Fatigue Crack Initiation in AA2024: a coupled in-situ mechanical testing and crystal ..... A python code was .... crystal plasticity code developed in this study.
Fatigue Crack Initiation in AA2024: a coupled in-situ mechanical testing and crystal plasticity study Panos Efthymiadis a*, Christophe Pinna b, John Yates c a
Warwick Manufacturing Group, International Manufacturing Centre, University of Warwick, Coventry
CV4 7AL, UK, previously in the Department of Mechanical Engineering, the University of Sheffield, Mappin Street, Sheffield S1 3JD, UK b
Department of Mechanical Engineering, The University of Sheffield, Mappin Street, Sheffield S1 3JD,
UK c
Simuline Ltd., Derbyshire S18 1QD, UK
Abstract A novel combined experimental and modeling approach has been developed in order to understand the physical mechanisms that lead to crack nucleation in polycrystalline metals undergoing cyclic loading. Out-ofplane four point bending low cycle fatigue tests were performed inside the chamber of a scanning electron microscope (SEM) on polycrystalline aluminium alloy AA2024. A specimen geometry with a central hole was used to localise stresses and strains in a relatively small region on the top surface of the sample. Fatigue crack initiation (FCI) and small crack growth (SCG) mechanisms were analysed through high resolution SEM images, local orientation measurements using electron-back-scattered-diffraction (EBSD) and local strain measurements using digital image correlation (DIC). The effect of crystal orientation on slip band formation and crack nucleation was studied by means of inverse pole figures (IPF) in order to identify soft (crack-prone) and hard (damage resistant) grains. A crystal plasticity finite element (CPFE) model was developed to simulate the cyclic deformation of the microstructure. A two dimensional Voronoi-based microstructure was generated with 80 grains. Boundary conditions were applied using DIC measurements and parameters for the constitutive equations including both isotropic and kinematic hardening were identified using fully reversed cyclic tests. FCI criteria found in the literature were evaluated but could not predict the formation of cracks in grains located away from the hole in
the bent specimen. A new criterion combining measures of maximum accumulated slip over each individual slip system and maximum accumulated slip over all slip systems has been formulated to reproduce experimental observations of crack nucleating in highly deformed grains showing intense slip bands formation.
1.
Introduction
Even though the fatigue of metals has been studied for more than two centuries [1], much research is still needed to understand and predict the conditions leading to crack formation in polycrystalline microstructures under cyclic loading. Fatigue crack initiation (FCI) and small crack growth (SCG) are the two phases of crack development which are most influenced by the microstructure [2]. As a result, significant scatter is commonly observed in experimental results [3] leading to high uncertainty in fatigue model prediction. Microstructureinformed models therefore need to be developed to improve the reliability of fatigue life calculations for engineering components [4]. Recent modelling studies on FCI and SCG have covered a range of polycrystalline metals, such as aluminium alloys [5-7], titanium [8,9], stainless steels [10] and nickel-based super-alloys [8,11]. The physical mechanisms that lead to FCI are highly localised. For instance, dislocation dipoles, interface decohesion, triple points, second-phase particles can lead to strain localisation followed by crack nucleation [1, 3, 8-9]. The propagation of a short crack depends on the resistance and strength of the matrix. Depending on the location of the crack, whether it is close or not to a grain boundary (GB), crack acceleration or deceleration can also occur [12]. Particles embedded in the matrix may facilitate further propagation if the particles break in a brittle manner or retard crack growth if the particles break in a ductile manner. As a result factors affecting localised damage nucleation and small crack growth can be very complex. A reliable physically-based FCI criterion must therefore capture the effect of various likely mechanisms leading to the formation of a fatigue crack at the scale of the microstructure. Various computational approaches, such as crystal plasticity finite element modeling, molecular dynamics, discrete dislocation dynamics and conventional continuum-based FEM modeling have been utilised with different criteria to model FCI by various researchers [3, 5-11, 13-16]. Crystal plasticity finite element modelling (CPFEM) simulations have been shown to model FCI efficiently [12]. However various crystal plasticity based criteria have been proposed in the literature. Cheong et al. used an energy based approach and defined the local energy dissipated (Ep) in terms of plastic work as an indicator for FCI [13]. Manonukul and Dunne reported that FCI can be accurately modelled by using a very simple crack
initiation criterion based on the critical accumulated slip (pcrit) [8]. In the work by Shenoy et al., five fatigue indicator parameters (FIPs) were introduced in order to account for the factors that influence FCI and SCG [3]. These parameters were: the accumulated plastic slip per cycle (P cyc), the impingement of slip due to pile-ups (Pr), the critical accumulated plastic slip which leads to crack initiation (p crit), the modified Fatemi-Socie parameter (PFS) which takes into account the tensile stress normal to the crack and the shear strain range acting along the crack, and the maximum range of cyclic shear strain (Pmps) [3]. Similarly to Shenoy et al., Bozek and Hochhalter et al. used five nucleation metrics to model FCI and SCG regimes [5-7]. These metrics were the maximum accumulated slip over each slip system (nucleation metric D1), the maximum accumulated slip over each slip plane (D2), the maximum accumulated slip over all slip systems (D3), the maximum energy dissipated along the critical slip plane (D4) and the modified Fatemi-Socie criterion (D5) [5-7]. Table 1 summarizes all the different FIPs used in the above papers. Table 1. Fatigue parameters for indicating damage initiation and small crack growth.
K.-S. Cheong, M. J Smillie & D.M Knowles13 Ep: local energy dissipation, Lp: plastic component of the velocity gradient, 𝑭̇𝒑 : rate of change of the plastic deformation gradient, 𝑭𝒑 : plastic deformation gradient
Manonukul and F. Dunne8 pcyc: accumulated slip per cycle, Nf: number of cycles to failure, pcrit: critical accumulated slip
M. Shenoy, J. Zhang & D. L. McDowell3
pcyc: accumulated slip per cycle, Dp: plastic rate of deformation tensor, Ninc: number of cycles to initiate a crack, pcrit: critical plastic strain, Pr: cumulative net plastic shear strain, 𝜺̇ 𝒑 : plastic shear strain rate along a specific slip system, t and n: the direction along a given slip plane and the normal to that plane, PFS: local Fatemi-Socie parameter which accounts for both the accumulated shear strain along the crack and the tensile stress normal to the crack, ∆𝜸𝑷𝒎𝒂𝒙 : maximum range of plastic shear strain, 𝝈𝒎𝒂𝒙 : peak tensile stress normal to the slip plane, 𝝈𝒚 : 𝒏 cyclic yield strength, K: material properties related to multiaxial strain state, Pmps: maximum range of cyclic plastic shear strain
J. D. Hochhalter, D. J. Littlewood, R. J. Christ Jr, M. G. Veilleux, J. E. Bozek, A. M. Maniatty & A. R. Ingraffea6
γa: accumulated slip for slip system a, D1, D2 & D3: the maximum accumulated slip over each slip system, over each slip plane & over all slip systems, respectively, D4: the maximum energy dissipated along the critical slip plane, 𝜏𝑝𝛼 : resolved shear stress over slip system a, 𝑵𝒅 : total number of slip systems on a given slip plane, D5: criterion which accounts for the tensile stress normal to the crack and the shear strain range acting along the 𝒑
crack, 〈𝝈𝒏 〉: tensile stress on slip plane p (〈… 〉: Macaulay brackets defined as 〈𝑥〉=0 if x≤0 and 〈𝑥〉=x if x>0 so that only tensile stress has an effect, g0: the initial hardness (resistance to slip) on the slip systems, and k dictates the importance of tensile stress relative to plastic slip.
A general conclusion from all these studies is that different FCI parameters are used for the same physical process, which means that a generic criterion has not yet been derived. Furthermore, the criteria used above have not been verified experimentally down to the microstructural level.
Micromechanical testing inside a SEM has been successfully applied to a range of alloys extending from high strength steels to titanium and nickel-based super-alloys [10, 13]. However, most studies have been focused on in situ tensile testing [10].
This work therefore aims at generating new experimental insight into the formation of fatigue cracks at the scale of microstructures by focusing on AA5052 aluminium alloy. Low cycle fatigue tests are conducted inside a scanning electron microscope (SEM) on bespoke bending specimens with high resolution images used to analyse intra-granular deformation leading to crack initiation and early propagation. Electron-back-scattereddiffraction (EBSD) and digital image correlation (DIC) measurements are also conducted to study crack development in relation to local orientations and strain fields in the microstructure. Experimental results are used in combination with a CPFE model of the cyclic deformation of the microstructure to infer a new generic microstructure-informed FCI criterion.
2. Materials & Methods AA5052 T3 aluminium alloy with a 207m average grain size was used in this study. A bending specimen was machined to a rectangle 50mm long, 7mm wide and 4mm thick with a 1mm-diameter hole drilled at the center of the specimen to promote early formation of a fatigue crack. The corresponding geometry is shown in Fig. 1(a). The top surface of the sample was then mechanically ground and polished down to 0.06μm (with Silico solution) followed by electro-polishing in a solution containing 30% Nitric acid and 70% Methanol, for 13 seconds, at -16o C, 15V and 10mA. EBSD measurements were carried out using a FEI-Sirion SEM, at 20kV and spot size 5 as operating conditions. The step size was 1.5μm, covering approximately a total area of 1.6mm x 3.0mm, as shown in the magnified EBSD map of Fig. 1(b). Following the EBSD measurements, the samples were re-polished with Silico solution for 4 minutes, to remove any Hydrocarbon (HC) deposition generated during the EBSD-measurement acquisition. The samples were then etched with Keller’s reagent (95ml water, 2.5ml Nitric acid, 1.0ml Hydrochloric Acid and 1.5ml Hydrofluoric acid) for approximately 4-5 seconds, to reveal the microstructural features such as GBs, particles and inclusions.
Low cycle fatigue tests were performed using a 2kN Deben out-of-plane four-point bending stage inside a CamScan SEM with a cross head speed of 2mm/min. The test configuration enables the top surface of the specimen to be under maximum tensile loading, promoting fatigue crack formation on the surface visualized using SEM imaging. Initially a static test was carried out up to a maximum strain value of 0.01 measured, using DIC, on left and right edges of the EBSD map shown in Fig. 1(b). The test was interrupted every 0.2 mm vertical displacement increments to acquire images. The local displacement and strain values were then obtained over the microstructure by means of DIC, using the natural features of the microstructure (precipitates, inclusions, GBs) for the correlation between two successive images, as explained in [17]. The local strain distributions were acquired at maximum displacement after static loading (first peak load). Subsequently, displacement controlled cyclic loading was initiated with a frequency of 0.5 Hz, from minimum displacement (zero load) up to maximum displacement. The test was interrupted after the 1 st, 10th, 100th, 200th, 300th, 400th and 500th cycle, at the maximum applied displacement and images were acquired. All images were correlated with respect to the image at the first peak load and hence to the image of the undeformed sample. Therefore the strain values and the plastic strain accumulation were obtained at the scale of the microstructure throughout the whole test. A low magnification (50x) was used to acquire images of relatively large microstructural areas with respect to the EBSD map shown in Fig. 1(b), in order to obtain representative results. The two regions of the microstructure analysed with DIC cover approximately a 2.25mm x 1.5mm area each and are highlighted with two yellow boxes shown in Fig. 1(b). The micrographs were analysed using the commercial DIC software, DaVIS 7.0, LAVision to compute the in-plane displacement field from which the major strain values were calculated, as explained in [17]. A sensitivity analysis was pursued relative to the interrogation window size. Instead of using a constant interrogation window defined by the number of pixels, a multi-pass procedure of four passes, two passes with 16x16 pixels and another two with 32x32 pixels was selected for all the experimental results presented of this work. A 25% overlap between windows has been used to achieve the best correlation results and a displacement accuracy of 0.01 pixel was obtained, with a strain resolution of about 0.1%. In the following section results for the two analysed regions of the EBSD map of Fig. 1(b) are presented.
4mm 1mm
7mm
a 50mm
RD
c
b
=1000µm
Loading direction
Fig. 1. (a) Specimen geometry, (b) the EBSD map at the centre of the sample in the region around the hole. The two regions of the EBSD map highlighted with yellow boxes were selected for DIC analysis and in-situ SEM observations during cyclic-bending, c) Inverse pole figures (IPF) for the EBSD map of Fig. 1(b).
3. Experimental Results 3.1.
FCI regime: in situ testing, SEM imaging and DIC strain analysis
SEM images recorded during the four-point bending cyclic tests were used to study microstructure evolution in the two regions of the microstructure highlighted by yellow boxes in Fig. 1(b). Cracks were found to initiate in
these two regions. After only 10 cycles, intense slip bands can be seen in the grains highlighted in red in Fig. 2(a) and (c) and in the grain highlighted in white in Fig. 2(c). It is interesting to note that the grains highlighted in red in Fig 2(a) and in white in Fig. 2(c), are located close to the hole, but not exactly at the geometrical edge of the hole, therefore suggesting the influence of the microstructure and, in particular, grain orientation on the local plastic deformation during cyclic loading. The DIC-computed εxx strain distributions in Fig 2(b) and (d) show the high heterogeneity of the deformation at the scale of the microstructure. A concentration of high strain values is clearly visible at the edge of the hole (regions highlighted with a white triangle) in Fig. 2(b) and (d) as expected from the presence of the hole. More localised regions of high strain values are also observed in these strain maps further away from the hole and suggest, again, the influence of grain orientations on local deformation. In the triangular regions of Fig. 2(b) and (d) the highest strain magnitude of 0.75 is located at the very edge of the hole as shown by the red arrows in Fig. 2(b) and (d). Although such location corresponds to the region of intense slip band formation in the grain highlighted in red in Fig. 2(c), such correlation is not always observed, as intense slip bands are also observed in the grain highlighted in red in Fig. 2(a) as well as in the grain highlighted in white in Fig. 2(c) located outside the triangular region of high strain values.
a
10 cycles
300 μm
b
10 cycles
300 μm
c
10 cycles
300 μm
d
10 cycles
300 μm
Fig. 2. (a) and (c) SEM images of the microstructural areas highlighted in Fig. 1(b) after 10 cycles and (b) and (d) the corresponding strain maps measured using DIC (the x-axis corresponds to the loading direction which is horizontal in these images) Fig. 3(a) and (b) show higher magnification SEM images of the grain highlighted in red in Fig. 2(a), after 1 and 10 cycles respectively. These close-up views of the microstructure reveal that, although the formation of intense slip bands can actually be observed after the 1st cycle, FCI can be seen after 10 cycles along one of the slip bands in Fig. 3 (b) as shown by the red arrow. Fig. 3(c) and (d) show high magnification SEM images of the grain highlighted in red in Fig. 2(c), after 1 and 10 cycles respectively. Crack initiation does not occur after the first cycle but micro-voids can be seen in the close-up view of Fig. 3(c) as highlighted by the red arrow. After 10 cycles, these micro-voids have joined up to form a crack along a particular slip band, as shown by the red arrow in Fig. 3(d). These results therefore show that FCI does not always take place at the very edge of the hole where strain magnitudes are the highest but rather in grains favourably orientated for intense slip band formation.
a
1st cycle
100 μm
b
10 cycles
30 μm
c
1st cycle
10 μm
dc
1010cycles cycles
10 μm
Fig. 3. SEM images of the grains highlighted in red in Fig. 2(a) and Fig. 2(c) under tensile loading after1 cycle (a) and (c) and 10 cycles (b) and (d) respectively. However FCI requires a certain level of strain intensity to generate a crack as can be seen in Fig. 4(a) and (b) with a close-up view of the grain highlighted in white in Fig. 2(c). Although intense slip bands can be observed after the first cycle in this grain located outside the triangular region of high strain values in Fig. 2(d), a crack does not form in this grain, even after 500 cycles (Fig. 4(b)). It is interesting to note that the morphology and density of slip bands in this grain hardly change between 1 and 500 cycles. FCI in AA2024 T3 is therefore dependent on both strain magnitude, grain orientation and propensity for intense slip band formation, with cracks originating along intense slip bands in highly deformed grains. The major strain is 0.75 and 0.55 at the location where crack initiation occurs in the two red-highlighted grains in Fig. 2c. The white-circled grain of Fig. 2c is strained between 0 up to 0.4 and therefore does not form any cracks in its matrix.
a
1st cycle
100 μm
bb
500 cycles cycles 500
100 μm
Fig. 4. SEM images of the grain highlighted in white in Fig. 2(c) under tensile loading after (a) 1 cycle and (b) 500 cycles.
3.2.
Comparison between crack nucleation along slip bands and at particle sites
The highlighted triangular area in Fig. 2(d) is shown in Fig. 5(a) and (b). The EBSD map is shown together with low and high magnification SEM images to clearly correlate the localised deformation and damage mechanisms with the underlying microstructure. In the two circled areas of Fig. 5(a) and (b), two different FCI mechanisms occur. In the white circled area, intense slip bands form from the very early stages of the test with a small crack initiating along one of these slip bands (Fig. 5(d) and 3(d)). Instead, in the black circled area, few, less intense, slip bands are mainly localised around a particle. Chemical analysis was not possible to be done as the tester was obstructing the EDS detector. However a detailed analysis was done in [4] and it can be stated that the broken and debonded from the matrix particle shown with black arrow in Fig. 5c is an Al-Cu
second phase particle. The area highlighted with a red-arrow in Fig. 5(c) shows a crack that has formed at an inclusion site [4]. Intense slip bands form at the neighbourhood of the inclusion, close to 45 o with respect to the loading axis (pointed with two red-arrowed lines). The crack propagation direction though is perpendicular to the loading direction. Damage development at the two microstructural sites shown in Figs. 5(c) and (d) is different. Fig. 5(d) shows the formation of a 15 m crack along intense slip bands after 10 cycles. The crack in Fig. 5(c) initiates after the first cycle, however it reaches only a 6 m length after 10 cycles by propagating perpendicularly to the tensile stress direction. Furthermore the propagation shown in Fig. 5(e) and (f) after 500 cycles confirms the different amount of plasticity involved that drives crack propagation for the two cracks. The crack initiated in the black circled area continues its propagation along slip bands through a mode II mechanism by branching from one slip plane to another at a much higher growth rate reaching a projected crack length of about 110 m in Fig. 5(f). The crack initiated at the particle in Fig. 5(e) has continued its propagation along the same direction, at about 45o with respect to the original slip band directions highlighted by red arrows, through a mode I mechanism, to reach a crack length of about 9 m. The difference in growth rate can be related to the different crack propagation mechanisms, controlled by dislocation movement, with extensive plasticity taking place ahead of the crack tip along the extended slip bands in Fig. 5(f) and a much smaller plastic zone size at the tip of the mode I crack, with hardly any evidence of slip bands at the crack tip in Fig. 5(e). These observations therefore show that FCI along slip bands generated due to Ainitial plastic deformation in the microstructure is the predominant mechanism for crack nucleation in the investigated AA2024 T3, and FCI due to second phase particles can be neglected.
Fig. 5. (a) EBSD map of the undeformed region of Fig. 2(c) and (b) SEM image at the same region under loading after 10 cycles. (c) - (f) Higher magnification images of the regions highlighted in Fig. 5(b) after 10 cycles (c)-(d) and after 500 cycles (e)-(f).
3.3.
Detrimental and preferential grain orientations
In this section grain orientations are analysed, through EBSD, to determine detrimental and beneficial crystal orientations in relation to FCI. Crystals that have the tendency to localise deformation, in the form of persistent slip bands, and nucleate fatigue cracks along specific slip systems were characterised as ‘weak’
grains, while crystals with unfavourable orientation for the formation of intense slip bands were named as “strong” crystals, following the nomenclature introduced by Cheong et al [13]. The Inverse Pole Figure (IPF) for the EBSD map of Fig. 1 is shown in Fig. 6 where the orientation of the rolling direction (RD) is plotted with respect to the crystal reference system. Results show that most grains lie along the 001-101 plane . In order to better evaluate the population of grains that lie on the 001-101 plane, the density contours for the population of grains in IPF figure are shown in Fig. 7b. Fig. 7b shows that there is a considerable amount of grains that lie away from the 001-101 plane which might be prone to damage nucleation, as these grains have not taken up adequate plastic deformation during the rolling process. High resolution SEM analysis was performed after cyclic testing and an analysis was carried out to correlate the grains (and crystal orientations) that are prone to slip band formation (damage prone crystals) with their corresponding location in the IPF map. The highlighted grains of Fig. 7(a) and (b) were also plotted in the IPF figure and are shown in Fig. 7(c) and (d), respectively. Fig. 7(a) shows grains (high contrast grains), lying along the 001-101 plane, that did not show localised plastic deformation, did not form any strong slip bands and did not contribute to damage nucleation and evolution (named as ‘hard’ grains). The highlighted grains of Fig. 7(b) lie mostly along the 001-111 plane, formed strong slip bands (and in some cases micro-cracks) and therefore are prone to FCI (named ‘soft’ grains). However two grains (grain highlighted with an arrow in Fig. 7(a) and grain highlighted with a red square in Fig. 7(b)) show orientations away from those of the other highlighted grains. This effect is attributed to the local texture effects and the orientations of the corresponding neighbouring grains. The data points highlighted with an arrow in Fig. 7(c) correspond to the grain highlighted with an arrow in Fig. 7(a). High misorientations were found with all the corresponding neighbouring grains, restricting plastic flow between the grains. As for the grain highlighted with a red square in Fig. 7(b), whose orientation is close to the 001-101 crystal orientation in Fig. 7d, intense slip bands were observed. However this grain showed slip bands only in the lower part of the grain, highlighted by the square in Fig. 7(b). The misorientations with the neighbouring grains were high, and the ‘hard’ grain neighbours did not show any slip bands. A sharp corner also exists at the grain boundary, which could possibly concentrate high local stresses
and strains. The two black-circled grains in Fig. 7(b) show the two grains with the most intense slip bands as explained in section 3.1, and they both lie far from the 001-101 orientation.
a
b
RD
RD
c
d
Fig. 7. (a) Hard grains, with no intense slip bands highlighted with high contrast, (b) Soft grains highlighted with high contrast, showing intense slip bands, c -d) The corresponding Inverse pole figures (IPF) for the hard and soft grains of Fig. 12a and b, respectively.
4.
Computational Study
The CPFE model is based on the work of Huang, Hill and Rice [18, 19 and 20]. However modifications have been made to the original hardening law in order to be able to model cyclic loading conditions. In detail, the total deformation gradient F of a crystalline material can be decomposed into an elastic (Fe) and plastic part (Fp):
𝐹 = 𝐹𝑒 𝐹𝑝 = 𝐹𝑒 𝐹∗
L F F 1
(2)
F Fe F p Fe F p
(3)
Where L is the velocity gradient. The plastic part of the velocity gradient, L p, and the resolved shear stress τ(α) are given by:
LP F P F P 1
( )
s ( ) m ( )
(4)
( ) : s ( ) m( ) sym
(5)
𝛾̇ is the slip rate along each slip system α, and s and m are the slip direction and the normal to the slip plane, respectively. 𝜎̃ is the Cauchy stress tensor. The velocity gradient can be decomposed into a symmetric rate of stretching tensor D and an antisymmetric spin tensor Ω:
𝐷 = 𝐷 𝑒 +𝐷𝑝 and Ω = Ω𝑒 +Ω𝑝 D e e F e F e 1 D P P ( ) s ( ) m ( )
(8)
The stretching tensor (D) and the Jaumann rate of Cauchy stress, 𝜎̃, are given by [18, 19]:
: De L : De
(9)
For monotonic loading conditions, the slip rate is related to the corresponding resolved shear stress as shown below (Schmid law):
n
( )
( )
( ) ( a ) sign ( ) g
(10)
Where 𝛾̇(𝜶) is the reference strain rate on the αth slip system, n is the strain rate exponent and g (α) determines the current strength of the αth slip system. For cyclic loading conditions in order to reflect the contribution from the backstresses, gα and χα are introduced to represent the isotropic and kinematic hardening [21, 22, 25]: n
( )
( )
( ) ( ) sign ( ) ( ) g (a)
(11)
The hardening law for gα and χα can be obtained by using the Armstrong-Frederick hardening rule:
g ( ) h ( )
(12)
( ) b ( a ) r ( ) ( a )
(13)
(A)
Where the self (hαα) and latent (hαβ) hardening moduli are given by:
h h ho sec h 2
ho , s o
h q h ,
(14)
(15)
b and r and q are material constants. h0 is the initial hardening modulus, τs is the stage I critical resolved shear stress and τ0 is the saturation critical resolved shear stress. While the cumulative shear strain on all slip systems can be obtained by: t
( ) dt a
4.1.
(16)
0
Finite Element Model of the microstructure
For this study 2D RVE was employed to model the macroscopic stress-strain behaviour in order to be able to obtain an efficient tool for modeling the cyclic deformation behaviour and be able to model the accumulation of plastic deformation within the individual grains using CPFEM. Instead of modeling the exact microstructure, the Voronoi technique was employed to create the (2D) microstructure geometry for the model, as described in more detail in [4]. This was done in order to obtain a tool that can enable to user to employ a modeling tool that is independent of experimental techniques (such as EBSD) but also in order to be able to create a more
efficient modeling technique. It has been widely reported the modeling efficiency of Voronoi-based crystal plasticity techniques with respect to experimentally-based (EBSD-generated) CPFEM. A homogenisation study was carried out with a 2D RVE with different number of grains and it was found that 80 crystals are adequate to yield and model efficiently the macroscopic response for the artificially created microstructure. Random grain orientations were assigned to the numerical microstructure by using three random number generators, through a Fortran subroutine, for the three Euler angles. A mesh sensitivity study was pursued. 4-node plane strain quadrilateral elements (CPE4) were used for the mesh. A python code was developed in order to optimize the mesh size (computational time versus homogenized stress-strain fields). The homogenised stress-strain fields over the whole area (all integration points) were obtained for a mesh size (edge size) of 0.01. A representative volume element (RVE) of 80 Voronoi polygons was created and meshed as shown in Figs. 8(a) and (b). The bottom edge of the polycrystal was gripped in the y-direction while the top edge was left free to move, the left-edge was gripped in the x-direction, and a constant displacement value 0f 0.01 was applied at the right edge to match the experimental strain value measured with DIC. Periodic boundary conditions were also applied to this RVE via Abaqus CAE (interface), but similar results were obtained for the macroscopic response.
b
a
Al-2024 T3 500
500 500
500450
Fig. 8. (a) The polycrystalline (b) the meshed microstructure. Al-2024 TensileT3 geometry of the 2-D RVE, andFully reversed fatigue
450 450
Al-2024 T3
The material parameters (elastic constants and hardening coefficients) for AA2024 aluminium alloy are shown
450400
400 400
in Table 350 350 3 and were obtained from [18, 20-22]. minimise the square of differences between experimental and
350300
300 300 modelling stress-strain values. The fitted values for the hardening parameters (h0, τs and τ0) of Asaro’s 400
250 250
Stress (MPa)
agreement between the experiment and the 150 150 Experimental experimental
50 50 0
00 0.02 00
c
0 =25,τsτ=100, s=100,τ0τ=82.5 0 =82.5 hh0=25, 25..100..82.5 25..100..82.5
-0.025
Experimental h0=50, τs=100, 50..100..82.5 experimental -0.02 -0.015 0 s 50..100..82.5
-0.01 0
37..100..82.5 h0=37, τs=100,
-0.005
0.04 0.02 0.02
0.06 0.04 0.04 Strain
τ0=82.5 -0.025
Strain
Series2
500 100 00 ss 50..100..82.5 50..100..82.5
h =50, τ =100, τ =82.5 τ0=82.5 37..100..82.5 h0=37, τs=100, τ0=82.5 h0=50, τs=100, τ0=82.5 50..100..82.5
100 100
100 50
0
τ0=82.5 =50,τ τ=100, =100,τ0τ=82.5 0 =82.5 hh=50, modelling results 400 obtained. 37..100..82.5 h0=25, τs=100, τ0=82.5 25..100..82.5 h0=37, τs=100, τ0was =82.5 37..100..82.5 37..100..82.5 0 =37,τsτ=100, s=100,τ0τ=82.5 0 =82.5 hh00=37,
h0=25, τs=100, 25..100..82.5
Stress (MPa)
150100
Series1
0 25..100..82.5
200 200
200150
0
300
Experimental experimental
Experimental Experimental experimental hardening rule are given by equation A and can hbe seen in experimental Table 3. As shown in Fig. 9a, a rather good 200 =25, τs=100, τ =82.5
250200
50 0
500
Stress (MPa)
Stress
(MPa) Stress Stress Stress
400350
300250
Stress
Al-2024 Al-2024T3 T3
0
0.005
0.01
0.015
Series5 Series6 Series7
0.02
0.025
300 -100
Series4 Log. (Series3) Series1 Experiment Poly. (Series3) Simulation Series2 Poly. (Series4) Series5
200 -200 100 -300
-0.02
-0.015
0.08 0.06 0.06
-0.01
d
-0.005
0 -400
Series3
Series6 Series7
0
0.1 -100 0.08 0.08 -500
Strain
0.005
0.01
0.015
0.1 0.1
Strain
0.02
0.025
Series3 Series4 Log. (Series3)
The values of parameters b and r, shown in Table 3, were calibrated using strain-controlled, fully reversed cyclic fatigue test with a strain amplitude of 0.02. Further details about the procedure to obtain the two parameters can be found in [18, 22]. Fig. 9(b) shows the good agreement between the curve obtained using the CPFE model and the experimentally obtained stabilised curve. Table 3. Material parameters for AA2024. Elastic moduli (GPa) C1111
C1122
C2222
C1133
C2233
C3333
C1112
C2212
112
59.5
114
59
57.5
114
1.67
-0.574
C3312
C1212
C1113
C2213
C3313
C1213
C1313
C1223
-1.09
26.7
1.25
-0.125
-1.12
-1.92
26.2
-0.125
C2223
C3323
C1123
C1323
C2323
1.86
0.068
-1.92
-1.09
24.7
Hardening exponent & Hardening coefficient 𝛾̇0
n
0.001 sec-1
10
Self & Latent Hardening ℎ0
τs
τ0
q
37
100
82.5
1
b
a
Al-2024 T3 500 500
500450
450 450
450400
400 400
400350
350 350
350300
τ0=82.5 =50,τ τ=100, =100,τ0τ=82.5 0 =82.5 hh=50, 400 37..100..82.5 h0=25, τs=100, τ0=82.5 25..100..82.5 h0=37, τs=100, τ0=82.5 37..100..82.5 37..100..82.5 0 =37,τsτ=100, s=100,τ0τ=82.5 0 =82.5 hh00=37,
Series5
Stress (MPa)
-0.025
-0.015 -0.02 0 s 50..100..82.5
-0.01 0
37..100..82.5 h0=37, τs=100,
00 0.02 00
0.04 0.02 0.02
ac
0.06 0.04 0.04 Strain
0.02
0.04
0.04
0.06 Strain
Strain 0.06 Strain
τ0=82.5 -0.025
0
-0.005
h =50, τ =100, τ =82.5 τ0=82.5 37..100..82.5 h0=37, τs=100, τ0=82.5 h0=50, τs=100, τ0=82.5 50..100..82.5 Stress (MPa)
Stress
Series1
0.01
0.015
Series6 Series7
0.02
0.025
-0.015
0.08 0.06 0.06
Strain Strain 0.08
-0.01
bd
-0.005
0 -400
Log. (Series3) Series1 Experiment (Series3) Poly. Simulation Series2 Poly. (Series4) Series5
Series6 Series7
0
0.1 -100 0.08 0.08 -500
Strain
0.1
Series3 Series4
200 -200 100 -300
-0.02
0.1
0.005
300 -100
-200
0.005
0.01
0.015
0.1 0.1
Strain
0.02
0.025
Series3 Series4 Log. (Series3) Poly. (Series3) Poly. (Series4)
-300
0.08
Series2
500 100 00 ss 50..100..82.5 50..100..82.5
Experimental h0=50, τs=100, 50..100..82.5 experimental
h0=25, τs=100, 25..100..82.5
50 50 0
Stress (MPa)
Stress
0.02
Experimental experimental
100 100
100 50
300
Experimental Experimental experimental experimental
200 τ0=82.5 0 =25,τsτ=100, s=100,τ0τ=82.5 0 =82.5 hh0=25, 25..100..82.5 25..100..82.5
h0=25, τs=100, 25..100..82.5
150 150
150100
400
Experimental experimental
200 200
200150
0
500
250 250
250200
0
Fully reversed fatigue
Al-2024 T3
300 300
300250
50 0
Al-2024 TensileT3
(MPa) Stress Stress Stress
500
Al-2024 Al-2024T3 T3
-400 -500
Strain
Fig. 9. Comparison between modeling and experimental results for (c) tensile testing (for a strain up to 0.02) and (d) fully reversed fatigue testing of AA2024 (strain amplitude of 0.02).
4.2.
Modelling FCI
The 4-point bending geometry outlined in section 3 was also modeled. The grain structure, orientation and morphology was not modeled explicitly in order to facilitate fast computational times and obtain a more general model applied also in other polycrystals. Instead following the concepts of the previous section (4.1) the Voronoi technique was employed to model the cyclic behaviour of a 2D RVE polycrystal, containing a 1mm hole, shown in Fig. 10a-b. The total number of grains was approximately 900. In section 2 and 3, the EBSD map contained 787 grains in a rectangular area of 4.8 mm2. In this case a larger number of grains was used in this study, as the shape of the grains was not accounted for. In the experiment in section 1 the grains were elongated due to the rolling process, while here the grains have a polygonal, rather equiaxed shape as shown in Fig. 10a-b. The critical parameter here was selected to be the grain diameter perpendicular to the loading direction (the grain size or diameter along which the crack propagates). The grain orientations in the RVE were random numbers selected from the sets of Euler angles obtained from the EBSD map of Fig. 1. Meshing of the geometry was done with 4-node bi-linear plane strain quadrilateral elements. 1% strain was applied to the right edge (an average of 1% strain was applied in the case study of section 3.1 and 3.2), with the bottom and left side being clamped in the y- and x- direction respectively. Following the concepts presented in Table 1, and the observations made in sections 3.1 and 3.2, an additional FIP parameter is introduced here, namely D*. Using the notation by Hochhalter et al [2] the D1 metric was employed, which represents the maximum accumulated slip over each slip system, and the D3 metric, which is the total accumulated slip over all slip systems. The new D* is a combination of the two metrics: 𝐷∗ =
(2𝛼+𝛽)
√𝐷12𝛼 ∙ 𝐷3𝛽
For this study α and β are 1. In this criterion a combination of D1 and D3 metrics was employed as it is the presence of critical grains (that lead to strong slip bands) within regions with high strains that leads to FCI. In areas where high strains are observed it will be preferentially the critical grains (that facilitate strong slip banding mechanisms) that will initiate cracks, and that is the reason why an exponent of 2 is used for D1 metric, while an exponent of 1 is used for D3 metric. The D3 metric is also important, as it was found that critical grains outside critically loaded areas formed slip bands but did not lead eventually to FCI (section 3.2),
as the applied stresses and strains were not high enough to facilitate FCI. This signifies the importance of both parameters: D1 and D3. The (2𝛼 + 𝛽) root is used in order to obtain similar scale with respect to D1 and D3 metric. Μore experiments are necessary in the future to validate the importance and the competition between the two parameters, namely D1 and D3. Furthermore, there could be an influence of the material investigated, Al2024 on the values for the two exponents. Different polycrystals, such as Nickel superalloys or Ti alloys or steels might yield different dependence between the two metrics. Fig. 10c shows the total accumulated slip over all slip systems, the D3 metric, after 1 cycle at maximum displacement. The total accumulated slip is highly non homogeneous. Shear bands have formed, extending from the edge of the hole at 45o towards the rest of the material. The highest strain localisations occur at the edge of the hole. The maximum value for D3 is 0.037 and therefore a criterion based on D3 predicts failure at the edge of the hole. Similar observation can be made at the lower semi-circle of the hole, with the highest shear strain values found at the edge; i.e. 0.033. Fig. 10d shows the newly introduced FIP, D*. The shear bands are still observed in the D* strain distribution. Yet the strains appear to be even more non -homogeneous in comparison with Fig. 10c, showing an increased influence of the individual grains. The highest values for D* are found to occur at the edge of the hole, as shown by the strain contours in Fig. 10c. For the lower semi-circle of the hole though, the highest D* values are observed to occur not exactly at the edge of the hole. The grain depiction of Fig. 10e shows the locations predicted to fail by using the D3 and D* metrics. The grains predicted to initiate a crack by using the D3 metric are coloured in grey and both correspond to locations at the edge of the hole. By using the newly introduced FIP, D*, the FCI locations (highlighted in red) are not located both at the edge of the hole. Only for the upper semi-circle of the hole, the two corresponding locations are at the same location. For the lower semi-circle of the hole, the two corresponding locations are far from each other which means that the two metrics, D3 and D*, can lead to a very different prediction of the FCI process.
a
b
D3
Max: 0.037
Max: 0.033
c D*
Max: 0.02
e
Max: 0.018
d
Fig. 10. a) The geometry used for Abaqus simulations. The Voronoi grains had random orientations, while grain boundaries were modelled explicitly, b) Meshed part. The deformed microstructure after 1 cycle; the contours
show the variations of the: c) D3 metric and d) the FIP introduced. e) Predicted location for damage initiation by using the D3 metric (grey grains), and by using the FIP criterion (light red coloured grains).
4.3.
Plastic deformation analysis
A numerical procedure was developed that calculates and quantifies the state dependent variables within the crystal plasticity code developed in this study. It was therefore possible to identify both the slip systems which accommodate high shear strain values (plastic slip), and the combination of slip systems that can contribute to the formation of strong slip bands. This procedure enables to identify which slip systems are responsible for the formation of strong slip bands and potential cracks for the specific alloy, i.e. AA2024 aluminium alloy. Initially the accumulated slip at each node for the RVE of Fig. 10 (a) and (b) was stored for each of the slip systems. So the total accumulated slip over the number of cycles was obtained at each node for each slip system. The shear strain values (plastic slip) at every node within each grain were added for each of the slip systems separately. So the total contribution of each slip system was obtained within each grain. Therefore it is possible to understand which are the major slip systems and which slip systems contribute to the plastic deformation within each grain. The slip systems were sorted in descending order, where the first slip system carries the highest shear strain values and the last one has minimum contribution to the plastic deformation of the specific grain. In literature it has been reported that the first two major slip systems are responsible for the formation of strong slip bands [4-7]. In this analysis the first four slip systems with major shear strain accumulation are stored for each grain to obtain the contributions from more slip systems. The number of times these combinations of slip systems appear in the polycrystal and in each crystal are counted and listed. Table 4 shows the combinations of critical slip systems that appear for the 2D RVE of Fig. 10 (a) and (b); but only for the grains that were plastically deformed; a strain cut-off value was used in order to eliminate the grains aligned along the hole, which were under low stress-strain conditions. The 1st and 2nd slip systems in Table 4 were found to carry strain values one order of magnitude larger than those for 3rd of 4th slip systems in Table 4. 92.5% of the total deformation is due to 3 combinations of slip systems; namely 1-4-9-10, 2-6-7-11 and 3-5-812. Most combinations of slip systems are inactive or have a very small effect (1st, 3rd - 6th, 8th - 9th and 11th 12th row in Table 4). The slip systems accumulating the highest shear strains values were the 1st, 2nd and 3rd, with the most critical one being the 3rd slip system. The most critical combination of slip systems was the 3-58-12, which appeared in 253 grains.
Table 4. Critical slip systems and the number of grains these appear for the RVE of Fig. 10a.
In how many grains appear
Combinations of critical slip systems st
nd
rd
these combinations? th
1 crit. slip
2 crit. slip
3 crit. slip
4 crit. slip
Nr of grains
system
system
system
system
1
2
10
11
27
1
4
9
10
132
1
5
8
10
15
2
3
5
11
2
2
4
6
7
1
2
5
8
11
10
2
6
7
11
146
3
4
9
12
17
3
5
6
7
1
3
5
8
12
263
3
6
7
12
11
4
6
7
9
26
5. Discussion High amplitude (low-cycle) four point bending cyclic tests were performed for an Al2024 T3 alloy (with specimen geometry as shown in Fig. 1) within a SEM chamber and the following subject areas were analysed for this work: the strain distributions and strain localisations during cyclic loading, the experimentally (and simulation verified) appropriate FCI criteria in relation to the literature available FIPs, the small crack growth regime and the occurring crack deflections, the crack growth rates at particle sites and along slip bands, and the characterisation of individual grains as hard (or damage resistant) and soft (or damage prone) crystals. The sample was initially grinded and polished. EBSD measurements were taken in the region around the hole to investigate all the above mechanisms.
5.1.
Local Strain distributions and FCI
Figs. 2 and 5 showed that the strains were highly non-homogeneous around the hole. During cyclic loading, strain localisation occurred close to the drilled-hole region. Highly strained regions were found to be next to
low strained regions, even within the same grains. A triangle of high strain localisation at the edge of the hole was drawn in Fig. 2 to indicate the characteristics of strain localisation around the hole. Yet, crack nucleation was found not to occur always at the edge of the hole. Figs. 2 - 5 show that slip band formation was the main mechanism that lead to FCI (for high amplitude loading). A damage prone grain, close to the edge of the hole in Fig. 2, forms strong slip bands, and a crack forms and propagates along the strongly deformed slip bands. Similarly, a damage prone grain, exactly at the edge of the hole in Fig. 3, leads to the formation of strong slip bands at the edge of the hole, and FCI, after just 10 cycles. The mechanism which leads to crack formation is highlighted in Fig. 2b, where small, sub-micron voids form prior to crack initiation. High magnification imaging revealed that the size of these voids was approximately 0.3 to 0.5 μm. It is the accumulation of high numbers of dislocations along the slip bands that leads to sub-micron void formation. Once this void density becomes critical, a crack will form, joining together the aligned voids. The appearance of the crack is relatively sudden, with the crack length already reaching 10 μm after 10 cycles. This is due to the presence of these voids, which suddenly join up forming a relatively large crack (Figs. 3 and 5). The FCI is a process involving a large amount of ductility for high amplitude loading. At the location where initiation occurred, elongated voids can be observed together with large out of plane deformation along the extensively deformed slip band. Yet crack propagation is a process with less involved ductility in comparison to crack initiation as. FCI involves large three dimensional displacements, while crack propagation was relatively shallow and this is related to the associated strain fields ahead of the crack tip.
5.2.
Fatigue Indicating Parameters
Several FIPs (Fatigue Indicating Parameters) have been suggested in the literature from a modelling point of view [2, 7, 9, 11, 24]. Most frequently, the maximum accumulated slip over all the slip systems, the maximum accumulated slip over each slip system, and a local Fatemi-Socie parameter have been used to describe both the fatigue initiation and small crack growth regimes. Yet, there is no published work suggesting which parameter can correctly describe the exact location where fatigue crack initiation will occur in a polycrystal. One important observation made in this study is that the DIC measured strain is not always an adequate parameter or indicator for the exact location where crack initiation will occur. Fig. 3 showed that for the 1st case study, the maximum DIC measured strain was at the edge of the hole. Yet, the specific location-crystal showed uniform plastic deformation without any strong slip bands formed. Therefore, the location where the
highest strains occurred did not lead to slip band formation and therefore to crack formation. Instead a ‘weak’ grain within (the triangle of) the highly strained region, forms strong slip bands in Fig. 3 and leads to Fatigue Crack Initiation. In contrast, the presence of a ‘weak’ grain at the edge of the hole in the 2 nd case study localises the highest strains at the edge of the hole. In this case, the location of the highest DIC-measured strain value coincides with the location where crack nucleation occurs. Therefore, the maximum accumulated strain can be an indicator for crack nucleation, but it is not always adequate. It is suggested in this work that a new alternative FIP (Fatigue Indicating Parameter) needs to be introduced, which incorporates both the effect of the total accumulated strain over all slip systems (the total plastic strain, D3 metric) and the effect of the maximum accumulated slip-strain over each slip system (plasticity along critical slip systems, D1 metric) in order to take into account grain orientation. This is because it is the presence of critical grains, at critical regions (regions of highly localised strains) which will facilitate fatigue crack initiation. Similarly the total accumulated slip over each slip system is not an adequate FIP parameter either. A ‘weak’ grain in Fig. 10 lies far away from the highly deformed triangle region, and while strong slip bands form just after the 1 st cycle, the local stress-strain conditions do not facilitate crack formation. Therefore it is the combination of the two FIPs that can correctly and geometrically predict fatigue crack initiation. Fig. 16 shows that the newly introduced FIP, D* metric, which is the multiplication of D1 and D3 metrics predicts damage nucleation both at the edge of the hole and at the region near the edge of the hole; in accordance with the experimental observations.
5.3.
Comparison for the crack growth rates at slip bands and particles
A comparison was also made between the crack growth rates at particle sites and along slip bands. Fig. 5 shows that crack nucleation at particle sites occurs earlier with respect to slip band cracking, the crack growth rates along slip bands are much higher. The crack growth rate at the particle site in Fig. 5 is approximately 0.032μm/cycle, while the slip band crack has an average crack growth rate of 0.3μm/cycle over the course of 500 cycles. Fig. 5e shows the two cracks together; and while the slip band crack has grown to a length of approximately 150μm, the particle crack has just grown to 16μm; approximately one tenth of the crack length due to slip banding. This is due to the associated plasticity involved in the two mechanisms for crack propagation. Slip band mechanisms are associated with high accumulation of plastic deformation along specific orientations, with the generation and movement of a high number of dislocations through a mode II mechanism. It is this high plastic deformation that drives accelerated crack propagation. In contrast, in the
neighbourhood of a particle, stress concentration occurs just during the first peak load. As the crack propagates in a direction perpendicular to the loading direction, through a mode I mechanism, the associated plasticity is relatively small. Miller stated that several cracks will form during the fatigue lifetime of a component, but only one critical will lead to the final failure of the component [8, 25]. Observations made here were in agreement with the above statement, as several cracks were observed at the areas of interest, but only the cracks that nucleated within highly strained regions (within the triangle highly strained regions of Figs. 2 and 5) and along critical slip bands within critical grains (grains that formed strong slip bands) had the necessary driving force to lead to the final failure of the component. The crack growth rates for the other cracks were considerably smaller even though they might have nucleated much earlier during cyclic loading.
5.4.
Hard and soft grains
The micromechanical behavior of individual grains was assessed. It was shown that specific grains have higher potency for slip band formation and therefore crack initiation. A correlation was established between the observed micromechanical behaviour and the local crystal orientation. It was found that the orientation of the individual grains greatly influences the occurring deformation mechanisms. Fig. 7 highlights the grains that showed resistance to slip banding and crack initiation. These grains were characterized as ‘hard’ grains, following the nomenclature of reference [11]. Fig. 7a shows the orientation-data points for the grains of Fig. 1 plotted over the IPF map. Most grains lie along the 001-101 plane, which suggests that the more the rolling of the material or the more the plastic deformation that is given to the material, the ‘harder’ the grains and the resulting microstructure. Therefore, further rolling of the material yields more data points closer to the 001101 plane, and therefore superior fatigue performance. The red-arrowed grain in Fig. 7, can be close to the 001-111 orientation plane (shown with red arrow in Fig. 7a), and yet have ‘hard’ grain characteristics. This effect was related to the local texture effect, and the orientation of the neighbouring grains. High misorientations (>500) were found for the specific grain with all its neighbouring grains, not facilitating the flow of plastic deformation from the neighbouring grains towards the grain matrix of the highlighted grain. There is not always a direct correlation between the location of the grain on the IPF map and the tendency for fatigue crack initiation in the corresponding grain. The IPF map can give an indication and a potency for fatigue cracking, yet the orientation of the neighbouring grains needs also to be taken into account for accurate
prediction of the occurrence of damage. Fig. 7 shows the grains which formed strong slip bands, named as ‘weak’ grains [11]. The location of the corresponding grains on the IPF map is shown in Fig. 7b. The grains lie along the 001-111 orientation plane. The weakest grain is red circled in Fig. 7 and 7b. For superior fatigue performance, grains with such orientation should be avoided upon manufacturing design. A grain with orientation far from the 001-111 plane was found and highlighted in Fig. 7, which showed strong slip banding characteristics, only in the circled area of Fig. 7. The rest of the matrix in this grain showed uniform plastic deformation characteristics. Apart from correlating the microstructure and the orientation of the individual grains with the mechanical performance of the alloy, the importance of this work is that a new FCI criterion has been proposed based on in-situ observations, that more accurately predicts the location where FCI occurs. This was incorporated into a Crystal Plasticity FE model, which can now be used to more accurately predict the total fatigue lifetime of engineering components [4, 7, 18, 22, 24].
6.
Conclusions
In situ four point bending fatigue testing was performed on a rectangle Al2024 specimen (with a 1mm machined hole at the top surface) within a SEM chamber in order to investigate the Fatigue Crack Initiation (FCI) and small crack growth mechanisms of the alloy. Digital Image Correlation (DIC) measurements were taken during cyclic loading and it was found that two triangle-shaped regions adjacent to the hole concentrated the highest strains. In all cases the critical crack initiated within these (critical-) triangle regions, even though damage was also observed to occur outside these regions as well. FCI for high amplitude loading occurred due to the formation of strong slip bands. It is the formation of small voids along a strong slip band that will later coalesce and form a crack during cyclic loading that leads to FCI. The cracks did not always initiate at the edge of the hole where the highest strains were measured but rather at a critical grain within the highly strained triangle regions. So it was stated that it was the presence of critical grains within critically strained regions that drives FCI. A new FCI criterion was thus proposed which is based on two parameters: the total accumulated slip which accounts for the applied strain and the accumulated slip along the critical slip system which accounts for the presence of critical grains which have the potency to form strong slip bands. A square 2D RVE with 80 grains was created and a crystal plasticity model was employed which was later
calibrated by performing tensile testing and fully reversed cyclic fatigue testing. A much larger 2D RVE geometry with 900 grains of a rectangle with a 1 mm hole in the middle was developed, which consisted of 900 grains, in an effort to simulate the experimental geometry. It was found that by using the total accumulated slip as a FCI criterion, damage is always predicted to occur at the edge of the hole due to the high strain concentration. Yet by using the modified FCI criterion, depending on the presence of a critical grain at the edge of the hole or not damage may initiate at a distance from the hole. So by using the newly introduced criterion the effect of the microstructure at the neighbourhood of the hole can be accounted for and a more accurate prediction for the location of FCI can be obtained. Furthermore the effect of grain orientation on strain localisation and damage nucleation was evaluated by coupled EBSD and high resolution SEM imaging. Grains with strong slip bands and cracks embedded were plotted over the Inverse Pole Figures. The overall behaviour of the grains with respect to the crystal orientation yielded that direct evaluation from the IPF is not always possible, as the effect of the neighbouring grains as well as the grains underneath the surface might need to be taken into account. The results in this paper provide useful insights about designing crystalline materials with superior fatigue performance and addresses the necessity of coupling in situ testing with microstructural modelling in order to obtain accurate models for the FCI regime.
References 1. 2.
3.
4. 5.
6.
Suresh, S. Fatigue of Materials. 1998; Available from: http://public.eblib.com/EBLPublic/PublicView.do?ptiID=1103761. Miller, K.J., Metal Fatigue—Past, Current and Future. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 1991. 205(5): p. 291-304. Shenoy, M., J. Zhang, and D.L. McDowell, Estimating fatigue sensitivity to polycrystalline Nibase superalloy microstructures using a computational approach. Fatigue & Fracture of Engineering Materials & Structures, 2007. 30(10): p. 889-904. Efthymiadis, P., Multiscale Expreimentation and Modeling of fatigue crack development in aluminium alloy Al2024. University of Sheffield, 2014. Bozek, J.E., et al., A geometric approach to modeling microstructurally small fatigue crack formation: I. Probabilistic simulation of constituent particle cracking in AA 7075-T651. Modelling and Simulation in Materials Science and Engineering, 2008. 16(6): p. 065007. Hochhalter, J.D., et al., A geometric approach to modeling microstructurally small fatigue crack formation: II. Physically based modeling of microstructure-dependent slip localization and actuation of the crack nucleation mechanism in AA 7075-T651. Modelling and Simulation in Materials Science and Engineering, 2010. 18(4): p. 045004.
7.
8.
9.
10.
11.
12. 13. 14. 15.
16. 17. 18. 19. 20.
21. 22. 23. 24. 25. 26.
Hochhalter, J.D., et al., A geometric approach to modeling microstructurally small fatigue crack formation: III. Development of a semi-empirical model for nucleation. Modelling and Simulation in Materials Science and Engineering, 2011. 19(3): p. 035008. Manonukul, A. and F.P.E. Dunne, High- and low-cycle fatigue crack initiation using polycrystal plasticity. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 2004. 460(2047): p. 1881-1903. Dunne, F.P.E., D. Rugg, and A. Walker, Lengthscale-dependent, elastically anisotropic, physically-based hcp crystal plasticity: Application to cold-dwell fatigue in Ti alloys. International Journal of Plasticity, 2007. 23(6): p. 1061-1083. I. Simonovski, K.-F.N., L. Cizelj Material properties calibration for 316L steel using polycrystalline model. In: . ICONE 13: proceedings of the 13th international conference on nuclear engineering. Beijing, 2005. Przybyla, C.P. and D.L. McDowell, Microstructure-sensitive extreme value probabilities for high cycle fatigue of Ni-base superalloy IN100. International Journal of Plasticity, 2010. 26(3): p. 372-394. A.J. McEvily, The growth of short fatigue cracks: a review, Materials Science Research International, Vol. 4, no. 1 (1998) pp. 3-11 Cheong, K.-S., M.J. Smillie, and D.M. Knowles, Predicting fatigue crack initiation through image-based micromechanical modeling. Acta Materialia, 2007. 55(5): p. 1757-1768. McDowell, D.L. and F.P.E. Dunne, Microstructure-sensitive computational modeling of fatigue crack formation. International Journal of Fatigue, 2010. 32(9): p. 1521-1542. Brinckmann, S. and E. Van der Giessen, A fatigue crack initiation model incorporating discrete dislocation plasticity and surface roughness. International Journal of Fracture, 2007. 148(2): p. 155-167. McDowell D.L. Top-down and bottom-up microstructure-sensitive modeling of inelasticity. International Symbosium on Plasticty, Hawaii, 2016. Ghadbeigi, H., et al., Local plastic strain evolution in a high strength dual-phase steel. Materials Science and Engineering: A, 2010. 527(18–19): p. 5026-5032. Hill, R. and J.R. Rice, Constitutive analysis of elastic-plastic crystals at arbitrary strain. Journal of the Mechanics and Physics of Solids, 1972. 20(6): p. 401-413. Rice, J.R., Inelastic Constitutive Relations for Solids: An Internal-Variable Theory and its Application to Metal Plasticity. J. Mech. Phys. Solids , 433-455. , 1971. Huang, Y., "A user-material subroutine incorporating single crystal plasticity in the ABAQUS finite element program, Mech Report 178". Division of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts. Luo, C. and A. Chattopadhyay, Prediction of fatigue crack initial stage based on a multiscale damage criterion. International Journal of Fatigue, 2011. 33(3): p. 403-413. Luo, C., et al., Fatigue Damage Prediction in Metallic Materials Based on Multiscale Modeling. AIAA Journal, 2009. 47(11): p. 2567-2576. Asaro, R.J., Crystal Plasticity. Journal of Applied Mechanics, 1983. 50(4b): p. 921-934. Asaro, R.J. and J.R. Rice, Strain localization in ductile single crystals. Journal of the Mechanics and Physics of Solids, 1977. 25(5): p. 309-338. Luo, C. Multiscale Modeling & Virtual Sensing for Structural Health Monitoring. 2011; Available from: http://hdl.handle.net/2286/7ozpxe85egh. Sweeney, C.A., et al., The role of elastic anisotropy, length scale and crystallographic slip in fatigue crack nucleation. Journal of the Mechanics and Physics of Solids, 2013. 61(5): p. 1224-1240.
