valley on the surface of the sample, as shown in Fig. 1 [16]. In the model the metal plate being SNH-processed is treated as a semi-infinite body and is the only body having the deformation during the ball impact (Fig. 2). The gravity of the ball is neglected since it is very small when compared with the impacting force. According to NewtonÕs second law, the equation of motion during contact is m
dv ¼ P m A dt
ð1Þ
where m and v represent the mass and velocity of the ball, respectively; t is time; Pm is the mean normal pressure on the plate; and A is the projected contact area. The negative sign in Eq. (1) is to account for the reduction of the ball velocity. Additionally, one has the following relations dv dZ dv dv ¼ ¼v dt dt dZ dZ
ð2Þ
PV
Fig. 1. The peak-to-valley (PV) value as an indicator of the maximum surface roughness.
v
P(r) r Z
2a Z Fig. 2. The illustration of deformation on a semi-infinite body being impacted by a rigid ball under fully plastic contact. The area within the dashed line represents the plastic zone in the deformed body.
K. Dai et al. / Scripta Materialia 52 (2005) 259–263
and 4 m ¼ pR3 qb 3
Z pv ¼ c2 R
A ¼ pa2 2pRZ
ð4Þ
where a is the radius of the projected contact area. Substituting Eqs. (2)–(4), into Eq. (1) and integrating each side over the indent depth from Z1 to Z2 during the impacting process, one obtains Z Z2 1 P m Z dZ ¼ qb R2 ðv21 v22 Þ ð5Þ 3 Z1 where v1 and v2 are the velocities of the ball corresponding to the indent depths Z1 and Z2, respectively. It is further assumed that the contact is fully plastic, i.e. plastic flow occurs on the entire contact surface [17– 19]. This assumption is reasonable for the SNH treatment of Al alloys because Al-5052 alloy is soft and the impacting energy is high. Thus, for a fully plastic contact, Pm can be approximately expressed by a mean contact pressure under static load by a spherical indenter [20] n 6k 8a ð6Þ Pm ¼ n þ 2 9pR where n and k are the strain hardening exponent and strength coefficient, respectively, in the following power-law strain hardening relation of the plate r ¼ ken
ð3Þ
where Z is the indent depth (Fig. 2), and qb and R are the density and radius of the ball, respectively. Based on the geometry (Fig. 2) and neglecting Z2 term (because Z R), one can also obtain
ð7Þ
where r and e are the true stress and strain, respectively. Note that using Eqs. (6) and (7) implies a low strain rate insensitivity of the Al-5052 plate. This is reasonable since previous studies [21] have indicated that many aluminum alloys are strain-rate insensitive. To integrate Eq. (5), let Z1 = 0, then v1 is the velocity of the impacting ball right before contacting the surface of the sample. Furthermore, choose Z2 as the total indent depth and thus v2 = 0. Substituting Eq. (6) and these boundary conditions into Eq. (5), one has n 2 9p ðn þ 2Þðn þ 4Þqb v21 ðnþ4Þ pffiffiffi Z2 ¼ R ð8Þ 36k 8 2 Note that Z2 is not only the total indent depth, but also the PV value, Zpv, when the piling-up or sinking-in is small enough to be neglected. When the effect of piling-up or sinking-in around the spherical indent is considered, the PV value is corrected as follows. For the material with n smaller than 0.213, piling-up occurs around the spherical indenter and the PV value is [22]
261
2 n 9p ðn þ 2Þðn þ 4Þqb v21 ðnþ4Þ pffiffiffi ðn < 0:213Þ 36kcnþ2 8 2 ð9aÞ
where c2 1 = 0.276 1.748n + 2.451n2 1.469n3 [22]. For the material with n larger than 0.213 (which leads to c2 1 < 0), sinking-in occurs around the spherical indent and the PV value is [22] n 2 9p ðn þ 2Þðn þ 4Þqb v21 ðnþ4Þ pffiffiffi Z pv ¼ R ðn > 0:213Þ 36kcnþ2 8 2 ð9bÞ
4. Results and discussion The parameters used in the experiment (Section 2) are listed as follows: for Al-5052 plates, n = 0.12 and k = 332 MPa [9]; for WC/Co balls, qb = 14.5 g/cm3 and v1 = 5 m/s. Substituting these parameters into Eq. (9a) (the piling-up effect is considered here since n is smaller than 0.213), one obtains Zpv = 50.7 lm when R = 2.5 mm, and Zpv = 79.8 lm when R = 3.95 mm. These data along with the maximum PV values measured experimentally and the predicted PV value from the previous finite element model [14] are listed in Table 1 for comparison. It can be seen that the finite element predictions are very close to the experimental maximum PV values with discrepancy less than 11%. In contrast, the discrepancy between the predictions from the analytical model and the measured maximum PV values is relatively large with 23% difference for the Al-plates treated with 5-mm balls and 27% difference for the Alplates treated with 7.9-mm balls. The large discrepancy between the experimental data and the analytical model predictions is due to the expression of the mean contact pressure, Pm, in Eq. (6), which is an empirical equation under static load by a spherical indenter [20]. As a result of the approximation and the assumed relationship between the mean contact pressure and the uniaxial yield strength [20], Pm is overestimated. Thus, it can be seen from Eq. (5) that the PV value is then underestimated. In spite of the relatively large discrepancy between the experimental data and the analytical model predictions, the analytical model is valuable in evaluating the dependency of surface roughness on the SNH parameters.
Table 1 The PV values measured experimentally and predicted from the models
5-mm Balls 7.9-mm Balls *
Ref. [14].
Experiment* (lm)
Finite element model* (lm)
Analytical model (lm)
65 110
72 105
50 80
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K. Dai et al. / Scripta Materialia 52 (2005) 259–263
Based on Eqs. (9a) and (9b), one can show that the surface roughness of the specimen treated with the SNH process has the following dependencies on the radius of the impacting ball, the impacting velocity, the ball density, and the strength coefficient of the specimen. 4
Z pv / R;
Z pv / v1nþ4 ;
2
2
Z pv / k nþ4
Z pv / qbnþ4 ;
ð10Þ
Eq. (10) shows that the surface roughness increases linearly with the radius of the ball, whereas the effects of the ball velocity and density on surface roughness are described by power laws. Substituting n = 0.12 (for Al5052) into Eq. (10), one has Z pv / v0:97 1
and
Z pv / q0:48 b
ð11Þ
Thus, for Al-5052 alloys the effect of reducing the radius of the impacting ball on reducing the surface roughness of Al-5052 is similar to that of reducing the impacting velocity, whereas reducing the ball density has less influence on the surface roughness than the ball radius and velocity. Eq. (10) also indicates that the surface roughness is inversely proportional to the strength coefficient of the material being treated. Again, assuming the material being treated is an Al-5052 alloy, one has Z pv / k 0:48
ð12Þ
which indicates that increasing the strength coefficient of the material being treated will reduce the surface roughness. In order to verify the predictability of Eq. (10), numerical simulations based on a finite element model established in the previous study [14] have been conducted. Fig. 3 shows the effect of the strength coefficient of the plate on the PV value. Clearly, the finite element analysis shows that Zpv is indeed varied with k defined by a power law with a power of 0.52 which is very close to the analytical prediction of 0.48 shown in Eq. (12). Effects of other parameters, R, v1 and qb, have also been checked using finite element modeling (not
shown here), and all are found to be consistent with the predictions of Eq. (10). It should be pointed out that the effect of some SNH parameters may not be able to be expressed explicitly, as those shown in Eq. (10). Under such conditions, curves showing the relationship between the surface roughness and the SNH parameter would be good choices. Shown in Fig. 4 are the curves showing the PV value as a function of the strain hardening exponent, n, of the material being treated. There are two curves in this figure, one corresponding to the strength coefficient, k, as a constant and the other to the yield strength, ry, as a constant. This is necessary because of the power-law strain hardening relationship we used, Eq. (7). This relationship dictates that the yield strength at e = 0.002 and k cannot be kept constant simultaneously if n is changing. Thus, the two curves have been evaluated. The physical meanings of the two curves are as follows. For the curve with ry = constant, increasing n increases the work hardening rate and the flow stress for a given strain. Thus, the PV value decreases as n increases because of the increased resistance to plastic deformation. For the curve with k = constant, increasing n decreases the yield strength of the material. As a result, although the material has a higher work hardening rate (larger n), the flow stress at a given strain (when e < 1) is lower for the material with higher n than that for the material with lower n. Consequently, the PV value increases as n increases because of the decreased resistance to plastic deformation. It is also noted from Fig. 4 that the PV value change is very small for the curve with ry = constant when sinking-in occurs (i.e. in the region where n > 0.213). This phenomenon is due to the fact that when sinking-in occurs, the contact area decreases as the flow stress increases. As a result, the PV value does not change much because the increased flow stress of the material (due to the increased n) is compensated by the decreased contact area.
180.00
5
160.00
y = -0.5156x + 7.6462
4.8
PV (µm)
ln (Zpv) (µm)
4.9
4.7 4.6
120.00
Sinking-in regime
k = constant
σy = constant
100.00 80.00
4.5 4.4 5.2
Piling-up regime
140.00
60.00 5.4
5.6
5.8
6
6.2
ln(k) (MPa)
Fig. 3. The PV value predicted from the finite element model as a function of the strength coefficient of the plate, k, predicting a relationship of Zpv / k0.52. Other parameters for the simulation are: (i) single impact; (ii) a 7.9-mm WC/Co ball; (iii) n = 0.12; and (iv) v1 = 5 m/s.
0
0.1
0.2
0.3
0.4
0.5
0.6
Strain hardening exponent, n
Fig. 4. The PV value predicted from the analytical model as a function of the strain hardening exponent, n, when k is constant (k = 332 MPa) or the yield strength is constant (194 MPa). Other parameters are: (i) single impact; (ii) a 7.9-mm WC/Co ball; and (iii) v1 = 5 m/s. The piling-up and sinking-in regimes are shown.
K. Dai et al. / Scripta Materialia 52 (2005) 259–263
Finally, it should be noted that in many cases the surface roughness is preferably characterized using the root-mean-square average (RMS) or arithmetic-mean value (AMV) rather than the PV value. In a previous study [14], it was shown that RMS and AMV exhibit the same trend as the PV value and all of them reach the maximum value at the same processing time. Thus, although the analytical model proposed here does not provide estimation of the root-mean-square average and arithmetic-mean value, it does offer the guideline on how to minimize the surface roughness since the PV value and RMS and AMV have the same dependence on experimental conditions.
5. Concluding remarks An analytical model based on the fully plastic contact theory has been developed to predict the surface roughness of Al-5052 plates treated with the SNH process. The predicted surface roughness compares reasonably well with the experimental measurement. The analytical model developed can be utilized to evaluate the dependence of the surface roughness on various SNH processing and material parameters. These dependences provide the fundamental knowledge for optimization of the SNH process to minimize the surface roughness.
Acknowledgment The authors acknowledge the financial support by the National Science Foundation through Grant No. DMR-0207729.
References [1] Valiev RZ, Islamgaliev RK, Alexandrov IV. Bulk nanostructured materials from severe plastic deformation. Prog Mat Sci 2000;45:103–89. [2] Vinogradov A, Hashimoto S, Kopylov VI. Enhanced strength and fatigue life of ultra-fine grain Fe–36Ni invar alloy. Mater Sci Eng A 2003;355:277–85. [3] Tao NR, Sui ML, Lu J, Lu K. Surface nanocrystallization of iron induced by ultrasonic shot peening. Nanostruct Mater 1999;11(4):433–40. [4] Wu X, Tao N, Hong Y, Xu B, Lu J, Lu K. Microstructure and evolution of mechanically induced ultrafine grain in surface layer of Al-alloy subjected to USSP. Acta Mater 2002;50:2075–84.
263
[5] Liu G, Wang SC, Lou XF, Lu J, Lu K. Low carbon steel with nanostructured surface layer induced by high-energy shot peening. Scripta Mater 2001;44:1791–5. [6] Tao NR, Wang ZB, Tong WP, Sui ML, Lu J, Lu K. An investigation of surface nanocrystallization mechanism in Fe induced by surface mechanical attrition treatment. Acta Mater 2002;50:4603–16. [7] Zhang HW, Hei ZK, Liu G, Lu J, Lu K. Formation of nanostructured surface layer on AISI 304 stainless steel by means of surface mechanical attrition treatment. Acta Mater 2003;51:1871–81. [8] Zhu KY, Vassel A, Brisset F, Lu K, Lu J. Nanostructure formation mechanism of a-titanium using SMAT. Acta Mater 2004;52:4101–10. [9] Villegas J, Dai K, Shaw L. Surface roughness evolution in the surface nanocrystallization and hardening (SNH) process. In: Srivatsan T, Varin R, editors. Processing and fabrication of advanced mateirials: XII. Materials Park, OH: ASM International; 2003. p. 358–72. [10] Villegas J, Dai K, Shaw L, Liaw P. Experiments and modeling of the surface nanocrystallization and hardening (SNH) process. In: Shaw L, Suryanarayana C, Mishra R, editors. Processing and properties of structural nanomaterials. Warrendale, PA: TMS; 2003. p. 61–8. [11] Yuan W, Villegas J, Tian JW, McDaniels R, Shaw L, Liaw PK, et al. Fatigue behavior of a Hastelloy C-2000 superalloy processed by the surface nanocrystallization and hardening process. Acta Mater. Submitted for publication. [12] Itoga H, Tokaji K, Nakajima M, Ko H-N. Effect of surface roughness on step-wise S–N characteristics in high strength steel. Int J Fatigue 2003;25:379–85. [13] Rodopoulos CA, Curtis SA, de los Rios ER, SolisRomero J. Optimization of the fatigue resistance of 2024-T351 aluminum alloys by controlled shot peening—methodology, results and analysis. Int J Fatigue 2004;26:849–56. [14] Dai K, Villegas J, Stone Z, Shaw L. Finite element modeling of the surface roughness of 5052 Al alloy subjected to a surface severe plastic deformation process. Acta Mater 2004;52(20): 5771–82. [15] Wang W. Available from: http://www.geocities.com/www67/ MA.html#figure4. [16] MetroPro Reference Guide OMP-0347D, Zygo Corp., Middlefield, CT, p. 3.51–3.52. [17] Al-Obaid YF. Shot peening mechanics: experimental and theoretical analysis. Mech Mat 1995;19:251–60. [18] Shaw MC, DeSalvo G. On the plastic flow beneath a blunt axisymmetric indentor. Trans ASME J Eng Ind 1970;92:480–94. [19] Ahn JH, Kwon D. Derivation of plastic stress–strain relationship from ball indentation: examination of strain definition and pileup effect. J Mat Res 2001;16(11):3170–8. [20] Matthews JR. Indentation hardness and hot pressing. Acta Metall 1980;28(3):311–8. [21] Pageau G, Vaziri R, Poursartip A. An evaluation of metal matrix composites as ballistic protection materials. In: 23rd International SAMPE Technical Conference. 1991. pp. 639–50. [22] Alcala´ J, Barone AC, Anglada M. The influence of plastic hardening on surface deformation modes around vickers and spherical indents. Acta Mater 2000;48(13):3451–64.