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Plastic strain gradients and transient fatigue crack growth: A computational study
International Journal of Fatigue 120 (2019) 283–293
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International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
Plastic strain gradients and transient fatigue crack growth: A computational study
T
Joshua D. Pribea, Thomas Siegmunda, , Vikas Tomarb, Jamie J. Kruzicc ⁎
a
School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907, USA c School of Mechanical and Manufacturing Engineering, University of New South Wales, Sydney, NSW 2052, Australia b
ARTICLE INFO
ABSTRACT
Keywords: Fatigue crack growth Plasticity Crack closure Overloads Damage accumulation
Fatigue crack growth is analyzed for steady state and overload conditions. The model combines an elastic-plastic strain gradient hardening material and an irreversible cohesive zone model. Plastic strain gradients are found not to affect steady-state fatigue crack growth but play a key role in the overload response. Plastic strain gradient hardening lowers plastic strain magnitudes and alters the spatial distribution of plastic strain. This leads to reduced crack closure and higher crack growth rates during and after the overload. Using classical plasticity alone to describe fatigue crack growth following an overload is thus nonconservative.
1. Introduction A key question in fatigue failure is how crack growth rates change in response to variable amplitude loads, where no steady state solutions exist for the fatigue crack growth problem. As a result, transients in fatigue crack growth require a detailed, often cycle-by-cycle analysis of the problem [1,2]. The use of cohesive zone models for fatigue crack growth has been shown to be highly relevant in such studies and has provided detailed insight into transient crack growth problems [3–8]. This is of particular relevance for cases where plastic deformation occurs during fatigue crack growth, since the cohesive zone model approach can account for the dissipation processes in both the formation of new fracture surfaces and bulk plastic deformation [4,7]. It is now well known that plasticity models accounting for only plastic strains, associated with statistically-stored dislocations (SSDs), provide an incomplete description of plastic deformation. Experiments have shown that plastic strain gradients, associated with geometrically necessary dislocations (GNDs), are important in the hardening response when there is significant nonuniform plastic deformation over a lengthscale on the order of micrometers [9,10]. Classical plasticity theories do not describe this behavior. Strain gradient plasticity models, on the other hand, account for both GNDs and SSDs by introducing the plastic strain gradient and an intrinsic material lengthscale to the constitutive law. These models include higher-order theories (e.g. [11]), in which higher-order stresses are conjugate to the plastic strain gradients, and lower-order theories, in which an effective plastic strain
⁎
gradient alters the tangent modulus during plastic deformation (e.g. [12]). Strong plastic strain gradients can develop in the plastic zone ahead of a crack tip, particularly under plane strain conditions [13]. Numerical studies of cracked bodies subjected to monotonic loading have shown that GNDs associated with plastic strain gradients reduce cracktip blunting and produce a region ahead of the crack tip with significantly higher stress than that predicted by either the classical HRR fields or computations with classical plasticity models [14,15]. As a result, strain gradient plasticity models predict lower fracture toughness for both sharp and initially blunted cracks [16]. In [17] an analytical model for the intrinsic stress intensity threshold for fatigue crack growth was derived based on the strain gradient in the crack-tip plastic zone. An overview of experimental results indicated that plastic strain gradients explained the existence of this intrinsic threshold, on which crack closure effects are superimposed in experimentally measured thresholds. At higher crack growth rates, plastic strain gradients were found to reduce crack closure in constant amplitude loading, which led to a slightly increased steady-state growth rate in a numerical study using strain gradient plasticity model based on dislocation densities [18]. Computations using a cohesive zone model coupled to the GND density predicted the delayed retardation phenomenon following an overload [19], well-known from numerous fatigue crack growth experiments [2,20,21]. Regions of enhanced and reduced GND densities in the computations corresponded to the overload-induced acceleration and retardation, respectively. Results in [19]
Corresponding author. E-mail address: [email protected] (T. Siegmund).
https://doi.org/10.1016/j.ijfatigue.2018.11.020 Received 28 August 2018; Received in revised form 7 November 2018; Accepted 21 November 2018 Available online 27 November 2018 0142-1123/ © 2018 Elsevier Ltd. All rights reserved.
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were limited, however, due in part to numerical difficulties that arose outside of a small range of applied loads. Experiments also provide evidence for the relevance of plastic strain gradients in fatigue crack growth. Electron backscatter diffraction measurements have been used by several researchers to infer grain misorientations and GND densities ahead of a fatigue crack [22–24]. These studies have generally observed GNDs at distances ahead of the crack close to classical estimates of the plastic zone size. Using X-ray microdiffraction, the authors in [25] observed large GND densities, along with small voids and cracks, near the surface of a fatigue crack subjected to an overload. However, the relative role of GNDs in comparison to SSDs remains an open question for both steady-state and transient fatigue crack growth. Further investigation is thus warranted to quantify the changes in crack tip fields and cyclic deformation predicted by strain gradient plasticity models relative to classical plasticity. This will in turn determine whether plastic strain gradients must be accounted for in practical fatigue assessments. This study addresses the hypothesis that lengthscale effects in plasticity would not affect predicted steady-state fatigue crack growth rates but would alter predicted crack growth rates when the crack is subjected to load transients, such as an overload. This hypothesis is motivated by the fact that steady-state fatigue crack growth is known to be essentially microstructure insensitive and can be described by linear elastic fracture mechanics under small-scale yielding conditions. On the other hand, the extent of retardation due to an overload depends on the well-known mechanisms of plasticity-induced crack closure and compressive residual stress ahead of the crack tip, both of which are closely tied to the crack-tip plastic zone size and plastic strain magnitudes [20,26,27]. These quantities will be influenced by plastic strain gradient hardening in the large monotonic plastic zone resulting from the overload. Furthermore, it has been argued that post-overload crack growth is governed by near-threshold micromechanisms [21], which, as noted previously, have been connected to GNDs at the crack tip and in the plastic zone. A computational model is used which employs a first-order strain gradient plasticity formulation [12] in conjunction with an irreversible cohesive zone model [28]. Crack growth rates, the evolution of cohesive zone tractions and separations, and plasticity in terms of strains and strain gradients in the bulk material are considered for both constant-amplitude loading and the transient period following a single overload. The effects of plastic strain gradients are then elucidated by analyzing the crack-tip stress, strain, and strain gradient fields.
displacement boundary conditions are applied on the outer nodes, as shown in Fig. 1a. Due to symmetry in mode I loading, a half-model is considered. Vertical displacements are constrained for the nodes on the crack plane ahead of the initial crack tip. Rigid body motion is prevented by Eq. (1), as the horizontal displacement at = is zero. The continuum consists of 3892 linear plane strain elements and 123 cohesive zone elements along the symmetry line. A structured quadrilateral mesh is used near the crack tip, extending in the crack growth direction. The MBL model radius (RMBL ) is 10,000 times larger than the minimum element size. 2.2. Strain gradient plasticity model The conventional theory of mechanism-based strain gradient plasticity (CMSG) [12] is used for the deformation of the solid. This formulation employs Taylor dislocation theory to define the flow stress flow accounting for both strain hardening and strain gradient hardening:
=
flow
f 2 ( p) + l
y
(3)
p.
Here, y is the initial yield strength, f is a non-dimensional function of the equivalent plastic strain p , p is the effective plastic strain gradient, and l is the intrinsic material length for strain gradient plasticity. For face-centered cubic polycrystals, l is given by
l = 18
µ
2
2
b,
(4)
y
where is an empirical prefactor, µ is the shear modulus, and b is the magnitude of the Burgers vector. A value of l on the order of microns corresponds to the experimental lengthscale where size effects are observed [12]. Following [11], p is given by p
=
1 4
p p ijk ijk
p ik , j
+
,
(5)
where p ijk
=
p jk , i
p ij, k .
+
(6)
For a power-law hardening material, f is defined as
f ( p) = 1 +
p
E
n
,
(7)
y
where n is the strain hardening exponent. The CMSG constitutive equation is developed using a viscoplastic limit approach, in which the effective plastic strain rate is given by
2. Model description 2.1. Model geometry
M p
Mode I cyclic loading under plane strain conditions is applied via a modified boundary layer (MBL) model [29]. In this approach, an initially sharp crack and the domain of crack extension are enclosed in a circular domain. Displacements corresponding to the asymptotic cracktip solution are imposed on the boundary, which remains elastic and surrounds the small crack-tip plastic zone. Given a prescribed mode I stress intensity factor KI , the boundary displacements u (t ) are determined from the Williams expansion [30] as
u x ( t , r , ) = K I (t )
r 1+2 cos (3 2 E 2
4
cos ),
(1)
u y ( t , r , ) = K I (t )
r 1+2 sin (3 2 E 2
4
cos ),
(2)
e
= y
f 2 ( p)
+l
=
p
e
M
,
(8)
flow
where is the effective strain rate, and e is the von Mises equivalent stress. A value of M = 75.0 was found to predict the same crack growth rates as a conventional rate-independent plasticity formulation when neglecting plastic strain gradients (i.e. setting l = 0 ). Therefore, M = 75.0 was used for all computations presented here. The CMSG User Material (UMAT) subroutine developed in [12] for the finite element solver ABAQUS was used for this study. 2.3. Irreversible cohesive zone model A cohesive zone model based on the potential in [31] is used to describe the material separation process. For mode I separation, the normal traction Tn is related to the normal separation n by
where E is Young’s modulus, is Poisson’s ratio, and r and are the radial and angular coordinates, with the initial crack tip as the origin. A triangular waveform is prescribed for the mode I stress intensity factor in order to simulate cyclic loading. The finite element mesh used in this study is shown in Fig. 1. The
Tn (
n)
where 284
=
max,0 exp
max,0
1
n
n
0
0
is the initial cohesive strength and
(9) 0
is the characteristic
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Fig. 1. (a) Global and (b) crack-tip mesh in the modified boundary layer model.
separation. The cohesive energy 0
=
0
is
max,0/ y influences the amount of plastic dissipation that accompanies crack growth. Two values of this ratio are considered: max,0/ y = 4.0 and max,0/ y = 4.5. These correspond to cohesive energies of 0 = 20.0 J/m2 and 0 = 25.0 J/m2 for constant characteristic separation 0 = 7.36 nm. These two cases are used to investigate the effect of plastic strain gradients relative to the amount of crack-tip plasticity. The cohesive zone endurance limit and characteristic fatigue separation are chosen as f / max,0 = 0.25 and / 0 = 4.0 , following previous studies of fatigue crack growth in the Paris regime [4,18]. The implications of these parameter choices are considered further in the Discussion. A reference stress intensity factor K 0 is defined from the linear elastic fracture mechanics relation between the energy release rate and mode I stress intensity:
(10)
max,0 0 exp(1).
A damage variable D is introduced to account for the decrease in load-carrying capacity following from cyclic loading [28]. Damage accumulation is modeled as a decrease in effective cross-sectional area corresponding to the current value of D. This can be described by introducing the current effective cohesive strength max : max
=
max,0 (1
D),
D=
Ddt .
(11)
Damage evolution is given by n
D=
Tn max
f
H(
max,0
0 ),
n,acc
n,acc
=
n
dt ,
(12)
where n,acc is the accumulated normal separation and H is the Heaviside function. Two material parameters are introduced in Eq. (12): the characteristic fatigue separation and the cohesive zone endurance limit f . The characteristic fatigue separation scales the increment in material separation and sets the accumulated separation at which the cohesive zone fails. The endurance limit determines the traction below which damage will not accumulate. Damage evolution does not begin until the accumulated material separation exceeds 0 . The damage increment then depends on the normalized separation increment n / , weighted by the traction reduced by the endurance limit. Damage is defined on an average-per-element basis, and the crack tip is taken to be the location where D = 1.0 [28]. When the crack faces are in contact, the cohesive tractions are negative, and no damage accumulation occurs. In this situation, Eq. (9) is modified with a penalty stiffness A = 30 [4] such that, in contact,
Tn (
n)
=A
max,0 exp
1
n
n
0
0
.
E
K0 =
0 2
1
.
(14) 2
K 0 = 1.50 MPa m and 1.68 For 0 = 20.0 J/m and 25.0 MPa m , respectively. The maximum applied stress intensity factor Kmax and stress intensity range K are related to the plane strain monotonic and cyclic plastic zone sizes Rp and R c , respectively [33]: Rp =
1 3
Kmax
Rc =
1 3
K
J/m2 ,
2
(15)
y
2
2
.
y
(16)
All computations presented here were completed under small-scale yielding conditions, with RMBL on the order of 100Rp or greater. The smallest element size in the structured quadrilateral mesh, equal to the minimum incremental crack advance, was 2.5 0 , while crack extension was restricted to the structured mesh region, resulting in a maximum crack extension of amax = 275 0 . Results are depicted for crack extension up to only a = 250 0 since this ensures the crack-tip plastic zone also remains contained in the structured mesh region.
(13)
The irreversible cohesive zone model was implemented as a user element (UEL) subroutine in ABAQUS.
3. Results
2.4. Model parameters
3.1. Steady-state fatigue crack growth
A model material is considered in order to provide general understanding of the role of plastic strain gradients in fatigue crack growth. For all computations, Young’s modulus and Poisson’s ratio are E = 100 GPa and = 0.34 , respectively. The yield strength is y = 0.0025E , with an isotropic hardening exponent of n = 0.1. Two intrinsic material length values are considered: l = 0 (predicts strain hardening only) and l = 10 µm (strain hardening and strain gradient hardening). The value l = 10 µm emerges from Eq. (4) given the properties above, with b on the order of a tenth of a nanometer and around 0.3, and represents an approximate upper bound on the value of l based on previous CMSGbased studies [12,32]. The ratio of the initial cohesive strength to the yield strength
First, steady-state crack growth is considered. Fig. 2 shows the dependence of the predicted normalized crack growth rate d(a/ 0)/dN on the normalized applied stress intensity factor range K / K 0 for combinations of the intrinsic material length l, load ratio R, and initial cohesive strength max,0 . In all cases, a Paris Law-type response emerges from the model, i.e. the predicted response can be described by the equation d(a/ 0)/dN = C ( K / K 0 )m . The exponent m characterizes the slope of the lines in Fig. 2. Table 1 summarizes the predicted m values for each combination of max,0/ y , R , and l. The exponent is found to be quite insensitive to the value of l, with small changes in the predicted steady-state crack growth rate only 285
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Fig. 2. Dependence of the normalized crack growth rate on the normalized stress intensity range for (a)
R
l (µ m)
m
4.0
0.1
0 10 0 10
3.23 3.27 3.19 3.23
0 10 0 10
3.03 3.09 3.03 3.08
0.5 4.5
0.1 0.5
= 4.0 and (b)
max,0 / y
= 4.5.
computations, the crack grows through the entire overload plastic zone, allowing for analysis of the region where the overload most significantly affects crack growth rates. The importance of plastic strain gradients becomes clear in the postoverload crack growth transients, where the computations predict less severe retardation for l = 10 µm than for l = 0 . The retardation was measured using the number of delay cycles Ndelay , defined as the number of cycles required to traverse the overload-affected region relative to a crack under only constant-amplitude loading [2,36]. The value of Ndelay was calculated by applying a linear fit to the region of crack extension far beyond the overload and extrapolating back to the crack length when the overload was applied, as depicted in Fig. 4a. The number of delay cycles for computations with l = 0 and l = 10 µm for each max,0/ y and several values of K / K 0 is shown in Fig. 4b. The predicted Ndelay is always smaller for l = 10 µm , with the effect becoming stronger for higher cohesive strength. Although Ndelay becomes smaller at higher values of K / K 0 , the difference between the predicted values for l = 0 and l = 10 µm remains very nearly constant. The severity of retardation can be further examined through the crack growth rates, shown in Fig. 5 for K / K 0 = 0.35, relative to the number of cycles and the crack extension. Fig. 5a shows that, for l = 10 µm , the model predicts a higher minimum fatigue crack growth rate than for l = 0 for both values of cohesive strength. Fig. 5b provides information about the spatial extent of crack retardation after the overload. Following the peak, the model predicts that the crack growth rate almost immediately falls to the minimum value for l = 0 while the crack growth rate for l = 10 µm remains elevated for the first few elements after the overload is applied. However, the maximum extent of the overload-affected region is not significantly affected by the value of l. The overload-affected region ends at about a/ 0 = 175 for all cases, with the higher cohesive strength ( max,0/ y = 4.5) leading to a small increase. Thus, for l = 10 µm , the model predicts a higher minimum crack growth rate and a lower number of delay cycles, but a similar maximum extent of the overload-affected region, relative to l = 0 . The transient processes in the cohesive zone and in the bulk plastic deformation give insight into the role of plastic strain gradients in postoverload crack growth. In the following results, these processes are analyzed using the computations with K / K 0 = 0.35 as a representative case. Figs. 6 and 7 show a comparison between the steady-state and post-overload traction distributions for and max,0/ y = 4.0 max,0/ y = 4.5, respectively. In each case, the steady-state traction distributions are nearly independent of the value of l, with the main difference being a small amount of crack closure—indicated by negative traction—for l = 0 and max,0/ y = 4.5. This occurs only at the minimum applied load, meaning that the crack remains open for nearly the entire
Table 1 Paris law exponents. max,0 / y
max,0 / y
becoming apparent for high values of K with max,0/ y = 4.5. This produces differences in the predicted value of m on the order of 10 2 . Changing the load ratio R shifts the data with respect to K / K 0 and slightly reduces the exponent m as well. This behavior is seen in experiments [34] and in other numerical studies with cohesive zone models [35]. 3.2. Fatigue crack growth following an overload Constant-amplitude loading interrupted by a single overload is now considered. Each computation was conducted with load ratio R = 0.1 and consisted of the following steps. First, constant-amplitude loading was applied with K / K 0 = 0.28, 0.35, or 0.40. The overload was applied in the cycle after the crack extension exceeded a/ 0 = 75, such that steady-state conditions developed prior to the overload. The overload ratio was Kmax,OL/ Kmax = 1.73, where Kmax,OL is the maximum applied stress intensity factor during the overload. This corresponds to applied energy release rate ratio Gmax,OL/ Gmax = 3.0 . Fig. 3 shows the predicted crack extension for the computations with a single overload and different steady-state K / K 0 values. Predictions for both l = 0 and l = 10 µm are shown. All computations predict a brief transient period while crack growth begins from the initial crack tip, followed by steady-state crack growth where the crack growth rate is insensitive to the value of l. During the overload, the computations predict a brief acceleration followed by crack growth retardation in the following cycles. For large crack extensions, the crack growth rate is predicted to nearly return to the original steady-state value. The slightly lower crack growth rate for longer crack lengths is a known feature of the modified boundary layer model [4], as the assumptions behind Eqs. (1) and (2), i.e. that the displacements correspond to a constant K , begin to be violated. In the present 286
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Fig. 3. Crack extension with a single overload for l = 0 (solid lines) and l = 10 µ m (dashed lines) with cohesive strength (a) ratio is R = 0.1, with several cases of steady-state K / K 0 .
cycle. During and after the overload cycle, the model predicts higher tractions and therefore more damage accumulation, per Eq. (12), and a lower magnitude of crack closure for l = 10 µm . This is most pronounced in the region where compressive tractions occur at minimum load, which corresponds to the overload-affected region where crack growth rates are significantly reduced. The evolution of cohesive zone tractions in the overload-affected region is summarized using the values of the maximum and minimum traction in the cohesive zone in each cycle, Fig. 8. For a given cycle, this corresponds to the maximum traction at point “a” and the minimum traction at point “c” as depicted in Figs. 6 and 7. The region of interest is restricted to the cohesive zone ahead of D = 1.0 in order to gain insight into the material separation process. The maximum traction provides a measure of Ktip , the effective maximum stress intensity at the crack tip, suggested by Willenborg et al. [27] as the controlling parameter for retardation following an overload. The minimum traction is a measure for the magnitude of crack closure, which is associated with K op , the stress intensity factor required to open the crack [5,20]. After the overload, for l = 10 µm , the model predicts higher maximum and minimum traction in each cycle. The predicted level of crack closure is therefore reduced when plastic strain gradients are accounted for, while
max/ y = 4.0
and (b)
max,0 / y
= 4.5. The load
the crack-tip stress intensity is increased. This effect persists for a larger number of cycles for the higher cohesive strength value due to the lower crack growth rates. The increased tractions are associated with a reduction in compressive residual stresses induced by the overload and correspond to the higher minimum crack growth rate seen for l = 10 µm . Fig. 9 shows representative separation distributions for steady-state crack growth and the overload cycle. Plastic strain gradients have virtually no effect on the separation during steady-state crack growth. At maximum load in the overload cycle for l = 10 µm , the model predicts elevated separation in the cohesive zone compared to l = 0 , which leads to additional damage accumulation. At minimum load, the crack faces are in contact ahead of the location where D = 1.0 . The length of the region where closure occurs is quite independent of the value of l, while the magnitude of closure is reduced for l = 10 µm . The predicted separation behind the crack tip is also lower for l = 10 µm at both maximum and minimum load in the overload cycle, which is consistent with previously reported computations with cracks in monotonically loaded structures where strain gradient plasticity models predict less blunting than classical plasticity models [14,37]. Figs. 10a and b show the spatial distribution of the accumulated
Fig. 4. (a) Sample calculation of number of delay cycles. (b) Number of delay cycles for l = 0 (solid lines) and l = 10 µm (dashed lines). 287
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Fig. 5. Dependence of normalized crack growth rates for l = 0 (solid lines) and l = 10 µ m (dashed lines) on (a) cycles after the overload (OL) cycle and (b) normalized crack extension. The load ratio is R = 0.1, with K / K 0 = 0.35 .
plastic strain in the structured-mesh region for each value of l. The outer contour, at p = 0.001, marks the edge of the plastic zone. Initially, a steady-state plastic zone forms as the crack grows at a constant rate. The overload then induces an increase in the plastic zone size and maximum plastic strain magnitude, which is mostly ahead of the location where D = 1.0 at the start of the overload cycle. No further crack-tip plasticity is generated for 40 cycles and 36 cycles after the overload for l = 0 and l = 10 µm , respectively, as the most significant crack growth retardation occurs. At the crack extension shown in Fig. 10, a/ 0 = 245, the steady-state plastic zone has re-formed. This occurs at 66 cycles and 59 cycles after the overload for l = 0 and l = 10 µm , respectively. While the plastic zone sizes in the steady state and at the overload for each value of l are quite similar, the maximum plastic strain magnitude in the overload plastic zone is reduced by a factor of about 1.6 for l = 10 µm relative to l = 0 . This reduced plastic strain magnitude in the bulk means that more separation occurs in the cohesive zone following the overload, leading to enhanced damage accumulation. This is responsible for the higher minimum crack growth rate and the reduced number of cycles required for the steady-state plastic zone to re-develop for l = 10 µm .
Fig. 10c shows the spatial distribution of plastic strain gradients for l = 10 µm . Similar to the plastic strain, the plastic strain gradients reach a steady-state distribution until the overload induces an instantaneous increase in the plastic strain gradient magnitude. The regions where plastic strain gradients are highest correspond to regions of restricted plastic deformation in Fig. 10b relative to Fig. 10a. In the overloadaffected region, the maximum value of p is 82 mm−1, while the maximum value of p during steady-state crack growth is 24 mm−1. A more quantitative description of the role of plastic strain gradients in the hardening response is based on Eq. (3). If isotropic hardening is neglected (i.e. the strain hardening exponent n is set to 0), the flow SG stress associated with strain gradient hardening alone, flow , is SG flow y
=
1+l
p.
(17)
SG / y = 1.05. During The outer contour in Fig. 10c corresponds to flow SG steady-state crack growth, the largest value of flow / y is about 1.11, SG / y is while, in the overload-affected region, the largest value of flow about 1.35.
Fig. 6. Traction distribution in the cohesive zone for l = 0 (solid lines) and l = 10 µ m (dashed lines) during (a) steady-state crack growth and (b) the overload cycle. The load ratio is R = 0.1, with K / K 0 = 0.35 and max,0/ y = 4.0 . The insets depict the points at which the traction distribution is shown. x = 0 corresponds to D = 1.0 for point “a”. 288
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Fig. 7. Traction distribution in the cohesive zone for l = 0 (solid lines) and l = 10 µ m (dashed lines) during (a) steady-state crack growth and (b) the overload cycle. The load ratio is R = 0.1, with K / K 0 = 0.35 and max,0/ y = 4.5. The insets depict the points at which the traction distribution is shown. x = 0 corresponds to D = 1.0 for point “a”.
Two measures that allow for comparisons across values of l, max,0/ y , and steady-state K / K 0 are now defined. Fig. 11 shows the maximum plastic strain magnitude in the steady-state and overload plastic zones for each computation. For l = 10 µm , the model predicts significantly reduced plastic strains in the overload plastic zone for all cases, with the effect becoming more significant for the higher cohesive strength. This leads to more deformation in the cohesive zone and corresponds to the higher separation, traction, and crack growth rates seen previously for l = 10 µm . While the absolute magnitude of the plastic strain increases with increasing K / K 0 , the relative effect of increasing l does not significantly change. The plastic strain in the steady-state plastic zone is also fairly insensitive to l. Fig. 12 shows the length of the cohesive zone that is closed at minimum load in the cycles following the overload for each computation, denoted by Lclosure . While this quantity increases with increasing cohesive strength and stress intensity range, it shows only a small dependence on the value of l. In fact, Lclosure differs between each value of l by a maximum of one element length. This is consistent with evidence from the crack growth rate, traction and separation distributions, and plastic zone size that the maximum extent of the overload-affected
region does not depend on l. 4. Discussion The computations presented here show that hardening due to plastic strain gradients significantly affects numerical predictions of fatigue crack growth following an overload. Due to restricted plastic deformation in the overload plastic zone, there is increased dissipation in the cohesive zone, seen in the elevated values of traction and separation predicted for l = 10 µm that persist throughout the overload-affected region and lead to higher crack growth rates relative to l = 0 . The relative increase in crack growth rates is amplified for computations with a higher cohesive strength, which is associated with more crack-tip plasticity. Changing the applied load does not significantly alter the model predictions for l = 10 µm relative to l = 0 . The computations thus support the hypothesis that plastic strain gradient hardening is important in transient fatigue crack growth situations but does not significantly affect steady-state fatigue crack growth. The irreversible cohesive zone model implemented in this study assumes that fatigue crack growth is a stress- and deformation-
Fig. 8. Maximum and minimum traction in the cohesive zone for each cycle relative to the overload (OL) with (a) is R = 0.1, with K / K 0 = 0.35 . 289
max,0 / y
= 4.0 and (b)
max,0 / y
= 4.5. The load ratio
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Fig. 9. Normalized separation distribution in the cohesive zone for l = 0 (solid lines) and l = 10 µ m (dashed lines) at (a) maximum applied load and (b) minimum applied load. The load ratio is R = 0.1, with K / K 0 = 0.35 and max,0/ y = 4.0 . The inset in (b) shows a zoom of the region where the cohesive surfaces are in contact, and x = 0 corresponds to D = 1.0 .
Fig. 10. (a) Accumulated plastic strain ( p ) contours for l = 0; (b) Accumulated plastic strain ( p ) contours for l = 10 µ m; and (c) Plastic strain gradient ( p) contours for l = 10 µ m. The superimposed solid lines mark the current crack extension ( a/ 0 = 245) while the dashed lines mark the crack extension at the start of the overload cycle ( a/ 0 = 75 ). The load ratio is R = 0.1, with K / K 0 = 0.35 and max,0/ y = 4.0 .
290
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Fig. 11. Maximum accumulated plastic strain in the steady-state and overload plastic zones for computations with (a) lines are for l = 0 ; dashed lines are for l = 10 µm .
max,0 / y
= 4.0 and (b)
max,0 / y
= 4.5. Solid
parameters are associated with experimental observations of crack retardation [2,20,27] and finite element studies of crack extension [13,45,46]. To determine K op , the time step at which the cohesive zone separation became positive (i.e. the crack opened) at all points ahead of D = 1.0 was recorded for each cycle after the overload. The applied stress intensity factor at this time step was then determined, which corresponds to an estimate of K op . Although the precise value of K op determined through this process is dependent on the time incrementation, trends with respect to the plastic strain gradient effects can still be observed. For each loading case in this study, the model predicts K op for l = 10 µm is 20% to 30% less than for l = 0 in the region of maximum crack growth retardation. For example, with K / K 0 = 0.35 and max,0/ y = 4.5, K op/ Kmax = 0.5 for l = 0 while K op/ Kmax = 0.4 for l = 10 µm at the minimum post-overload crack growth rate. This indicates that plastic strain gradient hardening has a meaningful effect on post-overload crack closure through its role in the plastic strain distribution. As a consequence of this analysis, a useful conceptual question emerges: can the effect of plastic strain gradients following the overload be replicated by increasing the bulk yield strength relative to the cohesive strength? The predicted crack growth rates in Fig. 13 indicate
Fig. 12. Length of the region where crack closure occurs following the overload. Solid lines are for l = 0 ; dashed lines are for l = 10 µm .
dependent process. The damage evolution law depends on scaled measures of both traction and separation, which is seen in discrete dislocation models of fatigue crack growth [38], other studies using cohesive zone models [6], and analytical models that include both K and Kmax in the crack growth law [39]. The Kmax dependence is associated with ductile damage in metals, such as micro- and nano-voids ahead of the crack tip, which have produced dimpled damage zones corresponding to the region where acceleration occurs after an overload [40,41]. The current study indicates that plastic strain gradients aid in understanding this process, as they promote the higher stresses required to activate ductile damage mechanisms during the overload cycle. This leads to elevated crack growth rates over a larger region relative to computations with l = 0 , as observed in Fig. 5b. This is similar to arguments for strain gradient hardening as the mechanism by which the stress ahead of cracks in monotonic loading situations can reach the high values required for cleavage cracking, even in ductile materials undergoing significant plastic deformation [14,42–44]. A key finding is that plastic strain gradients do not alter the overload plastic zone size, but are critical in predicting the plastic strain magnitude and distribution in the overload plastic zone. Along with the predicted tractions shown in Fig. 8, this indicates that plastic strain gradients should be considered in predictions of Ktip and K op . These
Fig. 13. Predictions of the crack growth rate for l = 10 µm compared with predictions for l = 0 with varied initial yield strength. The load ratio is R = 0.1, with K / K 0 = 0.28 . 291
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that the answer to this question is no—that is, the effect of plastic strain gradients is unique. Here, two additional computations were conducted with higher bulk yield strengths ( y = 260 MPa and y = 270 MPa) and no strain gradient hardening. The cohesive strength was fixed at 1125 MPa. While increasing the initial yield strength can capture either the minimum crack growth rate or the number of cycles in the overloadaffected region predicted by the strain gradient plasticity model, it cannot capture both. This is because hardening due to plastic strain gradients is highly localized at the crack tip, where the most extreme gradients appear as seen in Fig. 10c. The role of this localized hardening in predicted crack growth rates cannot be captured by simply recalibrating the parameters of the classical plasticity model or the cohesive zone. It should be noted that the magnitude of the predicted steady-state crack growth rates in this study is on the order of 10 8 to 10 7 m, which corresponds to the lower end of the Paris law regime. The applied K values necessary to achieve these crack growth rates for the model material considered here are similar to those found in discrete dislocation studies but are an order of magnitude below those of typical alloys [34,47,48]. This is due to the choice of cohesive zone damage parameters ( f , max,0 , and ), which allow for computations of the Paris law regime and of the response to load transients with reasonable computational expense. Extensions of the current study could involve calibrating these parameters to a specific fatigue threshold and crack growth rates. Such computations would face significantly increased cost: the finite element mesh must be small enough—on the order of the size used here and in strain gradient plasticity-based studies of fracture [32,43,49]—to resolve plastic strain gradients and the active cohesive zone, while the crack extension and number of cycles required to achieve a stabilized plastic wake scales with the increasing plastic zone size [46]. Possible roadmaps for this type of study are discussed for cohesive zone models in [35] and in the context of discrete dislocation models of fatigue crack growth in [47]. Further extensions of this study could consider large-scale plasticity or small cracks, in which plastic strain gradients may indeed be important even in constant-amplitude loading. In addition, the isotropic hardening law used here is expected to give acceptable results for growing cracks, but the model could be extended to kinematic hardening, which would be necessary for considering crack initiation.
References [1] Skorupa M. Load interaction effects during fatigue crack growth under variable amplitude loading—A literature review. Part I: Empirical trends. Fatigue Fract Eng Mater Struct 1998;22(8):987–1006. https://doi.org/10.1046/j.1460-2695.1998. 00083.x. [2] Bichler C, Pippan R. Effect of single overloads in ductile metals: A reconsideration. Eng Fract Mech 2007;74(8):1344–59. https://doi.org/10.1016/j.engfracmech. 2006.06.011. [3] Siegmund T. A numerical study of transient fatigue crack growth by use of an irreversible cohesive zone model. Int J Fatigue 2004;26(9):929–39. https://doi.org/ 10.1016/j.ijfatigue.2004.02.002. [4] Wang B, Siegmund T. A numerical analysis of constraint effects in fatigue crack growth by use of an irreversible cohesive zone model. Int J Fract 2005;132(2):175–96. https://doi.org/10.1007/s10704-005-0627-1. [5] Wang B, Siegmund T. Numerical simulation of constraint effects in fatigue crack growth. 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Mechanism-based strain gradient plasticity – I. Theory. J Mech Phys Solids 1999;47(6):1239–63. https://doi.org/10. 1016/S0022-5096(98)00103-3. [12] Huang Y, Qu S, Hwang KC, Li M, Gao H. A conventional theory of mechanism-based strain gradient plasticity. Int J Plast 2004;20(4-5):753–82. https://doi.org/10. 1016/j.ijplas.2003.08.002. [13] McClung RC, Thacker BH, Roy S. Finite element visualization of crack closure in plane stress and plane strain. Int J Fract 1991;50:27–49. https://doi.org/10.1007/ BF00035167. [14] Martínez-Pañeda E, Niordson CF. On fracture in finite strain gradient plasticity. Int J Plast 2016;80:154–67. https://doi.org/10.1016/j.ijplas.2015.09.009. [15] Komarigi U, Agnew S, Gangloff R, Begley M. The role of macroscopic hardening and individual length-scales on crack tip stress elevation from phenomenological strain gradient plasticity. J Mech Phys Solids 2008;56(12):3527–40. https://doi.org/10. 1016/j.jmps.2008.08.007. [16] Nielsen KL, Niordson CF, Hutchinson JW. Strain gradient effects on steady state crack growth in rate-sensitive materials. Eng Fract Mech 2012;96:61–71. https:// doi.org/10.1016/j.engfracmech.2012.06.022. [17] Gil Sevillano J. The effective threshold for fatigue crack propagation: a plastic size effect? Scr Mater 2001;44(11):2661–5. https://doi.org/10.1016/S1359-6462(01) 00948-4. [18] Brinckmann S, Siegmund T. Computations of fatigue crack growth with strain gradient plasticity and an irreversible cohesive zone model. Eng Fract Mech 2008;75(8):2276–94. https://doi.org/10.1016/j.engfracmech.2007.09.007. [19] Brinckmann S, Siegmund T. A cohesive zone model based on the micromechanics of dislocations. Model Simul Mater Sci Eng 2008;16(6):065003. https://doi.org/10. 1088/0965-0393/16/6/065003. [20] Elber W. Fatigue crack closure under cyclic tension. Eng Fract Mech 1970;2(1):37–45. https://doi.org/10.1016/0013-7944(70)90028-7. [21] Suresh S. Micromechanisms of fatigue crack growth retardation following overloads. Eng Fract Mech 1983;18(3):577–93. https://doi.org/10.1016/00137944(83)90051-6. [22] Wilkinson AJ, Meaden G, Dingley DJ. High resolution mapping of strains and rotations using electron backscatter diffraction. Mater Sci Technol 2006;22(11):1271–8. https://doi.org/10.1179/174328406X130966. [23] Glaessgen E, Saether E, Smith S, Hochhalter JD, Yamakov V, Gupta V. Modeling and characterization of damage processes in metallic materials. 52nd AIAA/ASME/ ASCE/AHS/ASC structures, structural dynamics and materials conference Reston, Virigina: American Institute of Aeronautics and Astronautics; 2011. p. 1–16. https://doi.org/10.2514/6.2011-2177. [24] Tucker AT, Wilkinson AJ, Henderson MB, Ubhi HS, Martin JW. Measurement of fatigue crack plastic zones in fine grained materials using electron backscattered diffraction. Mater Sci Technol 2000;16(4):457–62. https://doi.org/10.1179/ 026708300101507910. [25] Barabash RI, Gao Y, Sun Y, Lee SY, Choo H, Liaw PK, et al. Neutron and X-ray diffraction studies and cohesive interface model of the fatigue crack deformation behavior. Philos Mag Lett 2008;88(8):553–65. https://doi.org/10.1080/ 09500830802311080. [26] Salvati E, Zhang H, Fong KS, Song X, Korsunsky AM. Separating plasticity-induced closure and residual stress contributions to fatigue crack retardation following an overload. J Mech Phys Solids 2017;98:222–35. https://doi.org/10.1016/j.jmps. 2016.10.001. [27] Willenborg J, Engle RM, Wood HA. A crack growth retardation model using an effective stress concept. AFFDL TM-71-1-FBR. 1971.
5. Conclusions Accurate predictions of fatigue crack growth are essential to ensure the reliability and safety of engineering structures. Such computations are typically preformed with classical plasticity formulations. This study indicates that the use of classical plasticity alone for analyzing steady-state fatigue crack growth is sufficient for the small-scale yielding problem considered here. The Paris Law response predicted by the strain gradient plasticity model is virtually the same as that predicted by the classical plasticity model. However, the analysis of overloads and the subsequent fatigue crack growth transients would require the consideration of strain gradient plasticity. Large plastic strain gradients are predicted to significantly restrict plastic deformation in the overload plastic zone. This leads to reduced crack closure and a higher minimum crack growth rate during the overload-induced retardation, relative to the classical plasticity model alone. This indicates that classical plasticity models underpredict crack growth rates under transient conditions. Acknowledgment This research was supported by the U.S. Department of Energy, National Energy Technology Laboratory, under contract No. DEFE0011796. 292
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Fatigue 2001;23:39–53. https://doi.org/10.1016/S0142-1123(01)00171-2. [40] Pippan R, Bichler C, Tabernig B, Weinhandl H. Overloads in ductile and brittle materials. Fatigue Fract Eng Mater Struct Fract Eng Mater Struct 2005;28(11):971–81. https://doi.org/10.1111/j.1460-2695.2005.00934.x. [41] Daneshpour S, Dyck J, Ventzke V, Huber N. Crack retardation mechanism due to overload in base material and laser welds of Al alloys. Int J Fatigue 2012;42:95–103. https://doi.org/10.1016/j.ijfatigue.2011.07.010. [42] Martínez-Pañeda E, Betegón C. Modeling damage and fracture within strain-gradient plasticity. Int J Solids Struct 2015;59:208–15. https://doi.org/10.1016/j. ijsolstr.2015.02.010. [43] Jiang H, Huang Y, Zhuang Z, Hwang KC. Fracture in mechanism-based strain gradient plasticity. J Mech Phys Solids 2001;49(5):979–93. https://doi.org/10.1016/ S0022-5096(00)00070-3. [44] Wei Y, Hutchinson JW. Steady-state crack growth and work of fracture for solids characterized by strain gradient plasticity. J Mech Phys Solids 1997;45(8):1253–73. https://doi.org/10.1016/S0022-5096(97)00018-5. [45] Fleck NA. Finite element analysis of plasticity-induced crack closure under plane strain conditions. Eng Fract Mech 1986;25(4):441–9. https://doi.org/10.1016/ 0013-7944(86)90258-4. [46] Jiang Y, Feng M, Ding F. A reexamination of plasticity-induced crack closure in fatigue crack propagation. Int J Plast 2005;21(9):1720–40. https://doi.org/10. 1016/j.ijplas.2004.11.005. [47] Deshpande VS, Needleman A, Van der Giessen E. Scaling of discrete dislocation predictions for near-threshold fatigue crack growth. Acta Mater 2003;51(15):4637–51. https://doi.org/10.1016/S1359-6454(03)00302-1. [48] Curtin WA, Deshpande VS, Needleman A, Van der Giessen E, Wallin M. Hybrid discrete dislocation models for fatigue crack growth. Int J Fatigue 2010;32(9):1511–20. https://doi.org/10.1016/j.ijfatigue.2009.10.015. [49] Wei Y, Qiu X, Hwang KC. Steady-state crack growth and fracture work based on the theory of mechanism-based strain gradient plasticity. Eng Fract Mech 2004;71(1):107–25. https://doi.org/10.1016/S0013-7944(03)00065-1.
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Contents lists available at ScienceDirect
International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
Plastic strain gradients and transient fatigue crack growth: A computational study
T
Joshua D. Pribea, Thomas Siegmunda, , Vikas Tomarb, Jamie J. Kruzicc ⁎
a
School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907, USA c School of Mechanical and Manufacturing Engineering, University of New South Wales, Sydney, NSW 2052, Australia b
ARTICLE INFO
ABSTRACT
Keywords: Fatigue crack growth Plasticity Crack closure Overloads Damage accumulation
Fatigue crack growth is analyzed for steady state and overload conditions. The model combines an elastic-plastic strain gradient hardening material and an irreversible cohesive zone model. Plastic strain gradients are found not to affect steady-state fatigue crack growth but play a key role in the overload response. Plastic strain gradient hardening lowers plastic strain magnitudes and alters the spatial distribution of plastic strain. This leads to reduced crack closure and higher crack growth rates during and after the overload. Using classical plasticity alone to describe fatigue crack growth following an overload is thus nonconservative.
1. Introduction A key question in fatigue failure is how crack growth rates change in response to variable amplitude loads, where no steady state solutions exist for the fatigue crack growth problem. As a result, transients in fatigue crack growth require a detailed, often cycle-by-cycle analysis of the problem [1,2]. The use of cohesive zone models for fatigue crack growth has been shown to be highly relevant in such studies and has provided detailed insight into transient crack growth problems [3–8]. This is of particular relevance for cases where plastic deformation occurs during fatigue crack growth, since the cohesive zone model approach can account for the dissipation processes in both the formation of new fracture surfaces and bulk plastic deformation [4,7]. It is now well known that plasticity models accounting for only plastic strains, associated with statistically-stored dislocations (SSDs), provide an incomplete description of plastic deformation. Experiments have shown that plastic strain gradients, associated with geometrically necessary dislocations (GNDs), are important in the hardening response when there is significant nonuniform plastic deformation over a lengthscale on the order of micrometers [9,10]. Classical plasticity theories do not describe this behavior. Strain gradient plasticity models, on the other hand, account for both GNDs and SSDs by introducing the plastic strain gradient and an intrinsic material lengthscale to the constitutive law. These models include higher-order theories (e.g. [11]), in which higher-order stresses are conjugate to the plastic strain gradients, and lower-order theories, in which an effective plastic strain
⁎
gradient alters the tangent modulus during plastic deformation (e.g. [12]). Strong plastic strain gradients can develop in the plastic zone ahead of a crack tip, particularly under plane strain conditions [13]. Numerical studies of cracked bodies subjected to monotonic loading have shown that GNDs associated with plastic strain gradients reduce cracktip blunting and produce a region ahead of the crack tip with significantly higher stress than that predicted by either the classical HRR fields or computations with classical plasticity models [14,15]. As a result, strain gradient plasticity models predict lower fracture toughness for both sharp and initially blunted cracks [16]. In [17] an analytical model for the intrinsic stress intensity threshold for fatigue crack growth was derived based on the strain gradient in the crack-tip plastic zone. An overview of experimental results indicated that plastic strain gradients explained the existence of this intrinsic threshold, on which crack closure effects are superimposed in experimentally measured thresholds. At higher crack growth rates, plastic strain gradients were found to reduce crack closure in constant amplitude loading, which led to a slightly increased steady-state growth rate in a numerical study using strain gradient plasticity model based on dislocation densities [18]. Computations using a cohesive zone model coupled to the GND density predicted the delayed retardation phenomenon following an overload [19], well-known from numerous fatigue crack growth experiments [2,20,21]. Regions of enhanced and reduced GND densities in the computations corresponded to the overload-induced acceleration and retardation, respectively. Results in [19]
Corresponding author. E-mail address: [email protected] (T. Siegmund).
https://doi.org/10.1016/j.ijfatigue.2018.11.020 Received 28 August 2018; Received in revised form 7 November 2018; Accepted 21 November 2018 Available online 27 November 2018 0142-1123/ © 2018 Elsevier Ltd. All rights reserved.
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J.D. Pribe et al.
were limited, however, due in part to numerical difficulties that arose outside of a small range of applied loads. Experiments also provide evidence for the relevance of plastic strain gradients in fatigue crack growth. Electron backscatter diffraction measurements have been used by several researchers to infer grain misorientations and GND densities ahead of a fatigue crack [22–24]. These studies have generally observed GNDs at distances ahead of the crack close to classical estimates of the plastic zone size. Using X-ray microdiffraction, the authors in [25] observed large GND densities, along with small voids and cracks, near the surface of a fatigue crack subjected to an overload. However, the relative role of GNDs in comparison to SSDs remains an open question for both steady-state and transient fatigue crack growth. Further investigation is thus warranted to quantify the changes in crack tip fields and cyclic deformation predicted by strain gradient plasticity models relative to classical plasticity. This will in turn determine whether plastic strain gradients must be accounted for in practical fatigue assessments. This study addresses the hypothesis that lengthscale effects in plasticity would not affect predicted steady-state fatigue crack growth rates but would alter predicted crack growth rates when the crack is subjected to load transients, such as an overload. This hypothesis is motivated by the fact that steady-state fatigue crack growth is known to be essentially microstructure insensitive and can be described by linear elastic fracture mechanics under small-scale yielding conditions. On the other hand, the extent of retardation due to an overload depends on the well-known mechanisms of plasticity-induced crack closure and compressive residual stress ahead of the crack tip, both of which are closely tied to the crack-tip plastic zone size and plastic strain magnitudes [20,26,27]. These quantities will be influenced by plastic strain gradient hardening in the large monotonic plastic zone resulting from the overload. Furthermore, it has been argued that post-overload crack growth is governed by near-threshold micromechanisms [21], which, as noted previously, have been connected to GNDs at the crack tip and in the plastic zone. A computational model is used which employs a first-order strain gradient plasticity formulation [12] in conjunction with an irreversible cohesive zone model [28]. Crack growth rates, the evolution of cohesive zone tractions and separations, and plasticity in terms of strains and strain gradients in the bulk material are considered for both constant-amplitude loading and the transient period following a single overload. The effects of plastic strain gradients are then elucidated by analyzing the crack-tip stress, strain, and strain gradient fields.
displacement boundary conditions are applied on the outer nodes, as shown in Fig. 1a. Due to symmetry in mode I loading, a half-model is considered. Vertical displacements are constrained for the nodes on the crack plane ahead of the initial crack tip. Rigid body motion is prevented by Eq. (1), as the horizontal displacement at = is zero. The continuum consists of 3892 linear plane strain elements and 123 cohesive zone elements along the symmetry line. A structured quadrilateral mesh is used near the crack tip, extending in the crack growth direction. The MBL model radius (RMBL ) is 10,000 times larger than the minimum element size. 2.2. Strain gradient plasticity model The conventional theory of mechanism-based strain gradient plasticity (CMSG) [12] is used for the deformation of the solid. This formulation employs Taylor dislocation theory to define the flow stress flow accounting for both strain hardening and strain gradient hardening:
=
flow
f 2 ( p) + l
y
(3)
p.
Here, y is the initial yield strength, f is a non-dimensional function of the equivalent plastic strain p , p is the effective plastic strain gradient, and l is the intrinsic material length for strain gradient plasticity. For face-centered cubic polycrystals, l is given by
l = 18
µ
2
2
b,
(4)
y
where is an empirical prefactor, µ is the shear modulus, and b is the magnitude of the Burgers vector. A value of l on the order of microns corresponds to the experimental lengthscale where size effects are observed [12]. Following [11], p is given by p
=
1 4
p p ijk ijk
p ik , j
+
,
(5)
where p ijk
=
p jk , i
p ij, k .
+
(6)
For a power-law hardening material, f is defined as
f ( p) = 1 +
p
E
n
,
(7)
y
where n is the strain hardening exponent. The CMSG constitutive equation is developed using a viscoplastic limit approach, in which the effective plastic strain rate is given by
2. Model description 2.1. Model geometry
M p
Mode I cyclic loading under plane strain conditions is applied via a modified boundary layer (MBL) model [29]. In this approach, an initially sharp crack and the domain of crack extension are enclosed in a circular domain. Displacements corresponding to the asymptotic cracktip solution are imposed on the boundary, which remains elastic and surrounds the small crack-tip plastic zone. Given a prescribed mode I stress intensity factor KI , the boundary displacements u (t ) are determined from the Williams expansion [30] as
u x ( t , r , ) = K I (t )
r 1+2 cos (3 2 E 2
4
cos ),
(1)
u y ( t , r , ) = K I (t )
r 1+2 sin (3 2 E 2
4
cos ),
(2)
e
= y
f 2 ( p)
+l
=
p
e
M
,
(8)
flow
where is the effective strain rate, and e is the von Mises equivalent stress. A value of M = 75.0 was found to predict the same crack growth rates as a conventional rate-independent plasticity formulation when neglecting plastic strain gradients (i.e. setting l = 0 ). Therefore, M = 75.0 was used for all computations presented here. The CMSG User Material (UMAT) subroutine developed in [12] for the finite element solver ABAQUS was used for this study. 2.3. Irreversible cohesive zone model A cohesive zone model based on the potential in [31] is used to describe the material separation process. For mode I separation, the normal traction Tn is related to the normal separation n by
where E is Young’s modulus, is Poisson’s ratio, and r and are the radial and angular coordinates, with the initial crack tip as the origin. A triangular waveform is prescribed for the mode I stress intensity factor in order to simulate cyclic loading. The finite element mesh used in this study is shown in Fig. 1. The
Tn (
n)
where 284
=
max,0 exp
max,0
1
n
n
0
0
is the initial cohesive strength and
(9) 0
is the characteristic
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J.D. Pribe et al.
Fig. 1. (a) Global and (b) crack-tip mesh in the modified boundary layer model.
separation. The cohesive energy 0
=
0
is
max,0/ y influences the amount of plastic dissipation that accompanies crack growth. Two values of this ratio are considered: max,0/ y = 4.0 and max,0/ y = 4.5. These correspond to cohesive energies of 0 = 20.0 J/m2 and 0 = 25.0 J/m2 for constant characteristic separation 0 = 7.36 nm. These two cases are used to investigate the effect of plastic strain gradients relative to the amount of crack-tip plasticity. The cohesive zone endurance limit and characteristic fatigue separation are chosen as f / max,0 = 0.25 and / 0 = 4.0 , following previous studies of fatigue crack growth in the Paris regime [4,18]. The implications of these parameter choices are considered further in the Discussion. A reference stress intensity factor K 0 is defined from the linear elastic fracture mechanics relation between the energy release rate and mode I stress intensity:
(10)
max,0 0 exp(1).
A damage variable D is introduced to account for the decrease in load-carrying capacity following from cyclic loading [28]. Damage accumulation is modeled as a decrease in effective cross-sectional area corresponding to the current value of D. This can be described by introducing the current effective cohesive strength max : max
=
max,0 (1
D),
D=
Ddt .
(11)
Damage evolution is given by n
D=
Tn max
f
H(
max,0
0 ),
n,acc
n,acc
=
n
dt ,
(12)
where n,acc is the accumulated normal separation and H is the Heaviside function. Two material parameters are introduced in Eq. (12): the characteristic fatigue separation and the cohesive zone endurance limit f . The characteristic fatigue separation scales the increment in material separation and sets the accumulated separation at which the cohesive zone fails. The endurance limit determines the traction below which damage will not accumulate. Damage evolution does not begin until the accumulated material separation exceeds 0 . The damage increment then depends on the normalized separation increment n / , weighted by the traction reduced by the endurance limit. Damage is defined on an average-per-element basis, and the crack tip is taken to be the location where D = 1.0 [28]. When the crack faces are in contact, the cohesive tractions are negative, and no damage accumulation occurs. In this situation, Eq. (9) is modified with a penalty stiffness A = 30 [4] such that, in contact,
Tn (
n)
=A
max,0 exp
1
n
n
0
0
.
E
K0 =
0 2
1
.
(14) 2
K 0 = 1.50 MPa m and 1.68 For 0 = 20.0 J/m and 25.0 MPa m , respectively. The maximum applied stress intensity factor Kmax and stress intensity range K are related to the plane strain monotonic and cyclic plastic zone sizes Rp and R c , respectively [33]: Rp =
1 3
Kmax
Rc =
1 3
K
J/m2 ,
2
(15)
y
2
2
.
y
(16)
All computations presented here were completed under small-scale yielding conditions, with RMBL on the order of 100Rp or greater. The smallest element size in the structured quadrilateral mesh, equal to the minimum incremental crack advance, was 2.5 0 , while crack extension was restricted to the structured mesh region, resulting in a maximum crack extension of amax = 275 0 . Results are depicted for crack extension up to only a = 250 0 since this ensures the crack-tip plastic zone also remains contained in the structured mesh region.
(13)
The irreversible cohesive zone model was implemented as a user element (UEL) subroutine in ABAQUS.
3. Results
2.4. Model parameters
3.1. Steady-state fatigue crack growth
A model material is considered in order to provide general understanding of the role of plastic strain gradients in fatigue crack growth. For all computations, Young’s modulus and Poisson’s ratio are E = 100 GPa and = 0.34 , respectively. The yield strength is y = 0.0025E , with an isotropic hardening exponent of n = 0.1. Two intrinsic material length values are considered: l = 0 (predicts strain hardening only) and l = 10 µm (strain hardening and strain gradient hardening). The value l = 10 µm emerges from Eq. (4) given the properties above, with b on the order of a tenth of a nanometer and around 0.3, and represents an approximate upper bound on the value of l based on previous CMSGbased studies [12,32]. The ratio of the initial cohesive strength to the yield strength
First, steady-state crack growth is considered. Fig. 2 shows the dependence of the predicted normalized crack growth rate d(a/ 0)/dN on the normalized applied stress intensity factor range K / K 0 for combinations of the intrinsic material length l, load ratio R, and initial cohesive strength max,0 . In all cases, a Paris Law-type response emerges from the model, i.e. the predicted response can be described by the equation d(a/ 0)/dN = C ( K / K 0 )m . The exponent m characterizes the slope of the lines in Fig. 2. Table 1 summarizes the predicted m values for each combination of max,0/ y , R , and l. The exponent is found to be quite insensitive to the value of l, with small changes in the predicted steady-state crack growth rate only 285
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Fig. 2. Dependence of the normalized crack growth rate on the normalized stress intensity range for (a)
R
l (µ m)
m
4.0
0.1
0 10 0 10
3.23 3.27 3.19 3.23
0 10 0 10
3.03 3.09 3.03 3.08
0.5 4.5
0.1 0.5
= 4.0 and (b)
max,0 / y
= 4.5.
computations, the crack grows through the entire overload plastic zone, allowing for analysis of the region where the overload most significantly affects crack growth rates. The importance of plastic strain gradients becomes clear in the postoverload crack growth transients, where the computations predict less severe retardation for l = 10 µm than for l = 0 . The retardation was measured using the number of delay cycles Ndelay , defined as the number of cycles required to traverse the overload-affected region relative to a crack under only constant-amplitude loading [2,36]. The value of Ndelay was calculated by applying a linear fit to the region of crack extension far beyond the overload and extrapolating back to the crack length when the overload was applied, as depicted in Fig. 4a. The number of delay cycles for computations with l = 0 and l = 10 µm for each max,0/ y and several values of K / K 0 is shown in Fig. 4b. The predicted Ndelay is always smaller for l = 10 µm , with the effect becoming stronger for higher cohesive strength. Although Ndelay becomes smaller at higher values of K / K 0 , the difference between the predicted values for l = 0 and l = 10 µm remains very nearly constant. The severity of retardation can be further examined through the crack growth rates, shown in Fig. 5 for K / K 0 = 0.35, relative to the number of cycles and the crack extension. Fig. 5a shows that, for l = 10 µm , the model predicts a higher minimum fatigue crack growth rate than for l = 0 for both values of cohesive strength. Fig. 5b provides information about the spatial extent of crack retardation after the overload. Following the peak, the model predicts that the crack growth rate almost immediately falls to the minimum value for l = 0 while the crack growth rate for l = 10 µm remains elevated for the first few elements after the overload is applied. However, the maximum extent of the overload-affected region is not significantly affected by the value of l. The overload-affected region ends at about a/ 0 = 175 for all cases, with the higher cohesive strength ( max,0/ y = 4.5) leading to a small increase. Thus, for l = 10 µm , the model predicts a higher minimum crack growth rate and a lower number of delay cycles, but a similar maximum extent of the overload-affected region, relative to l = 0 . The transient processes in the cohesive zone and in the bulk plastic deformation give insight into the role of plastic strain gradients in postoverload crack growth. In the following results, these processes are analyzed using the computations with K / K 0 = 0.35 as a representative case. Figs. 6 and 7 show a comparison between the steady-state and post-overload traction distributions for and max,0/ y = 4.0 max,0/ y = 4.5, respectively. In each case, the steady-state traction distributions are nearly independent of the value of l, with the main difference being a small amount of crack closure—indicated by negative traction—for l = 0 and max,0/ y = 4.5. This occurs only at the minimum applied load, meaning that the crack remains open for nearly the entire
Table 1 Paris law exponents. max,0 / y
max,0 / y
becoming apparent for high values of K with max,0/ y = 4.5. This produces differences in the predicted value of m on the order of 10 2 . Changing the load ratio R shifts the data with respect to K / K 0 and slightly reduces the exponent m as well. This behavior is seen in experiments [34] and in other numerical studies with cohesive zone models [35]. 3.2. Fatigue crack growth following an overload Constant-amplitude loading interrupted by a single overload is now considered. Each computation was conducted with load ratio R = 0.1 and consisted of the following steps. First, constant-amplitude loading was applied with K / K 0 = 0.28, 0.35, or 0.40. The overload was applied in the cycle after the crack extension exceeded a/ 0 = 75, such that steady-state conditions developed prior to the overload. The overload ratio was Kmax,OL/ Kmax = 1.73, where Kmax,OL is the maximum applied stress intensity factor during the overload. This corresponds to applied energy release rate ratio Gmax,OL/ Gmax = 3.0 . Fig. 3 shows the predicted crack extension for the computations with a single overload and different steady-state K / K 0 values. Predictions for both l = 0 and l = 10 µm are shown. All computations predict a brief transient period while crack growth begins from the initial crack tip, followed by steady-state crack growth where the crack growth rate is insensitive to the value of l. During the overload, the computations predict a brief acceleration followed by crack growth retardation in the following cycles. For large crack extensions, the crack growth rate is predicted to nearly return to the original steady-state value. The slightly lower crack growth rate for longer crack lengths is a known feature of the modified boundary layer model [4], as the assumptions behind Eqs. (1) and (2), i.e. that the displacements correspond to a constant K , begin to be violated. In the present 286
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Fig. 3. Crack extension with a single overload for l = 0 (solid lines) and l = 10 µ m (dashed lines) with cohesive strength (a) ratio is R = 0.1, with several cases of steady-state K / K 0 .
cycle. During and after the overload cycle, the model predicts higher tractions and therefore more damage accumulation, per Eq. (12), and a lower magnitude of crack closure for l = 10 µm . This is most pronounced in the region where compressive tractions occur at minimum load, which corresponds to the overload-affected region where crack growth rates are significantly reduced. The evolution of cohesive zone tractions in the overload-affected region is summarized using the values of the maximum and minimum traction in the cohesive zone in each cycle, Fig. 8. For a given cycle, this corresponds to the maximum traction at point “a” and the minimum traction at point “c” as depicted in Figs. 6 and 7. The region of interest is restricted to the cohesive zone ahead of D = 1.0 in order to gain insight into the material separation process. The maximum traction provides a measure of Ktip , the effective maximum stress intensity at the crack tip, suggested by Willenborg et al. [27] as the controlling parameter for retardation following an overload. The minimum traction is a measure for the magnitude of crack closure, which is associated with K op , the stress intensity factor required to open the crack [5,20]. After the overload, for l = 10 µm , the model predicts higher maximum and minimum traction in each cycle. The predicted level of crack closure is therefore reduced when plastic strain gradients are accounted for, while
max/ y = 4.0
and (b)
max,0 / y
= 4.5. The load
the crack-tip stress intensity is increased. This effect persists for a larger number of cycles for the higher cohesive strength value due to the lower crack growth rates. The increased tractions are associated with a reduction in compressive residual stresses induced by the overload and correspond to the higher minimum crack growth rate seen for l = 10 µm . Fig. 9 shows representative separation distributions for steady-state crack growth and the overload cycle. Plastic strain gradients have virtually no effect on the separation during steady-state crack growth. At maximum load in the overload cycle for l = 10 µm , the model predicts elevated separation in the cohesive zone compared to l = 0 , which leads to additional damage accumulation. At minimum load, the crack faces are in contact ahead of the location where D = 1.0 . The length of the region where closure occurs is quite independent of the value of l, while the magnitude of closure is reduced for l = 10 µm . The predicted separation behind the crack tip is also lower for l = 10 µm at both maximum and minimum load in the overload cycle, which is consistent with previously reported computations with cracks in monotonically loaded structures where strain gradient plasticity models predict less blunting than classical plasticity models [14,37]. Figs. 10a and b show the spatial distribution of the accumulated
Fig. 4. (a) Sample calculation of number of delay cycles. (b) Number of delay cycles for l = 0 (solid lines) and l = 10 µm (dashed lines). 287
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Fig. 5. Dependence of normalized crack growth rates for l = 0 (solid lines) and l = 10 µ m (dashed lines) on (a) cycles after the overload (OL) cycle and (b) normalized crack extension. The load ratio is R = 0.1, with K / K 0 = 0.35 .
plastic strain in the structured-mesh region for each value of l. The outer contour, at p = 0.001, marks the edge of the plastic zone. Initially, a steady-state plastic zone forms as the crack grows at a constant rate. The overload then induces an increase in the plastic zone size and maximum plastic strain magnitude, which is mostly ahead of the location where D = 1.0 at the start of the overload cycle. No further crack-tip plasticity is generated for 40 cycles and 36 cycles after the overload for l = 0 and l = 10 µm , respectively, as the most significant crack growth retardation occurs. At the crack extension shown in Fig. 10, a/ 0 = 245, the steady-state plastic zone has re-formed. This occurs at 66 cycles and 59 cycles after the overload for l = 0 and l = 10 µm , respectively. While the plastic zone sizes in the steady state and at the overload for each value of l are quite similar, the maximum plastic strain magnitude in the overload plastic zone is reduced by a factor of about 1.6 for l = 10 µm relative to l = 0 . This reduced plastic strain magnitude in the bulk means that more separation occurs in the cohesive zone following the overload, leading to enhanced damage accumulation. This is responsible for the higher minimum crack growth rate and the reduced number of cycles required for the steady-state plastic zone to re-develop for l = 10 µm .
Fig. 10c shows the spatial distribution of plastic strain gradients for l = 10 µm . Similar to the plastic strain, the plastic strain gradients reach a steady-state distribution until the overload induces an instantaneous increase in the plastic strain gradient magnitude. The regions where plastic strain gradients are highest correspond to regions of restricted plastic deformation in Fig. 10b relative to Fig. 10a. In the overloadaffected region, the maximum value of p is 82 mm−1, while the maximum value of p during steady-state crack growth is 24 mm−1. A more quantitative description of the role of plastic strain gradients in the hardening response is based on Eq. (3). If isotropic hardening is neglected (i.e. the strain hardening exponent n is set to 0), the flow SG stress associated with strain gradient hardening alone, flow , is SG flow y
=
1+l
p.
(17)
SG / y = 1.05. During The outer contour in Fig. 10c corresponds to flow SG steady-state crack growth, the largest value of flow / y is about 1.11, SG / y is while, in the overload-affected region, the largest value of flow about 1.35.
Fig. 6. Traction distribution in the cohesive zone for l = 0 (solid lines) and l = 10 µ m (dashed lines) during (a) steady-state crack growth and (b) the overload cycle. The load ratio is R = 0.1, with K / K 0 = 0.35 and max,0/ y = 4.0 . The insets depict the points at which the traction distribution is shown. x = 0 corresponds to D = 1.0 for point “a”. 288
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Fig. 7. Traction distribution in the cohesive zone for l = 0 (solid lines) and l = 10 µ m (dashed lines) during (a) steady-state crack growth and (b) the overload cycle. The load ratio is R = 0.1, with K / K 0 = 0.35 and max,0/ y = 4.5. The insets depict the points at which the traction distribution is shown. x = 0 corresponds to D = 1.0 for point “a”.
Two measures that allow for comparisons across values of l, max,0/ y , and steady-state K / K 0 are now defined. Fig. 11 shows the maximum plastic strain magnitude in the steady-state and overload plastic zones for each computation. For l = 10 µm , the model predicts significantly reduced plastic strains in the overload plastic zone for all cases, with the effect becoming more significant for the higher cohesive strength. This leads to more deformation in the cohesive zone and corresponds to the higher separation, traction, and crack growth rates seen previously for l = 10 µm . While the absolute magnitude of the plastic strain increases with increasing K / K 0 , the relative effect of increasing l does not significantly change. The plastic strain in the steady-state plastic zone is also fairly insensitive to l. Fig. 12 shows the length of the cohesive zone that is closed at minimum load in the cycles following the overload for each computation, denoted by Lclosure . While this quantity increases with increasing cohesive strength and stress intensity range, it shows only a small dependence on the value of l. In fact, Lclosure differs between each value of l by a maximum of one element length. This is consistent with evidence from the crack growth rate, traction and separation distributions, and plastic zone size that the maximum extent of the overload-affected
region does not depend on l. 4. Discussion The computations presented here show that hardening due to plastic strain gradients significantly affects numerical predictions of fatigue crack growth following an overload. Due to restricted plastic deformation in the overload plastic zone, there is increased dissipation in the cohesive zone, seen in the elevated values of traction and separation predicted for l = 10 µm that persist throughout the overload-affected region and lead to higher crack growth rates relative to l = 0 . The relative increase in crack growth rates is amplified for computations with a higher cohesive strength, which is associated with more crack-tip plasticity. Changing the applied load does not significantly alter the model predictions for l = 10 µm relative to l = 0 . The computations thus support the hypothesis that plastic strain gradient hardening is important in transient fatigue crack growth situations but does not significantly affect steady-state fatigue crack growth. The irreversible cohesive zone model implemented in this study assumes that fatigue crack growth is a stress- and deformation-
Fig. 8. Maximum and minimum traction in the cohesive zone for each cycle relative to the overload (OL) with (a) is R = 0.1, with K / K 0 = 0.35 . 289
max,0 / y
= 4.0 and (b)
max,0 / y
= 4.5. The load ratio
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Fig. 9. Normalized separation distribution in the cohesive zone for l = 0 (solid lines) and l = 10 µ m (dashed lines) at (a) maximum applied load and (b) minimum applied load. The load ratio is R = 0.1, with K / K 0 = 0.35 and max,0/ y = 4.0 . The inset in (b) shows a zoom of the region where the cohesive surfaces are in contact, and x = 0 corresponds to D = 1.0 .
Fig. 10. (a) Accumulated plastic strain ( p ) contours for l = 0; (b) Accumulated plastic strain ( p ) contours for l = 10 µ m; and (c) Plastic strain gradient ( p) contours for l = 10 µ m. The superimposed solid lines mark the current crack extension ( a/ 0 = 245) while the dashed lines mark the crack extension at the start of the overload cycle ( a/ 0 = 75 ). The load ratio is R = 0.1, with K / K 0 = 0.35 and max,0/ y = 4.0 .
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Fig. 11. Maximum accumulated plastic strain in the steady-state and overload plastic zones for computations with (a) lines are for l = 0 ; dashed lines are for l = 10 µm .
max,0 / y
= 4.0 and (b)
max,0 / y
= 4.5. Solid
parameters are associated with experimental observations of crack retardation [2,20,27] and finite element studies of crack extension [13,45,46]. To determine K op , the time step at which the cohesive zone separation became positive (i.e. the crack opened) at all points ahead of D = 1.0 was recorded for each cycle after the overload. The applied stress intensity factor at this time step was then determined, which corresponds to an estimate of K op . Although the precise value of K op determined through this process is dependent on the time incrementation, trends with respect to the plastic strain gradient effects can still be observed. For each loading case in this study, the model predicts K op for l = 10 µm is 20% to 30% less than for l = 0 in the region of maximum crack growth retardation. For example, with K / K 0 = 0.35 and max,0/ y = 4.5, K op/ Kmax = 0.5 for l = 0 while K op/ Kmax = 0.4 for l = 10 µm at the minimum post-overload crack growth rate. This indicates that plastic strain gradient hardening has a meaningful effect on post-overload crack closure through its role in the plastic strain distribution. As a consequence of this analysis, a useful conceptual question emerges: can the effect of plastic strain gradients following the overload be replicated by increasing the bulk yield strength relative to the cohesive strength? The predicted crack growth rates in Fig. 13 indicate
Fig. 12. Length of the region where crack closure occurs following the overload. Solid lines are for l = 0 ; dashed lines are for l = 10 µm .
dependent process. The damage evolution law depends on scaled measures of both traction and separation, which is seen in discrete dislocation models of fatigue crack growth [38], other studies using cohesive zone models [6], and analytical models that include both K and Kmax in the crack growth law [39]. The Kmax dependence is associated with ductile damage in metals, such as micro- and nano-voids ahead of the crack tip, which have produced dimpled damage zones corresponding to the region where acceleration occurs after an overload [40,41]. The current study indicates that plastic strain gradients aid in understanding this process, as they promote the higher stresses required to activate ductile damage mechanisms during the overload cycle. This leads to elevated crack growth rates over a larger region relative to computations with l = 0 , as observed in Fig. 5b. This is similar to arguments for strain gradient hardening as the mechanism by which the stress ahead of cracks in monotonic loading situations can reach the high values required for cleavage cracking, even in ductile materials undergoing significant plastic deformation [14,42–44]. A key finding is that plastic strain gradients do not alter the overload plastic zone size, but are critical in predicting the plastic strain magnitude and distribution in the overload plastic zone. Along with the predicted tractions shown in Fig. 8, this indicates that plastic strain gradients should be considered in predictions of Ktip and K op . These
Fig. 13. Predictions of the crack growth rate for l = 10 µm compared with predictions for l = 0 with varied initial yield strength. The load ratio is R = 0.1, with K / K 0 = 0.28 . 291
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that the answer to this question is no—that is, the effect of plastic strain gradients is unique. Here, two additional computations were conducted with higher bulk yield strengths ( y = 260 MPa and y = 270 MPa) and no strain gradient hardening. The cohesive strength was fixed at 1125 MPa. While increasing the initial yield strength can capture either the minimum crack growth rate or the number of cycles in the overloadaffected region predicted by the strain gradient plasticity model, it cannot capture both. This is because hardening due to plastic strain gradients is highly localized at the crack tip, where the most extreme gradients appear as seen in Fig. 10c. The role of this localized hardening in predicted crack growth rates cannot be captured by simply recalibrating the parameters of the classical plasticity model or the cohesive zone. It should be noted that the magnitude of the predicted steady-state crack growth rates in this study is on the order of 10 8 to 10 7 m, which corresponds to the lower end of the Paris law regime. The applied K values necessary to achieve these crack growth rates for the model material considered here are similar to those found in discrete dislocation studies but are an order of magnitude below those of typical alloys [34,47,48]. This is due to the choice of cohesive zone damage parameters ( f , max,0 , and ), which allow for computations of the Paris law regime and of the response to load transients with reasonable computational expense. Extensions of the current study could involve calibrating these parameters to a specific fatigue threshold and crack growth rates. Such computations would face significantly increased cost: the finite element mesh must be small enough—on the order of the size used here and in strain gradient plasticity-based studies of fracture [32,43,49]—to resolve plastic strain gradients and the active cohesive zone, while the crack extension and number of cycles required to achieve a stabilized plastic wake scales with the increasing plastic zone size [46]. Possible roadmaps for this type of study are discussed for cohesive zone models in [35] and in the context of discrete dislocation models of fatigue crack growth in [47]. Further extensions of this study could consider large-scale plasticity or small cracks, in which plastic strain gradients may indeed be important even in constant-amplitude loading. In addition, the isotropic hardening law used here is expected to give acceptable results for growing cracks, but the model could be extended to kinematic hardening, which would be necessary for considering crack initiation.
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5. Conclusions Accurate predictions of fatigue crack growth are essential to ensure the reliability and safety of engineering structures. Such computations are typically preformed with classical plasticity formulations. This study indicates that the use of classical plasticity alone for analyzing steady-state fatigue crack growth is sufficient for the small-scale yielding problem considered here. The Paris Law response predicted by the strain gradient plasticity model is virtually the same as that predicted by the classical plasticity model. However, the analysis of overloads and the subsequent fatigue crack growth transients would require the consideration of strain gradient plasticity. Large plastic strain gradients are predicted to significantly restrict plastic deformation in the overload plastic zone. This leads to reduced crack closure and a higher minimum crack growth rate during the overload-induced retardation, relative to the classical plasticity model alone. This indicates that classical plasticity models underpredict crack growth rates under transient conditions. Acknowledgment This research was supported by the U.S. Department of Energy, National Energy Technology Laboratory, under contract No. DEFE0011796. 292
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