Jun 22, 2018 - mechanical fatigue life for metals and alloys just based on their Burgers ... to fatigue damage without timeâdependent deformation, ie, creep,.
Received: 15 January 2018
Revised: 22 June 2018
Accepted: 25 July 2018
DOI: 10.1111/ffe.12907
TECHNICAL NOTE
A fatigue crack nucleation model for anisotropic materials Xijia Wu Structures, Materials and Manufacturing Laboratory, Aerospace, National Research Council Canada, 1200 Montreal Rd, Ottawa, ON, Canada, K1A 0R6 Correspondence Xijia Wu, Structures, Materials and Manufacturing Laboratory, Aerospace, National Research Council Canada, 1200 Montreal Rd., Ottawa, ON, Canada K1A 0R6. Email: xijia.wu@nrc‐cnrc.gc.ca
Abstract Based on Tanaka‐Mura's dislocation‐dipole pile‐up configuration, a formulation of mixed mode fatigue crack nucleation is derived using the Stroh formalism for generally anisotropic materials. The fatigue life is explicitly expressed as inversely proportional to the stored plastic energy, depending on the material's anisotropic elastic matrix F−1, surface energy, Burgers vector, and lattice resistance. The model has been shown to agree with experimental observation on critical slip planes in PWA 1493 single crystal Ni‐base superalloy. KEYWORDS anisotropy, fatigue crack initiation, fatigue life prediction
1 | INTRODUCTION It has been well understood that fatigue cracks often nucleate from surface marks, ie, intrusions and extrusions of persistent slip bands.1-4 Tanaka and Mura originally proposed a model of fatigue crack nucleation in terms of dislocation dipole pile‐ups, where vacancy dipoles represent intrusions and interstitial dipoles represent extrusions.5 However, Tanaka and Mura evaluated the plastic strain by integration of the displacement function, which resulted in a dimensional error. Recently, the Tanaka‐Mura model has been revisited with a revised formula of dislocation pile‐up strain. The corrected model has been shown to be capable of predicting low‐cycle mechanical fatigue life for metals and alloys just based on their Burgers vector, surface energy, elastic modulus, and surface conditions without resorting to fatigue experiments.6 Here, the mechanical fatigue refers to fatigue damage without time‐dependent deformation, ie, creep, Nomenclature: Cijkl, the fourth‐order tensor of elastic stiffness; pα, complex eigenvalues (α = 1, 2, 3); A, eigenvector; L, complex solution matrix; h (z), dislocation field potential, z = x1 + pα x2 is the complex coordinate; F, anisotropic material matrix; B, dislocation density vector; b, Burgers vector; d, grain size; ws,, surface energy; μ, shear modulus; v, Poisson's ratio; τ, shear stress on slip system; τF, friction resistance of slip system; γ, shear strain; ε, normal strain; U, plastic strain energy; Rs, surface roughness factor
Fatigue Fract Eng Mater Struct. 2018;1–7.
and in absence of environmental effects. The above theoretical treatment is given, assuming isotropic material properties. As one understands, fatigue crack nucleation from persistent slip bands always occur on the most favorable slip systems at the microstructure level, which obviously sees the effect of crystalline anisotropy. Therefore, it is also important to give a theoretically precise treatment for fatigue crack nucleation in anisotropic materials, in order to understand the microstructural effects in fatigue design of engineering materials. In this paper, we will first develop a theoretical formulation for dislocation‐dipole pile‐ups in generally anisotropic materials, using the Stroh formalism, and then derive the fatigue crack nucleation life based on the Griffith energy criterion. In the end, the correlation for a single crystal Ni‐base superalloy is given as an example.
2 | CONTINUOUSLY DISTRIBUTED DI S LO C A T I O N PI L E‐U P A N D CRACK N UCLEATION Ideally, a crystalline solid only deform elastically, except the moving of dislocations within the body. For a generally anisotropic material, the elastic behavior is described
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1
2
TECHNICAL NOTE
by the generalized Hooke's law, as: σ ij ¼ Cijkl εkl
det∣Ci1k1 þ pC i1 k2 þ pC i2 k1 þ p2 Ci2k2 ∣ ¼ 0: (1)
where Cijkl is the fourth‐order tensor of elastic stiffness. When dislocations are present in a homogeneous anisotropic elastic material, each creates a local stress and displacement field, and they add up to generate an overall effect. The theoretical treatment of continuously distributed dislocations was first given by Eshelby‐Read‐ Shockley7 and later developed by Stroh into an elegant formulation, called the Stroh formalism,8 which has been widely used to deal with anisotropic elastic problems of various points and line defects, eg, 9-15. In the Stroh formalism, the elementary field potential of a unit line dislocation without the line tension force is given by h ðz Þ ¼
1 < lnz>LT b 2πi
(2)
where b is the Burgers vector, L is a matrix to be determined by satisfying the equilibrium condition. The function denotes a 3 × 3 diagonal matrix with ln (z1), ln (z2), and ln (z3) at each diagonal position; zα = x1 + pαx2 (Im (pα) > 0, α = 1, 2, 3) are complex variables with the origin of the coordinates set at the dislocation core, and pα is the complex variable eigenvalue with a positive imaginary pffiffiffiffiffiffi part, i = −1 is the imaginary number. The coordinate axes are chosen to represent edge glide dislocation with Burgers vector in the x1 directions, edge climb dislocation with Burgers vector in the x2 direction, and screw dislocation with Burgers vector in the x3 direction, respectively. By the Stroh formalism, the displacement vector u, and the stress vectors, s and t, are expressed in terms of the complex field potential vector h (z), as
(5)
For each eigenvalue, its unit eigenvector is obtained from Equation (4). Then, the matrix L is constructed with the material's stiffness constants, and eigenvalues/eigenvectors. Then, for the convenience of defining dislocation‐generated stress, we now define the matrix F as F ¼ −2iLLT :
(6)
When a dislocation with Burgers vector b lying on a gliding plane that extends in the x1‐direction (the plane normal is in the x2‐direction), the stress component on the gliding plane can be expressed as t i ðx 1 Þ ¼
F ij bj 2πx 1
(7)
Equation (7) reduces to the solution for isotropic materials with F 11 = F 22 = μ/(1 − ν) and F 33 = μ, where μ is the shear modulus and ν is the Poisson's ratio, and the rest of F ijare zero. Here again, as the original Tanaka‐Mura model, we consider two‐layer inverted dislocation pile‐ups forming along two adjacent slip planes, as shown in Figure 1, with a distribution function Bðx Þ ¼ fB1 ðx Þ; B2 ðx Þ; B3 ðx ÞgT
(8)
where B1(x), B2(x), and B3(x) are the distributions of edge‐glide, edge‐climb, and screw dislocations, corresponding to mode‐II, mode‐I, and mode‐III fracture, respectively.
(A)
u ¼ fu1 ; u2 ; u3 g ¼ AhðzÞ þ A hðzÞs ¼ fσ 11 ; σ 12 ; σ 13 gT ¼ LPh′ ðzÞ−LPh′ ðzÞt T
¼ fσ 21 ; σ 22 ; σ 23 gT ¼ Lh′ ðzÞ−Lh′ ðzÞ
(3)
(B)
where Aiα is a matrix of equilibrium eigenvectors; Liα= [Ci2k1 + pαCi2 k2]Akα; P is a diagonal matrix of the three complex eigenvalues pα(α= 1, 2, 3); A, L, and P are their conjugates, respectively. It can be shown that the displacement and stress functions given in Equation (3) satisfy the equilibrium condition, when
(C)
Ci1k1 þ pα C i1 k2 þ pα Ci2 k1 þ pα 2 Ci2k2 ÞAkα ¼ 0; α ¼ 1; 2; 3: (4)
The eigenvalue equation for pα(α=1, 2, 3) is obtained by solving the sextic equation:
FIGURE 1 Dislocations in (A) vacancy dipoles (forming an intrusion), (B) interstitial dipoles (forming an extrusion), and (C) tripoles (forming an intrusion‐extrusion pair) at the surface
TECHNICAL NOTE
3
Under the first loading of stress ti0 greater than the lattice frictional stress ti F , a dislocation pile‐up occurs in layer I, Bjð1Þ (j = 1, 2, 3), satisfying d
∫
F ij Bjð1Þ ðξ Þdξ 2π ðx 1 −ξ Þ
−d
þ t 0i −t Fi ¼ 0
(9)
where the superscript 1 denotes the first loading segment, ξ is the integration variable, and d is the distribution length ~ gain size. The solution of Equation (9) can be obtained following the Muskhelishvili inversion formula,16 as 2x 1 F −1 t 0j − Fj ij qffiffiffiffiffiffiffiffiffiffiffiffiffi when t 0j > Fj : Bið1Þ ðx 1 Þ ¼ d2 −x 21
(10)
Note that the dislocation distribution is asymmetrical about x = 0. This is typical of Bilby‐Cottrell‐Swinden distribution with equal number but opposite sign of dislocations distributed on the two sides, which would lead to formation of a centre crack or a surface edge crack (with half of the configuration for real). The total number of dislocations on layer I in the interval of [0, d] is given by
d
Ni ¼
d
1 ∫ B ð1Þ ðx Þdx ¼ 2F −1 t 0j −t Fj ij bi 0 i bi
0 F 2x 1 F −1 Δt −2t ij j j qffiffiffiffiffiffiffiffiffiffiffiffiffi Bið2Þ ðx 1 Þ ¼ when t 0j >t Fj d2 −x 21
(15)
where Δtj0 = 2tj0 is the fully reversed stress range. Such two‐layer inverted dislocation pile‐ups constitute either (1) vacancy dipoles, (2) interstitial dipoles, or (3) a combination of both, eg, tripoles, as schematically shown in Figure 1. It should be noted that such dislocation pile‐ups may appear as discontinuities under metallurgical examination, but the dislocation distribution planes are considered to be continuum planes before crack nucleation. Similar to the solution above, the plastic strain associated with the pile‐up in layer II is given by 1 0 F Δt −2t γ ið2Þ ¼ ∫ Bið2Þ ðx Þdx ¼ −2F −1 ij j j d0 d
(16)
and hence, the stored energy associated with the dislocation pile‐up in layer II is given by 1 U ð2Þ ¼ − Δt 0i −2t Fi γ ið2Þ 2
(11)
By definition, strain is displacement over the distance it is measured. In this case, the slip distance is evaluated as the number of dislocations times the Burgers vector, ie, Δl = Nibi, which occurs over the distance d. Thus, the plastic strain due to the dislocation pile‐up should be equal to Nibi/d. Rearranging Equation (11), we have 1 ¼ ∫ Bið1Þ ðx Þdx ¼ 2F −1 t 0j −tFj ij d0
The distribution function for the pileup in layer II is thus obtained as
(17)
On the k‐th reversal, the increment of dislocation ΔBi(k) (x1), the strain Δγi(k), and the energy ΔUi(k) are obtained in a similar manner: 0 F 2x 1 F −1 Δt −2t ij j j qffiffiffiffiffiffiffiffiffiffiffiffiffi when t 0j >t Fj ΔBiðkÞ ðx 1 Þ ¼ ð−1Þkþ1 d2 −x 21 (18)
d
γ ið1Þ
(12)
The stored energy associated with the dislocation pile‐ up is given by U ð1Þ ¼
1 2
F
t 0i −t i γ ið1Þ
Δγ iðkÞ ¼ ð−1Þkþ1 Δγ i
(19a)
0 F Δt −2t Δγ i ¼ 2F −1 ij j j
(19b)
with
(13) and
On loading reversal (the second load segment), instead of these pile‐up dislocations moving back on the original pile‐up plane, another pile‐up of opposite sign occurs in layer II, very close to layer I, Bjð2Þ (j = 1, 2, 3), satisfying
d
∫
−d
F ij Bjð2Þ ðξ Þdξ 2π ðx 1 −ξ Þ
d
þ ∫
−d
F ij Bjð1Þ ðξ Þdξ 2π ðx 1 −ξ Þ
−t 0i þ tFi ¼ 0
(14)
1 1 ΔU ðkÞ ¼ ð−1Þkþ1 Δt0i −2t Fi Δγ ðkÞ ¼ Δt0i −2t Fi Δγ i 2 2 ¼ ΔU (20) The index k takes 2 N at the minimum and 2 N + 1 at the maximum stress after N cycles. The entire dislocation pile‐up may burst into a crack once the stored energy in the strip of dislocation pile‐up (bd) becomes equal to the energy to form new crack
4
TECHNICAL NOTE
surfaces. The latter condition can be described by the energy balance as 2NΔUbd ¼ 4dws
(21)
where ws is the surface energy (J/m2). Then, the number of cycles to crack nucleation is given by Nc ¼
2ws ðΔt 0i −2t Fi ÞF −1 Δt 0j −2tFj b ij
(22)
or in terms of strain: Nc ¼
8ws Δγ i F ij Δγ j b
(23)
Equations (22) and (23) reduce to the stress/strain‐based forms for single mode‐II (shear) crack nucleation in an isotropic materials,2 when F11 = μ / (1 − ν). For engineering application, ws may be replaced with w′s = Rsws, where Rs is the surface roughness factor.6
3 | DISC USS I ON A general formulation has been derived for mixed mode fatigue crack nucleation in anisotropic materials. As implied by Equation (10), even application of a single stress component may induce dislocation distributions of all modes, depending on the coupling terms in the F matrix. As cyclic loading continues, the accumulation of these mixed mode pile‐ups will eventually burst into a crack when the stored strain energy is enough to create a pair of crack surfaces with the life as given by Equation (22) or Equation (23). In this sense, metal fatigue is almost unavoidable under repeated loading of any kind exceeding the lattice frictional resistance ti F . In materials with sufficient number of active slip systems such as face‐ centered cubic and body‐centered cubic materials under large plastic deformation, dislocation pile‐ups would exist in all possible slip systems such that the material appears to be almost “isotropic.” In the previous paper,6 the author showed that the isotropic formulation can satisfactorily predict the low‐cycle fatigue life of polycrystalline materials with appreciable plastic deformation. However, it can be expected that this is not the case when the load amplitude is relatively small. Then, only a few dislocation pile‐ups exist in the most favorable slip systems, depending on the microstructure, while the rest of the body remains elastic. This situation corresponds to high‐cycle fatigue. Therefore, the solution for fatigue crack nucleation in anisotropic materials is needed in dealing with fatigue problems at the microstructural level. In doing
that, all the constituents of microstructure need to be modeled using the finite element method. Before endeavoring into that effort, we shall try Equation (22) or Equation (23) on some simple real life cases as follows. Fine and Bhat studied fatigue crack nucleation in single crystal iron and copper.17 They remarked that “It is hard to see definitive cracks, but the slip bands are very intense and perhaps represent the stage just before cracks nucleate.” So, they treated fatigue crack nucleation process as random fluctuations in a metastable assembly of defect structures accumulated over prior cycles. They considered this was in analogy to phase transformation, where an energy barrier must be overcome during fatigue crack initiation due to the energy required to create new surfaces. This model employs an energy efficiency factor f . Because f is unknown, instead of predicting fatigue nucleation life, they estimated f based on the metallurgical examination of crack nucleation events. For example, as given in the Table 2 of their paper,17 they showed the results for two orientations of single crystal iron, orientation 1 and orientation 3, with Young's modulus of 207 GPa cycled at a shear plastic strain amplitude of 0.125%. They observed crack‐like intrusion marks in orientation 1 at 20 000 cycles and in orientation 3 at 1000 cycles. Using the surface energy value of 2.3729 J/ m2 and Burgers vector of 2.48 × 10−10m for pure iron,6 and the Young's modulus of the orientation (for simple estimation of the single mode II crack nucleation, it is assumed that the F matrix takes the “isotropic” form but with the directional modulus, as reasoned from Ting13), Equation (23) then predicts the fatigue crack nucleation life (applying the same surface roughness factor as in Wu6) is calculated to be 35 963 cycles. This theoretical life is almost twice of the observed crack nucleation life for orientation 1. But, there is quite a large disparity between crack nucleation in different orientations. Polák et al also observed that persistent slip marks could form at very early stage of fatigue life, sometimes as early as 5% of the fatigue life, and they considered the rest fatigue process as short crack growth.18 Persistent slip marks are certainly discontinuities, but in physical metallurgy there is no uniform definition about the width and depth of the intrusion mark that can be called as a crack, especially with regards to its surface traction, which has to be zero in the fracture mechanics sense. On the other hand, as implied by the original Tanaka‐Mura model,5 dislocation accumulate cycle by cycle on both layer I and II, forming an increasingly dense dislocation dipole pile‐up, which resembles a growing crack‐like discontinuity. However, in continuum mechanics, the dislocation distribution plane remains to be a continuum plane, which transmits stresses in full magnitude, until the stored strain energy reaches a critical value to create
TECHNICAL NOTE
traction‐free surfaces, which is called crack. Under low cycle fatigue (LCF) conditions, where the displacement (strain) is held constant, creation of such stress‐free surfaces will cause sudden load drop, as often observed at the very last stage of LCF close to fracture. Therefore, for description of LCF with uniform plastic deformation throughout the body, progression of vacancy dipoles indeed represents the accumulation of fatigue damage, which is consistent with the above metallurgical observations. In fracture mechanics, crack nucleation is defined as the event of creation of stress‐free surfaces within a solid body. This is a clear‐cut definition, but perhaps it may also limit its description for atomistic events where physical crack nucleation may indeed be random fluctuations in a metastable assembly of defect structures, which will need atomistic simulations to address. Nevertheless, it is still meaningful to use the fracture mechanics definition to deal with crack initiation at the component level, where the crack does not transmit loads, macroscopically. With that in mind, we shall proceed to analyse LCF data on Ni‐base superalloys. On the continuum scale, the effect of γ′ strengthening is present as the back stress to resist dislocation motion, which corresponds to t F . Still, characterization of fatigue with regards to specific slip systems is very difficult, because microstructural stress‐strain measurements are difficult to obtain on slip systems. Arakere and Orozco analyzed low‐cycle fatigue of single crystal Ni‐base turbine blade superalloy PWA 1493 tested at 650°C.19 They applied various multi‐axial fatigue failure theories on the active slip systems of this material, including (1) the critical plane theory where shear and normal strain on a critical plane are combined in the failure equation,20 (2) the modified critical plane theory where the mean stress is added to the critical plane failure equation,21 (3) the alternate shear model,22 and (4) Smith‐Watson‐Topper model,23 because the critical slip systems in an face‐centered cubic material do operate under multi‐axial stresses even the material is under uniaxial loading. However, none of the above theories were satisfactory in correlating the experimental data; neither did the uniaxial Coffin‐Manson strain equation. Instead, the maximum shear stress amplitude on the critical slip system appeared to provide a better correlation. To provide the rationale for the above correlation, here we use Equation (22) to re‐analyze the experimental data from Arakere and Orozco for the , , and < 011> orientations. At this temperature under LCF conditions, single crystal Ni‐base superalloys exhibit predominantly plastic behavior without significant creep and oxidation. Therefore, the case suits the premise of the present model. The dislocation pile‐ups are assumed to reside on the {111} octahedral slip system in the and < 011> orientations, and on the {100}
5
cube slip system in the orientations, to be consistent with Arakere and Orozco's analysis. The F−1 matrices for the octahedral and cube slip systems are obtained, respectively, as14,15: 2 6 F−1 f111g ¼ 4
8:89
0
−2:67
0
8:03
0
−2:67
0
13:97
3 7 −12 −1 (24) 510 ðPaÞ
and 2
5:45
6 F−1 f100g ¼ 4 0
0 5:45
0
0
0
3
7 0 510−12 ðPaÞ−1 8:44
(25)
Suppose that the dislocation pile‐up dipoles are formed only by edge‐glide dislocations on these slip systems. Using the surface energy value of Ni, ws = 2.38 J/m2, and assume that the machining surface roughness introduces a factor of Rs = 1/3,6 we rearrange Equation (22) as 1 Δτ ¼ τ þ 2 F
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ws Rs F −1 11 Nb
(26)
Note here the stress amplitude Δτ is half of the fully reversed stress range in Equation (22). Then, the slip system shear stresses are calculated using Equation (26), based on the observed life, as given in Table 1, assuming the lattice resistance τ F = 345 MPa and for the orientation, andτ F = 550 MPa for and orientations. The difference in the lattice resistances is perhaps due to orientation dependence of γ‐γ′ interfacial properties that affect the dislocation pile‐up as opposed to in a single‐phase material. Because of lacking the cyclic stress‐strain information for this material, the lattice resistant τ F values are the fitting values to match the asymptotic fatigue endurance limits for each orientation. The model calculated results are shown in Figure 2 in comparison with Arakere and Orozco's data. It is seen that the model description agrees well with the experimental observations. Especially, there is a strong dependence of fatigue life on the material's orientation via the anisotropic elasticity matrix F−1. It seems that simple projections of stress and strain on slip systems as in the other multi‐axial criteria do not truly reflect this anisotropic effect. Of course, the physical properties such as the Burgers vector, the surface energy, and lattice resistance of a single crystal material may also be orientation dependent, which has not considered fully in this study. It is noted that with the advance of ab initio density functional theory, elastic, thermal, and surface properties can be calculated from the first principles of physics.24,25
6
TECHNICAL NOTE
TABLE 1
LCF of PWA 149319
Orientation
εmax
εmin
Δγ
Δτ, MPa
Δτ (Predicted)
Life
oct
0.01509
0.00014
0.0198
745.2
713.3531
1326
oct
0.0174
0.0027
0.0194
731.4
681.0688
1593
oct
0.0112
0.0002
0.014728
550.62
546.8924
4414
oct
0.01202
0.00008
0.016
602.37
523.086
5673
oct
0.00891
0.00018
0.012
446.43
423.0742
29516
cube
0.01219
−0.006
0.02106
2311.5
3909.712
26
cube
0.0096
0.0015
0.00924
1028.1
1140.031
843
cube
0.00809
0.00008
0.009406
1021.2
1087.455
1016
cube
0.006
0
0.0076
759
843.3672
3410
cube
0.00291
0.00284
0.0067
738.3
753.2961
7101
cube
0.00591
0.00015
0.006724
731.4
749.7413
7356
cube
0.01205
0.00625
0.007
793.5
742.6927
7904
oct
0.0092
0.0004
0.01435
814.2
809.4884
2672
oct
0.00896
0.00013
0.015
848.7
704.5543
7532
oct
0.00695
0.00019
0.01069
624.45
627.1594
30220
FIGURE 2 LCF life of PWA 1493 as function of the critical shear stress range at 650°C [Colour figure can be viewed at wileyonlinelibrary.com]
Atomistic calculation of Peierls‐Nabarro stress for single phase materials has also been attempted.26 These computational methods have opened the door for computational material design, and with the present fatigue crack nucleation model and furthermore finite element method, computational evaluation may be extended to complex engineering alloys, through a multi‐scale approach. This will greatly reduce the cost of fatigue testing, as experiments will only serve the purpose for validation.
materials as described by Equation (22) or Equation (23), which depends on the material's anisotropic elasticity matrix F−1, surface energy, Burgers vector, and lattice resistance. The model has been shown to agree with experimental observation on critical slip planes in PWA 1493 single crystal Ni‐base superalloy.
ORCID Xijia Wu
4 | CONCLUSION Based on Tanaka‐Mura's dislocation‐dipole pile‐up configuration, a formulation of mixed mode fatigue crack nucleation has been derived for generally anisotropic
http://orcid.org/0000-0002-0250-112X
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How to cite this article: Wu X. A fatigue crack nucleation model for anisotropic materials. Fatigue Fract Eng Mater Struct. 2018;1–7. https://doi.org/ 10.1111/ffe.12907
