Characterization of plastically graded nanostructured material: Part II. The experimental validation in surface nanostructured material

Mechanics of Materials 42 (2010) 698–708

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Characterization of plastically graded nanostructured material: Part II. The experimental validation in surface nanostructured material H.H. Ruan, A.Y. Chen, H.L. Chan, J. Lu * Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

a r t i c l e

i n f o

Article history: Received 31 August 2009 Received in revised form 1 April 2010

Keywords: Plastically graded material (PGM) Deformation twins Nanograin Nanoindentation Surface mechanical attrition treatment (SMAT)

a b s t r a c t The computational algorithm developed in Part I was utilized to quantify the depth-dependent constitutive law of the plastically graded material (PGM) – the surface mechanical attrition treated AISI304 stainless steel. This material possesses high yield stress due to the hardened surface layers, while the necking of the hardened surface layers was retarded in a tensile test due to the more ductile core layers. A number of cross-sectional nanoindentations were conducted and the curves were processed to remove the effect of the strain rate, the tip blunting as well as the adhesion. The yield stress and the hardening coefficient could then be, respectively, calculated from the loading curvature and energy recovery ratio of a single indentation. By taking Bauschinger effect into account, the integrated stress–strain curve over the whole thickness replicates that from the tensile test well. It is noted that the subsurface layers have substantial hardening rates, which is mainly attributed to the existence of the dense nanotwins. A further discussion in this paper is addressed on the relation between the measured depth-dependent constitutive laws and the corresponding microstructures. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction The framework for exploring the flow behaviour of the linear hardening material from the loading and unloading indentation curves is presented in Part I. Herein we present experimental results that correlate the graded mechanical properties and the microstructures. The target material, AISI304 stainless steel (304SS) sheet, is subjected to ultrasonic surface mechanical attrition treatment (SMAT). Such process provides high-speed (10 m/s; Chan et al., submitted for publication) local impact of hard spherical shots, which generate a high strain rate and large resultant strain in the subsurface layers. Technically, this process is different from the treatment used by Zhang et al. (2003). A detailed study of the motion and impact of the balls and their effect on the treatment efficiency are presented by * Corresponding author. E-mail address: [email protected] (J. Lu). 0167-6636/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2010.04.007

Chan et al. (submitted for publication). The impressive result of the treated 304SS is the excellent combination of the strength (>800 MPa) and ductility (>40%) as well as the high hardening rate. Such tensile property of the monolithic 304SS sheet was not reported in the literature. The treated material is characterized by the graded microstructures and mechanical properties, which is resulted from the distribution of the strain and average strain rate from the surface to interior layers during the treatment process (Chan et al., submitted for publication; Lu and Lu, 2003). SMAT is essentially a strain hardening process: the elastic modulus of different layers could thus be regarded as uniform and the treated material is exactly plastically graded material (PGM). Such PGM possesses large variations of yield stress and hardening rate through the thickness (1 mm), which will be demonstrated in this paper. In order to make this paper more self-contained, the equations that will be used to analyze the indentation curves are listed as follows:

H.H. Ruan et al. / Mechanics of Materials 42 (2010) 698–708

C

rr

1 1 1 þ þ me E =rr me2 ðE =rr Þ2 mp

¼

H ¼ ½wE =18:67rr 

10:7

!1 ð1Þ

 1:34



699

er ðrr =E ; HÞ ¼ K e1 ðHÞðrr =E Þ þ K e2 ðHÞðrr =E Þ2 þ

ð2Þ 1 1 eRPP r ðHÞ ð3Þ

where

E ¼

 1 1  m2 1  m2i þ E Ei

ð4Þ

is the reduced Young’s modulus; C is the curvature of the indentation loading curve; w is the recovery-absorption energy ratio; H is dimensionless hardening coefficient; rr and er are the representative stress and strain, respectively. The other symbols are fitting parameters, whose magnitudes are given in Part I. As discussed in Part I, the flow behaviour of the treated 304SS is assumed to be linear hardening, which is given by

rr ¼ ry ð1 þ Her Þ

ð5Þ

where rr , ry and er , respectively, pertain to the representative stress, yield stress, and representative strain. 2. The treated and untreated material – optical micrograph The as-received and treated 304SS were sectioned, polished and etched to visualize the boundaries of micronscale grains under the optical microscope. Fig. 1(a) shows the optical micrograph of the etched cross-sections of the materials before and after treatment. It is clearly observed that the subsurface region (within 100 lm from the surface) of the treated sample possesses much greater numbers of deformation bands in each grain than the asreceived material. From the selected area electron diffraction (SAED) pattern in TEM observations of the thinned cross-sectional sample, shown in Fig. 1(b), it can be confirmed that these deformation bands are mainly deformation twins. The density of the deformation twins decreases as the depth increases. A similar result can be found in the explosively loaded 304SS in Firrao et al. (2006), in which the abundance of deformation twins in the subsurface region can be observed if the impulsive force is sufficiently large. It is noted that the SMAT process applies impulsive force on the sample surface, which thus bears analogy to the explosive load. The high twinability of the 304 austenite steel stems from its low stacking fault energy (21 J/m2; Murr, 1975) and the twinning mediated plastic deformation in the SMAT process is attributed to the high strain rate (104 s1; Chan et al., submitted for publication). Deformation twinning generally requires higher resolved stress at each grain (Meyers et al., 2001) and its multiplication renders higher strain hardening rates than slip does (Chichili et al., 1998; Lu et al., 2009a,b). Therefore, in the subsequent nanoindentation experiments of treated 304SS, one will note remarkable increases of the yield

Fig. 1. Microscope images of the cross-section of the AISI 304SS before and after SMAT: (a) optical micrograph and (b) TEM image with SAED pattern inset.

stress and hardening rate, which is significantly different from the ultrafine-grained material. 3. The mechanical property – nanoindentation characterization 3.1. Experimental setup The displacement-controlled nanoindentations were performed with the approaching speed of 50 nm/s by using a TriboScope™ nanoindenter (Hystron, USA). A Berkovich indenter was adopted. Every indentation is initiated by a contact search process, which located the surface of the sample by slowly moving the indenter tip towards the sample and monitoring the contact force. If a 100 lN or larger contact force is recorded, the contact search finished and the sample surface is located. Afterwards, the thermal drifting is calibrated in every indentation under a very low contact force (100 lN) and is automatically eliminated from the subsequent load–displacement data. At the maximum indentation depth of 1400 nm, the indenter was held for 10 s before unloading. The indentation pattern was a 5  13  40 lm array for each set of indentations. A total of three sets of indentations were conducted. For the three sets, the indentation spots closest to the boundary are, respectively, 15 lm, 20 lm and 25 lm from the treated surface. Some indentation curves were discarded since their initial parts did not

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follow Kick’s law, which was due to the failure (over contact) of the contact search process and can be identified by the extra large imprint area (note that due to displacement control, the imprint areas should be uniform for all the good indentations). 3.2. Processing of experimental data Application of Eqs. (2)–(4) to the virtual indentation of linear hardening material is generally reliable from the trial numerical cases in Part I. However, the real challenge will be the extension to the actual nanoindentation. The applications of the reverse algorithms, as can be observed from the literature, are quite limited. It may be partly attributed to the fact that the nanoindentation experiment, unlike the numerical simulation, contains errors and uncertainties, which are hardly excludable. Such errors come from geometrical error of the indentation tip, the obliquity of the sample surface, artifacts due to metallographic processes, thermal drifting and strain rate effect, etc. In order to partly account for these experimental disturbances on the indentation curves, the experimental data must be processed before application.

Fig. 2. The cavity expansion model (Johnson, 1985).

Therefore, a = htan a. For indentations with self-similar tip, 1=3 c=a is constant and was found to be ðE cot a=rr Þ . Suppose the material is subjected to mainly plastic deformation while being sharply indented, the volume remains unchanged. From Hill’s solution (Hill, 1950), the radial, longitude and latitude plastic strain rates, representatively denoted by e_ r ; e_ h ; e_ / , are given by:

1 2

e_ h ðrÞ ¼ e_ / ðrÞ ¼  e_ r ðrÞ ¼ 3.2.1. Strain rate effect The strain rate effect, which can significantly elevate the indentation force while the indenter is pressed into the rate-sensitive material, can be excluded if the rate sensitivity of the material is prescribed. A rate sensitivity material can be modeled by the phenomenological Johnson–Cook model (Johnson and Cook, 1983), that is:

e_ r ¼ ry ð1 þ He Þ 1 þ m ln _ e0 n

! ð6Þ

where e_ 0 , generally taking 103, is the reference strain rate for a quasi-static test; m is known as the rate sensitivity parameter. From the references, e.g. Lichtenfeld et al. (2006) and Mousavi and Al-Hassani (2006), the parameter m for the 304SS is around 0.02–0.025. Noting that the rate sensitivity for the fcc metals generally increases as the grains are refined (e.g. Wei et al., 2004), we take 0.025 for m. Directly using the explicit dynamic finite element code to simulate the strain rate effect would be extremely timeconsuming since the characteristic time step for the typical mesh shown in Fig. 3 of Part I is 2  1011 s (the minimum element size over the speed of stress wave) while the whole indentation process, with the approaching speed 50 nm/s in the experiment, is 28 s. Alternatively, we resort to the cavity expansion model (Johnson, 1985) to account for the strain rate enhancement of the contact force. Johnson (1985) gave the relationship between the indentation load P and depth h as

P¼

2 Arr ½1 þ 3 lnðc=aÞ 3

ð7Þ

where A = 24.5h2 is the nominal contact area for the Berkovich indenter; c is the hemispherical boundaries of the plastic deformation zone (Fig. 2); and a is the boundary of the expanding core which encompasses the indented tip.

_ uðrÞ r

ð8Þ

_ where uðrÞ is the radial displacement rate of the material (Fig. 2). The equivalent plastic strain rate is then expressed as:

_ 2 2uðrÞ e_ ðrÞ ¼ je_ r  e_ h j ¼ 3 r

ð9Þ

_ The displacement rate of the core boundary uðaÞ follows the simple geometric relationship to accommodate the fur_ ther infinitesimal indentation hdt, which renders:

_ _ uðaÞ ¼ h=2

ð10Þ

Substituting Eqs. (9) and (10) into Eq. (6) by noting   _ of the reprer = a = hcot a, the enhancement ratio W h=h sentative stress at the core boundary is obtained as:

  _ W h=h ¼

h_ 1 þ m ln h cot ae_ 0

! ð11Þ

Noting that the strain rate effect quickly diminishes as the radius r increases to c, we assume c/a remains unchanged for the rate-sensitive material. Thus, the corresponding contact force P atthe  indentation depth h is also elevated _ . That is, to remove the strain by the ratio of W h=h   rate _ . effect, the contact force should be divided by W h=h Since the thermal drifting has been automatically removed from every load–displacement curve, the decline of contact force at the 10-s rest period should mainly be attributed to the strain rate effect. Fig. 3 plots the contact force versus time for the indentation at the subsurface layer (depth = 20 lm). The ascending portion of the force–time curve was then modified to remove the strain rate effect. The result is shown by the solid line. The modified maximum contact force is very close to the force at the end of the rest period, which confirms the validity of Eq. (11). For the loading speed (50 nm/s) adopted in our

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1.6 (Oliver and Pharr, 2004)) due to the radial motion of the unloaded material. It should be noted that Eq. (13) perfectly fits the unloading curves of all simulated cases, which do not involve adhesion. Fig. 4(a) and (b) shows the typical loading and unloading curve fittings for the cross-sectional indentation at the subsurface layer (depth = 20 lm) of the 304SS sheet subjected to SMAT.

Contact force (μN)

250000 200000 150000 100000 50000 0 0

20

40

60

time (s) Fig. 3. Experimental load–time curve for the indentation at the subsurface layer (depth = 20 lm).

experiments, the loading curvature C, calculated from the modified loading part of the indentation curve, is around 5% smaller than the one from the original data. 3.2.2. Tip blunting and tip-sample adhesion The effect of tip blunting can be eliminated by fitting the loading curve with the modified Kick’s law, which is:

P ¼ Cðh  h0 Þ2

ð12Þ

where h0 is the offset of indentation depth due to tip blunting. The adhesion between the tip and the sample, which bends the final portion of the unloading curve, can be excluded by using Oliver and Pharr’s fitting equation (Oliver and Pharr, 1992), which is given by:

P ¼ C 0 ðh  hf Þg

ð13Þ

where hf is the fitting parameter corresponding to the final imprint depth, g is generally smaller than 2 (typically 1.2–

3.2.3. Quantify Young’s modulus All experimental data is processed with Eqs. (1)–(3) to evaluate C and w for each indentation. In order to apply the inverse algorithm, the Young’s modulus of the PGM sample must be quantified. Although the nominal Young’s modulus of the wrought 304SS is generally 200 GPa, the actual value may be smaller. For example, ASM Handbook (2002) indicates the elastic modulus of the annealed 304SS is 193 GPa. The measured Young’s modulus of the as-received stainless steel from the tensile tests varies between 180 and 190 GPa. The effect of texture is very weak and the material can be regarded isotropic. For the 304SS sheet after SMAT, the Young’s modulus measured from the initial loading and unloading parts in a tensile test is even smaller (172 GPa). Fig. 5(a) and (b) illustrates such result. The reduction of elastic modulus is also found in nickel-based alloy subjected to surface severe plastic deformation (Tian et al., 2008). Tian et al. (2008) ascribed this phenomenon to the large tensile residual stresses in the interior layers. Consequently, as the sample is further strained, the core layers yield at infinitesimal strain, which curves the initial part of the tensile curve with the manifestation of the reduced elastic modulus. However, it is noted that for the unloading curve (Fig. 5(b)), which is theoretically regarded as pure elastic and free from the effect of residual stresses, the elastic modulus is also smaller than the as-received sample. This should be related to the annihilation of dislocations as the resolved stress on each grain decreases. Although the grain orientations also affect the Young’s modulus, this is not reason for the present case since SMAT

250000

250000 200000 Force (μN)

Force (μN)

200000 150000 100000 50000 experimental data fitted curve

0 0

400 800 1200 Indentation depth (nm)

(a)

1600

experimental data fitted curve

150000 100000 50000 0 1100

1200 1300 Indentation depth (nm)

1400

(b)

Fig. 4. The indentation curves and the fitted curves for the cross-sectional indentation at the subsurface layer (20 lm): (a) loading curve and (b) the unloading curve.

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1000

E=185GPa E=172GPa

True Tensile Stress (MPa)

True Tensile Stress (MPa)

600

400

200 As-received 10min SMAT

0 0

0.2 0.4 True Tensile Strain (%)

As-received E=172GPa 10min SMAT

800

E=185GPa

600 400 200 0

0.6

(a)

5.2

5.4 5.6 5.8 True Tensile Strain (%)

6

(b)

Fig. 5. Elastic modulus from the tensile curves for the as-received and treated materials: (a) loading curve and (b) unloading curve.

tends to make grain orientations more random (Zhang et al., 2003). For the treated material, although the linearly fitted relationship between flow stress and flow strain indicates a very high yield stress at zero plastic strain, small plastic deformation actually arises at the stress lower than the particular yield stress as determined by the linear hardening model. However, precisely accounting for such elastoplastic stress history would cause great difficulty in appreciating the indentation data, so we simply adopt the value of 172 GPa evaluated from the unloading curve (Fig. 5(b)) as the Young’s modulus for calculation purpose. The reduced modulus E* is thus 162 GPa. 3.3. The plastically graded properties 3.3.1. The representative stresses Fig. 6(a) plots the representative stresses against the depth from the surface. A clear descending trend can be observed from the experimental data, which demonstrates a steep decrease of representative stress with depth within the first 200 lm and a gentle change in the core layers (depth > 200 lm). Such trend is fitted with the bilinear curves in Fig. 6(a). It is noted that the highest representative stress at the subsurface layer (depth = 15 lm) is around 1.5 GPa, which is consistent with the tensile test result of nanocrystallized 316SS by Chen et al. (2005). The representative stress rr of the as-received 304SS is also plotted in Fig. 6(a). The resulting rr slightly varied around the average value of 556 MPa, which indicates that the boundary effect is not prominent for the indentation started from the depth of 15 lm. For the as-received 304SS, if the Ludwik’s constitutive model as shown in Fig. 1 of Part I is used for the nanoindentation simulation, the representative stress evaluated from Eq. (1) is 460 MPa. This indicates that the representative stress from the indentation experiment is 20% higher than that of the numerical simulation with the stress–strain relation determined by the tensile test. The discrepancy could partly be attributed to the notable difference between the plastic

response underneath the indenter tip and that in the tensile test: for example, the martensitic phase transformation may develop and propagate at a shallow indentation and the incipient plasticity could be induced by deformation twinning (Tadmor et al., 1999), whereas the most possible cause would be the difference in the constitutive behaviour between compression and tension (Meyers et al., 2006). Noting that the 304SS sheet has been subjected to severe rolling, the remarkable difference in the compressive and tensile response could be the consequence of the Bauschinger effect. For the 304SS sheet subjected to SMAT, it will be shown that the difference between compression and tension is consistent with the as-received one. 3.3.2. The hardening rate and the yield stress To extract the hardening coefficient from the indentation curves, Eq. (2) is applied. However, as indicated in Part I, the high sensitivity of Eq. (2) to experimental errors would make the trend of the hardening coefficients less appreciable. Therefore, we made a number of indentations in order to cancel the positive and negative experimental error by calculating the average value of H at different depths. Some of measurement data are shown in Fig. 6(b) in grey crosses. The average H at different depths, indicated by the circle in Fig. 6(b), still demonstrates the trend of the hardening coefficient, which matches the overall picture of the graded 304SS: the layer close to the surface has less hardening coefficient due to the enhanced yield stress. The error bars associated with the circles represent ±75% resultant error. This error range is quite consistent for all depths. Following the distribution of the representative stress, we also use the bilinear curve to fit all data. It is shown that the changing slope of H drops significantly for the core layers with depth > 200 lm. By applying the Considere criterion, it is elementary to show that for the linear hardening material, the onset of necking occurs at the true strain eneck ¼ ð1  1=HÞ. From the bilinear curve in Fig. 6(b), the necking of the standalone surface layer of

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Hardeing coefficient H

Representative stress σr (MPa)

1600

1200

800 556

400 304SS subjected to SMAT As-received 304SS

0 0

100

200 300 Depth (μm)

400

H=1.18+0.0039h (h<200μm) 6 H=1.96+4.65×10-4(h-200) (500μm>h>200μm) 4

2

0

500

0

100

200 300 Depth h (μm)

(a)

400

500

(b)

Yield stress σy (MPa)

1600

1200

800 σy=1441-2.57h (h<200μm) σy=927-0.46(h-200) (500μm>h>200μm)

400

0 0

100

200 300 Depth h (μm)

400

500

(c)

1400

Ture stress (MPa)

1200 1000

The curve from tensile test

800

Tensile curve of layer remonved sample (remove150um from both surfaces)

600

Calculated from nanoinent

400

Calculated from nanoindentation (20% reduction)

200

Calculated from nanoindent (20% reduction, 150um removed)

0 0

10

20 30 True strain (%)

40

50

(d) Fig. 6. Distribution of the (a) representative stresses, (b) hardening coefficients and (c) yield stresses of the 304SS sheet subjected to SMAT, and (d) the integrated stress–strain curves.

the treated 304SS should commence at the strain of 15%, while the standalone core layers should neck at a strain larger than 50%. As demonstrated by Xue and Hutchinson

(2007) and Li and Suo (2006) for the metal–elastomer bilayers, the necking of surface material is retarded by more ductile substrate. The overall necking of the present

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treated 304SS would then start at the strain larger than 15%. This is corroborated by the tensile curve shown in Fig. 1 of Part I. The representative strains and yield stresses were evaluated by Eqs. (3) and (5). The latter is plotted in Fig. 6(c). The variation of the yield stress is similar to that of the representative stress, which is also fitted by the bilinear curve. By integrating each layer’s constitutive response to a given strain, the macroscopic stress–strain curve over the whole thickness (1 mm) can then be obtained and represented by the dashed line in Fig. 6(d). By comparing the stress– strain curves obtained from the nanoindentation test and the tensile test, the 20% discrepancy is noted, which is consistent with the as-received material. This discrepancy should be attributed to Bauschinger effect. If we assume such difference is a constant of 20% for every layer of the treated 304SS sheet, the reduced stress–strain curve

replicates the tensile one well with a slight underestimate of overall yield stress and slight overestimate of hardening rate. For layer-removed sample, the constitutive curve calculated from the nanoindentation tests also agrees the experimental one very well (Fig. 6(d)). 3.3.3. Remarks The depth of 200 lm is chosen as the turning point of the variation rates of the representative and yield stresses. Such decision is made partly due to the observation of the trend of the data. On the other hand, it is attributed to the distribution of the plastic strain and strain rate of a 304SS subjected to the impact of a rigid ball. In the simulation of the ball-plate impact, the bearing steel ball used in SMAT is assumed to be rigid and the initial velocity is 10 m/s, which is the maximum velocity of the balls in the treatment chamber (Chan et al., submitted for publication). The

Equivalent plastic strain

0.08

0.06

0.04

0.02

0 0

200

(a)

400 600 Depth (mm) (b)

800

1000

strain rate (s-1)

1.2x105

8.0x10

h=50μm simulated h=100μm simulated h=200μm simulated h=400μm simulated

4

4.0x104

0.0x10

0

0x100

2x10-6

4x10-6

6x10-6

time (s) (c) Fig. 7. The distribution of equivalent plastic strain in the 304SS subjected impact by a 3 mm rigid ball with initial velocity 10 m/s. (a) The contour map, (b) the distribution along the central axis, and (c) the strain rate history at several depths.

H.H. Ruan et al. / Mechanics of Materials 42 (2010) 698–708

whole model was assumed to be axisymmetric. Fig. 7(a) shows the contour map of plastic strains in the stainless steel. Fig. 7(b) plots the equivalent plastic strain distribution along the centre line. It is noted that within the depth of 200 lm, the plastic strain remains at a very high magnitude. The plastic strain at the depth of 200 lm is close to the one at the layer several microns depth from the surface. The maximum strain occurs at the depth of 70–80 lm rather then at the surface. After 200 lm, the plastic strain decreases quickly with increase of depth. On the other hand, the peak strain rate changes sharply within the first 200 lm as shown in Fig. 7(c). This indicates that the high strain rate is much more efficient in the enhancement of the material strength than the high strain, which corroborates the general consensus that the rate of cell size (grain size or twin spacing) refinement depend mainly on strain rate and temperature (e.g. Molinari and Ravichandran, 2005). It is worthwhile pointing out that, although the deeper layer has higher H, the hardening rate, which is given by Hry, may reach the maximum at the subsurface layer (depth < 200 lm). For example, at the depth of 100 lm, 200 lm and 400 lm, the hardening rate, Hry, calculated from the fitted bilinear curves (Fig. 6(b) and (c)) are 1858 MPa, 1817 MPa and 1714 MPa, respectively. The material subjected to grain refinement will be strengthened based on the well-known Hall–Petch relation, while the hardening rate is generally diminished (Meyers et al., 2006) with the consequence of lowered ductility. This is explained by the deviation of plastic deformation mechanism from the dislocation-mediate process to the grain-boundary-mediate one (Schiøtz and Jacobsen, 2003). However, for the present 304SS sheet after SMAT, excellent ductility was preserved (as shown in Fig. 1 of Part I) and the subsurface layers still have a significant hardening rate. In order to support this result, the transmission electron microscopy (TEM) observation of different layers is presented in the subsequent section. The TEM images of the subsurface layer demonstrate the high density of deformation twins. The spacing between the twins can be as small as 10 nm. This nanostructure was not observed by Zhang et al. (2003) in which a different SMAT process (with low impact speed) was adopted. For the present SMAT process with ultrasonic vibration source, the impact velocities of the balls were as high as 10 m/s, which renders a strain rate of more than 104 at the subsurface layers (Fig. 7(c)). The high strain rate is the driving force for the twinning induced plasticity (Meyers et al., 2001). Consequently, the treated 304SS is full of nanoscale spaced deformation twins especially at the subsurface layers, which should be the main cause for the high hardening rate of these layers. In the recent publication of nano-twinned copper, Lu et al. (2009a,b) confirmed the trend that the ductility and hardening rate increase with the twin density. The higher hardening rate generally ensures the higher ductility for the material without catastrophic defect.

microscope with an operating voltage of 200 kV. The plane-view TEM foils of the layers from different depths were obtained first by mechanical polishing from both surfaces until the sample reached about 40 lm thickness and then by ion-thinning at low temperature. We made a series of observations at different depths as shown in Fig. 8(a)– (e). The microstructures in these TEM bright field images exhibit corresponding features to the distribution of the yield stresses and hardening rates evaluated from the nanoindentation curves. It should be noted that the microstructures observed from our sample are different from the relevant paper on SMAT 304SS (Zhang et al., 2003). Nanoscale spaced twins are found at the subsurface layers and the nanograins can be identified in layers as deep as 200 lm. This is because our treatment process rendered a much higher strain rate. Fig. 8(a) shows the microstructures of the layer very close to the surface (within 10 lm), which demonstrates the coexistence of nanograins (20–30 nm) and ultra fine grains (100–200 nm). The subsurface layers, represented by Fig. 8(b) from the depth of 50 lm, are characterized by the abundance of nanotwins with spacing in the range of 10–100 nm. In this layer nanograins can also be found between deformation twins. The hardening coefficients calculated from the nanoindentation curves imply that the subsurface layers (depth < 200 lm) have substantial hardening rates, which can be even more significant than the core layers (depth > 200 lm). This is indeed supported by the observed dense nanotwins. Actually, the abundance of deformation twins in these subsurface layers has been directly visualized by the optical microscope and is shown in Fig. 1. As demonstrated by Lu et al. (2009b), denser twins are always positive to both the hardening rate and the yield stress. This is also the most promising toughening mechanism for the nanostructured material (Lu et al., 2009a; Ma, 2006). The layer at around 200 lm from the surface is found to be a transition layer, in the vicinity of which, we reported the change of the slopes of the graded yield stresses and hardening coefficients. Accordingly, we observed different microstructures in this layer, as shown in Fig. 8(c) and (d), in which twins, nanograins, ultrafine grains and dislocation cells in large grains coexist. As the depth increases, grain size continuously increases and the main microstructure is dislocation tangles and dislocation cells as shown in Fig. 8(e). 5. Discussion The overall necking strain of the plastically graded material can be evaluated by Xue and Hutchinson’s treatment of the bilayered material (Xue and Hutchinson, 2007). By simply adding each layer’s contribution to the overall stress–strain curve, the overall linear hardening coefficient is given by

R h 4. The microstructures – TEM characterization Transmission electron microscopy (TEM) observations were made with a JEOL JEM 2010 transmission electron

705

H¼

0

rY ðhÞHðhÞ dh rY ðhÞ dh 0

R h

ð14Þ

Putting the fitting equations of rY ðhÞ and H(h) (shown in Fig. 6(b) and (c)) to Eq. (14) produces H ¼ 1:86, which indi-

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Fig. 8. The microstructures at different depth: (a) surface layer, (b) depth = 50 lm, (c) and (d) depth = 200 lm, and (e) depth = 400 lm.

cates a necking strain of 0.46. The calculated hardening rate and the associated necking strain are slightly larger than the result of tensile experimental: this is due to the error in the measurement of the each layer’s hardening coefficient by nanoindentations. The difference between compressive and tensile behaviour of the material also contributes to such discrepancy.

The main advantage of the present plastically graded material after SMAT is the elevated strength over the original monolithic material and the delocalized deformation of the hardened surface layers due to the existence of the ductile core layers. For a more general consideration, we may suppose that a material, by engineering nanostructures, can have linearly distributed yield stress and

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hardening coefficients with the depth from the surface, resembling what has been shown in Fig. 6(b) and (c). The graded linear hardening constitutive model is then given by

ry ðnÞ ¼ ry0 UðnÞ ¼



PGM materials, some intriguing phenomena can be addressed. One can infer that the material, after certain thermo-mechanical treatment, has elevated strength while reduced necking strain or ductility. Therefore, in terms of energy absorption capacity, there may be an optimum combination of the coefficients in Eq. (17). Fig. 9 illustrates such a case, in which H0 = 2 and Sr = 4. It is noted that, if the hardened surface has no strain hardening (i.e. Hr = 0), the monolithic material (i.e. the material before treatment) still holds the maximum energy absorption capacity. However, if the surface layer can have non-zero hardening coefficient through engineering coherent twin boundaries (Lu et al., 2009a) or controlling distributions of different sized grains or multiple phases (Ma, 2006), the optimum point will shift. For example, as shown in Fig. 9, if Hr = 0.4, the maximum energy absorption capacity X is achieved at hr = 0.4, indicating that the plastically graded layers, which occupy 40% of the whole thickness, could render the best performance in terms of energy absorption capacity as well as elevated yield stress over the original material. It should be further noted that the plastically graded material would be more appealing than the monolithic material when it is applied in the bending dominated scenario, since the outer surface has a higher yielding strength. For the present treated 304SS, one is able to find that the fully plastic (i.e. all the layers yield) bending moment is 30% larger than that of the monolithic material having even the same overall yield stress as the graded one.

ry0 ð1 þ ðSr  1Þð1  n=hr ÞÞ; n 6 hr ry0 ; n > hr ð15Þ

HðnÞ ¼ H0 PðnÞ ¼



H0 ð1  ð1  Hr Þð1  n=hr ÞÞ; n 6 hr H0 ;

n > hr ð16Þ

where ry0 and H0 are, respectively, the yield stress and linear hardening coefficient of the as-received or annealed material that is to be treated to enhance its strength; Sr is the ratio of the maximum yield stress that can be achieved to the original one; Hr is the ratio of the minimum hardening coefficient to the original one; n is the nondimensional depth; hr is the ratio of the plastically graded portion to the whole thickness. Similar to the discussion in Xue and Hutchinson (2007), in order to evaluate the overall performance of the treated material, we investigate the energy absorption capacity of the material before it reaches the necking strain in a tensile test. The energy absorption capacity is then given by:

  D hr ; Sr ; Hr ; H0 ; ry0 ¼ ry0 Xðhr ; Sr ; Hr ; H0 Þ Z 11=H   ¼ ry 1 þ Hep dep

ð17Þ

0

R  y ¼ ry0 01 UðnÞ dn is the overall yielding stress and where r H is given by Eq. (14). Although the above presumption on the graded constitutive behaviour would be too simple to model the realistic

6. Concluding remarks

1

H r=0.5

2.8

0.8

Hr =0.4

2.4

0.6 Ω

H

r

0.4

H r

.1 =0 Hr =0 Hr

0 /σ y

0.2

σy

2

=0 .3

=0 .2

σy/σy0

The material subjected to study is 304SS sheet after surface mechanical attrition treatment. From the indentations

1.6

1.2

0 0

0.2

0.4

0.6

0.8

1

hr Fig. 9. The curves show the influence of the minimum hardening coefficient Hr and the graded portion hr on the energy absorption capacity.

708

H.H. Ruan et al. / Mechanics of Materials 42 (2010) 698–708

at different depths, we obtained the distribution of yield stresses as well as the hardening coefficients. These data points were fitted by bilinear curves, demonstrating decreasing yield stresses as well as increasing hardening coefficients from the treated surface to the core. However, the resultant hardening rate, which is the product of yield stress and the hardening coefficient, approach a maximum value at the subsurface layer. This phenomenon was attributed to the great amount of nanoscale spaced deformation twins in the subsurface layer. By taking Bauschinger effect into account, the integrated stress–strain curve over the whole thickness replicates that from the tensile experiment well. The collected data can then be utilized to construct the constitutive models of the plastically graded material for studying its behaviour in complex stress states, which is also the motivation for this paper. The proposed computation scheme to utilize the nanoindentation curves is believed to be applicable for other materials, although the means to extract the hardening rate needs further investigation to improve the accuracy. Acknowledgements The work described in this paper was supported by the Hong Kong Innovation and Technology Commission (ITC) under Project No. ITP/004/08NP and by the Hong Kong Research Grant Council (RGC) under Project No. PolyU 5189/07E and Project No. PolyU7/CRF/08. The authors express their gratitude to these supports. References ASM Handbook Online, 2002. ASM International, Materials Park, OH. Chan, H.L., Ruan, H.H., Chen, A.Y., Lu, J., submitted for publication. Optimization of strain-rate to achieve exceptional mechanical properties of 304 stainless steel using high-speed ultrasonic SMAT. Acta Materialia. Chen, X.H., Lu, J., Lu, L., Lu, K., 2005. Tensile properties of a nanocrystalline 316L austenitic stainless steel. Scripta Materialia 52, 1039–1044. Chichili, D.R., Ramesh, K.T., Hemker, K.J., 1998. The high-strain-rate response of alpha-titanium: experiments, deformation mechanics and modeling. Acta Materialia 46, 1025–1043. Firrao, D., Matteis, P., Scavino, G., Ubertalli, G., Ienco, M.G., Pellati, G., Piccardo, P., Pinasco, M.R., Stagno, E., Montanari, R., Tata, M.E., Brandimarte, G., Petralia, S., 2006. Mechanical twins in 304 stainless steel after small-charge explosions. Materials Science and Engineering A 424, 23–32. Hill, R., 1950. The Mathematical Theory of Plasticity. Clarendon Press, Oxford.

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