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Crack-tip-opening-displacement-based description of creep crack border fields in specimens with different geometries and thicknesses
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Crack-tip-opening-displacement-based description of creep crack border fields in specimens with different geometries and thicknesses Pengfei Cui, Wanlin Guo∗ State Key Laboratory of Mechanics and Control of Mechanical Structures, Key Laboratory of Intelligent Nano Materials and Devices, Institute of Nanoscience of Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
a r t i c l e
i n f o
Article history: Received 21 June 2019 Revised 18 August 2019 Accepted 1 October 2019 Available online xxx Keywords: Creep crack border fields Creep stress intensity factor Kδ ( t ) -Tz Geometry effect Constraints δ (t)-Tz description
a b s t r a c t Creep crack growth is accompanied by strong nonlinear deformation and stress relaxation. Finding an appropriate parameter to characterize the crack tip fields, as well as the fracture resistance, has long been a challenge. Most previous studies were performed within the framework of the C(t)-integral, which is limited to the assumptions of small deformation and simple proportional loading. By using the crack tip opening displacement (CTOD), we propose a new creep stress intensity factor Kδ ( t ) -Tz consisting of the time-dependent CTOD, δ (t) and the out-of-plane stress constraint factor Tz to characterize the three-dimensional creep crack tip fields. Four typical specimens, single-edge cracked tension specimens, compact specimens, centre-cracked tension specimens and single-edge-notched bending specimens under three-point bending are comprehensively analysed using the power-law creeping model and threedimensional finite element analyses. It is found that under both small-scale and large-scale creep conditions, the change in Kδ ( t ) -Tz along the thickness direction for different specimens is within 8.6%, whereas the change in C(t) can exceed 400%, showing that Kδ ( t ) -Tz is a stable parameter that governs the creep crack tip fields. With the exception of the centre-cracked tension specimens under large-scale creep conditions, good agreements are obtained between the two-parameter description δ (t)-Tz of crack border stress fields with the three-dimensional finite element results under small-scale and large-scale creep conditions. These results indicate that the CTOD-based two-parameter description δ (t)-Tz can be taken as the basis of creep fracture criteria. © 2019 Published by Elsevier Ltd.
1. Introduction In the last several decades, many efforts have been made to find an appropriate parameter to characterize the creep cracktip fields, as well as the fracture resistance. Although many achievements for the creep fracture of materials have been made, most studies were performed within the framework of the C(t)integral (C(t)), which is analogous to the J-integral (J) in elasticplastic solids (Riedel and Rice, 1980; Budden and Ainsworth, 1999; Nguyen et al., 20 0 0a, 20 0 0b; Chao et al., 20 01; Wang et al., 2010; Xiang et al., 2011; Shlyannikov et al., 2015; Shlyannikov and Tumanov, 2018a,2018b). With the C(t) as the dominating parameter, Riedel and Rice (1980) first described the singular cracktip fields under plane stress and ideal plane strain conditions for power-law creeping solids, resulting in the well-known RR solution. Considering the in-plane constraint effect, Budden and Ainsworth (1999) and Chao et al. (2001) further introduced the
∗
Corresponding author. E-mail address: [email protected] (W. Guo).
higher-order terms to the RR asymptotic solution in the form of the parameters Q and A2 , respectively. Similarly, A∗ 2 and σ ∞ were also introduced by Nguyen et al. (20 0 0a, 20 0 0b) to consider the effects of specimen geometry and loading modes at the steady power-law creeping stage. In recent works, Wang et al. (2010) proposed a tensile stress related parameter R to study the effect of constraint caused by crack depth in compact tension specimens. The three-dimensional (3D) asymptotic solution for creep crack border fields has been obtained by Xiang et al. (2011), with the out-of-plane stress constraint factor Tz (Guo, 1993a, 1993b, 1995) being considered. By investigating the RR-type singular solution, Shlyannikov et al. (2015) proposed a single parameter, the stress intensity factor Kcr , to characterize the crack growth resistance in creeping solids. As C(t) is based on deformation theory, it has a limited ability to describe the large deformation near the crack border under large-scale creep yielding conditions and with strong accompanying stress relaxation. Considering these properties of creeping solids, a comprehensive study of crack tip deformation is important to understand creep crack border fields under 3D constraints.
https://doi.org/10.1016/j.ijsolstr.2019.10.001 0020-7683/© 2019 Published by Elsevier Ltd.
Please cite this article as: P. Cui and W. Guo, Crack-tip-opening-displacement-based description of creep crack border fields in specimens with different geometries and thicknesses, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019.10.001
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Fig. 1. Theory and fitting results of (f-2) for different n and Tz .
Crack tip opening displacement (CTOD) can quantify large local deformations without the assumptions of small deformation and simple proportional loading for C(t) and J (Newman et al., 2003; Han et al., 2014; Antunes et al., 2016; Zhu, 2017). Newman et al. (2003) showed that the K-factor, J-integral and CTOD have equivalence relations in linear-elastic or small-scale elastic-plastic solids, while CTOD can be further extended into large-scale plastic yielding conditions. To quantify the crack-tip
constraint of elastic-plastic solids, Han et al. (2014) introduced the parameter CTODC through a Gurson-type void model. By detailed finite element (FE) analysis, the parameter CTOD is proven to be independent of the stress ratio to characterise the fatigue crack growth (Antunes et al., 2016). Zhu (2017) measured the lowconstraint fracture toughness of pipeline steels experimentally using the CTOD approach. An appropriate governing parameter should quantify the deterioration of creeping solids near the crack tip in both simulations and experiments. Due to its ability to represent the nonlinear deformation ahead of the crack tip, CTOD has the potential to characterise the creep crack border fields. Analogous to the relation between J and C(t), some modifications are made for CTOD with the creep time term. In addition, the factor Tz proposed by Guo (1993a, 1993b, 1995) to consider the out-of-plane stress constraint effects in elastic-plastic solids has been applied effectively to characterise the 3D crack border fields together with C(t) and Q∗ in creeping solids (Xiang et al., 2011; Xiang and Guo, 2013; Guo et al., 2018). Thus, to describe the 3D creep crack border fields, Guo’s constraint theory can be further developed with the CTOD approach. In this work, the CTOD approach and Guo’s constraint theory are extended to characterise the crack border fields in creeping solids and validated by comprehensive 3D FE analyses of the typical single-edge-notched bending specimens under three-point bending (SENB), compact specimens (CT), centre-cracked tension specimens (CCT) and single-edge cracked tension specimens (SECT) using the power-law creeping model. To characterise the 3D creep crack-tip fields, a creep stress intensity factor Kδ ( t ) -Tz consisting of
Fig. 2. Schematic of specimen (a) CCT; (b) SECT; (c) SENB; (d) CT.
Please cite this article as: P. Cui and W. Guo, Crack-tip-opening-displacement-based description of creep crack border fields in specimens with different geometries and thicknesses, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019.10.001
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Fig. 3. (a) FE models of a CCT (1/8 model), SENB (1/4 model) and SECT (1/4 model) specimens; (b) FE model of a CT specimen (1/4 model).
Fig. 4. Variations in CTOD along thickness direction for different specimens with creep time. (a) CCT specimens; (b) SECT specimens; (c) SENB specimens; (d) CT specimens.
Please cite this article as: P. Cui and W. Guo, Crack-tip-opening-displacement-based description of creep crack border fields in specimens with different geometries and thicknesses, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019.10.001
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Fig. 5. Variations in Tz on the ligament (θ = 0) along the thickness direction for different specimens with creep time at t∗ = 0.1 (nontransparent) and t∗ = 10 (transparent).
the time dependent CTOD, δ (t), and the out-of-plane stress constraint Tz is proposed and systematically evaluated. The results show that the distributions of Kδ ( t ) -Tz along the thickness direction are nearly uniform in all specimens under both small-scale and large-scale creeping conditions. With the exception of CCT specimens under large-scale creep conditions, the two-parameter δ (t)-Tz description of the creep crack border stress fields is proven to be highly efficient for different specimens, especially for large-scale creeping deformation. The results show that the CTOD-based 3D stress intensity factor Kδ ( t ) -Tz and the two-parameter δ (t)-Tz description have advantages over the C(t)-based approaches.
2. Constraint parameters for three-dimensional creeping problems
C (t ) =
n+1
σ33 (1, 2, 3 ) = (x, y, z )or (r, θ , z ), σ11 + σ22
Tz =
(3)
3 2
(4)
σi j ε˙ i j n1 − σi j n1 u˙ i,1 ds,
u˙ i j =
3 n n( f −2)+1 ˜ AK r u˙ i j (θ , Tz , n ), 2
(5)
where
(1)
where and ds are the integral path and its length, respectively; n and n1 are the creep exponent constant and unit outward normal to , respectively; and σ ij , u˙ i,1 and ε˙ i j are the components of stress, displacement gradient rate and strain rate, respectively. Under ideal plane strain and plane stress conditions, C(t) is pathindependent and can quantify the intensity of crack tip fields effectively (Riedel and Rice, 1980; Budden and Ainsworth, 1999; Chao et al., 2001).
(2)
where, as shown in Figs. 2 and 3, o is a point of the crack front line, and the subscripts (1, 2, 3) stand for the (x, y, z) Cartesian coordinates or the (r, θ , z) Cylindrical coordinates, respectively. z is the tangent to the crack front line at the point o, and the corresponding normal plane is the x-y plane in the Cartesian coordinate system or the r-θ plane in the Cylindrical coordinate system. By using the amplitude coefficient K, Xiang et al. (2011) developed the functions of stress, strain rate and displacement rate in the forms of
ε˙ i j = AK n rn( f −2) ε˜˙ i j (θ , Tz , n ),
C(t) is the governing parameter for characterisation of the intensity of creep crack-tip fields (Bassani and McClintock, 1981) and can be evaluated in the form of
n
Tz is the out-of-plane stress constraint factor proposed by Guo (1993a, b, 1995)
σi j = K r f −2 σ˜ i j (θ , Tz , n ),
2.1. C(t)-integral
2.2. Creep stress intensity factor kδ ( t ) -tz
K=
C (t ) AI (Tz , n )r ( f −2)(n+1)+1
1/(n+1) ,
(6)
r and θ are polar coordinates centred at the crack border, and θ = 0 corresponds to the ligament directly ahead of the crack border; A is a creep material constant and A = ε˙ 0 /σ0n , with ε˙ 0 and σ 0 being the initial yield strain rate and stress, respectively; K is an amplitude coefficient; (f-2) is the order of stress singularity; I(Tz ,n) can be seen as the function of Tz and n; and σ˜ i j , u˜˙ θ and ε˜˙ θ are the angular distribution functions of stress, displacement rate and strain rate, respectively.
Please cite this article as: P. Cui and W. Guo, Crack-tip-opening-displacement-based description of creep crack border fields in specimens with different geometries and thicknesses, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019.10.001
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Fig. 6. Creep stress intensity factors Kδ ( t ) -Tz and C(t) with creep time for different specimens.
In the previous work of Xiang et al. (2011), the order of creep time t is 1/(n + 1) for displacement near the crack-tip. Analogous to the displacement rate in Eq. (5), the displacement near the crack tip can be assumed in the form of
ui j =
3 n n( f −2)+1 ˜ AK r u˙ i j (θ , Tz , n )(n + 1 )t 1/(n+1) . 2
(7)
By using the 90° definition, CTOD can be expressed as
δt = 2uθ |θ =π ; r=uθ .
(8)
Substituting Eq. (7) into Eq. (8), the amplitude coefficient K is obtained as
K=
δ (t )
1/n ,
3(n + 1 )Au˜˙ θ (π , Tz , n )
(9)
Please cite this article as: P. Cui and W. Guo, Crack-tip-opening-displacement-based description of creep crack border fields in specimens with different geometries and thicknesses, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019.10.001
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P. Cui and W. Guo / International Journal of Solids and Structures xxx (xxxx) xxx Table 1 Variations in Kδ ( t ) -Tz and C(t) along thickness direction at t∗ = 0.1 for different specimens.
. ∗
C(t) and Kδ ( t ) -Tz both are decreasing monotonically from the mid-plane (z/B = 0) to the surface plane (z/B = 0.5) as shown in Fig. 7. Thus, the midplane and surface plane are chosen to calculate the change of C(t) and Kδ ( t ) -Tz along the thickness direction. The first column is the change along
−C (t )min × 100%; the thickness direction and the second column is the change for different specimens. The change is defined as change|C (t ) = C (t )max C (t )
change|Kδ (t )−T z =
Kδ (t )−T z,max −Kδ (t )−T z,min Kδ (t )−T z,min
min
× 100%.
where
δ (t ) = t˜ =
Analogously, Kδ ( t ) -Tz under large-scale creep conditions can be obtained as follows:
(δt )−( f −2) t˜1/n
.
(10)
t t1 , t1
is unit time (h). Rearranging Eq. (9) further, the dimensionless creep stress intensity factor Kδ ( t ) -Tz consisting of δ (t) and Tz can be defined as follows:
Kδ (t )−Tz =
δ (t )
1/n , σ0 L−( f −2) 3(n + 1)Au˜˙ θ (π , Tz , n )
where L is the unit length (mm).
(11)
Kδ (t )−Tz =
σ0
L−( f −2)
δ (t )
1/n .
3Au˜˙ θ (π , Tz , n )
(12)
In this work, we focus on characterization of the 3D stationary crack border fields for the SECT, CCT, CT and SENB specimens for undamaged power-law creeping solids. To estimate the scale during creeping, Ehlers and Riedel (1981) proposed the parameter transition time tT under plane strain conditions in the form of
tT =
J
(n + 1 )C ∗
,
(13)
where C∗ is a constant, approximately equal to the value of C(t) under large-scale creep conditions. The J-integral is obtained
Please cite this article as: P. Cui and W. Guo, Crack-tip-opening-displacement-based description of creep crack border fields in specimens with different geometries and thicknesses, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019.10.001
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Table 2 Variations in Kδ ( t ) -Tz and C(t) along thickness direction at t∗ = 20 for different specimens.
B
z/B
Kδ(t)-Tz
0.5
0.282
0
0.306
10
0
0.291
20
0
0.290
0.5
0.226
0
0.237
8
0
0.227
10
0
0.223
0.5
0.223
0
0.230
2
0
0.239
10
0
0.220
0.5
0.245
0
0.266
4
0
0.272
16
0
0.254
(mm) 4 CCT
4 SECT
5 SENB
10 CT
C(t)
change*
(%)
(MPa·mm·h -1)
(%)
9.13E-4
8.5
2.39E-3 5.5
161.8
2.21E-3
24.5
1.92E-3 7.97E-5
4.9
395
3.94E-4 6.3
2.91E-4
57
2.51E-4 3.39E-5
3.1
1.62E-4 8.6
378.5
1.95E-4
68.1
1.16E-4 1.43E-4
8.6
7.22E-4 7.1
405.1
7.91E-4
36.6
5.79E-4
from the linear elastic analysis step. Based on the works of Xiang et al. (2011), the explicit formula of u˜˙ θ (π , Tz , n )at n = 5 and (f-2) for 2< n <13 can be obtained as follows:
u˜˙ θ (π , Tz , n ) = −1.52 − 5 × 10−4 Tz −3.24(Tz )2 +19.1(Tz )3 − 35.87(Tz )4 +20.93(Tz )5 ,
change*
C3 = −8.15 + 5.65n − 0.75n2 + 3.69 × 10−2 n3 − 5.77 × 10−4 n4 , C4 = 21.2 − 14.3n+1.89n2 + 9.36 × 10−2 n3 +1.47 × 10−3 n4 , C5 = −18.9 + 11.8n − 1.53n2 + 7.48 × 10−2 n3 − 1.14 × 10−3 n4 . (16)
(14) 3. Characterisation of the 3D creep crack border fields
f − 2 = C0 + C1 Tz + C2 (Tz ) + C3 (Tz ) + C4 (Tz ) + C5 (Tz ) , 2
3
4
5
(15)
where the adjusted coefficient of determination (Adj. R-Square) of Eq. (14) is 0.9997. f-2 obtained by Eq. (15) and a 3D asymptotic solution is shown in Fig. 1. The fitting results are in good agreement with the theory results.
C0 = −0.53 + 0.14n − 1.85 × 10−2 n2 + 1.2 × 10−3 n3 − 2.99 × 10−5 n4 , C1 = −3.14 × 10−2 + 4.38 × 10−2 n − 6.93 × 10−3 n2 + 4.34 × 10−4 n3 − 1.1 × 10−5 n4 , C2 = 1.19 − 0.82n+0.11n2 − 5.27 × 10−3 n3 +8.47 × 10−5 n4 ,
By replacing K with Kδ (t )−Tz in Eqs. (4), (5) and (6), the 3D creep crack border fields can be obtained as follows:
σi j = Kδ (t )−Tz r˜ f −2 σ˜ i j (θ , Tz , n ), σ0 3 2
(18)
3 n AK r n( f −2)+1 u˜˙ i j (θ , Tz , n ), 2 δ (t )−Tz
(19)
ε˙ i j = AKδn(t )−Tz r˜n( f −2) ε˜˙ i j (θ , Tz , n ), u˙ i j =
(17)
where r˜ = Lr .
Please cite this article as: P. Cui and W. Guo, Crack-tip-opening-displacement-based description of creep crack border fields in specimens with different geometries and thicknesses, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019.10.001
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Fig. 7. Variations in Kδ ( t ) -Tz and C(t) along the thickness direction for different specimens at different creep time.
For mode I crack, the stress fields on the ligament (θ = 0) near the creep crack border are our concerns. Based on the previous works of Xiang et al. (2011) and Xiang and Guo (2013), the explicit formulae of σ˜ θ θ (n,θ ,Tz ) at n = 5 and θ = 0 have been further obtained:
σ˜ θ θ = 1.14 + 1.39Tz − 1.57(Tz )2 + 59.79(Tz )3 − 200.42(Tz )4
+ 186.48(Tz )5 ,
σrr = (0.301Tz +0.609 )σθ θ ,
(20)
(21)
where the adjusted coefficient of determination (Adj. R-Square) of Eq. (20) is 0.9998.
Please cite this article as: P. Cui and W. Guo, Crack-tip-opening-displacement-based description of creep crack border fields in specimens with different geometries and thicknesses, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019.10.001
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Fig. 8. Comparisons of tensile stress and radial stress obtained by δ (t)-Tz solution, δ (t)-Tz -Q∗ solution, 3D FE results and RR solution for SECT specimens at t∗ = 0.1 and t∗ = 10.
Analogous to C(t)-Tz -Q∗ theory proposed by Xiang et al. (2011), the three-parameter δ (t)-Tz -Q∗ description is proposed as follows:
where
Q∗ =
σi j = Kδ (t )−Tz r˜ f −2 σ˜ i j (θ , Tz , n )+Q ∗ σ0 δi j , σ0
(22)
σi j
FE
−
σi j
σ0
δ (t )−Tz
atθ = 0andr = 2.
(23)
Xiang et al. (2011) derived the dimensionless length consisting of C(t), the initial yield stress σ 0 and creep t in creeping solids under small-scale and large-creep conditions. The dimensionless
Please cite this article as: P. Cui and W. Guo, Crack-tip-opening-displacement-based description of creep crack border fields in specimens with different geometries and thicknesses, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019.10.001
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Fig. 9. Variations in Q∗ along the thickness direction for SECT specimens at t∗ = 0.1 and t∗ = 10.
length can be unified by the CTOD (δ t ) and dn at different creep times.
⎧ ⎨ C (t )t (n + 1 )(n+1)/nt˜1/n /σ0 , δt = = dn ⎩ C ∗ t t˜1/n /σ , 0
(small − scalecreep), (large − scalecreep), (24)
dn = 3
3 1/n 2
ε˜˙ 01/n
(n+1 )/n
u˜˙ θ (π , Tz , n ) I (Tz , n )
,
(25)
ε˜˙ 0 is the dimensionless initial yield strain rate. I(Tz ,n) is obtained by Xiang et al. (2011). For better applications, here I(Tz ,n) is further formulated in the form of
I (Tz , n ) =
I1 + I2 Tz + I3 (Tz )2 I4 + I5 Tz + I6 (Tz )2 + I7 (Tz )3
(0 ≤ Tz ≤ 0.3 ) (0.3 ≤ Tz ≤ 0.5 ),
(26)
where
I1 I2 I3 I4 I5 I6 I7
= 5.16 − 0.6n + 0.058n2 − 0.002n3 , = −0.89 + 3.82n − 0.44n2 + 0.015n3 , = 20.52 − 9.9n+0.66n2 − 0.016n3 , = −136.57+67.37n − 8.1n2 +0.286n3 , = 1092.49 − 505.78n+60.08n2 − 2.1n3 , = −2696.23 + 1217.42n − 144.15n2 + 5.02n3 , = 2164.17 − 954.96n+113.15n2 − 3.92n3 .
(27)
σ 0 = 417 MPa, (Yang et al., 1996). Three different thicknesses are adopted for four typical specimens, and the other geometrical parameters are W = 20 mm, a/W = 0.5, and H/W = 2, as shown in Fig. 2. The values of remote tensile stress, which are constant, are 55 MPa, 20 MPa, and 30 MPa for CCT, SECT and CT specimens, respectively. The applied loadings are 0.1, 0.25, and 0.5 kN for the 2-mm, 5-mm and 10-mm thick SENB specimens by an analytical rigid body, respectively. Further, the entire creep time and maximum time increment for every step in the simulation are 50 0 0 h and 100 h, respectively. C(t) of the twentieth contour at t = 50 0 0 h is deemed to be steady (C∗ ). The creep scale conditions along the thickness direction are estimated by the mid-plane (z/B = 0) with the transition time tT . Due to the geometrical symmetry, one eighth of the CCT specimen and one fourth of the SECT, CT and SENB specimens are modelled, respectively, as shown in Figs. 2 and 3. In the x-y plane, the mesh size gradually decreases in size with the decreasing of the radial distance (r). In the x-z plane, the mesh number is 10 along the thickness direction for different specimens and increases in size when the mid-plane (z/B = 0) is approaching, where the bias radio is 10 (Xiang et al., 2011; Xiang and Guo, 2013; Guo et al., 2018), more details about mesh sensitivity are shown in Appendix A. Considering the stress and strain gradients near the crack tip, 20-node 3D quadratic brick elements are adopted to construct the mesh, and an initial notch with root radius ρ = 0.001 mm is used to consider the large creep deformation.
At n = 5, the related parameters I1 ∼I7 are 3.36, 9.08, −14.43, 33.53, 196.3, 414.62 and −271.88, respectively.
5. Result and analyses
4. Finite element models and numerical procedures
5.1. Creep stress intensity factor kδ ( t ) -tz
In this work, CCT, SECT, SENB and CT specimens are modelled using ABAQUS 6.12. The power-law creeping material is superalloy Inconel 800H at 650 C , and the related material parameters are E = 154 GPa, ν = 0.33, n = 5, A = 1.348 × 10–16 (MPa)- n h- 1 ,
5.1.1. Creep constraint parameters Fig. 4 shows the variations in CTOD along the thickness direction with creep time for different specimens. It is found that CTOD increases with creep time and decreases gradually when the free
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Fig. 10. Comparisons of tensile stress and radial stress obtained by δ (t)-Tz solution, δ (t)-Tz -Q∗ solution, 3D FE results and RR solution for CT specimens at t∗ = 0.1 and t∗ = 10.
surface is approaching. Fig. 5 shows the variations in Tz on the ligament (θ = 0) along the thickness direction for different specimens with creep time at t∗ = 0.1 and t∗ = 10. Tz sharply decreases with increasing radial distance (r) from the upper limit of (Tz = 0.5) to the lower limit (Tz = 0). Near the free surface, the stress state has an ephemeral transition period from plane strain state to plane stress state, which is usually ignored in most studies of creeping
constraint parameters. The 3D constraint zone increases with t∗ and becomes steady under large-scale creep conditions, being approximately half of the specimen thickness. Variations in Kδ ( t ) -Tz and C(t) for different specimens are shown in Fig. 6 with t∗ . It is found that at the early creep stage, Kδ ( t ) -Tz decreases and gradually becomes constant at the large-scale creep stage. The thicker the specimen is, the smaller the constant is.
Please cite this article as: P. Cui and W. Guo, Crack-tip-opening-displacement-based description of creep crack border fields in specimens with different geometries and thicknesses, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019.10.001
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Fig. 11. Variations in Q∗ along the thickness direction for CT specimens at t∗ = 0.1 and t∗ = 10.
Fig. 12. Comparisons of tensile stress and radial stress obtained by δ (t)-Tz solution, δ (t)-Tz -Q∗ solution, 3D FE results and RR solution for SENB specimens at t∗ = 0.1 and t∗ = 10.
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Fig. 13. Variations in Q∗ along the thickness direction for SENB specimens at t∗ = 0.1 and t∗ = 10.
Fig. 14. Comparisons of tensile stress and radial stress obtained by δ (t)-Tz solution, δ (t)-Tz -Q∗ solution, 3D FE results and RR solution for CCT specimens at t∗ = 0.1 and t∗ = 10.
However, the change is rather small compared to C(t); the maximum change in Kδ ( t ) -Tz is only 8.6%, whereas the change in C(t) is 280% at mid-plane (z/B = 0) for different specimens during creeping, as shown in Tables 1 and 2.
5.1.2. Comparisons Variations in C(t) and Kδ (t)-Tz along the thickness direction are shown in Fig. 7 for different specimens. It is found that C(t) decreases sharply near the free surface, while Kδ ( t ) -Tz changes quite gradually along thickness direction. During creeping, Kδ ( t ) -Tz and
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Fig. 15. Variations in Q∗ along the thickness direction for CCT specimens at t∗ = 0.1 and t∗ = 10.
C(t) first decrease and gradually become nearly constant under large-scale creep conditions. Variations in Kδ ( t ) -Tz and C(t) along and with the thicknesses under small-scale and large-scale creep conditions are shown in Tables 1 and 2, respectively. It is found that the change of C(t) at t∗ = 0.1 can be up to 379%, whereas the value of Kδ ( t ) -Tz is only 8.0%. At t∗ = 20 the values of C(t) and Kδ ( t ) -Tz are 405.1% and 8.6%, respectively, showing that to characterise the creep crack-tip constraints, C(t) sharply changes along the thickness direction, while the geometric thickness has a slight influence on Kδ ( t ) -Tz for different specimens. The slight changes of Kδ ( t ) -Tz along and with the thicknesses can be attributed to ignorance of in-plane constraint. It is found from Figs. 9, 11, 13 and 15 that with the exception of the CCT specimen under large-scale creep condition, the maximum of Q∗ /σ 3D FE for the radial stress and tensile stress is within 14.8% along the thickness direction for different specimens, which means Kδ ( t ) -Tz has a sufficient accuracy for high temperature engineering structures in situation without significant loss of in-plane constraint. 5.2. Characterisation of the 3D creep crack border stress fields The radial stress and tensile stress on the ligament along the thickness direction for different specimens are obtained by various approaches. The radial stress and tensile stress obtained by the δ (t)-Tz solution, δ (t)-Tz -Q∗ solution, 3D FE results and RR solution for SECT, CT, SENB and CCT specimens are shown in Figs. 8, 10, 12 and 14, respectively. Comparisons of the results for SECT, CT, SENB and CCT specimens at z/B = 0.45 are shown in Appendix B. It is found that the results by the two-parameter description δ (t)Tz are in good agreement with the 3D FE results at t∗ = 0.1 and t∗ = 10, except for the CCT specimens under large-scale creep conditions. For the CCT specimen, loss of in-plane constraint becomes strong with increasing creep time, so that the higher order terms should be introduced to the leading δ (t)-Tz singular term.
Variables of the parameter Q∗ along the thickness direction for SECT, CT, SENB and CCT specimens are shown in Figs. 9, 11, 13 and 15, respectively, at t∗ = 0.1 and 10. It is found that with the exception of the CT specimen, the absolute Q∗ is getting smaller from the free surface to the mid-plane for different specimens. At t∗ = 0.1 and 10 for the SECT, CT and SENB specimens, absolute Q∗ is smaller than 0.1 and 0.2, respectively, and for the CCT specimen, absolute Q∗ is smaller than 0.2 and 0.3, respectively. This phenomenon can determine the scope of application of CTOD, originally being introduced for the high-constraint testing specimens in elastic-plastic solids. Therefore, the CTOD-based two-parameter description δ (t)-Tz can characterise crack border fields with sufficient accuracy for common testing specimens, such as the SECT, SENB and CT specimens.
6. Conclusion Considering the large nonlinear deformation near the 3D creep crack tip, a CTOD-based description has been developed to describe the 3D crack border fields in creeping solids. First, a CTODbased 3D stress intensity factor Kδ ( t )- Tz consisting of the time dependent CTOD, δ (t), and the out-of-plane stress constraint factor Tz is proposed to describe the asymptotic crack tip fields and then validated through comprehensive 3D FE analyses of typical centrecracked tension, single-edge cracked tension, single-edge-notched bending and compact tension specimens under small-scale and large-scale creep conditions. In contrast to the C(t)-integral, which decreases sharply along the thickness direction (over 400%) when the free surface is approaching, the change in Kδ ( t )- Tz along the thickness direction remains within 8.6% in all the analysed situations in this study, indicating that Kδ ( t )- Tz is a stable parameter. In addition, the two-parameter δ (t)-Tz description of the 3D crack border stress fields is proven to be highly efficient, except for centrecracked tension specimens under large-scale creep conditions.
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7. Prospective for applications of kδ ( t ) -tz in experiments For engineering applications, the dominated parameters should be convenient enough to be determined experimentally and numerically. As the 3D numerical results reveal that these parameters, such as C(t) or CTOD, are strongly thickness-dependent as shown in Figs. 5 and 7, determining how to evaluate their local values through far field displacements remains a challenge. Here, we have shown that Kδ ( t ) -Tz is stable along the thickness direction and should be more accurate and reasonable to connect with the measured loading point displacement. The analogy between elastic-plastic solids and creeping solids suggests that the method to obtain CTOD in elastic-plastic solids can be extended into applications in creeping solids. By separating the elastic and inelastic parts from the deformation, BS 74481 proposed an experimental approach to characterise the crack growth as well as the fracture toughness under plane strain conditions. Here, take a CT specimen as an example:
δtEX =
rpl bVpl K 2 (1 − ν 2 ) + , 2σYS E rpl b + a + z
(28)
where σ YS is the 0.2% offset yield strength; K is the elastic stress intensity factor; rpl and Vpl are the plastic rotational factor and the plastic part of the clip gauge opening displacement, respectively; b is the uncracked ligament width; and z is the distance of the knife edge measurement point from the load line. Analogous to rpl and Vpl in elastic-plastic solids, rcl and Vcl are introduced into creeping solids in the forms of the creeping rotational factor and the creeping part of the clip gauge opening displacement, respectively, and Eq. (28) can be extended as follows:
δtEX =
K 2 (1 − ν 2 ) rcl bVcl + . 2σYS E rcl b + a + z
15
Substituting Eq. (29) into Eq. (11), the creep stress intensity factor KEX δ ( t ) -Tz under small-scale creep conditions for experiments can be expressed as follows:
K EX δ (t )−T z =
δ EX (t )
1/n .
3A(n + 1 )u˜˙ θ (π , Tz , n )
(30)
Analogously, KEX δ ( t ) -Tz under large-scale creep conditions for experiments can be obtained in the form of
K EX δ (t )−T z =
δ EX (t )
1/n .
3Au˜˙ θ (π , Tz , n )
(31)
The relation between the local CTOD and the far field displacement has been obtained by Kumar et al. (1981) under plane strain conditions. Thus, the stable governing parameter KEX δ ( t ) -Tz can be easily measured with high accuracy by the far field displacement. The crack growth resistance of power-law creeping materials can be performed numerically and experimentally within the framework of Kδ ( t ) -Tz in the future. Acknowledgements This work was supported by National Natural Science Foundation of China (51535005), the Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures (MCMS-I0418K01, MCMS-I-0419K01), the Fundamental Research Funds for the Central Universities (NC2018001, NP2019301, NJ2019002), A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions. The authors would like to thank Mr. Wei Guo for helpful discussions
(29)
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Appendix A Comparisons along the thickness direction with creep time for different N: (a) C(t); (b) radial stress, t = 40 h; (c) radial stress, t = 50 0 0 h; (d) tensile stress, t = 40 h; (e) tensile stress, t = 50 0 0 h.
Appendix B Comparisons of tensile stress and radial stress obtained by the Kδ ( t ) -Tz solution and other solutions at different positions along the thickness direction for different specimens
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Fig. B.1. Comparisons of tensile stress and radial stress obtained by δ (t)-Tz solution, δ (t)-Tz -Q∗ solution, 3D FE results and RR solution for SECT specimens at z/B = 0.45.
Fig. B.2. Comparisons of tensile stress and radial stress obtained by δ (t)-Tz solution, δ (t)-Tz -Q∗ solution, 3D FE results and RR solution for CT specimens at z/B = 0.45.
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Fig. B.3. Comparisons of tensile stress and radial stress obtained by δ (t)-Tz solution, δ (t)-Tz -Q∗ solution, 3D FE results and RR solution for SENB specimens at z/B = 0.45.
Fig. B.4. Comparisons of tensile stress and radial stress obtained by δ (t)-Tz solution, δ (t)-Tz -Q∗ solution, 3D FE results and RR solution for CCT specimens at z/B = 0.45.
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References Antunes, F.V., Rodrigues, S.M., Branco, R., Camas, D., 2016. A numerical analysis of CTOD in constant amplitude fatigue crack growth. Theor. Appl. Fract. Mech. 85, 45–55. Bassani, J.L., McClintock, F.A., 1981. Creep relaxation of stress around a crack tip. Int. J. Solids Struct. 17, 479–492. Budden, P.J., Ainsworth, R.A., 1999. The effect of constraint on creep fracture assessments. International Journal of Fracture 97, 237–247. BS 7448-1, 1991. Method of deformation of KIC, critical CTOD and critical J values of metallic materials. British Standards Institution, London. Chao, Y.J., Zhu, X.K., Zhang, L., 2001. Higher-order asymptotic crack-tip fields in a power-law creeping materials. Int. J. Solids Struct. 38, 3853–3875. Ehlers, R., Riedel H., 1981. A finite element analysis of creep deformation in a specimen containing macroscopic crack. Cannes: The French Association of Metallurgy. Guo, W., 1993a. Elastoplastic three dimensional crack border field- I. singular structure of the field. Eng. Fract. Mech. 46, 93–113. Guo, W., 1993b. Elastoplastic three dimensional crack border field-II. asymptotic solution for the field. Eng. Fract. Mech. 46, 105–113. Guo, W., 1995. Elasto-plastic three-dimensional crack border field-III. fracture parameters. Eng. Fract. Mech. 51, 51–71. Guo, W., Chen, Z., She, C., 2018. Universal characterization of three-dimensional creeping crack-front stress fields. Int. J. Solids Struct. 50, 3067–3088. Han, K., Shuai, J., Deng, X., Kong, L., Zhao, X., Sutton, M., 2014. The effect of constraint on CTOD fracture toughness of API X65 steel. Eng. Fract. Mech. 124-125, 167–181. Kumar V., German M.D., Shih C.F., 1981. An engineering approach for elastic–plastic fracture analysis, EPRI NP-1931.
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Newman, Jr., James, M.A., Zerbst, U., 2003. A review of the CTOA/CTOD fracture criterion. Eng. Fract. Mech. 70, 371–385. Nguyen, B., Onck, P., van der Giedden, E., 20 0 0a. Crack-tip constraint effects on creep fracture. Eng. Fract. Mech. 65, 467–490. Nguyen, B., Onck, P., van der Giedden, E., 20 0 0b. On higher-order crack-tip fields in creeping solids. J. Appl. Mech. 67, 372–382. Riedel, H., Rice, J., 1980. Tensile cracks in creeping solids. In: Fracture Mechanics: Twelfth Conference, ASTM STP, pp. 112–130 700. Shlyannikov, V.N., Tumanov, A.V., Boychenko, N.V., 2015. A creep stress intensity factor approach to creep-fatigue crack growth. Eng. Fract. Mech. 142, 201–219. Shlyannikov, V.N., Tumanov, A.V., 2018a. Creep-fracture resistance parameters determination based on stress and ductility damage models. Fatigue Fract. Eng. Mater. Struct. 41, 2110–2129. Shlyannikov, V.N., Tumanov, A.V., 2018b. Creep damage and stress intensity factor assessment for plane multi-axial and three-dimensional problems. Int. J. Solids Struct. 150, 166–183. Wang, G.Z., Liu, X.L., Xuan, F.Z., Tu, S.T., 2010. Effect of constraint induced by crack depth on creep crack-tip stress field in CT specimens. Int. J. Solids Struct. 47, 51–57. Xiang, M., Yu, Z., Guo, W., 2011. Characterization of three-dimensional crack border fields in creeping solids. Int. J. Solids Struct. 48, 2695–2705. Xiang, M., Guo, W., 2013. Formulations of the stress fields in power law solids ahead of three-dimensional tensile cracks. Int. J. Solids Struct. 48, 2695–2705. Yang, L., Sutton, M., Deng, X., Lyons, J., 1996. Finite element analysis of creep fracture initiation in a model superalloy material. Int. J. Fract. 81, 299–320. Zhu, X., 2017. Progress in development of fracture toughness test methods for SENT specimens. Int. J. Press. Vessels Pip. 156, 40–58.
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Contents lists available at ScienceDirect
International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr
Crack-tip-opening-displacement-based description of creep crack border fields in specimens with different geometries and thicknesses Pengfei Cui, Wanlin Guo∗ State Key Laboratory of Mechanics and Control of Mechanical Structures, Key Laboratory of Intelligent Nano Materials and Devices, Institute of Nanoscience of Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
a r t i c l e
i n f o
Article history: Received 21 June 2019 Revised 18 August 2019 Accepted 1 October 2019 Available online xxx Keywords: Creep crack border fields Creep stress intensity factor Kδ ( t ) -Tz Geometry effect Constraints δ (t)-Tz description
a b s t r a c t Creep crack growth is accompanied by strong nonlinear deformation and stress relaxation. Finding an appropriate parameter to characterize the crack tip fields, as well as the fracture resistance, has long been a challenge. Most previous studies were performed within the framework of the C(t)-integral, which is limited to the assumptions of small deformation and simple proportional loading. By using the crack tip opening displacement (CTOD), we propose a new creep stress intensity factor Kδ ( t ) -Tz consisting of the time-dependent CTOD, δ (t) and the out-of-plane stress constraint factor Tz to characterize the three-dimensional creep crack tip fields. Four typical specimens, single-edge cracked tension specimens, compact specimens, centre-cracked tension specimens and single-edge-notched bending specimens under three-point bending are comprehensively analysed using the power-law creeping model and threedimensional finite element analyses. It is found that under both small-scale and large-scale creep conditions, the change in Kδ ( t ) -Tz along the thickness direction for different specimens is within 8.6%, whereas the change in C(t) can exceed 400%, showing that Kδ ( t ) -Tz is a stable parameter that governs the creep crack tip fields. With the exception of the centre-cracked tension specimens under large-scale creep conditions, good agreements are obtained between the two-parameter description δ (t)-Tz of crack border stress fields with the three-dimensional finite element results under small-scale and large-scale creep conditions. These results indicate that the CTOD-based two-parameter description δ (t)-Tz can be taken as the basis of creep fracture criteria. © 2019 Published by Elsevier Ltd.
1. Introduction In the last several decades, many efforts have been made to find an appropriate parameter to characterize the creep cracktip fields, as well as the fracture resistance. Although many achievements for the creep fracture of materials have been made, most studies were performed within the framework of the C(t)integral (C(t)), which is analogous to the J-integral (J) in elasticplastic solids (Riedel and Rice, 1980; Budden and Ainsworth, 1999; Nguyen et al., 20 0 0a, 20 0 0b; Chao et al., 20 01; Wang et al., 2010; Xiang et al., 2011; Shlyannikov et al., 2015; Shlyannikov and Tumanov, 2018a,2018b). With the C(t) as the dominating parameter, Riedel and Rice (1980) first described the singular cracktip fields under plane stress and ideal plane strain conditions for power-law creeping solids, resulting in the well-known RR solution. Considering the in-plane constraint effect, Budden and Ainsworth (1999) and Chao et al. (2001) further introduced the
∗
Corresponding author. E-mail address: [email protected] (W. Guo).
higher-order terms to the RR asymptotic solution in the form of the parameters Q and A2 , respectively. Similarly, A∗ 2 and σ ∞ were also introduced by Nguyen et al. (20 0 0a, 20 0 0b) to consider the effects of specimen geometry and loading modes at the steady power-law creeping stage. In recent works, Wang et al. (2010) proposed a tensile stress related parameter R to study the effect of constraint caused by crack depth in compact tension specimens. The three-dimensional (3D) asymptotic solution for creep crack border fields has been obtained by Xiang et al. (2011), with the out-of-plane stress constraint factor Tz (Guo, 1993a, 1993b, 1995) being considered. By investigating the RR-type singular solution, Shlyannikov et al. (2015) proposed a single parameter, the stress intensity factor Kcr , to characterize the crack growth resistance in creeping solids. As C(t) is based on deformation theory, it has a limited ability to describe the large deformation near the crack border under large-scale creep yielding conditions and with strong accompanying stress relaxation. Considering these properties of creeping solids, a comprehensive study of crack tip deformation is important to understand creep crack border fields under 3D constraints.
https://doi.org/10.1016/j.ijsolstr.2019.10.001 0020-7683/© 2019 Published by Elsevier Ltd.
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Fig. 1. Theory and fitting results of (f-2) for different n and Tz .
Crack tip opening displacement (CTOD) can quantify large local deformations without the assumptions of small deformation and simple proportional loading for C(t) and J (Newman et al., 2003; Han et al., 2014; Antunes et al., 2016; Zhu, 2017). Newman et al. (2003) showed that the K-factor, J-integral and CTOD have equivalence relations in linear-elastic or small-scale elastic-plastic solids, while CTOD can be further extended into large-scale plastic yielding conditions. To quantify the crack-tip
constraint of elastic-plastic solids, Han et al. (2014) introduced the parameter CTODC through a Gurson-type void model. By detailed finite element (FE) analysis, the parameter CTOD is proven to be independent of the stress ratio to characterise the fatigue crack growth (Antunes et al., 2016). Zhu (2017) measured the lowconstraint fracture toughness of pipeline steels experimentally using the CTOD approach. An appropriate governing parameter should quantify the deterioration of creeping solids near the crack tip in both simulations and experiments. Due to its ability to represent the nonlinear deformation ahead of the crack tip, CTOD has the potential to characterise the creep crack border fields. Analogous to the relation between J and C(t), some modifications are made for CTOD with the creep time term. In addition, the factor Tz proposed by Guo (1993a, 1993b, 1995) to consider the out-of-plane stress constraint effects in elastic-plastic solids has been applied effectively to characterise the 3D crack border fields together with C(t) and Q∗ in creeping solids (Xiang et al., 2011; Xiang and Guo, 2013; Guo et al., 2018). Thus, to describe the 3D creep crack border fields, Guo’s constraint theory can be further developed with the CTOD approach. In this work, the CTOD approach and Guo’s constraint theory are extended to characterise the crack border fields in creeping solids and validated by comprehensive 3D FE analyses of the typical single-edge-notched bending specimens under three-point bending (SENB), compact specimens (CT), centre-cracked tension specimens (CCT) and single-edge cracked tension specimens (SECT) using the power-law creeping model. To characterise the 3D creep crack-tip fields, a creep stress intensity factor Kδ ( t ) -Tz consisting of
Fig. 2. Schematic of specimen (a) CCT; (b) SECT; (c) SENB; (d) CT.
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Fig. 3. (a) FE models of a CCT (1/8 model), SENB (1/4 model) and SECT (1/4 model) specimens; (b) FE model of a CT specimen (1/4 model).
Fig. 4. Variations in CTOD along thickness direction for different specimens with creep time. (a) CCT specimens; (b) SECT specimens; (c) SENB specimens; (d) CT specimens.
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Fig. 5. Variations in Tz on the ligament (θ = 0) along the thickness direction for different specimens with creep time at t∗ = 0.1 (nontransparent) and t∗ = 10 (transparent).
the time dependent CTOD, δ (t), and the out-of-plane stress constraint Tz is proposed and systematically evaluated. The results show that the distributions of Kδ ( t ) -Tz along the thickness direction are nearly uniform in all specimens under both small-scale and large-scale creeping conditions. With the exception of CCT specimens under large-scale creep conditions, the two-parameter δ (t)-Tz description of the creep crack border stress fields is proven to be highly efficient for different specimens, especially for large-scale creeping deformation. The results show that the CTOD-based 3D stress intensity factor Kδ ( t ) -Tz and the two-parameter δ (t)-Tz description have advantages over the C(t)-based approaches.
2. Constraint parameters for three-dimensional creeping problems
C (t ) =
n+1
σ33 (1, 2, 3 ) = (x, y, z )or (r, θ , z ), σ11 + σ22
Tz =
(3)
3 2
(4)
σi j ε˙ i j n1 − σi j n1 u˙ i,1 ds,
u˙ i j =
3 n n( f −2)+1 ˜ AK r u˙ i j (θ , Tz , n ), 2
(5)
where
(1)
where and ds are the integral path and its length, respectively; n and n1 are the creep exponent constant and unit outward normal to , respectively; and σ ij , u˙ i,1 and ε˙ i j are the components of stress, displacement gradient rate and strain rate, respectively. Under ideal plane strain and plane stress conditions, C(t) is pathindependent and can quantify the intensity of crack tip fields effectively (Riedel and Rice, 1980; Budden and Ainsworth, 1999; Chao et al., 2001).
(2)
where, as shown in Figs. 2 and 3, o is a point of the crack front line, and the subscripts (1, 2, 3) stand for the (x, y, z) Cartesian coordinates or the (r, θ , z) Cylindrical coordinates, respectively. z is the tangent to the crack front line at the point o, and the corresponding normal plane is the x-y plane in the Cartesian coordinate system or the r-θ plane in the Cylindrical coordinate system. By using the amplitude coefficient K, Xiang et al. (2011) developed the functions of stress, strain rate and displacement rate in the forms of
ε˙ i j = AK n rn( f −2) ε˜˙ i j (θ , Tz , n ),
C(t) is the governing parameter for characterisation of the intensity of creep crack-tip fields (Bassani and McClintock, 1981) and can be evaluated in the form of
n
Tz is the out-of-plane stress constraint factor proposed by Guo (1993a, b, 1995)
σi j = K r f −2 σ˜ i j (θ , Tz , n ),
2.1. C(t)-integral
2.2. Creep stress intensity factor kδ ( t ) -tz
K=
C (t ) AI (Tz , n )r ( f −2)(n+1)+1
1/(n+1) ,
(6)
r and θ are polar coordinates centred at the crack border, and θ = 0 corresponds to the ligament directly ahead of the crack border; A is a creep material constant and A = ε˙ 0 /σ0n , with ε˙ 0 and σ 0 being the initial yield strain rate and stress, respectively; K is an amplitude coefficient; (f-2) is the order of stress singularity; I(Tz ,n) can be seen as the function of Tz and n; and σ˜ i j , u˜˙ θ and ε˜˙ θ are the angular distribution functions of stress, displacement rate and strain rate, respectively.
Please cite this article as: P. Cui and W. Guo, Crack-tip-opening-displacement-based description of creep crack border fields in specimens with different geometries and thicknesses, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019.10.001
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Fig. 6. Creep stress intensity factors Kδ ( t ) -Tz and C(t) with creep time for different specimens.
In the previous work of Xiang et al. (2011), the order of creep time t is 1/(n + 1) for displacement near the crack-tip. Analogous to the displacement rate in Eq. (5), the displacement near the crack tip can be assumed in the form of
ui j =
3 n n( f −2)+1 ˜ AK r u˙ i j (θ , Tz , n )(n + 1 )t 1/(n+1) . 2
(7)
By using the 90° definition, CTOD can be expressed as
δt = 2uθ |θ =π ; r=uθ .
(8)
Substituting Eq. (7) into Eq. (8), the amplitude coefficient K is obtained as
K=
δ (t )
1/n ,
3(n + 1 )Au˜˙ θ (π , Tz , n )
(9)
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. ∗
C(t) and Kδ ( t ) -Tz both are decreasing monotonically from the mid-plane (z/B = 0) to the surface plane (z/B = 0.5) as shown in Fig. 7. Thus, the midplane and surface plane are chosen to calculate the change of C(t) and Kδ ( t ) -Tz along the thickness direction. The first column is the change along
−C (t )min × 100%; the thickness direction and the second column is the change for different specimens. The change is defined as change|C (t ) = C (t )max C (t )
change|Kδ (t )−T z =
Kδ (t )−T z,max −Kδ (t )−T z,min Kδ (t )−T z,min
min
× 100%.
where
δ (t ) = t˜ =
Analogously, Kδ ( t ) -Tz under large-scale creep conditions can be obtained as follows:
(δt )−( f −2) t˜1/n
.
(10)
t t1 , t1
is unit time (h). Rearranging Eq. (9) further, the dimensionless creep stress intensity factor Kδ ( t ) -Tz consisting of δ (t) and Tz can be defined as follows:
Kδ (t )−Tz =
δ (t )
1/n , σ0 L−( f −2) 3(n + 1)Au˜˙ θ (π , Tz , n )
where L is the unit length (mm).
(11)
Kδ (t )−Tz =
σ0
L−( f −2)
δ (t )
1/n .
3Au˜˙ θ (π , Tz , n )
(12)
In this work, we focus on characterization of the 3D stationary crack border fields for the SECT, CCT, CT and SENB specimens for undamaged power-law creeping solids. To estimate the scale during creeping, Ehlers and Riedel (1981) proposed the parameter transition time tT under plane strain conditions in the form of
tT =
J
(n + 1 )C ∗
,
(13)
where C∗ is a constant, approximately equal to the value of C(t) under large-scale creep conditions. The J-integral is obtained
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Table 2 Variations in Kδ ( t ) -Tz and C(t) along thickness direction at t∗ = 20 for different specimens.
B
z/B
Kδ(t)-Tz
0.5
0.282
0
0.306
10
0
0.291
20
0
0.290
0.5
0.226
0
0.237
8
0
0.227
10
0
0.223
0.5
0.223
0
0.230
2
0
0.239
10
0
0.220
0.5
0.245
0
0.266
4
0
0.272
16
0
0.254
(mm) 4 CCT
4 SECT
5 SENB
10 CT
C(t)
change*
(%)
(MPa·mm·h -1)
(%)
9.13E-4
8.5
2.39E-3 5.5
161.8
2.21E-3
24.5
1.92E-3 7.97E-5
4.9
395
3.94E-4 6.3
2.91E-4
57
2.51E-4 3.39E-5
3.1
1.62E-4 8.6
378.5
1.95E-4
68.1
1.16E-4 1.43E-4
8.6
7.22E-4 7.1
405.1
7.91E-4
36.6
5.79E-4
from the linear elastic analysis step. Based on the works of Xiang et al. (2011), the explicit formula of u˜˙ θ (π , Tz , n )at n = 5 and (f-2) for 2< n <13 can be obtained as follows:
u˜˙ θ (π , Tz , n ) = −1.52 − 5 × 10−4 Tz −3.24(Tz )2 +19.1(Tz )3 − 35.87(Tz )4 +20.93(Tz )5 ,
change*
C3 = −8.15 + 5.65n − 0.75n2 + 3.69 × 10−2 n3 − 5.77 × 10−4 n4 , C4 = 21.2 − 14.3n+1.89n2 + 9.36 × 10−2 n3 +1.47 × 10−3 n4 , C5 = −18.9 + 11.8n − 1.53n2 + 7.48 × 10−2 n3 − 1.14 × 10−3 n4 . (16)
(14) 3. Characterisation of the 3D creep crack border fields
f − 2 = C0 + C1 Tz + C2 (Tz ) + C3 (Tz ) + C4 (Tz ) + C5 (Tz ) , 2
3
4
5
(15)
where the adjusted coefficient of determination (Adj. R-Square) of Eq. (14) is 0.9997. f-2 obtained by Eq. (15) and a 3D asymptotic solution is shown in Fig. 1. The fitting results are in good agreement with the theory results.
C0 = −0.53 + 0.14n − 1.85 × 10−2 n2 + 1.2 × 10−3 n3 − 2.99 × 10−5 n4 , C1 = −3.14 × 10−2 + 4.38 × 10−2 n − 6.93 × 10−3 n2 + 4.34 × 10−4 n3 − 1.1 × 10−5 n4 , C2 = 1.19 − 0.82n+0.11n2 − 5.27 × 10−3 n3 +8.47 × 10−5 n4 ,
By replacing K with Kδ (t )−Tz in Eqs. (4), (5) and (6), the 3D creep crack border fields can be obtained as follows:
σi j = Kδ (t )−Tz r˜ f −2 σ˜ i j (θ , Tz , n ), σ0 3 2
(18)
3 n AK r n( f −2)+1 u˜˙ i j (θ , Tz , n ), 2 δ (t )−Tz
(19)
ε˙ i j = AKδn(t )−Tz r˜n( f −2) ε˜˙ i j (θ , Tz , n ), u˙ i j =
(17)
where r˜ = Lr .
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Fig. 7. Variations in Kδ ( t ) -Tz and C(t) along the thickness direction for different specimens at different creep time.
For mode I crack, the stress fields on the ligament (θ = 0) near the creep crack border are our concerns. Based on the previous works of Xiang et al. (2011) and Xiang and Guo (2013), the explicit formulae of σ˜ θ θ (n,θ ,Tz ) at n = 5 and θ = 0 have been further obtained:
σ˜ θ θ = 1.14 + 1.39Tz − 1.57(Tz )2 + 59.79(Tz )3 − 200.42(Tz )4
+ 186.48(Tz )5 ,
σrr = (0.301Tz +0.609 )σθ θ ,
(20)
(21)
where the adjusted coefficient of determination (Adj. R-Square) of Eq. (20) is 0.9998.
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Fig. 8. Comparisons of tensile stress and radial stress obtained by δ (t)-Tz solution, δ (t)-Tz -Q∗ solution, 3D FE results and RR solution for SECT specimens at t∗ = 0.1 and t∗ = 10.
Analogous to C(t)-Tz -Q∗ theory proposed by Xiang et al. (2011), the three-parameter δ (t)-Tz -Q∗ description is proposed as follows:
where
Q∗ =
σi j = Kδ (t )−Tz r˜ f −2 σ˜ i j (θ , Tz , n )+Q ∗ σ0 δi j , σ0
(22)
σi j
FE
−
σi j
σ0
δ (t )−Tz
atθ = 0andr = 2.
(23)
Xiang et al. (2011) derived the dimensionless length consisting of C(t), the initial yield stress σ 0 and creep t in creeping solids under small-scale and large-creep conditions. The dimensionless
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Fig. 9. Variations in Q∗ along the thickness direction for SECT specimens at t∗ = 0.1 and t∗ = 10.
length can be unified by the CTOD (δ t ) and dn at different creep times.
⎧ ⎨ C (t )t (n + 1 )(n+1)/nt˜1/n /σ0 , δt = = dn ⎩ C ∗ t t˜1/n /σ , 0
(small − scalecreep), (large − scalecreep), (24)
dn = 3
3 1/n 2
ε˜˙ 01/n
(n+1 )/n
u˜˙ θ (π , Tz , n ) I (Tz , n )
,
(25)
ε˜˙ 0 is the dimensionless initial yield strain rate. I(Tz ,n) is obtained by Xiang et al. (2011). For better applications, here I(Tz ,n) is further formulated in the form of
I (Tz , n ) =
I1 + I2 Tz + I3 (Tz )2 I4 + I5 Tz + I6 (Tz )2 + I7 (Tz )3
(0 ≤ Tz ≤ 0.3 ) (0.3 ≤ Tz ≤ 0.5 ),
(26)
where
I1 I2 I3 I4 I5 I6 I7
= 5.16 − 0.6n + 0.058n2 − 0.002n3 , = −0.89 + 3.82n − 0.44n2 + 0.015n3 , = 20.52 − 9.9n+0.66n2 − 0.016n3 , = −136.57+67.37n − 8.1n2 +0.286n3 , = 1092.49 − 505.78n+60.08n2 − 2.1n3 , = −2696.23 + 1217.42n − 144.15n2 + 5.02n3 , = 2164.17 − 954.96n+113.15n2 − 3.92n3 .
(27)
σ 0 = 417 MPa, (Yang et al., 1996). Three different thicknesses are adopted for four typical specimens, and the other geometrical parameters are W = 20 mm, a/W = 0.5, and H/W = 2, as shown in Fig. 2. The values of remote tensile stress, which are constant, are 55 MPa, 20 MPa, and 30 MPa for CCT, SECT and CT specimens, respectively. The applied loadings are 0.1, 0.25, and 0.5 kN for the 2-mm, 5-mm and 10-mm thick SENB specimens by an analytical rigid body, respectively. Further, the entire creep time and maximum time increment for every step in the simulation are 50 0 0 h and 100 h, respectively. C(t) of the twentieth contour at t = 50 0 0 h is deemed to be steady (C∗ ). The creep scale conditions along the thickness direction are estimated by the mid-plane (z/B = 0) with the transition time tT . Due to the geometrical symmetry, one eighth of the CCT specimen and one fourth of the SECT, CT and SENB specimens are modelled, respectively, as shown in Figs. 2 and 3. In the x-y plane, the mesh size gradually decreases in size with the decreasing of the radial distance (r). In the x-z plane, the mesh number is 10 along the thickness direction for different specimens and increases in size when the mid-plane (z/B = 0) is approaching, where the bias radio is 10 (Xiang et al., 2011; Xiang and Guo, 2013; Guo et al., 2018), more details about mesh sensitivity are shown in Appendix A. Considering the stress and strain gradients near the crack tip, 20-node 3D quadratic brick elements are adopted to construct the mesh, and an initial notch with root radius ρ = 0.001 mm is used to consider the large creep deformation.
At n = 5, the related parameters I1 ∼I7 are 3.36, 9.08, −14.43, 33.53, 196.3, 414.62 and −271.88, respectively.
5. Result and analyses
4. Finite element models and numerical procedures
5.1. Creep stress intensity factor kδ ( t ) -tz
In this work, CCT, SECT, SENB and CT specimens are modelled using ABAQUS 6.12. The power-law creeping material is superalloy Inconel 800H at 650 C , and the related material parameters are E = 154 GPa, ν = 0.33, n = 5, A = 1.348 × 10–16 (MPa)- n h- 1 ,
5.1.1. Creep constraint parameters Fig. 4 shows the variations in CTOD along the thickness direction with creep time for different specimens. It is found that CTOD increases with creep time and decreases gradually when the free
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Fig. 10. Comparisons of tensile stress and radial stress obtained by δ (t)-Tz solution, δ (t)-Tz -Q∗ solution, 3D FE results and RR solution for CT specimens at t∗ = 0.1 and t∗ = 10.
surface is approaching. Fig. 5 shows the variations in Tz on the ligament (θ = 0) along the thickness direction for different specimens with creep time at t∗ = 0.1 and t∗ = 10. Tz sharply decreases with increasing radial distance (r) from the upper limit of (Tz = 0.5) to the lower limit (Tz = 0). Near the free surface, the stress state has an ephemeral transition period from plane strain state to plane stress state, which is usually ignored in most studies of creeping
constraint parameters. The 3D constraint zone increases with t∗ and becomes steady under large-scale creep conditions, being approximately half of the specimen thickness. Variations in Kδ ( t ) -Tz and C(t) for different specimens are shown in Fig. 6 with t∗ . It is found that at the early creep stage, Kδ ( t ) -Tz decreases and gradually becomes constant at the large-scale creep stage. The thicker the specimen is, the smaller the constant is.
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Fig. 11. Variations in Q∗ along the thickness direction for CT specimens at t∗ = 0.1 and t∗ = 10.
Fig. 12. Comparisons of tensile stress and radial stress obtained by δ (t)-Tz solution, δ (t)-Tz -Q∗ solution, 3D FE results and RR solution for SENB specimens at t∗ = 0.1 and t∗ = 10.
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Fig. 13. Variations in Q∗ along the thickness direction for SENB specimens at t∗ = 0.1 and t∗ = 10.
Fig. 14. Comparisons of tensile stress and radial stress obtained by δ (t)-Tz solution, δ (t)-Tz -Q∗ solution, 3D FE results and RR solution for CCT specimens at t∗ = 0.1 and t∗ = 10.
However, the change is rather small compared to C(t); the maximum change in Kδ ( t ) -Tz is only 8.6%, whereas the change in C(t) is 280% at mid-plane (z/B = 0) for different specimens during creeping, as shown in Tables 1 and 2.
5.1.2. Comparisons Variations in C(t) and Kδ (t)-Tz along the thickness direction are shown in Fig. 7 for different specimens. It is found that C(t) decreases sharply near the free surface, while Kδ ( t ) -Tz changes quite gradually along thickness direction. During creeping, Kδ ( t ) -Tz and
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Fig. 15. Variations in Q∗ along the thickness direction for CCT specimens at t∗ = 0.1 and t∗ = 10.
C(t) first decrease and gradually become nearly constant under large-scale creep conditions. Variations in Kδ ( t ) -Tz and C(t) along and with the thicknesses under small-scale and large-scale creep conditions are shown in Tables 1 and 2, respectively. It is found that the change of C(t) at t∗ = 0.1 can be up to 379%, whereas the value of Kδ ( t ) -Tz is only 8.0%. At t∗ = 20 the values of C(t) and Kδ ( t ) -Tz are 405.1% and 8.6%, respectively, showing that to characterise the creep crack-tip constraints, C(t) sharply changes along the thickness direction, while the geometric thickness has a slight influence on Kδ ( t ) -Tz for different specimens. The slight changes of Kδ ( t ) -Tz along and with the thicknesses can be attributed to ignorance of in-plane constraint. It is found from Figs. 9, 11, 13 and 15 that with the exception of the CCT specimen under large-scale creep condition, the maximum of Q∗ /σ 3D FE for the radial stress and tensile stress is within 14.8% along the thickness direction for different specimens, which means Kδ ( t ) -Tz has a sufficient accuracy for high temperature engineering structures in situation without significant loss of in-plane constraint. 5.2. Characterisation of the 3D creep crack border stress fields The radial stress and tensile stress on the ligament along the thickness direction for different specimens are obtained by various approaches. The radial stress and tensile stress obtained by the δ (t)-Tz solution, δ (t)-Tz -Q∗ solution, 3D FE results and RR solution for SECT, CT, SENB and CCT specimens are shown in Figs. 8, 10, 12 and 14, respectively. Comparisons of the results for SECT, CT, SENB and CCT specimens at z/B = 0.45 are shown in Appendix B. It is found that the results by the two-parameter description δ (t)Tz are in good agreement with the 3D FE results at t∗ = 0.1 and t∗ = 10, except for the CCT specimens under large-scale creep conditions. For the CCT specimen, loss of in-plane constraint becomes strong with increasing creep time, so that the higher order terms should be introduced to the leading δ (t)-Tz singular term.
Variables of the parameter Q∗ along the thickness direction for SECT, CT, SENB and CCT specimens are shown in Figs. 9, 11, 13 and 15, respectively, at t∗ = 0.1 and 10. It is found that with the exception of the CT specimen, the absolute Q∗ is getting smaller from the free surface to the mid-plane for different specimens. At t∗ = 0.1 and 10 for the SECT, CT and SENB specimens, absolute Q∗ is smaller than 0.1 and 0.2, respectively, and for the CCT specimen, absolute Q∗ is smaller than 0.2 and 0.3, respectively. This phenomenon can determine the scope of application of CTOD, originally being introduced for the high-constraint testing specimens in elastic-plastic solids. Therefore, the CTOD-based two-parameter description δ (t)-Tz can characterise crack border fields with sufficient accuracy for common testing specimens, such as the SECT, SENB and CT specimens.
6. Conclusion Considering the large nonlinear deformation near the 3D creep crack tip, a CTOD-based description has been developed to describe the 3D crack border fields in creeping solids. First, a CTODbased 3D stress intensity factor Kδ ( t )- Tz consisting of the time dependent CTOD, δ (t), and the out-of-plane stress constraint factor Tz is proposed to describe the asymptotic crack tip fields and then validated through comprehensive 3D FE analyses of typical centrecracked tension, single-edge cracked tension, single-edge-notched bending and compact tension specimens under small-scale and large-scale creep conditions. In contrast to the C(t)-integral, which decreases sharply along the thickness direction (over 400%) when the free surface is approaching, the change in Kδ ( t )- Tz along the thickness direction remains within 8.6% in all the analysed situations in this study, indicating that Kδ ( t )- Tz is a stable parameter. In addition, the two-parameter δ (t)-Tz description of the 3D crack border stress fields is proven to be highly efficient, except for centrecracked tension specimens under large-scale creep conditions.
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7. Prospective for applications of kδ ( t ) -tz in experiments For engineering applications, the dominated parameters should be convenient enough to be determined experimentally and numerically. As the 3D numerical results reveal that these parameters, such as C(t) or CTOD, are strongly thickness-dependent as shown in Figs. 5 and 7, determining how to evaluate their local values through far field displacements remains a challenge. Here, we have shown that Kδ ( t ) -Tz is stable along the thickness direction and should be more accurate and reasonable to connect with the measured loading point displacement. The analogy between elastic-plastic solids and creeping solids suggests that the method to obtain CTOD in elastic-plastic solids can be extended into applications in creeping solids. By separating the elastic and inelastic parts from the deformation, BS 74481 proposed an experimental approach to characterise the crack growth as well as the fracture toughness under plane strain conditions. Here, take a CT specimen as an example:
δtEX =
rpl bVpl K 2 (1 − ν 2 ) + , 2σYS E rpl b + a + z
(28)
where σ YS is the 0.2% offset yield strength; K is the elastic stress intensity factor; rpl and Vpl are the plastic rotational factor and the plastic part of the clip gauge opening displacement, respectively; b is the uncracked ligament width; and z is the distance of the knife edge measurement point from the load line. Analogous to rpl and Vpl in elastic-plastic solids, rcl and Vcl are introduced into creeping solids in the forms of the creeping rotational factor and the creeping part of the clip gauge opening displacement, respectively, and Eq. (28) can be extended as follows:
δtEX =
K 2 (1 − ν 2 ) rcl bVcl + . 2σYS E rcl b + a + z
15
Substituting Eq. (29) into Eq. (11), the creep stress intensity factor KEX δ ( t ) -Tz under small-scale creep conditions for experiments can be expressed as follows:
K EX δ (t )−T z =
δ EX (t )
1/n .
3A(n + 1 )u˜˙ θ (π , Tz , n )
(30)
Analogously, KEX δ ( t ) -Tz under large-scale creep conditions for experiments can be obtained in the form of
K EX δ (t )−T z =
δ EX (t )
1/n .
3Au˜˙ θ (π , Tz , n )
(31)
The relation between the local CTOD and the far field displacement has been obtained by Kumar et al. (1981) under plane strain conditions. Thus, the stable governing parameter KEX δ ( t ) -Tz can be easily measured with high accuracy by the far field displacement. The crack growth resistance of power-law creeping materials can be performed numerically and experimentally within the framework of Kδ ( t ) -Tz in the future. Acknowledgements This work was supported by National Natural Science Foundation of China (51535005), the Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures (MCMS-I0418K01, MCMS-I-0419K01), the Fundamental Research Funds for the Central Universities (NC2018001, NP2019301, NJ2019002), A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions. The authors would like to thank Mr. Wei Guo for helpful discussions
(29)
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Appendix A Comparisons along the thickness direction with creep time for different N: (a) C(t); (b) radial stress, t = 40 h; (c) radial stress, t = 50 0 0 h; (d) tensile stress, t = 40 h; (e) tensile stress, t = 50 0 0 h.
Appendix B Comparisons of tensile stress and radial stress obtained by the Kδ ( t ) -Tz solution and other solutions at different positions along the thickness direction for different specimens
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Fig. B.1. Comparisons of tensile stress and radial stress obtained by δ (t)-Tz solution, δ (t)-Tz -Q∗ solution, 3D FE results and RR solution for SECT specimens at z/B = 0.45.
Fig. B.2. Comparisons of tensile stress and radial stress obtained by δ (t)-Tz solution, δ (t)-Tz -Q∗ solution, 3D FE results and RR solution for CT specimens at z/B = 0.45.
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Fig. B.3. Comparisons of tensile stress and radial stress obtained by δ (t)-Tz solution, δ (t)-Tz -Q∗ solution, 3D FE results and RR solution for SENB specimens at z/B = 0.45.
Fig. B.4. Comparisons of tensile stress and radial stress obtained by δ (t)-Tz solution, δ (t)-Tz -Q∗ solution, 3D FE results and RR solution for CCT specimens at z/B = 0.45.
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Please cite this article as: P. Cui and W. Guo, Crack-tip-opening-displacement-based description of creep crack border fields in specimens with different geometries and thicknesses, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019.10.001