International Journalof Fatigue - Library

International Journal of Fatigue 29 (2007) 666–676

International Journalof Fatigue www.elsevier.com/locate/ijfatigue

Microstructure-based multistage fatigue modeling of a cast AE44 magnesium alloy Y. Xue b

a,*

, M.F. Horstemeyer a, D.L. McDowell b, H. El Kadiri a, J. Fan

c

a Center for Advanced Vehicular Systems, Mississippi State University, 200 Research Building, Starkville, MS 39759, USA GWW School of Mechanical Engineering, Georgia Institute of Technology, 801 Ferst Drive N.W., Atlanta, GA 30332-0405, USA c Mechanical Engineering, Seidlin Hall Annex 204, Alfred University, Alfred, NY 14802, USA

Received 11 April 2006; received in revised form 12 June 2006; accepted 2 July 2006 Available online 26 September 2006

Abstract The multistage fatigue model developed by McDowell et al. was modified to study the fatigue life of a magnesium alloy AE44 for automobile applications. The fractographic examination indicated three distinct stages of fatigue damage in the high cycle fatigue loading regime: crack incubation, microstructurally small crack growth, and long crack growth. Cracks incubated almost exclusively at the cast pores that were near the free surface, located near sharp geometry changes of the test specimen, or at extremely large pores inside the specimens. Microstructurally small cracks grew in the eutectic region along the weak boundaries of the grains and dendrites or at very closely packed microstructural discontinuities. Long cracks were observed to grow in a transgranular fashion. Specimens fabricated from as-cast bars and extracted from a cast engine cradle were tested at room temperature and an elevated temperature typically required for automotive powertrain applications. A large variation of fatigue life in the high cycle fatigue region was observed in specimens from both conditions due to the sensitivity from microstructural discontinuities. The microstructure-based multistage fatigue model was generalized for the AE44 magnesium alloy to capture the network of porosity and temperature dependence. The modified multistage fatigue model was also used to estimate the upper and lower bounds of the strain-life curves based on the extreme microstructural discontinuities. 2006 Elsevier Ltd. All rights reserved. Keywords: Cast magnesium alloy; Fatigue modeling; Fatigue experiments; Fractography; Temperature effects

1. Introduction Magnesium alloys have received increased interest for automotive applications because of their attractive low density, the possible benefit of weight reduction, and consequent reduction in fuel consumption. The rather high specific strength, the good castability, the improved corrosion resistance of high purity alloys, and the possibility of recycling are additional advantages of these materials [1– 6]. Cast magnesium alloys are used in several automotive applications such as covers, door structures, and heavier load-bearing components like wheels, housings and frames. Vehicle components are subjected to variable and repeated *

Corresponding author. Tel.: +1 662 325 5450; fax: +1 662 325 5433. E-mail address: [email protected] (Y. Xue).

0142-1123/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2006.07.005

mechanical straining, where the number of load cycles may be on the order of 109 cycles in car wheels. When magnesium alloys are considered for powertrain components, such as crankcases or gearbox casings, fatigue properties at elevated temperatures are of great interest. The effects of the processing methods on the microstructures, mechanical properties, fracture and fatigue behavior of cast Mg alloys have been reported in the open literature [7–14]. Earlier fatigue investigations have shown that porosity plays a key role in crack initiation locations in cast Mg [15–17]. Fatigue studies of AM50 Mg alloy samples obtained from different locations of prototype control arms and wheels showed that the fatigue performance was significantly affected by shrinkage porosity and gas pores, depending on the location on the components where samples were extracted [18]. Design related fatigue experiments and

Y. Xue et al. / International Journal of Fatigue 29 (2007) 666–676

667

Nomenclature a l, E D c, e r S K b v a Cinc

crack length shear and Young’s modulus increment of a quantity shear and normal strain stress tensor engineering stress stress intensity factor non-local micronotch-root maximum plastic shear strain amplitude linear proportional constant between crack opening displacement and crack growth rate ductility exponent in modified microscale incubation C–M relation ductility coefficient in modified microscale incubation C–M relation

estimations for typical applications were conducted on smooth and notched specimens [19] and under constant and variable loading cases [20]. In this paper, the fatigue behavior of an AE44 Mg alloy is explored using test samples extracted from as-cast bars and cast automotive components. The strain-life behavior of the AE44 magnesium alloy was studied for two types of specimens at room temperature and at 121 C, a temperature range typical of automotive powertrain applications. The as-cast bars have a more consistent porosity level from specimen to specimen but can have large casting pores. The average dendrite cell size (DCS) for as-cast bars is small and relatively uniform from specimen to specimen. The engine cradle specimens have a wider variety of defect discontinuities from the thin section to the thick section of the cradle and can have very large pores. The microstructure-based multistage fatigue (MSF) model of McDowell et al. [21] was generalized for the AE44 Mg alloy. This model recognized the multiple inclusion severity scales for crack formation and addressed the role of constrained microplasticity around debonded particles or casting pores in forming and growing microstructurally small fatigue cracks. The ‘‘base pore size’’ for incubation for AE44 Mg alloy was estimated as the average pore size of one of a million pores in the large tail end of the pore distribution. Similar to the MSF model, the incubation was assumed to consume over 90% of the total cycles to failure in the high cycle fatigue (HCF) regime and approximately 30% of the total cycles to failure at the low cycle

A m

linear coefficient in the Paris’ Law exponent in the Paris’ Law

Superscript and subscript p plastic a amplitude th threshold per percolation limit y yield ut ultimate f life in terms of cycles eff effective op opening relation quantity

fatigue (LCF) regime. A few numerical constants in the MSF model were estimated based on their physical significance in the incubation model. A new factor is introduced to capture the effect of local porosity on acceleration of the microstructurally small crack (MSC) growth. The modified microstructure-based multistage fatigue model was employed to capture the upper and lower bounds of the fatigue life based on the limits of the microstructural discontinuities. 2. Material and experiments 2.1. Materials The Mg AE44 alloy was fabricated using a high-pressure die cast technique. Test specimens were fabricated from ascast bars and from cast engine cradles. The chemical composition was measured by means of an Electron Probe MicroAnalysis (EPMA) and the results are reported in Table 1. Also shown is the nominal chemical composition of the alloy provided by General Motors (GM) Corporation. The distribution of the elements along the microstructural features of the alloy was examined using X-ray mapping of the EPMA. The 4% Al was primarily concentrated in the eutectic region as a compound with rare earth elements of the form AlxREy at the dendrite boundaries inclosing the a-phase magnesium matrix. Mn and Si were in the formations of intermetallic particles scattered in the a-phase magnesium matrix.

Table 1 Composition of cast AE44 magnesium alloy provided by GM and measured by means of EPMA Resource

Mg

Al

Mn

Zn

Si

Fe

RE

O

GM EPMA

Bal. 87.88

3.5–4.5 3.44

0.15–0.5 0.18

0.2max 0.015

0.1max 0.005

0.005max 0.0038

3.5–4.5 1.9 (Ce)

None 0.88

All quantities are in weight percent.

668


2.2. Specimen

2.3. Experimental procedures

Smooth cylindrical dog-bone shaped fatigue specimens were employed based on ASTM standards (E-747-99). Standard sub-size specimens were fabricated with a gauge length of 25 mm and a uniformed gage cross-section diameter of 4.7 mm. The cross-section at the grip was 9.4 mm in diameter, and the total length of the specimens was 90 mm. The specimen cross-section was evaluated using an optical microscope and a backscatter field emission gun (FEG) scanning electron microscope (SEM). Fig. 1 shows the optical micrographs of polished sample cross-sections that illustrate the size and distribution of gas pores, which demonstrates that the gas pore size was larger and the shrinkage porosity was higher in the sample obtained from the engine cradle as compared to the as-cast bars. A high magnification optical micrograph of the same polished cross-sections revealed the morphology of the dendrite cells and boundaries, as shown in Fig. 2. A distinct dendrite size variation from the as-cast bar to thin sections of the engine cradle and thicker sections of the engine cradle was evident. The dendrite cell size (DCS) was measured employing the linear intercept method (ASTM E112) using AxioVision Grains software. The DCS of the as-cast bars varied from 2 to 10 lm over the whole cross-section with an average of 5 lm. The DCS of samples machined from the engine cradle increased from the thinner to thicker cross-sections varied from 9–15 to 12–17 lm with an average of 15 lm. The primary DCS in a thin section of the cast engine cradle was about 13% less than the DCS in a thick section.

The experiments were conducted using an MTS servocontrolled electro-hydraulic system. Monotonic and cyclic hardening experiments were conducted using the same specimens and test setup as the fatigue tests. The tests were conducted under strain-controlled, constant strain amplitude conditions, with the strain measured using a 0.5 in. axial fatigue rated extensometer attached within the gage length. Cyclic loading was applied at frequencies of 0.5 Hz for the first 43,000 cycles and 20 Hz, thereafter. A 50% drop in the peak cyclic load was used to determine the final failure of the specimens according to ASTM standards. A temperature chamber was employed for a constant elevated temperature during the fatigue test at 121 C. The strain amplitudes ranged 0.05% strain to above the yield strength 0.6%. The remotely applied strain ratio of Re = emin/emax = 1 was employed in all experiments. At least three replicated tests were conducted for each loading condition. 3. Experimental results 3.1. Static and fatigue Uniaxial tension tests were conducted at room temperature and 121 C with a loading rate of 0.005 min1. The stress–strain curves are shown in Fig. 3. The elastic modulus, yield strength, ultimate strength and strain-to-failure of the as-cast AE44 bars were higher than those extracted from the engine cradle. The static and cyclic properties are shown in Table 2. The Young’s modulus and the yield

Fig. 1. Optical images from polished cross-sections present an overview of size and distribution of casting pores in the AE44 magnesium alloy: (a) as-cast bar, (b) extracted from an engine cradle.


669

Table 2 Mechanical properties of AE44 magnesium alloy in (1) as-cast bar and (2) extracted from engine cradle at room and 121 C

Fig. 2. Optical images from polished cross-sections present an overview of morphologies of dendrite cells and boundaries in the AE44 magnesium alloy: (a) as-cast bar and (b) from an engine cradle.

Properties

AE44-Bar

AE44-Cradle

AE44-Bar at 121 C

Elastic modulus (MPa) Yield strength (MPa) Ultimate strength (MPa) Cyclic strength Coef. (K 0 – MPa) Cyclic strain hardening exp. (n 0 ) Cyclic yield strength (MPa) Fatigue ductility exp. (c) Fatigue ductility coef.

44,740 135 244 624

38,620 109 182 685

40,830 112 160 635 0.351

89 0.62 0.26

106 0.52 0.10

75 0.505 0.1

0.6

Strain Amplitude (Δε/2, %)

Bar.RT

250

True Stress (MPa)

0.30

strength of the as-cast bars were approximately 14% and 19% higher than the corresponding properties of the cradle, due to the cradle’s large pore sizes and high porosity. The uniaxial strain-life results are shown in Fig. 4. A large variation of fatigue lives in the HCF regime was observed on specimens from both conditions, due to the sensitivity to microstructure. The large scatter in the HCF regime could be induced by the sub-sized ASTM specimens in which the above stated detrimental pore was almost one tenth of the specimen diameter. The fatigue lives at 121 C are longer, in general, than those at the room temperature under the constant strain amplitude loading condition with the strain-controlled experiments, which maybe is induced by the increasing ductility at the elevated temperature. In addition, the specimens experienced large cyclic plastic strains when the strain amplitudes were greater than 0.28%. As expected, the cyclic plastic strain amplitudes in the cradle specimens were higher than those in the as-cast bars for the same applied strain amplitudes. However, the cradle specimens exhibited higher fatigue lives than those from the as-cast bars at the LCF region.

300

200 Cradle.RT

Bar.HT

150 Cradle.HT 100

As-cast Bar.RT Cradle.RT As-cast Bar.HT Cradle.HT

0.4

0.2

Runout

50 0 0.00

0.32

R = -1 0.0 1e+2 0.05

0.10

0.15

0.20

0.25

True Strain Fig. 3. Stress–strain curves of the AE 44 magnesium alloy: (1) as-cast bar, (2) extracted from engine cradle tested at room temperature (RT) and 121 C (HT).

1e+3

1e+4

1e+5

1e+6

1e+7

Cycles to failure, Nf Fig. 4. Uniaxial strain-life of AE44 Mg alloy specimens machined from as-cast bars and from an engine cradle tested at room temperature (RT) and 121 C (HT) under strain-controlled, constant amplitude with completely reversed strain amplitude experiments.

670


3.2. Fractographic analyses Typical specimens loaded in the strain amplitude of 0.075% (HCF) and 0.3% (LCF) were selected for fractographic analyses. At De/2 = 0.075%, some specimens did not fail at 107 cycles. A sample of medium life, S1 (Nf = 1.3 · 106, at 121 C), and samples with the shortest lives, S2 (Nf = 1.78 · 105, at 121 C), and S3 (Nf = 1.83 · 105, at 25 C) were examined under the SEM, which are shown in Fig. 5. An arrow indicates the location of the fatigue crack formation site. Most cracks formed at casting pores at or near the specimen surfaces or at a very large pore within the specimen as shown in Fig. 6. A strong inter-

action of two pores that were about 25 lm in diameter near the free surface formed a fatigue crack that produced the fatigue failure. A few relatively larger pores were scattered inside this specimen but no damage was formed. Even though S2 and S3 had similar low fatigue lives, the life incubation sites were quite different: the damage in S2 formed at the interaction of two pores that were 65 lm in diameter with one right at the edge while the damage in S3 formed at a large pore of 500 lm in diameter in the middle of the specimen. Therefore, the pore sizes of 25 lm and 500 lm were used to demonstrate the upper and lower bounds of the fatigue lives.

Fig. 5. SEM micrographs on the overall fracture surfaces of samples under 0.075% strain amplitude: (a) S1: Nf = 1.3 · 106 at 121 C; (b) S2: Nf = 1.78 · 105 at 121 C; (c) S3: Nf = 1.83 · 105 at room temperature. The locations of the incubation sites are indicated by arrows.


671

Fig. 6. SEM images of fatigue crack formation sites for samples: (a) S1: interaction of two casting pores of approximately 25 lm diameter near the edge; (b) S2: interaction of two cast pores of approximately 60 lm diameter at the edge; (c) S3: a large casting pore of approximately 0.5 mm in a linear dimension at the center of the specimens.

The backscatter electron images on the sample S1 on the earlier crack growth region indicated AlxREy-rich dendrite boundaries with the heavier AlxREy compound marked as white in Fig. 7. Fig. 7a shows that the fracture surface comprised regions of dendrite boundaries, which manifested the crack propagating and meandering along the dendrite boundaries with the combination of interdendritic and intradendritic crack growth. The large striations on the dendrite boundaries indicate the fast segregation along the boundaries MSC and earlier LC growth. Fig. 8 showed the fractographics of a bar specimen under De/2 = 0.3% with a life of 1.3 · 104. The fatigue damage formed at a region of casting pores at the surface with pore diameters ranging from 30 to 70 lm. The crack propagating approximately 140 lm into the specimens left a region that was very flat, which corre-

sponds to the microstructurally small crack growth region. Further into the specimens, the fracture surface became rough and intragranular and intergranalar crack growth was observed as shown in Fig. 8b. The MSC grew in the eutectic region along the weak grain and dendrite boundaries under the influences of very closely packed microstructural discontinuities. The eutectic region comprises weak bonding regions with accompanied cast pore and shrinkage porosity. In this region, the load bearing capability drops due to the large porosity, and simultaneous crack propagation is accelerated by the weak bonding between dendrites, which also enhances the scatter in fatigue life. Long cracks were observed to grow in a transgranular fashion and the fracture surfaces are relatively flat scattered with dimples formed from the pores.

672


Fig. 8. SEM micrographs of a specimen extracted from an as-cast bar loaded in strain amplitude at 0.3% with Nf = 1.3 · 104 at room temperature: (a) fatigue damage incubation region and (b) microstructurally and physically small crack growth region. Fig. 7. SEM backscatter electron imaging (BEI) of the fracture surface indicated interdendritic segregation in MSC/LC growth regime. The heavier AlxREy compounds at the dendrite cell boundaries are shown as white in BEI. (a) The crack propagate from one dendrite boundary at the lower right meandered into the other dendrite boundaries at the center of the image, and (b) the striation indicating fast crack propagation along the dendrite boundary.

4. Multistage fatigue model A microstructure-based multistage fatigue (MSF) model that incorporates different microstructural discontinuities (pores, inclusions, etc.) on physical damage progression was implemented to model the fatigue life of the AE44 Mg alloy. This model partitions the fatigue life into three stages based on the fatigue damage formation and propagation mechanisms: crack incubation, microstructurally small crack (MSC) and physically small crack (PSC) growth, and long crack (LC) growth. This model was initially developed for a cast A356-T6 [21] Al alloy and modified and generalized for a variety of aluminum alloys

[22,23]. As stated above, three types of defects were listed in descending order of severity with regard to detriment of fatigue life: (1) gas pores greater than 25 lm in diameter; (2) shrinkage porosity clusters with special dendrite features greater than 300 · 300 lm2 in size that demonstrated locally high porosity, and (3) extreme large pores of 500 lm or greater in diameter. In most cases, the shrinkage porosity was located in the eutectic region, weakening the load bearing capability and thereby enhancing the propagation rate of interdendritic cracks, leading to a greater reduction in fatigue lives. A brief recast of the multistage fatigue model is described in the following accompanied by the detailed explanation of model parameters for the AE44 Mg alloy. The total fatigue life is decomposed into the cumulative number of cycles spent in several consecutive stages as follows: N Total ¼ N Inc þ N MSC=PSC þ N LC ;

ð1Þ


where NInc is the number of cycles to incubate a crack at a micronotch that includes the nucleation of crack-like damage and early crack propagation through the region of the micronotch root influence. NMSC is the number of cycles required for propagation of a microstructurally small crack with the crack length, ai < a < kDCS, with the DCS defined as the dendrite cell size, and k as the non-dimensional factor that is representative of a saturation limit for the encountering of a 3-D crack front with sets of microstructural discontinuities. NPSC is the number of cycles required for propagation of a physically small crack (1–2) DCS < a < O (10DCS), during the transition from MSC status to that of a dominant LC. Depending on the microstructural inclusion morphology and texture of the matrix, the PSC regime may conservatively extend to 300–800 lm. Also, the scale of the crack tip plastic zone is typically so small for the PSC regime that it samples individual microstructural elements; hence, there is an insufficient volume of material to achieve the length scale-independent afforded by homogeneous DK solutions in long crack fracture mechanics. In Eq. (1), the MSC and PSC regimes were combined into one mathematical form. NLC is number of cycles required for LC propagation for crack length a > (10–20) DCS, depending on the amplitude of loading and the corresponding extent of microplasticity ahead of the crack tip. This stage of crack extension is commonly characterized using standard fatigue crack growth experiments, da/dN versus DK. 4.1. Incubation The incubation of a fatigue crack at a local discontinuity or a stress raiser was induced by the notch root microplasticity. Gas pores are life-limiting inclusions for AE44 Mg alloy, being relatively large in size and located near the free surface or near a sharp geometric change in a specimen. Using digital image analysis, the probability distribution of pore size was estimated in a one-millimeter by one-millimeter square area. The estimation of the life-limiting pore size corresponds to only one of a million pores in the large tail end of the distribution that will contribute to the incubation of the fatigue crack for smooth specimens considered with a uniform stress state. The correlation of the local plastic zone size and the macroscopic applied strain amplitude can be obtained using micromechanical finite element simulations. The correlation was obtained for the A356 Al alloy by Gall et al. [24], i.e., l hea eth i l ¼ glim 6 glim ; ; D eper eth D r l eper l ¼ 1 ð1 glim Þ ; glim < 6 1; D D ea

ð2aÞ ð2bÞ

where function Æfæ = f if f P 0; Æfæ = 0 otherwise. Also, ‘‘r’’ is a shape constant for the transition to limit plasticity and glim is the linear factor [21]. Parameters eth and eper are

673

material constants designating the fatigue damage threshold and the strain percolation limits, respectively, for microplasticity. The two parameters can be estimated using macroscopic material properties. ea is the macroscopic remote cyclic strain amplitude. l is the plastic zone size in front of the inclusion projected to the direction perpendicular to the loading, and D is the linear dimension of the inclusion particle that incubated the crack that caused the fatigue failure. The ratio l/D was used to estimate of the notch root plasticity at the influence of the particle, with the limit factor glim indicating the transition from constrained microplasticity to unconstrained microplasticity at the micronotch. r = 0.2, glim = 0.3 [21] were found also suitable for the AE44 Mg alloy. To study the damage incubation life as a function of local plastic deformation, a modified Coffin–Mansion law was implemented based on the non-local maximum plastic shear strain, i.e., Dcpmax ; ð3Þ 2 where b is the non-local maximum plastic shear strain amplitude around the inclusion calculated using an average of maximum plastic shear strain over an area approximately one percent of the inclusion area; Cinc and a are the linear and exponential coefficients in the modified Coffin–Mansion law at the micronotch. Exponent a is similar to that in the macroscopic Coffin–Mansion Law for strain life; coefficient Cinc is highly dependent on the remote load ratio in the CHF region. It is assumed that Cinc = c(1 R) [21], where c is material dependent constant that equals 0.24 for AE44 Mg alloy. The coefficient is dependent on the local plastic zone size in the transition to the unconstrained microplasticity, thus it becomes a function of the remote strain amplitude. The non-local maximum shear strain at the micronotch can be estimated using micromechanics analysis of a representative volume cell with a non-local averaging procedure [24] C inc N ainc ¼ b ¼

Dcpmax l q 6 glim ; ¼ Y ½ðea eth Þ ; ð4aÞ D 2 p Dc l l b ¼ max ¼ Y 1 þ n ½ðea eth Þq ; glim < 6 1; ð4bÞ D D 2 b¼

where Y, q, glim are material and microstructure-related parameters obtained using micromechanical simulation at the inclusion. Parameter Y depends on the remote load ratio; based on finite element analysis A356 Al alloy, it for d0 was assumed Y ¼ ½y 1 þ ð1 þ RÞy 2 DD0 , with y1 and y2 as constants. D0 is the diameter of the pore in the specimens that caused fatigue failure and the exponential d 0 depicts the effect of pore size to local plastic strain, which is taken as 0.1 < d 0 < 0.6. The strain threshold can be estimated using the fatigue endurance limit that is measured using the standard stress-life fatigue experiment in the HCF regime. The ultimate strength of the metallic alloys can be used to estimate ut =E the strain threshold eth ¼ 0:29S . The fatigue strength at ð1Re Þ

674


107 under completely reversed strains can also be used to estimate the strain threshold as well [25]. To evaluate the fatigue strength amplitude at 107 cycles, a procedure recommended by Maenning [26] gave the probability for failure, i.e., ð3i 1Þ ; ð5Þ P F ðea Þ ¼ 3n þ 1 where i is the number of specimens failed below 107 cycles, and n is the number of specimens tested at the particular strain amplitude. Stress controlled tests are usually used with at least 25 specimens to estimate the fatigue strength. The percolation limit for microplasticity denotes that the incubation becomes insignificant compared to the total fatigue life as extensive shear localization becomes dominant in the region near the inclusions, which can be estimated using eper ¼ ð0:5–0:9Þecyc where the cyclic yield strain can y cyc be obtained by ecyc y ¼ ry =E. As stated earlier, the incubation life is more important in the HFC regime than in the LCF regime [21]. To this end, the ratio of incubation life to total life is proposed as N inc ¼ N total ½1 6:4ðN total Þ0:34 ; N inc ¼ 0:3N total ;

N total < 103 ;

N total P 103 ;

ð6aÞ ð6bÞ

which gives Ninc = 0.94Ntotal, when Ntotal = 106 and Ninc = 0.3Ntotal, when Ntotal = 103. 4.2. Microstructurally small crack growth and long crack growth The growth of a MSC/PSC is governed by the crack tip displacements, DCTD, i.e., da ¼ vðDCTD DCTDth Þ; ð7Þ dN MSC where v is a material constant pertinent to given microstructures in which the crack growth rate is linearly proportional to the crack driving force, i.e., the crack tip open displacement. v is typically less than unity and is usually taken as 0.32 for aluminum alloys [27]; it is also taken as 0.32 for the AE44 Mg alloy. The Burgers vector for pure HCP magnesium was assigned as the threshold value for crack tip displacement in analogy to the cast A356-T6 Al alloy [21]. The crack tip opening displacement is related to the remote loading in the manner stated in [21], i.e., DCTD ¼ f ð/Þgð/ÞC 11

DCS DCS0

0:1

f p 2 UD^ r Dcmax a þ C1 ; 2 macro S ut ð8Þ

where C1, CII and f are material dependent parameters to be determined based on fatigue crack growth experiments in the MSC regime. The equivalent stress is defined as a linear combination h of ivon Mises Stress and maximum stresses, D^ r ¼ h1

0 0 0:5 3 rij rij 2 2 2

þ h2 Dr1 , with h1 and h2 as load

path dependent and combined stress state dependent parameters [21]. Similar to MSF for A356 Al alloy, f = 4.2 [21] was found suitable for Mg alloy. Here, two porosity related functions are considered: (1) the variation ofn average in the specimens: porosity

o ¼ 1 þ x 1 exp / , and the porosity threshf ð/Þ 2/th old (considered as 0.0001); (2) the porosity in front of the microstructurally small crack that reduces the load bearing capacity of the matrix, g(/) = (1 /)k and 0 < k < 1. The parameter U is defined as the load ratio parameter in the 1 form of U ¼ 1R . More detailed dependence on microstructure can be built into the MSC growth relation as required. The macroscopic maximum plastic shear strain amplitude p Dr 10 p Dcmax n ¼ 32 De2 ¼ 32 2K for the uniaxis calculated using 2 0 macro ial loading case. Finally, the LC regime is represented by linear elastic fracture mechanics (LEFM) growth, da ¼ A½ðDK eff Þm ðDK eff;th Þm ; ð9Þ dN LC where the intrinsic threshold and the effective stress intensity factor range are used, and A and m are the crack growth material constants in the Paris Law [27]. The effective stress intensity factor range is defined by DKeff = Kmax Kop if Kmin < Kop, DKeff = Kmax Kmin if Kmin P Kop. The opening stress intensity factor level can be calculated using finite element analysis based on opening stresses at the crack tip. The stress intensity factor is given by a pffiffiffiffiffiffi K¼f r pa; ð10Þ d where a is the radius of the semi-circular crack, and d is the diameter of the cylindrical specimen. The geometry function for a semi-circular crack at the surface of a cylindrical specimen for mode I fracture can be written as f ðxÞ ¼ 0:67 1:24ðxÞ þ 28:0ðxÞ2 162:4ðxÞ3 4

5

6

þ 472:2ðxÞ 629:6ðxÞ þ 326:1ðxÞ :

ð11Þ

The transition between the MSC and LC growth was governed by selecting the maximum of either of the two rates, i.e., da da da ¼ max ; : ð12Þ dN dN MSC dN LC 5. Multistage fatigue model correlation and fatigue life estimation The threshold strain amplitudes for incubating fatigue damage were estimated using the ultimate strength of the alloys that are listed in Table 2, i.e., eth = 0.08%, 0.07%, respectively, for samples from the as-cast bar and the cradle at room temperature, and eth = 0.06% for the as-cast bar at 121 C. The threshold strain amplitude calculated from fatigue strength at 107 cycles with 50% probability as estimated using Eq. (5) was estimated to be between




MSF-BAR.RT EXP-Bar.RT EXP-Bar.HT MSF-BAR.HT

0.4

0.2

10 2

10 3

10 4

10 5

10 6

10 7

Cycles to failure, Nf Fig. 10. The multistage fatigue model correlation for magnesium alloy AE44 of as-cast bar specimens and specimens extracted from an engine cradle at high temperature. 0.6

EXP-Bar.RT MSF. MSF-upper.B MSF-lower.B

0.4

0.2

10 2

10 3

10 4

10 5

10 6

10 7

Cycles to failure, N f Fig. 11. Multistage fatigue model estimation for the upper bound (UB) and lower bound (LB) of fatigue lives of the AE44 magnesium alloys machined from as-cast bars.

6. Conclusions

0.6

As-cast Bar Cradle MSF.BAR MSF.Cradle

0.4

0.2

0.0 10 2

0.6


0.05% and 0.075% for the AE44 Mg alloys, based on a limited number of experiments. These values are much less than those obtained for other similar cast Mg alloys [29,30], which could be caused by the large casting pores in the sub-sized ASTM specimens. The percolation limit for microplasticity at the notch was estimated directly using the cyclic yield strengths that are listed in Table 2. The cycles spent incubating a crack and propagating it through the micronotch root plasticity affected zone was estimated using Eq. (6). It was assumed that the non-local plastic shear strain amplitudes and local plastic zone size holds the same form as a function of applied strain amplitude and the R-ratio obtained for A356 [21]. The material constants for incubation were correlated using a mean square root regression scheme. In the MSC growth region, the crack tip driving force was estimated using the crack tip displacement that included two parts of contribution: the first term in Eq. (8) is primarily dominant in the HCF regime and the second term is dominant in the LCF regime. The primary contributions and the interaction of the two terms were treated as weighted constraint factors in the model correlation to obtain the constants CI and CII. Figs. 9 and 10 show the multistage fatigue model correlations for AE44 magnesium alloy in the as-cast and cradle specimens at room and elevated temperatures, respectively. Assuming a large pore size of 500 lm as the site of damage incubation and high porosity near the pore, the lower bound of fatigue life of AE 44 in the as-cast condition was estimated. Assuming an extremely small pore size of 10 lm as the site of damage incubation and no abnormal porosity near the pore, the upper bound for fatigue lives of AE44 as-cast bar specimens were estimated as shown in Fig. 11. The capability of estimating correct upper and low bounds demonstrates the robustness of the multistage fatigue model, which includes the key microstructure features that affect the fatigue life.

675

10 3

10 4

10 5

10 6

10 7

Cycles to failure, N f Fig. 9. The multistage fatigue model (MSF) correlation for magnesium alloy AE44 of as-cast bar specimens and specimens extracted from an engine cradle.

Samples of high-pressure die cast AE44 magnesium alloys extracted from as-cast bars and from an engine cradle were tested until failure under strain control at room and elevated temperatures. The initial microstructure and fatigue fractographs were examined using optical microscopy and SEM. A multistage fatigue model originally proposed by McDowell et al. [21], was modified for greater porosity interaction and temperature dependence. The model captured the following microstructural phenomena in fatigue crack growth: (1) Fatigue damage formed at large cast pores near the free surfaces with an average size greater than 25 lm; the most detrimental microstructural discontinuity to fatigue life were the largest pores greater than 400 lm and the interaction between large pores.

676


(2) The multistage fatigue model was able to explain the extreme limiting conditions for the strain-life of the AE44 Mg alloy. The upper and lower bounds for fatigue life were estimated using observed extreme microstructure features that coincided with the experimental results.

Acknowledgements The authors would like to express appreciation for support from the United States Automotive Materials Partnership (USAMP) under the United States Council for Automotive Research (USCAR) and the Center for Advanced Vehicular Systems at Mississippi State University. We would like to extend our thanks to Westmoreland Mechanical Testing and Research Inc., who conducted the static and fatigue experiments for AE44. We would like to thank Andrew Oppedal and Thomas Williams at Mississippi State University for conducting the measurements on the DCS and pore size distributions. References [1] Mordike BL, Ebert T. Magnesium properties – applications – potential. J Mater Sci Eng A 2001;302:37–45. [2] Aghion E, Eliezer D, editors. Proceedings of the first Israeli international conference on magnesium science and technology. Dead Sea, Israel: Magnesium Research Institute; 1997. [3] Mordike BL, Kainer KU, editors. Magnesium alloys and their applications. Wolfsburg, Germany: Werkstoff-Info GmbH Frankfurt; 1998. [4] Kainer KU, editor. Magnesium alloys and their applications. Weinheim, Germany: Wiley–VCH Verlag GmbH; 2004. [5] Goodenberger DL, Stephens RI. J Eng Mater Technol 1993;115:391–7. [6] Stephens RI, Schrader CD, Lease KB. Corrosion-fatigue of AZ91ET6, Cast magnesium alloy in A 3.5 percent NaCl aqueous environment. J Eng Mater Technol 1995;117(3):293–8. [7] Neite G, Kubota K, Higashi K, Hehmann F. Magnesium based alloys. In: Cahn RW, Haasen P, Kramer EJ, editors. Materials science and technology, vol. 8. Weinheim: VCH; 1996. p. 13–212. [8] Dahle AK, Lee YC, Nave MD, Schaffer PL, StJohn DH. Development of the as-cast microstructure in magnesium–aluminium alloys. J Light Met. 2001;1:61–72. [9] Kleiner S, Beffort O, Wahlen A, Uggowitzer PJ. Microstructure and mechanical properties of squeeze cast and semi-solid cast Mg–Al alloys. J Light Met. 2002;2:277–80. [10] Lu YZ, Wang OD, Zeng XQ, Ding WJ, Zhai CQ, Zhu YP. Effects of rare earths on the microstructure, properties and fracture behavior of Mg–Al alloys. Mater Sci Eng A 2000;278:66–76. [11] Gall K, Biallas G, Maier HJ, Gullett P, Horstemeyer MF, McDowell DL. In situ observations of high cycle fatigue mechanisms in cast AM60B magnesium in vacuum and water vapor environments. Int J Fatigue 2004;26:59–70.

[12] Horstemeyer MF, Yang N, Gall K, McDowell DL, Fan J, Gullet PM. High cycle fatigue mechanism in a cast AM60B magnesium alloy. Fatigue Fract Eng Mater Struct 2002;25:1045–56. [13] Horstemeyer MF, Yang N, Gall K, McDowell DL, Fan J, Gullet PM. High cycle fatigue of a die cast AZ91E-T4 magnesium alloy. Acta Mater 2004;52:1327–34. [14] Tokaji K, Kamakura M, Ishiizumi Y, Hasegawa N. Fatigue behaviour and fracture mechanism of rolled AZ31 magnesium alloy. Int. J. Fatigue 2004;26:1217–24. [15] Bhambri AK, Kattamis TZ. Cast microstructure and fatigue behavior, of a grain-refined Mg–Zn–Zr alloy. Met Trans 1971;2:1869–74. [16] Chaudhuri SK, Nair K. Influence of microshrinkage on mechanical properties of a magnesium–zinc–zirkonium–rare earth alloy. Mater Sci Eng 1979;37:159–64. [17] Mayer H, Papakyriacou M, Zettl B, Stanzl-Tschegg SE. Influence of porosity on the fatigue limit of die cast magnesium and aluminum alloys. Int J Fatigue 2003;25:245–56. [18] Donlon WT, Paige C, Morris CJ, Allison JE. The effects of casting defects and microstructure on the mechanical properties of die cast AM50 magnesium and 356 aluminum. In: Proceedings of the aluminum and magnesium for automotive applications, TMS/AIME, 1996. p. 17–27. [19] Berger C, Pyttel B, Trossmann T. Very high cycle fatigue tests with smooth and notched specimens and screws made of light metal alloys. Int J Fatigue 2006;28:1640–6. [20] Sonsino CM, Dieterich K. Fatigue design with cast magnesium alloys under constant and variable amplitude loading. Int J Fatigue 2006;28:183–93. [21] McDowell DL, Gall K, Horstemeyer MF, Fan J. Microstructurebased fatigue modeling of cast A356-T6 alloy. Eng Fract Mech 2003;70:49–80. [22] Xue Y, Burton CL, Horstemeyer MF, McDowell DL. Multistage fatigue modeling of cast A356-T6 and A380-F aluminum alloys. In: Wang QG, Krane MJM, Lee PD, editors, Simulation of aluminum shape casting processing, TMS, 2006. [23] Xue Y, Jordon B, Horstemeyer MF. Multiscale fatigue modeling and life prediction for 7075-T651 aluminum alloy. In: Larsen James M, Christodoulou Leo, Calcaterra Jeffrey R, Dent Michael L, Derriso Mark M, Hardman William J, Hoffman Paul, Jones J Wayne, Russ Stephan M, editors. Materials damage prognosis. TMS 2004, p. 1443– 50. [24] Gall K, Horstemeyer MF, McDowell DL, Fan J. Finite element analysis of the stress distributions near damaged Si particle clusters in case Al–Si alloys. Mech Mater 2000;32(5):277–301. [25] Renard AS, Cheng R, Veaux DL, Laird C. The cyclic stress–strain response of polycrystalline Al–Zn–Mg alloy and commercial alloys based on this system. Mater Sci Eng 1983;60:113–20. [26] Maenning WW. Planning and evaluation of fatigue tests. In: Lampman SR, Davidson GM, Reidenbach F, et al., editors. ASM Handbook fatigue and fracture, vol. 19. Materials Park (OH): ASM International; 1997. p. 303–13. [27] McClintock FA. Considerations for fatigue crack growth relative to crack tip displacement. In: Beynon JH, Brown MW, Lindley TC, Smith RA, Tomkins B, editors. Engineering against fatigue. Balkema Press; 1999. p. 227–41. [29] Henry SD, Davidson GM, Lampman SR, Reidenbach F, Boring RL, Scott WW, editors. Fatigue data book: light structural alloys. New Jersey: ASM International; 1995. [30] Mayer H, Papakyriacou M, Zettl B, Vacic S. Endurance limit and threshold stress intensity of die cast magnesium and aluminum alloys at elevated temperatures. Int J Fatigue 2005;27:1076–88.