Fatigue crack growth in a material with coarse brittle phases

International Journal of Fatigue 131 (2020) 105332

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Fatigue crack growth in a material with coarse brittle phases

T

Hermann Riedel Fraunhofer Institute for Mechanics of Materials IWM, Woehlerstraße 11, 79108 Freiburg, Germany

A R T I C LE I N FO

A B S T R A C T

Keywords: Fatigue crack growth Wavy crack front Crack pinning Aluminum alloys Silicon precipitates

The paper explores the effect of brittle particles on fatigue crack growth. A model based on the theory of slightly curved crack fronts predicts that fatigue crack growth is enhanced if the applied stress intensity factor is greater than the fracture toughness of the particles. In the other case, the crack front advances in the matrix before it can traverse the particles (crack pinning). Below a certain stress intensity factor, the crack is completely arrested by the particles causing a fatigue threshold. The predictions are compared with experiments on Al alloys. Limitations of the linear perturbation theory are discussed.

1. Introduction

alloys. In the Al alloy with 12% Si shown in Fig. 1 the size and spacing of the elemental Si particles are typically 40 µm and 120 µm, respectively. However, one might suspect that the brittle particles deteriorate the fatigue properties by enhancing the growth of fatigue cracks. On the other hand, the particles might be beneficial by pinning the growing fatigue crack front. These questions will be addressed in the present paper. The final goal of the analysis is to model fatigue crack growth in a material with circular particles. However, the numerical method developed for that purpose is not straightforward and needs confirmation by an independent, possibly analytical method. Therefore the first modeling step is to develop a closed-form solution for a stripe model (Section 4), in which the brittle phase forms periodically spaced stripes. In Section 6 the numerical method is described. After the validation of the numerical approach for the stripe model, it is used to simulate crack growth in the circular-particle model (Section 7). Another reason to study the stripe model is that there are materials with a lamellar microstructure of brittle and ductile phases, e.g. TiAl [11]. Further Patinet et al. [12] test and analyze a stripe geometry as a model system of an interface in a multilayer structure.

The present paper examines the influence of brittle particles on fatigue crack growth in a ductile base material. If the fracture toughness, KIc, of the particles is small, one expects that the growth rate is enhanced, while particles with high fracture toughness will retard the fatigue crack growth rate (‘crack pinning’). In both cases the crack front will not remain straight, but it will advance ahead in a brittle phase with low KIc, or lag behind in a phase with high KIc. To describe the evolution of a curved crack front, a theory is needed which relates the local stress intensity factor to the shape of the crack front. In the present work, the classical model of Rice [1] and Gao and Rice [2] is used, which is linear in the perturbation of the crack front, and therefore limited to only slightly curved crack fronts. Extensions to a quadratic approximation were worked out by Leblond et al. [3], Vasoya et al. [4], Willis [5] and others, but the quadratic theory shares certain qualitative shortcomings with the linear theory (see the Discussion section). A logical extension of the quadratic theory is the finite perturbation method proposed by Rice [6] and worked out by Bower and Ortiz [7,8], which yields accurate results for arbitrarily large nonlinearity. A different approach, which also covers the whole range of geometrical nonlinearities, is the boundary element method (Fares [9]). Lazarus [10] reviews the literature on the subject. The main motivation for the above-mentioned investigations was the toughening of brittle ceramics by ceramic particles with higher fracture toughness, e.g. ZrO2 particles in Al2O3. Unlike the work on brittle particles in a brittle matrix, the present work is motivated by the question how brittle particles influence the fatigue crack growth rate in ductile alloys. Aluminum alloys with high silicon content, can serve as an example. With increasing Si content, Al alloys contain more and more Si particles, which improve the wear resistance, e.g. in piston

2. Description of the model Fig. 2 shows two types of geometrical models considered in this paper. The brittle phase can either form periodically spaced stripes with width d and spacing λ (Fig. 2a), or it may form circular particles with diameter d and spacing λ (Fig. 2b). A crack lying in the y = 0 plane grows in the positive x direction. In the example shown here the crack advances faster in the brittle phase and stays behind in the ductile phase, i.e. the fracture toughness is assumed to be small.

E-mail address: [email protected]. https://doi.org/10.1016/j.ijfatigue.2019.105332 Received 8 August 2019; Received in revised form 8 October 2019; Accepted 10 October 2019 Available online 12 October 2019 0142-1123/ © 2019 Elsevier Ltd. All rights reserved.

International Journal of Fatigue 131 (2020) 105332

H. Riedel

4. Steady-state solutions for the stripe model In the brittle stripes the stress intensity factor must be constant and equal to the fracture toughness of the brittle phase, KIc. If we consider a steady-state situation in which the crack preserves its shape while it advances, the stress intensity factor should be constant also in the ductile phase with the still unknown value K duc . Hence

pore

Si particle

K (z ) =

⎧ KIc

for 0 < z <

⎨ K duc ⎩

for

d 2 d 2

and λ −
d 2 d 2


For n ≥ 1 the Fourier coefficients for this piecewise constant function are

kn =

Fig. 1. Microstructure of AlSi12CuNiMg-T7 with Si particles (dark grey), fine intermetallic phases in the Al grains and pores (black). Examples are encircled. Courtesy C. Fischer, Fraunhofer-IWM, Freiburg.

K duc − KIc 2 πdn ⎞ sin ⎛ K 0 (a0 ) πn ⎝ λ ⎠

(8)

The stress intensity factor in the ductile phase follows from the requirement that the average of K(z) should be equal to K 0 (a) :

K 0 (a) − KIc d λ 1−d λ

3. Governing equations

K duc =

Rice [1] derived the following relation between the shape of a crack front, a(z), and the resulting stress intensity factor, K(z),

The crack shape corresponding to the piecewise constant stress intensity factor is obtained from Eq. (2) together with the Fourier coefficients an from Eq. (6) using Eq. (8). The summation is carried out numerically. Fig. 3 shows results for two different volume fractions of the brittle phase (32% and 8%). The coordinate axes represent the piecewise constant stress intensity factor (right axis) and the normalized crack length A (left axis) with

K (z ) = K 0 (a (z )) +

K 0 (a (z )) 2π

∞

'

∫ a (z(z)'−−z )a2(z ) dz'

−∞

(1)

The integral is understood as its principal value, z is the coordinate along the straight reference crack front, z′ is the corresponding integration variable, and K 0 (a (z )) is the stress intensity factor for the straight reference crack having length a. Compared to Rice’s result, a possible explicit dependence of K 0 on z′ is ignored here. The formula is accurate to first order in a(z)-a(z′), i.e. for only moderately curved crack fronts. For a periodic structure with wavelength λ in z direction one expands a(z) in a Fourier series

A = (a − a0) [λ (1 − KIc K 0)]

∑ an cos(2πnz

λ) (2)

n=1

The principal-value integral in Eq. (1) with Eq. (2) inserted can be evaluated using tabulated solutions with the result ∞

πnz λ ) ' 2π dz = − ∫ cos(2πnz' λ(z)'−−z )cos(2 2 λ

2

−∞

2πnz ⎞ ncos ⎛ ⎝ λ ⎠

5. Crack growth rate in the stripe model

(3)

The reference stress intensity factor K0(a) can be expanded to linear accuracy in a-a0

K 0 (a) = K 0 (a0) +

dK 0 (a0 ) (a (z ) − a0) da0

Since the local stress intensity factor varies in proportion to the applied stress intensity factor for the reference crack, the amplitude of the stress intensity factor in the ductile phase is

(4)

ΔK duc =

Combining the preceding equations leads to the solution for the stress intensity factor ∞

K (z ) = K 0 (a0 ) + K 0 (a0 )

2πnz ⎞ ⎝ λ ⎠

∑ kn cos ⎛

(10)

It should be noted that the plot for the crack front shape is independent of KIc K 0, if the normalization in Eq. (10) is used. Hence an arbitrary value can be chosen for KIc K 0 to plot the crack front shape in Fig. 3. Since the denominator in Eq. (10) changes its sign when KIc K 0 = 1, the result correctly describes that the crack lags behind in the ductile phase when KIc K 0 < 1, while it lags behind in the brittle phase when KIc K 0 > 1 (‘crack pinning’). Fig. 4 shows the effect of the term dK 0 (a 0) da0 in Eq. (6), which was set equal to zero in Fig. 3. The term enhances the curvature of the crack front up to the point where the first Fourier coefficient of the crack length, a1, becomes infinite when (dK da) (πK λ ) = 1. As Rice [1] points out, this means that the crack front becomes unstable.

∞

a (z ) = a 0 +

(9)

0 1 − (KIc Kmax )(d λ ) ΔK 0 1−d λ

(11)

0 0 Kmax and the local R ratio is equal to the applied R = Kmin . If a Paristype crack growth law obtains, the steady-state crack growth rate is

da dN = A (ΔK duc )m

(12)

(5)

Here da dN is the crack growth increment per loading cycle, and A and m are parameters. This result will be further evaluated in the following sections.

(6)

6. Numerical transient solutions - validation for the stripe model

Eq. (6) represents the relation between the stress intensity factor and the shape of the crack front in terms of the Fourier coefficients kn and an. In the following, this relation will be inverted and used to calculate the crack front shape from a given distribution of the stress intensity factor.

The closed-form steady-state solution presented above was possible because the stress intensity factor is piecewise constant during steadystate crack growth. This simplification is no longer valid for describing the time dependent evolution of the crack. The time dependent solutions developed now are based on the crack growth laws

n=1

with the Fourier coefficients

π dK 0 (a 0) da 0 ⎞ kn = − ⎛ − nan nK 0 (a0 ) ⎠ ⎝λ ⎜

⎟

2

International Journal of Fatigue 131 (2020) 105332

H. Riedel

a)

b)

x, a

x, a

2

crack front

crack front d

d

0

z

0

z

Fig. 2. Stripe model (a) and particle model (b). The curves represent the growing crack front a(z) in the ductile matrix and the brittle phase (light blue). The particles may be circular cylindrical or spherical. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

(14)

a) d/ = 0.32

1.5

0.4 1.0

0.3 0.2

0.5 0.1 0.0

-3

-2

-1

0

1

2

3

1.5

0.8 0.2

1.0

0.6 0

0.4

0.5

0.2 0.0

-3

-2

-1

0

1

2

3

0.0

2 z/ Fig. 4. The crack front protrusion is enhanced by positive dK/da.

the stress intensity factor is discontinuous at the phase boundaries, and such step functions are difficult to approximate by Fourier series. Moreover, oscillations in the stress intensity factor are enhanced by the nonlinearity of the Paris law, so that also the crack length a(z) develops oscillations with Fourier coefficients beyond the original limit nmax. For reasons that are not completely understood by the author the following procedure leads to solutions which perfectly approach the closed-form steady-state solutions: The program works with a fixed number of Fourier coefficients nmax, thus ignoring the higher terms that develop during crack growth. Fig. 5 illustrates the result after a steady state has been reached. If the crack length is calculated strictly by integration of Eqs. (13) and (14), the solution exhibits strong oscillations near the phase boundary. However, if the Fourier spectrum of the crack length a (z) is truncated at n = nmax, the red curve in Fig. 5 is obtained, which is smooth and indistinguishable from the closed-form steady-state solution. While Fig. 5 was calculated with nmax = 150, nmax = 10–20 is

normalized crack length, A

0.5

K/K0

normalized crack length, A

Eq. (13) represents the Paris law for fatigue crack growth, now at each point of the crack front. Eq. (14) is not a physics based law, but it guarantees that the stress intensity factor remains close to the fracture toughness KIc ; Kmax is the maximum stress intensity factor within the load cycle. By choosing the parameter B large enough, the requirement that the stress intensity factor should be constant in the brittle phase can be fulfilled to a desired accuracy. The stability of the numerical scheme is improved if negative values of da dN are ignored. Eqs. (13) and (14) are first applied to the stripe model and, after validation of the approach, to the circular-particle model in Section 7. To reduce the number of parameters the numerical calculations are carried out in dimensionless form: stress intensity factors are normal0 ized by Kmax , lengths by the spacing λ, e.g. , a ̂ = a λ ; and the nor0  = NA (Kmax )m λ . The normalized crack malized number of cycles is N 0  = (da dN ) [A (Kmax )m ]. growth rate is then da ̂ dN The Fortran program for the time evolution starts from a given crack shape, a(z), for example a(z) = 0, calculates the Fourier coefficients an, uses Eq. (6) to calculate the Fourier coefficients kn, and inverts the Fourier transformation to obtain K(z). From Eqs. (13) and (14) one obtains the increment da (z ) = (da dN ) dN , which is used to update a(z) providing the starting value for the next time step. This explicit time integration scheme is stable for sufficiently small increments dN; in terms of the dimensionless number of cycles, increments of typically 1E−4 lead to stable runs with computing times of a few seconds on an 0 )m − 1 ] ≈ 100 ordinary PC. Values of the dimensionless factor B [A (Kmax guarantee that the stress intensity factor in the brittle phase is sufficiently close to KIc . The number of Fourier coefficients taken into account in the numerical scheme, nmax, plays a special role. One of the problems is that

d/ = 0.32

(dK/da) / K) = 0.5

1.0

K/K0

da dN = B (Kmax (z ) − KIc ) in the brittle phase

1.2

0.0

0.20 1.2

b) d/ = 0.08

1.0

0.15

0.8 0.10

0.6

K/K0

(13)

normalized crack length, A

da dN = A (ΔK (z ))m in the ductile phase

0.4

0.05

0.2 0.00

2 z/

-3

-2

-1

0

1

2

3

0.0

2 z/

Fig. 3. Crack front shape (red) for a given piecewise constant stress intensity factor (cyan), KIc legend, the reader is referred to the web version of this article.) 3

K0

= 0.2 . (For interpretation of the references to colour in this figure

International Journal of Fatigue 131 (2020) 105332

H. Riedel

10

d/ = 0.3

1.5

8 1.0

KIc/K0 = 1.4

KIc/K0 = 0.6

a/

a/

6 4

0.5

2

KIc/K0 = 2

0.0 0.0

0.1

0.2

0.3

0.4

0

0.5

0

z/

4

100 10 m (da/dN)/(AKIc )

7. Circular particles To model crack growth through an array of circular cylindrical particles, the same model equations are used as in the stripe model. Also the filtering of the spectrum of a(z) is applied. Fig. 6 shows three 0 > 1 (crack pinning), the other for results, two of them for KIc Kmax 0 KIc Kmax < 1 (crack growth enhancement). In both cases the crack front becomes nearly straight soon after having passed a row of particles. The pictures also illustrate that the curvature of the crack front increases 0 with KIc Kmax , so that the requirement of small curvatures, on which Eq. (1) is based, is increasingly violated. Fig. 7 shows the evolution of the average crack length. For a/λ > 3 there are no brittle particles in the geometrical model, so that the crack  = 1. While the crack advances freely with a normalized rate da ̂ dN traverses the array of particles, the average growth rate is  = 0.275, 0.66 and 1.35 for the cases shown in Figs. 6 and 7, i.e. da ̂ dN 0 = 2, 1.4 and 0.6, respectively. for KIc Kmax

1

no particles

0.1 0.01

circular particles

1E-3 1E-4

stripe model 0.3

0.6

In Fig. 8 normalized crack growth rates are plotted as a function of the amplitude of the normalized applied stress intensity factor. For the stripe model the crack growth rates are given by Eqs. (11) and (12); for

2.5

2.5

2.0

2.0

2.0

1.5

1.5

1.5

a/

a/ 1.0

1.0

1.0

0.5

0.5

0.5

0.0

0.0

0.0

2.0

-0.5 0.0

3

Fig. 8. Normalized da/dN curves. For the particle-free material a Paris law with m = 4 is assumed. The stripe model predicts a fatigue threshold at ΔK 0 KIc = 0.3.

2.5

KIc/K0 = 2

1

K0/KIc

c) 3.0

1.5

12

d/ = 0.3 m=4 R=0

3.0

1.0

10

a/

b)

3.0

0.5

8

0 Fig. 7. Evolution of the average crack length for various KIc Kmax ratios.

usually sufficient for an acceptable accuracy.

-0.5 0.0

6

N_hat

Fig. 5. Crack front shape; oscillating solution (black); smooth solution (red) if Fourier series is truncated at nmax. d/λ = 0.32, nmax = 150. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

a)

2

Kc/K0 = 1.4 0.5

1.0

z/

z/

1.5

2.0

KIc/K0 = 0.6 -0.5 0.0

0.5

1.0

1.5

2.0

z/

Fig. 6. Crack pinning (a and b) and growth enhancement by brittle particles (c). The difference in dimensionless number of cycles between consecutive crack front  = 0.18 for the cases with pinning and ΔN  = 0.06 for growth enhancement. The calculation is done in 0 < z/λ < 0.5; the rest is complemented by positions is ΔN symmetry. 4

International Journal of Fatigue 131 (2020) 105332

H. Riedel

Vasoya et al. [4] give second-order solutions for a stripe model with piecewise constant stress intensity factor, which is directly comparable with the one presented here. The greatest nonlinearity which they consider corresponds to KIc K 0 = 1.41. In this case the crack front shapes calculated from the linear and the second-order theories differ by 9%. The condition that the protrusion of the crack front should be limited to 2a1/λ = 0.6 implies that acceptable accuracy of the linear 0 ≤ 1.6. The solution in Fig. 6a for theory can be expected if KIc Kmax 0 KIc Kmax = 2 is just outside this range, while the solution in Fig. 9 with 0 KIc Kmax = 3.7 obviously violates the requirement that the crack front should be only slightly curved. In that case, neither the linear nor the quadratic theory would be appropriate. Rather, the vertical parts of the crack front would move in horizontal direction thus circumventing the particle. This process was modelled for one particular case by Fares [9] numerically using the boundary element method. The finite perturbation method of Bower and Ortiz [7,8], like the boundary element method, provides accurate results for arbitrarily large nonlinearity including the case that the crack circumvents a particle. After the particles have been circumvented, they serve as crack bridges behind the crack front, until they finally break as the crack advances. This bridging effect will reduce the crack growth rate compared to the material without particles, but probably to a smaller extent than predicted by the present model, which predicts crack arrest for sufficiently small load amplitudes. The present model does not allow the crack to leave the x-z-plane. For circular cylindrical particles, there would be no reason for the crack to leave the x-z-plane. However, if spherical particles act as strong obstacles, the crack would tend to circumvent them in the third dimension. Attempts to model both, crack deflection and crack pinning were made by Kim and Kishi [13] using results of Cotterell and Rice [14] on stress intensity factors at deflected cracks. To validate the predictions of the present model, fatigue crack growth rates in materials with different volume fractions of brittle particles should be compared experimentally. To the author’s knowledge this has rarely been done systematically to an extent that the present theory could be critically checked. In a few publications fatigue crack growth rates in Al alloys with different Si contents have been compared [15,16]. Generally the growth rates in Stage 2 of the da/dN curve are larger in the high-Si alloys containing coarse brittle Si particles, the difference between the alloys being moderate, in agreement with the prediction in Fig. 8. Further, Moffat [16] observes the predicted cross-over of the da/dN curves of materials with and without particles. This confirms a characteristic feature of the model, but in other cases such a cross-over is not observed [15], possibly since other factors have a stronger influence on the fatigue threshold thus masking the crack pinning effect.

Fig. 9. The crack is arrested at the first two rows of particles if KIc ΔK 0 > λ d = 1 0.3.

the circular particle model they are taken from the slopes of the curves 0 in Fig. 7. (Note that ΔK 0 = Kmax for R = 0). As Fig. 8 shows, brittle particles accelerate the crack growth rate compared to the particle-free material if the stress intensity factor is greater than the fracture toughness. However, the effect is not large. In 0 = 0 , the maximum enhancement factor for the extreme case KIc Kmax the stripe model is 1 (1 − d λ )m , i.e. = 4.2 for d λ = 0.3, m = 4 . For the circular particle model, the maximum enhancement factor is only 1.45. On the other hand, the pinning effect of the brittle phase may be large. In the stripe model a fatigue threshold occurs when ΔK 0 KIc = d λ (=0.3 in Fig. 8). Also, in the circular particle model crack growth comes to a standstill if ΔK 0 KIc < d λ = 0.3. This is illustrated in Fig. 9, where consecutive crack front positions are piled up against the first two rows of particles. 8. Discussion Rice’s [1] theory, and thus the present analysis, is based on the assumption that the crack front is only moderately curved. To estimate the inaccuracy of this linearized theory, second-order theories [3–5], finite perturbation theories [7,8] or numerical boundary element calculations [9] can be used. Fig. 10 shows a comparison of the present first-order solutions (black lines) with second-order solutions of Leblond et al. [3] (red lines). In this analysis the crack front is prescribed as a single cosine wave with amplitude a1, and the resulting stress intensity factor is calculated for various values of the crack front protrusion, 2a1/λ. One notes that the total amplitude of the stress intensity factor between its extremes is the same for the first- and the secondorder theory, and that the differences between first- and second-order results remain moderate up to a protrusion 2a1/λ = 0.6. For larger protrusions, 2a1/λ > 2/π ≈ 0.637, the first- and second-order formulae predict that the stress intensity factor would become negative at z = 0. As Gao and Rice [2] remark, Fares’ numerical solutions [9] do not become negative, which means that both, the linear and the quadratic solutions become qualitatively wrong when 2a1/λ > 2/π.

9. Conclusions Brittle particles may accelerate or impede the growth of a fatigue crack in the ductile matrix. If the fracture toughness of the particles, KIc , is smaller than the maximum stress intensity factor in the loading cycle 0 Kmax , the presence of the particles accelerates crack growth, but quantitatively the effect remains small, in agreement with experimental results in the literature. For tough particles, or in other words, for small 0 < KIc , the pinning effect of the applied stress intensity factors, Kmax particles may become large. The linear perturbation theory predicts a fatigue threshold at ΔK 0 KIc = d λ (=0.3 for the geometry studied in the present paper). Qualitative arguments based on singular numerical results of nonlinear theories suggest that, instead of a true threshold a substantial drop of the crack growth rate will occur. In real materials, such as Al alloys with brittle Si particles, this drop of the crack growth rate by particle pinning is observed in some cases [16], but may be overlain by other, more common causes for a fatigue threshold in other cases.

Fig. 10. First- and second-order solutions for the stress intensity factor at a crack front with a prescribed cosine shape with amplitude a1 and wavelength λ (after Leblond et al. [3]). 5

International Journal of Fatigue 131 (2020) 105332

H. Riedel

Declaration of Competing Interest None.

[7] [8]

Acknowledgement

[9]

The author would like to thank his colleague Carl Fischer for sharing unpublished experimental results. This research did not receive any specific grant from funding agencies in the public, commercial, or notfor-profit sectors.

[10]

[11]

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