An eigenstrain-based finite element model and the evolution of shot peening residual stresses during fatigue of GW103 magnesium alloy

International Journal of Fatigue 42 (2012) 284–295

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An eigenstrain-based finite element model and the evolution of shot peening residual stresses during fatigue of GW103 magnesium alloy X. Song a, W.C. Liu a,b,⇑, J.P. Belnoue a, J. Dong b, G.H. Wu b, W.J. Ding b,c, S.A.J. Kimber d, T. Buslaps d, A.J.G. Lunt a, A.M. Korsunsky a a

Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK National Engineering Research Centre of Light Alloy Net Forming, School of Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, China Key State Laboratory of Metal Matrix Composites, School of Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, China d ID15B, ESRF, 6 Rue Jules Horowitz, Grenoble, France b c

a r t i c l e

i n f o

Article history: Received 15 October 2010 Received in revised form 26 October 2011 Accepted 18 January 2012 Available online 8 February 2012 Keywords: Shot peening Ageing Magnesium alloy Residual stress Eigenstrain

a b s t r a c t Magnesium alloy GW103 samples were heat treated to different ageing conditions and then shot peened using process parameters that deliver optimized high cycle fatigue (HCF) life. Significant HCF life improvements were observed in all samples, with a peak-aged sample showing the biggest increase. In order to simulate the effect and evolution of residual stresses during low cycle fatigue (LCF), a Finite Element (FE) model was employed, taking into account both the shot-peening-induced plastic strains and the influence of hardening on subsequent deformation. Experimental and modelling results offer a basis for explaining the observed fatigue performance improvement due to shot peening. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Due to its low density and intrinsic high specific strength and stiffness, magnesium alloys have recently attracted a lot attention in the automobile and aerospace industries. [1,2]. Recently developed Mg–Gd–Y magnesium alloys show considerable precipitation hardening, leading to improved specific strengths at both room temperature and at elevated temperatures, and better creep resistance than the conventional Al and Mg alloys (including alloy WE54 (Mg–5Gd–2Nd–2HRE, wt.%) reported to possess the greatest high temperature strength of all commercial magnesium alloys [3– 5]). In the past few years, a number of investigations related to the microstructure and mechanical properties of the Mg–Gd–Y system have been reported [3–5]. According to the results reported in the literature [5–8], the Mg–10Gd–3Y alloy exhibits higher strength and better ductility compared with the other Mg–Gd–Y alloys. In a previous study [2,9], the high cycle fatigue (HCF) properties of hot-extruded and heat-treated Mg–10Gd–3Y alloys were investigated. T5 heat treatments that involved under-, peak- or over-ageing improved the HCF life of hot-extruded Mg–10Gd–3Y alloy, ⇑ Corresponding author. Address: National Engineering Research Center of Light Alloy Net Forming, Shanghai Jiao Tong University, Shanghai 200240, China. Tel.: +86 21 54742630; fax: +86 21 34202794. E-mail address: [email protected] (W.C. Liu). 0142-1123/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2012.01.019

while the effects of T4 and T6 heat treatments were found to be detrimental. The improvement in fatigue life of the hot-extruded Mg–10Gd–3Y alloy by T5 treatment was attributed to the retardation of crack nucleation by increased precipitation [2,9]. Mechanical surface treatments such as shot peening and roller burnishing are widely considered as simple and comparatively cheap methods to enhance the fatigue properties of lightweight alloys, such as Mg–Al alloys AZ31, AZ80 [10,11] and A8 [12]. These results effectively demonstrated that it is also possible to use shot peening surface treatment to improve the fatigue properties of magnesium alloys, which is also the case for Mg–10Gd–3Y alloy [2]. Improving the fatigue strength of the Mg–10Gd–3Y alloy can help broaden its applications. However, to date, the literature available on the subject remains scarce. In this study, the influence of shot peening on the surface characteristics and high cycle fatigue properties of the as-extruded and T5 aged specimens are examined. The residual stress profiles within the as-extruded and peak-aged specimens created by shot peening have been reconstructed using the 1D inverse eigenstrain Finite Element (FE) method. It is now generally accepted that eigenstrains are the source of residual stresses [13]. Using a known residual stress field to deduce the unknown eigenstrain field is termed the inverse eigenstrain problem and numerous studies have been carried out using this method [14,15]. However, so far, no numerical analysis has been

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Nomenclature A1, A2, l1, l2, r1, r2 Gaussian peak fitting parameters: A – amplitude; l – mean; r – variance. b, c, h and k1, k2 isotropic and kinematic hardening law fitting parameters ck kth Discretized eigenstrain coefficient d/0 stress-free lattice spacing at angle of / d/w(hkl) lattice spacing for hkl plane at angles of / and w E Young’s modulus Ek(x, y) kth Discretized eigenstrain base function G Shear modulus I and I second-order and fourth-order unit tensors k back stress/plastic strain component fitting coefficient vector n normal vector

reported of the eigenstrain field evolution subsequently to it being introduced into the model, or of the experimental observation of this phenomenon. In this work, particular focus is placed on the novel approach of employing the eigenstrain method to study low cycle fatigue (LCF) behavior following shot peening. 2. Experimental technique The magnesium alloy used in the study was hot-extruded Mg– 10Gd–3Y (GW103) with the nominal composition (wt.%): 10-Gd, 3-Y, 0.5-Zr, balance Mg. Based on the experimental results reported previously [16–18], T5 ageing treatments at 225 °C for 4 h (under-ageing), 10 h (peak-ageing) and 250 h (over-ageing) were applied to three of the four as-extruded cylindrical bars. For convenience, the samples in the four conditions are referred to as as-extruded, under-aged, peak-aged and over-aged, respectively. Their microstructures have been investigated in a previous study [9]. The microstructures of the longitudinal sections of both the as-extruded and aged GW103 alloys consist of not-fullyrecrystallized elongated a-Mg grains, Mg-base solid solution (aMg grains, average diameter about 11 lm) and precipitates (MgGdY intermetallic compounds). In comparison, the X-ray diffraction patterns of the as-extruded sample show that it is mainly composed of a-Mg solid solution with an Mg24Y5 secondary phase, where Gd probably substitutes Y. In the aged samples, there are Mg5(GdY) precipitates besides the a-Mg and Mg24Y5 intermetallics, and the Mg5(GdY) precipitates are found to increase in number with the ageing time [9]. Tensile properties of the tested alloys in the four conditions were determined using sheet specimens with marked dimensions of 15-mm gauge length, 3.5-mm width and 2-mm thickness (per ISO 6892: 1998, see Fig. 1a) using the Zwick/Roell Z020 tensile testing machine at room temperature. The initial strain rate was 5  10–4 s–1 (per ASTM E8204). Ageing treatments (under-, peakand over-ageing) can significantly improve the yield and ultimate tensile strengths of the as-extruded sample, but at the cost of reduced ductility (see Table 1). It was shown that the peak-aged sample exhibits the highest yield strength (ry = 339 MPa) and ultimate tensile strength (rUTS = 445 MPa) of the four, as well as the lowest performance in terms of elongation (9.1%). The details of the tensile properties of the GW103 alloy in these four conditions can be found elsewhere [9]. The hour-glass shaped round specimens (see Fig. 1b) were used for high cycle fatigue tests. The dimensions of the specimens were as per the ASTM E 466 specification with a gauge diameter of 5.8 mm. The S–N curves measured by testing electro-polished specimens were taken as reference. Shot peening was performed

q R T tq wq X Y0

equivalent plastic strain isotropic hardening stress temperature experimental residual stresses data at qth point the weighting factor at qth point kinematic hardening stress initial yield stress a thermal expansion coefficient e⁄ eigenstrain m Poisson’s ratio r, rvol, rdev stress tensor, volumetric stress tensor, deviatoric stress tensor r/ stress in / direction UY yield condition

with an injector type machine using glass beads (shot size: 0.35 mm, shot hardness: HRC48). Based on previous studies [2,10,19] that determined the optimal shot peening conditions for the improvement of HCF properties, the specimens were shot peened to full coverage using Almen intensities in the range of 0.05–0.40 mmN. The surface roughness of the peened specimens was determined using profilometry. The measurements of the microhardness/depth profiles were also carried out, as well as residual stress measurement by means of successive surface layer removal from the top of the peened specimen. High cycle fatigue tests were performed on the rotating bending fatigue machine (see Fig. 1d, stress ratio R = 1) at a frequency of about 100 Hz in air. The test method followed the procedure described in Refs. [10] and [19]. The ‘‘Standard Method’’ of fatigue testing was utilized, which is the most common method typically used when only a few test specimens are available [20]. In this method, a low number of tests (1–3) are conducted at a set of stress amplitudes that span the expected stress range of the material; the stress amplitude and number of cycles to failure are then recorded for each specimen. Run-outs (i.e. specimens that do not fail after 108 cycles) are noted. Often these are re-run at a higher stress level to maximize the data from the limited specimen set; however, this was not done in the present study. The fracture surfaces after fatigue failure were investigated by using scanning electron microscopy (SEM, Philips 505, Holland). In order to simulate the evolution of shot peening residual stresses of the studied alloys during LCF, hysteresis loops were obtained using an Instron-8801 universal testing machine. The specimens shown in Fig. 1c were machined from the extruded bar. The data were collected at frequency 0.5 Hz with strain ratio Re = 1 and a constant strain rate of 1  103 s1. 3. Results 3.1. Surface characteristics after shot peening Fig. 2 shows the surface roughness Ra of the tested samples at different Almen intensities. Shot peening increases the surface roughness proportionally to the Almen intensity. The T5 ageing treatment appears to have insignificant influence on the surface roughness. Fig. 2 illustrates that after shot peening, the roughness values of Ra in the as-extruded and the three ageing treated GW103 alloys were similar. Fig. 3 shows the microhardness-depth profile after shot peening at Almen intensities 0.10 and 0.20 mmN. The shot peening has resulted in a significant increase in the microhardness of the deformation layer as seen in the near-surface region. Increasing the

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Fig. 2. Surface roughness values vs. Almen intensity.

Fig. 3. Microhardness vs. depth for various Almen intensities.

precipitation hardening effect introduced by the T5 ageing treatment. It leads to a relatively high bulk hardness, thus reducing the plastic deformation (eigenstrain) introduced by shot peening.

3.2. High cycle fatigue

Fig. 1. Shape and size of (a) sheet tensile specimen for tensile test, (b) hour-glass shaped round fatigue specimen for high cycle fatigue test, (c) low cycle fatigue specimen, and (d) the schematic figure of the rotating bending fatigue machine. (high cycle fatigue specimen was carried out on the rotating bending fatigue machine, whereas low cycle fatigue specimen was conducted on a Instron-8801 testing machine under uni-axial loading).

Almen intensity leads to greater depths of plastic deformation. In comparison, as seen in Fig. 3, the magnitude of plastic deformation in as-extruded specimen is slightly higher than in the three ageing treated alloys with the same Almen intensity. This is due to the

Table 1 Tensile results of the studied GW103 alloys.* Material

As-extruded Under-aged Peak-aged Over-aged *

ry (MPa)

rUTS (MPa)

Elongation (%)

m

S

m

S

m

S

265 316 339 325

2.3 3.7 4.9 5.2

344 408 445 409

5.2 6.1 5.3 3.8

20.9 13.9 9.1 14.7

1.8 2.1 1.3 1.3

m and S refer to average and standard deviation respectively.

The effect of Almen intensity on the fatigue life of the as-extruded and peak-aged GW103 alloys at different stress amplitudes is illustrated in Fig. 4. From Fig. 4, it is very clear that the fatigue life of the peak-aged specimens is first dramatically increased with the increase in the Almen intensity, and then drastically decreased as the Almen intensity is increased further. Obviously, the significant increase in surface defects such as microcracks and surface spalls at higher Almen intensities outweighs the beneficial effects produced by work hardening and compressive residual stresses. With regard to the fatigue performance, the optimal Almen intensity for all the as-extruded and peak-aged samples was 0.10 mmN. For convenient comparison between the as-extruded and aged samples, 0.10 mmN was also selected as the optimum Almen intensity for the under-aged and over-aged specimens. The stress–life (S–N) curves of the studied alloys at the optimum Almen intensity of 0.10 mmN are shown in Fig. 5. Compared to the unpeened specimens, the fatigue lives of peened samples were improved at all stress amplitudes. The fatigue strengths (at 107 cycles) before and after shot peening at optimal intensity are listed in Table 2. As seen from Fig. 5 and Table 2, for the as-extruded, under-aged, peak-aged and over-aged GW103 alloys, the fatigue strengths of the peened specimens were 65, 70, 75 and 45 MPa higher than those of the unpeened specimens, with increases of about 43%, 44%, 45% and 28%, respectively.

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X. Song et al. / International Journal of Fatigue 42 (2012) 284–295 Table 2 Fatigue strengths of the studied alloys before and after shot peening.

r1 (MPa)

As-extruded

Under-aged

Peak-aged

Over-aged

Electropolished Shot peened

150 215

160 230

165 240

160 205

The fracture surfaces in both the as-extruded and three ageingtreated alloys without shot peening can be subdivided into three regions (see Fig. 6a–d): Region 1 (crack initiation region), Region 2 (fatigue crack propagation region) and Region 3 (unstable fracture region). The crack initiation and propagation mechanisms at each stage have been discussed in the work published previously [2]. Similarly, the fracture surfaces of the peened specimens also display three distinct regions: Region 1 (crack initiation in the subsurface), Region 2 (steady crack propagation region) and Region 3 (unstable fracture region). There is no significant difference in the appearance of Region 2 (or Region 3) of the fatigue fracture surfaces between the peened and unpeened samples. 3.4. Low cycle fatigue

Fig. 4. Fatigue life vs. Almen intensity curves of (a) as-extruded and (b) peak-aged alloys.

Fig. 5. S–N curves of the tested alloys after shot peening at optimal intensity.

3.3. Fractography The fracture surfaces of both the as-extruded and heat-treated specimens before and after shot peening are shown in Fig. 6. As seen from Fig. 6a–d, the fatigue cracks in both as-extruded and heat treated specimens in the electropolished condition nucleated at the specimen surface. Compared to the electropolished condition, the fatigue crack nucleation site in the peened specimens shifted to the sub-surface regions (see Fig. 6e–h). Magnified images of the crack nucleation sites for the four alloys with optimal shot peening intensity show that the crack initiates at the facets of the cleavage planes (see Fig. 6e–h).

It is well documented in literature that residual stresses relaxation may take place under HCF conditions [21–24], although full understanding of the mechanism remains the subject of debate. Understanding of the mechanisms that govern the evolution of residual stresses during LCF, on the other hand, may be achieved through a simple analytical approach that could be adopted as a first-order approximation. For LCF, the stress amplitude is high enough for plastic deformation to occur. This leads to significant modification of the ‘‘initial’’ plastic strain introduced by shot-peening. The initial plastic strain, well known to be the source of the residual stresses, can be modified even during the first loading cycle of LCF [23,25]. The residual stresses within the as-extruded and peak-aged samples were measured at the sample middle cross-section by conventional X-ray diffraction. This approach used the sin2 w method in combination with electropolishing layer removal technique to obtain the residual stresses values at different depths below the sample surface. The measurements were carried out in a Bruker AXS D8 ADVANCE machine with a Cu Ka X-ray anode tube source. The (3 0 0) family of lattice planes was selected for the d/ w(hkl) measurement with a 2h angle of around 112.5°. Elastic properties were assumed to be E(3 0 0) = 45 GPa and m(3 0 0) = 0.35. By varying the angle w, the linear relationship between d/w(hkl) and sin2 w can be established. Based on Eq. (1), the slope of the linear curve obtained from the experiment can be used to calculate the stress r/ in the / direction, where d/0 is the stress-free lattice spacing, and r1 and r2 are the principal stresses. Fig. 7 provides an illustration of the sin2 w method setup [26,27].

d/w  d/0 ¼ d/0

" #     m 1þm 2 r/ sin w  ðr1 þ r2 Þ E E ðhklÞ ðhklÞ

ð1Þ

Fig. 8a and c illustrate the residual stress profiles obtained from the experiments after shot-peening at the optimum Almen intensities (0.10 mmN under HCF for both samples). 4. Numerical modelling To capture the evolution of residual stresses in LCF, the inverse eigenstrain method was employed to incorporate residual stresses in finite element simulation. The inverse eigenstrain method was introduced by Korsunsky et al. [28] with the piecewise linear description of the inelastic strain distribution in either 1D [29] or 2D [30] obtained by the minimization of the difference between

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Fig. 6. Overall fracture surface and crack initiation site at surfaces for unpeened specimens and subsurfaces for peened specimens: (a) unpeened as-extruded, 155 MPa; (b) unpeened under-aged, 180 MPa; (c) unpeened peak-aged, 170 MPa; (d) unpeened over-aged, 165 MPa; (e) peened as-extruded, 0.10 mmN, 230 MPa; (f) peened under-aged, 0.10 mmN, 250 MPa; (g) peened peak-aged, 0.10 mmN, 250 MPa; (h) peened over-aged, 0.10 mmN, 220 MPa.

the residual stresses obtained from an FE model and the experimental data. The implementation of this method is as follows: Considering the inelastic strain (the so-called eigenstrain e ) as the main source of residual stress, the additive decomposition of total strain is assumed to apply, in the form:

etotal ¼ ee þ e

ð2Þ

For the purposes of modelling, the inelastic strain, i.e. eigenstrain (in the case of shot-peening, plastic strain), is represented by pseudo-thermal-expansion:

e ¼ ethermal ¼ aðT final  T initial Þ

ð3Þ

As described in [29,30], an arbitrary eigenstrain distribution can be represented by thermal strains caused by the difference in expansion coefficients across the solid body and subjected to a uniform temperature increment. Therefore, the eigenstrain distribution can be conveniently implemented in the Finite Element model, e.g. ABAQUS™ model with the subroutine UEXPAN. Furthermore, it is convenient to express eigenstrain as a linear combination of known functions Ek(x, y) with unknown coefficients ck, i.e., as

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Fig. 6 (continued)

e ðx; yÞ ¼

K X

ck Ek ðx; yÞ

ð4Þ

k¼1

The determination of ck can be readily accomplished by leastsquares minimization of the difference between experimental residual stresses data tq, and the sum of individual elastic responses skq from the eigenstrain basis functions Ek, where k is the number of the eigenstrain base functions, and q is the number of the experimental data points, wq are the weighting factors, i.e. by seeking

0 !2 1 X X @ minðJÞ ¼ min wq ck skq  t q A q

ð5Þ

k

The minimization procedure can be readily implemented in ABAQUSTM via model pre- and post-processing [30]. In this way

the ‘‘most likely’’ distribution of eigenstrain (in our case, plastic strain) is obtained within the shot-peened sample based on the knowledge of the residual stress distribution. Fig. 9 illustrates the axisymmetric FE model for the inverse eigenstrain method, representing the LCF specimen (Fig. 1c) middle cross-section. Axisymmetric quadratic 2D element with reduced integration points (ABAQUSTM type CAX8R) was used in the model with denser mesh (20 lm in element size) near the sample surface to capture the localized residual stresses distribution. The reconstructed residual stresses showed good agreement with the experimental data (see Fig. 8a and c, experiment vs. eigenstrain model). Hence, the plastic strain distributions within the samples were reconstructed, as plotted in Fig. 8b and d. It is worth noting that, although both samples were subjected to the same shotpeening process (recall that the optimized Almen intensities were the same for both), the plastic strains created were different in terms of their maximum magnitudes (0.25% vs. 0.3%) and spreads (the distribution widths at around 50 lm depth). This is mainly due to the difference in material properties and microstructure between the as-extruded and peak-aged samples, as confirmed by the microhardness test (see Fig. 3). In order to have a better understanding and a more compact description of the eigenstrain distribution, a concise and easy-toimplement functional expression for the eigenstrain profile is proposed here (Eq. (6)). This function is a sum of two Gaussian peaks with different central positions and widths:

e ðxÞ ¼ A1 exp  2

Fig. 7. Illustration of the sin w method and the coordinate system.

ðx  l1 Þ2 2r21

! þ A2 exp 

ðx  l2 Þ2 2r22

! ð6Þ

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Eigenstrain (inelastic) in as-extruded

Residual Stress in as-extruded sample 40

0.35

(a)

0

0

50

(b)

0.3 100

150

200

250

300

350

400

-20 -40 experiment

-60

0.25

Strain (%)

stress (MPa)

20

0.15 0.1

eigenstrain model

-80

0.2

0.05 0

-100

0

Depth (um)

50

100

150

200

250

300

350

400

450

Depth (um)

Eigenstrain (inelastic) in peak-aged

Residual stress in peak-aged sample 40

0.3

(c)

(d)

0.25

0 0

50

100

150

200

250

300

350

400

450

-20 -40

experiment

-60

eigenstrain model

Strain (%)

stress (MPa)

20

0.2 0.15 0.1 0.05

-80

0 0

Depth (um)

50

100

150

200

250

300

350

400

450

Depth (um) Fig. 8. (a) Residual stress profiles in the as-extruded sample with optimal Almen intensity and eigenstrain model approximation; (b) reconstructed eigenstrain (plastic strain) in as-extruded sample due to shot peening; (c) residual stress profiles in the peak-aged sample with optimal Almen intensity and eigenstrain model approximation, and (d) reconstructed eigenstrain (plastic strain) in peak-aged sample due to shot peening.

An axisymmetric model of the sample middle cross section

R=3.0 cm

Fig. 9. Highlight of the LCF sample middle cross section and the line along the radius, and the axisymmetric FE model employed to represent the specimen middle cross section with a height of 4 mm.

The parameters found by least square fitting for the peak-aged sample were: A1 = 0.19 (%), A2 = 0.08 (%), l1 = 55 (lm), l2 = 110

(lm), r1 = 30 (lm), r2 = 100 (lm). This description captures not only the primary plastic strain peak which creates the maximum

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Discrete and function eigenstrain in peak-aged sample

LCF of peak-aged GW103 alloy

Difference in Yield

Fig. 10. Comparison between the discrete eigenstrain reconstruction plastic strain profile and the functional description.

compressive residual stress in the sample, but also the ‘‘hump’’ behind it that controls the plastic strain penetration depth (see Fig. 10). By employing this function, the plastic strain and the residual stress state created by shot peening can be readily incorporated in the FE model as the initial step of the simulation using the eigenstrain method. The next stage in our simulation of the shot peening residual stress evolution is to consider the unknown material hardening history that is experienced by the sample surface. The hardening history is important to our modelling approach, because the plastic strain created by shot peening is accompanied by material hardening that modifies the shape and extent of the yield surface in the stress space. This effect is not accounted for in the simple eigenstrain reconstruction, but may in fact have a significant impact on the material behavior during subsequent loading [31,32]. A novel approach is proposed below which can be used to capture this history-dependent hardening effect. First, a parameter-based hardening continuum plasticity model is chosen and implemented in ABAQUSTM using the user subroutine UMAT. The general framework was proposed by Mahnken [33] with isotropic hardening stress of Chaboche type [34] augmented by a linear evolution term (Eq. (7)), and a kinematic hardening term of Armstrong and Frederick type [34] (Eq. (7)), where c, b, h, k1, k2 are the material parameters to be calibrated. A summary of the constitutive equations is given below:

r ¼ rv ol þ rdev ; where rv ol ¼ Ktreel I;rdev ¼ : eel and eel ¼ e  epl

1: Hooke’s law

2GIdev 2: Isotropic hardening stress 3: 4: 5: 6:

Fig. 11. Low cycle fatigue curve of peak-aged GW103 magnesium alloy: experimental data in comparison with the curve from the calibrated FE model.

Parameter calibration was carried out by fitting the experimental LCF data (see Fig. 11) of the unpeened peak-aged GW103 magnesium alloy sample. The fitting was carried out using least square method to minimize the difference between the experimental and simulation data by varying the parameters in the ranges [33–38]: c = 0–10,000 MPa, b = 0–1, h = 0–10,000 MPa, k1 = 0–20,000 MPa and k2 = 0–100 MPa, in steps of 1% of their individual max values. This fitting exercise returned the parameter values of c = 0 MPa, b = 0, h = 0 MPa, k1 = 5300 MPa and k2 = 45 MPa, which is essentially a kinematic hardening condition. The fit achieved was satisfactory except for some early yielding that was observed in the experiment. The next challenge was to relate the plastic strain (eigenstrain) with the back stresses created by shot peening, so that the hardening history was captured. This was achieved by considering spherical indentation as representative of the elementary act involved in the shot peening treatment. In the current study, the aim of the Finite Element simulation of the elasto-plastic indentation by a rigid spherical punch is to develop a prediction of the depth-varying mechanical states. More specifically, two aspects were sought: (i) the distribution of eigenstrain (and hence residual stress), and (ii) the distribution of the material hardening parameters (combined kinematic and isotropic description). In practical shot peening treatment, surface coverage of 200% is typically applied. This means that multiple impacts occur in the vicinity of any given point on the sample surface. The resulting mechanical states of eigenstrain and hardening parameters are

RðqÞ ¼ cð1  expðbqÞÞ þ hq rffiffiffi 2 ðY 0 þ RÞ; where b ¼ rdev  X Yield condition UY ¼ kbk  3 _ where n ¼ b e_ pl ¼ kn; Flow rule kbk rffiffiffi 2 Equivalent plastic strain evolution q_ ¼ k_ 3 Kinematic hardening evolution X_ ¼ k1 e_ pl  k2 Xq_

Vertical Impact from direct above: First Order

ð7Þ Here r denotes the stress tensor with volumetric/hydrostatic component rvol and deviatoric component rdev. eel, e and epl are the elastic, total and plastic strain tensors, respectively. In addition, Idev is defined as Idev ¼ I  13 I  I, where I and I are second-order and fourth-order unit tensors The notation k  k in Eqs. (3),(4) and (7) represents the norm of a second-order tensor, and in Eqs. (4) and (7) n is used to signify a normal vector.

Second or higher Orders

Second or higher Orders A Representative Point

Fig. 12. Illustration of the shot peening process and the first order approximation.

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Fig. 13. FE continuum modelling of a sphere indentation experiment: (a) model setup and (b) simulation result.

likely to be a complex consequence of these multiple impacts. The subject has attracted considerable attention and modelling effort [31,32,36]. The results were reported, e.g. in the regular series of International Conferences on Shot Peening. However, no consensus presently exists on the acceptable modelling strategy. Therefore, for the present study we chose to use, as a reasonable first order approximation, the assumption that both the eigenstrain depth profile and hardening parameter profile after shot peening are approximately given by the variation along the surface normal directly below the axis of a spherical indentation (Fig. 12). It is authors’ belief that this approach provides a more convenient tool for the prediction of the residual stresses profile and material hardening history than those of explicit dynamic models, for which the simulation and optimization process could be time-consuming [31,36]. The indentation load and penetration in the current study were adjusted to provide the best match with the experimentally measured residual stress profiles. The authors are fully aware of the fact that real shot peening involves multiple or even repeated impact which is different from the single spherical indentation model utilized here. However, we view this step as a useful ‘‘first order’’ approximation, as opposed to the ‘‘zero order’’ treatment that ignores the hardening effects altogether. In order to satisfy ourselves that spherical indentation is a satisfactory selection of the reference process, we considered the plastic strain depth distribution and found it to be close to that after shot peening. Fig. 13 shows the axisymmetric continuum model of spherical indention carried out using the material parameters from the calibrated model of Fig. 11. The indenter was chosen to be rigid, and spherical, so that the axial symmetry conditions could be applied to the model. CAX8R element was again employed in the study, along with local mesh refinement (4 lm element size) near the indenter tip. The indentation process was displacementcontrolled, with the ultimate depth prescribed by the user. The elasto-plastic mechanical properties determined from the LCF tests were prescribed for the substrate material that was encastred remotely during the indentation. Different combinations of sphere radius and indentation depth were tested, and the plastic strain distribution perpendicular to the indentation direction was extracted from the model to compare with the shot peening eigenstrain profile. The combination of indenter radius and depth was chosen that gave the best approximation to the eigenstrain profile (see Fig. 14). The model error can be evaluated in terms of the percentage difference between the predicted maximum plastic strain from indentation simulation and that from the eigenstrain reconstruction, and also in terms of the difference in the depth below the sample sur-

Fig. 14. Optimization of the indentation parameters against the eigenstrain profile, three representative parameter combinations are plotted.

face where this maximum is achieved. These respective errors were given as 7.7% and 2.1% in Fig. 14. It should be noted that the errors (especially for depth >0.3 mm) are likely to be reduced further by the optimization and refinement of the modelling approach presented here, i.e. taking into consideration the higher orders effect. However, for the purposes of quick evaluation and prediction, this would not appear necessary. Authors are fully aware of the approximate nature of the proposed framework in its present formulation. The next step of analysis was to establish a relationship between the back stress (the kinematic hardening effect) and the plastic strain. Once such a relationship is established, it can be applied to the modelling of residual stress evolution during LCF of shot peened samples. It was found from the series of simulations that the back stress and plastic strain component perpendicular to the indentation direction follow a simple linear relationship:

X ¼ kepl xx

ð8Þ

This remarkably simple (but accurate) result allows us to link the experimentally found eigenstrain (plastic strain) distribution to the ‘initial’ back stresses. We have thus derived a way of incorporating the effect of the material hardening history during shot peening in the simulation of subsequent sample deformation. With all the mechanical consequences of the shot peening process now incorporated in the model (residual elastic strain, plastic

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strain, and plasticity-induced yield surface change) in the initial step, the evolution of the residual stresses can then be studied.

Shot peened, peak aged Mg alloy sample

Incident Beam 87.11 keV

5.1. The effect of shot peening on high cycle fatigue behavior

5.2. The evolution of shot-peening induced residual stresses in LCF Above, it has already been established that the eigenstrain distribution within a sample determines its residual stress profile. Therefore, any change in the eigenstrain will in turn modify the residual stress field. LCF, by its very nature, involves plastic flow and results in the modification of the eigenstrain field through plastic deformation, this in turn has an effect the residual stress field, even in the first cycle of loading.

Monochromatic 2D diffraction setup

Sample-Detector distance, 1108.467 mm

Fig. 15. Monochromatic 2D polycrystalline diffraction setup in ID15B, ESRF for residual stresses measurement in shot peened peak-aged GW103 sample.

1000

residual elastic strain (μ ε )

The fatigue performance of the shot-peened GW103 alloy depends on the combined effects of surface roughness, strain hardening and the compressive residual stresses induced by shot peening. The surface roughening accelerates fatigue crack propagation, while strain hardening retards the propagation of cracks by increasing the resistance to plastic deformation. Compressive residual stresses introduce a corresponding crack closure stress that reduces the driving force required for crack propagation [2,10–12]. In the present study, the improvement of fatigue life by shot peening has been attributed to the retardation of microcrack growth. This has been demonstrated by showing that the positive effect of the compressive residual stress field on fatigue life is greater than the negative effects of high surface roughness introduced by shot peening [2,10–12]. Since the compressive residual stresses introduced into the surface and subsurface layers by shot peening usually act to decrease the tensile stress in the component, fatigue cracks do not easily initiate or propagate. Thus, improvements in fatigue strength can be achieved. In addition, it is also well-known that shot peening with excessively high Almen intensities may not only result in small near-surface compressive residual stresses, but this process can also increases surface roughness and induce microcracks [2]. This problem can be even more prominent in magnesium alloys, due to the limited deformability of HCP crystal structure at room temperature. With ever increasing Almen intensity, more severe defects (such as microcracks) occur and the life improvement effect of the shot peening process is dramatically decreased. The effect of shot peening on the fatigue strength of the underaged and peak-aged alloys is greater than that seen in the asextruded alloy, which in turn shows an improvement in fatigue strength over that seen in the over-aged alloy. The reasons for this effect is as follows; firstly, it should be considered GW103 is a precipitation-hardened alloy [4,5,16], with a precipitation sequence consisting of the following steps: a-Mg Super-Saturated Solid Solution (S.S.S.S.) ? metastable b00 (D019) ? metastable b0 (cbco) ? metastable b1 (fcc) ? stable b (fcc) [16]. In comparison, the b0 phase (or b) in the peak-aged alloy (or under-aged alloy) is conducive to dislocation tangling and dislocation accumulation during shot peening [16], while the coarse b1 phase in the over-aged alloy is not. Therefore, the b0 phase (or b and b1), as well as dislocations and twins in the ageing-treated alloy act as barriers to dislocation motion and fatigue crack growth. It should be noted that the stability of work hardening effects is desirable for fatigue life enhancement, especially for smooth and mechanically surface treated samples made from light metallic materials such as magnesium alloys. This is because the enhancement in fatigue properties through shot peening is governed by the stability of near-surface work hardening [37].

Mar2300 2D Detector

5. Discussion

Debye-Scherrer rings

500 0 0

100

200

300

400

500

-500 0

-1000

0.003

-1500

0.018 0.035

-2000

0.059

-2500

position from the edge (µm) Fig. 16. The evolution of residual strain of the shot peened peak-aged sample. The legend shows the values of total applied strain.

In the present study, this residual stress evolution phenomenon was studied by synchrotron X-ray diffraction in the experiment carried out on beamline ID15B at the ESRF (Grenoble, France). A monochromatic 2D diffraction set-up was used to collect diffraction patterns (Fig. 15). The incident beam had the photon energy of 87.11 ± 0.01 keV and was collimated to the spot size of 0.02  0.02 mm2. The beam traversed the entire thickness of the sample, and scattered from the Mg alloy sample to form a set of diffraction cones. A cross-section through this set of cones was recorded by the two-dimensional detector MAR 2300 with the pixel matrix of 2300(H)  2300(V) and pixel size of 150 lm. The diffraction patterns (Debye–Scherrer rings) registered by the detector were analyzed using Fit2D via the procedures of ‘‘caking‘‘ and binning to obtain the equivalent 1D profiles [38,39]. Then Rietveld refinement [40] was carried out on the binned 1D diffraction patterns using GSAS (General Structure Analysis System) software to determine the apparent value of the lattice parameter within each gauge volume. The shot-peened, peak-aged sample was then subjected to in situ step loading during X-ray diffraction data collection. The residual elastic strain profiles for the loading steps of 0%, 0.3%, 1.8%, 3.5% and 5.9% (total strain) were collected, and are plotted in Fig. 16. It is clear from Fig. 16 that at around 0.3% of total strain, the residual stresses within the sample already show a dramatic decrease from the original values seen at the first loading increment. At a strain of 1.8%, the residual stress profile flattens out, with the maximum value not exceeding 500 le. The ‘‘peak’’ near the sample surface created by shot peening completely disappears at this stage. For strain levels between 3.5% and 5.9% or higher, the residual strain magnitudes remain almost unchanged, implying that the plastic strain introduced by shot peening has been ‘washed out’ by the overwhelming plastic deformation in the LCF process. This

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3500

eigenstrain (με)

3000

0 0.003

2500

0.018

2000

0.035 0.059

1500 1000 500 0 0

100

200

300

400

500

-500

position from the edge (µm) Fig. 17. The evolution of eigenstrain of the shot peened peak-aged sample. The legend shows the values of total applied strain.

Fig. 18. The evolution of plastic strain under compression of the peened peak-aged sample. Total compressive strain of 0.33% caused the first yield of the residually stressed sample, while the total compressive strain of 1% completely annihilates the residual stress effects.

claim is further corroborated by the eigenstrain profiles extracted from Fig. 16, which have been plotted in a separate graph in Fig. 17. The experimental observations and the interpretation of residual stresse evolution in the shot-peened, peak-aged GW103 sample can be explained using our eigenstrain model. Fig. 18 illustrates our prediction for the evolution of plastic strain under compressive loading following shot peening. It is apparent that the shot-peening-induced plastic strain remains unchanged until the total compressive strain reaches the level of about 0.33%. This demonstrates the existence of a threshold value oftensile strain that must be applied before residual stress shake-down begins under LCF conditions. This can be expected to be the case for both tensile and compressive loading, albeit with different threshold strain values1. Note that the above threshold value is significantly smaller than the monotonic (nominal) yield strain of 0.55%. This is the result of the compressive residual stress combining with the applied compressive stress so that yielding occurs sooner. The 1 This reasoning is corroborated by the results of XRD measurement, in which 0.3% of total strain is seen to cause a change in the residual stress profile.

non-uniform plastic strain distribution after shot peening changes due to plastic straining under compressive loading, and disappears entirely after 1% total strain (equivalent to 0.45% applied plastic strain). This explains why in the experiment at 1.8% strain the residual stress profile became flat. This was due to the gradual elimination of the eigenstrain distribution inherited from the shot peening process. The eigenstrain evolution history under compressive loading can be summarized as follows: At 0.33% applied compressive strain (approximately corresponding to 2=3 of the monotonic yield strain), the primary peak of the residual plastic strain begins to flatten, i.e. the maximum compressive residual stresses begin to decrease. At 1% total applied compressive strain, the secondary hump of the eigenstrain distribution disappears completely. Further application of compressive stress causes additional yielding that is approximately uniform across the sample, and therefore this process does not give rise to any eigenstrain or residual stress modification. The experimental observations and numerical modelling combine to support the conclusion that residual stress evolution under LCF conditions can be captured by monitoring the eigenstrain variation due to applied loading. The modelling results described above provide the basis for some indirect observations on HCF performance enhancement by shot peening. Fig. 5 demonstrates that the effect of the shot peening treatment is to displace the S–N curve upwards by about 100 MPa. This indicates that the presence of shot peening offers added resistance to plastic yielding and its associated damage in HCF. The S–N curve shift appears to be approximately constant across the range of applied stresses and fatigue lives. The positive influence of the presence of shot peening plastic strain is particularly significant in the tensile part of the loading cycle. However, the beneficial effect of shot peening disappears when the load is increased by about 100 MPa compared to unpeened samples. This agrees with the fatigue improvement observed in the experimental S–N curves: the model described above predicts that shot peening leads to an improvement in fatigue resistance of the sample that is equivalent to the reduction of applied stress by about 100 MPa. Thus, it is consistent with the effects observed experimentally and documented in the form of S–N curves. 6. Conclusions Shot peening can improve significantly the high cycle fatigue behavior of both as-extruded and aged Mg–10Gd–3Y alloys. Under the optimum Almen intensity of 0.10 mmN, the high cycle fatigue strengths increase from 150, 160, 165 and 160 MPa, to 215, 230, 240 and 205 MPa in the as-extruded, under-aged, peak-aged and over-aged alloys, corresponding to improvements of about 43%, 44%, 45% and 28%, respectively. The peak ageing sample shows the biggest increase in the fatigue life and also possessed the highest fatigue strength. A novel approach to the simulation of the residual stress evolution in LCF has been proposed here. From the model, it is observed that under the subsequent compressive loading, the shot peening plastic strain that causes early yield eventually becomes ‘‘washed out’’. Therefore, the effect of shot peening induced residual stresses on LCF can be seen as minimal, due to the fact that during the first loading cycle the eigenstrain within the sample has been significantly modified and flattened by the overwhelming plastic deformation. Acknowledgments X. Song would like to thank Institute of Physics (IOP) for its Research Student Conference Fund support and China Scholarship

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Council for its National Award for Outstanding Self-Financed Students Abroad. W.C. Liu would like to acknowledge the financial sponsorship of the Aerospace Science and Technology Innovation Fund of China, Aerospace Science and Technology Corporation (No. 0502), the National Natural Science Foundation of China (No. 50901045) and Shanghai Rising-Star Program (A type, 09QA1403100). References [1] Bae DH, Kim SH, Kim DH, Kim WT. Deformation behavior of Mg–Zn–Y alloys reinforced by icosahedral quasicrystalline particles. Acta Mater 2002;50:2343–56. [2] Liu WC, Dong J, Zhang P, Yao ZY, Zhai CQ, Ding WJ. High cycle fatigue behaviour of as-extruded ZK60 magnesium alloy. J Mater Sci 2009;44:2916–24. [3] Tokaji K, Kamakura M, Ishiizumi Y, Hasegawa N. Statistical error in crack growth parameters deduced from dynamic fatigue tests. Int J Fatigue 2004;26:1217–24. [4] Anyanwu IA, Kamado S, Kojima Y. Aging characteristics and high temperature tensile properties of Mg–Gd–Y–Zr alloys. Mater Trans 2001;42:1206–11. [5] Honma T, Ohkubo T, Kamado S, Hono K. Effect of Zn additions on the agehardening of Mg–2.0Gd–1.2Y–0.2Zr alloys. Acta Mater 2007;55:4137–50. [6] Liu WC, Dong J, Song X, Belnoue JP, Hofmann F, Ding WJ, et al. Effect of microstructures and texture development on tensile properties of Mg–10Gd– 3Y alloy. Mater Sci Eng A 2011;528:2250–8. [7] Chang JW, Guo XW, He SM, Fu PH, Peng LM, Ding WJ. Investigation of the corrosion for Mg–xGd–3Y–0.4Zr (x = 6%, 8%, 10%, 12%, mass fraction) alloys in a peak-aged condition. Corros Sci 2008;50:166–77. [8] Wang J, Meng J, Zhang DP, Tang DX. Effect of Y for enhanced age hardening response and mechanical properties of Mg–Gd–Y–Zr alloys. Mater Sci Eng A 2007;456:78–84. [9] Dong J, Liu WC, Song X, Zhang P, Ding WJ, Korsunsky AM. Influence of heat treatment on fatigue behaviour of high-strength Mg–10Gd–3Y alloy. Mater Sci Eng A 2010;725:6053–63. [10] Zhang P, Lindemann J. Influence of shot peening on high cycle fatigue properties of the high-strength wrought magnesium alloy AZ80. Scripta Mater 2005;52:485–90. [11] Wagner L. Mechanical surface treatments on titanium, aluminum and magnesium alloys. Mater Sci Eng A 1999;263:210–6. [12] Barry N, Hainsworth SV, Fitzpatrick ME. Effect of shot peening on the fatigue behaviour of cast magnesium A8. Mater Sci Eng A 2009;507:50–7. [13] Withers PJ, Bhadeshia HKDH. Residual stress Part 1—measurement techniques. Mater Sci Technol 2001;17:355–65. [14] Song X, Kyriakoglou I, Korsunsky AM. Analysis of residual stresses around welds in a combustion casing. Procedia Eng 2009;1:189–92. [15] Dewald AT, Hill MR. Eigenstrain-based model for prediction of laser peening residual stresses in arbitrary three-dimensional bodies, Part 1: model description. J Strain Anal Eng Des 2009;44:1–11. [16] He SM, Zeng XQ, Peng LM, Gao X, Nie JF, Ding WJ, et al. J Alloy Compd 2007;427:316–23. [17] Gao Y, Wang QD, Gu JH, Zhao Y, Tong Y. Behavior of Mg–15Gd–5Y–0.5Zr alloy during solution heat treatment from 500 to 540 °C. Mater Sci Eng A 2007;459:117–23. [18] Li DJ, Zeng XQ, Dong J, Zhai CQ, Ding WJ. Microstructure evolution of Mg– 10Gd–3Y–1.2Zn–0.4Zr alloy during heat treatment at 773 K. J Alloy Compd 2009;468:164–9.

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