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Fatigue shape analysis for internal surface flaw in a pressurized hollow cylinder
International Journal of Pressure Vessels and Piping 77 (2000) 227–234 www.elsevier.com/locate/ijpvp
Fatigue shape analysis for internal surface flaw in a pressurized hollow cylinder V.N. Shlyannikov* Kazan State Power Engineering Institute, 51 Krasnoselskaya Street, 420066 Kazan, Russia Received 2 August 1999; received in revised form 20 October 1999; accepted 3 February 2000
Abstract Fatigue crack shape is simulated through a theoretical model for part-through cracks in a hollow cylinder under cyclic pressure. The defect is assumed to have an elliptical-arc shape. The propagation path of the surface flaw is obtained as a diagram of aspect ratio e against relative depth b=t: The numerical procedure calculates the local growth increments at a set of points defining a crack front by employing a fracture damage zone size model, that can directly predict the shape development of propagating cracks. The aspect ratio changes during crack growth are obtained and compared with the limited experimental data found in the literature. Thick- and thin-walled cylinders, and also semielliptical internal surface flaws are considered. The influence of the elasto-plastic characteristics of the steels’ different properties, related to test temperature, on the aspect ratio variation is studied. 䉷 2000 Elsevier Science Ltd. All rights reserved. Keywords: Pressurized cylinder; Surface flaw; Fatigue growth simulation; Aspect ratio; Fracture process zone; Elasto-plastic behavior and properties
1. Introduction The existence of crack-like flaws cannot be excluded in pressured vessels and pipings. Therefore, the interdisciplinary technology related to pressurized components is of considerable importance in many branches of industry, such as power engineering, petrochemicals, process plants, transport and space communications. In order to provide service in a safe condition, it is important to perform fracture mechanics assessment. During the manufacture and industrial applications of metallic cylinders there often appear some part-through surface defects. The defects may appear in different ways. Thus, defects are approximately considered as internal semi-elliptical cracks. In most cases, part-through cracks appear in the center of the cylinder on the inner surface. The assessment of both the form and size changes of the surface crack during propagation is an essential element for structural integrity prediction of the pressured vessels and pipelines in the presence of initial and accumulated operation damages. This problem has been studied by several investigators [1–3]. However, these results have no systematic or generalized character, concerning the influence of the geometric parameters of * Tel.: ⫹ 7-8432-438-634; fax: ⫹ 7-8432-438-634. E-mail address: [email protected] (V.N. Shlyannikov).
cylinder, elasto-plastic behavior and properties, changes of local fracture characteristics along the curvilinear crack front. The objectives of this paper are to study and present the behavior of internal surface flaws in steel cylinders subjected to fluctuating pressures, due to refilling operations. As shown in Fig. 1, a long tube of internal radius R and external radius R0 contains a radial crack of length 2a and depth b. In the present work, semi-elliptical internal surface flaws in a metallic cylinder subjected to the cyclic pressure P are considered. The elliptical-arc part-through defect is described by the flaw aspect ratio e b=a (a, b are the ellipse semi-axis), and the relative depth b=t of the deepest point B (Fig. 1) on the front is equal to the ratio between the maximum crack depth b, and the cylinder wallthickness t. The initial value of the parameter e0 ranges from 0.1 to 1.0, whereas the relative crack depth b0 =t is made to vary from 0.01 to 0.3. The dimensionless wall-thickness t=R ranges from 0.1 (thin-walled cylinder) to 1.0 (thick-walled cylinder). In each individual case of loading these steel cylinders are subjected to constant amplitude cyclic pressure. Hereby in all the considered cases the total variation of the maximum hoop stress s u s u =s 0 s u is normalized by the yielding stress of the material s 0 that is perpendicular to the crack growth direction, ranges from 0.2 to 0.7.
0308-0161/00/$ - see front matter 䉷 2000 Elsevier Science Ltd. All rights reserved. PII: S0308-016 1(00)00014-4
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Fig. 1. Basic geometry of problem. Simulating fatigue crack growth of crack front with varying defect sizes.
In accordance with the experimental observations on crack shape development in Refs. [4–7], the possibility of approximation by semi-elliptical shape for the front of growing part-through cracks during the fatigue process is proved. The marginal crack shape deviation from the canonical semi-elliptical form allows the use of well-known numerical solutions for stress intensity factors (SIFs) for the hollow cylinder loaded by internal pressure. For most practical purposes, the SIF’s of surface defects are sufficiently well described by solutions for idealized geometries and load configurations as found in handbooks such as Refs. [8,9]. Following the fundamentals of fracture mechanics, it may be assumed that the position of the front of a growing part-through defect will be determined by the magnitude of the SIF at each point of the curvilinear crack contour. Hence, it will be described by an infinite number of degrees of freedom. However, the authors [6] by means of numerical investigations have shown for semi-elliptical crack a possibility of the reduction of the number of degrees of freedom from 21 to 2 without a major loss of accuracy. Thus, the prediction of shape development of propagating cracks may be described by a model with two degrees of freedom, each of which determines the position of semi-axes tips a and b of an ellipse at any instant. The stress–strain-state of the pressurized cylinder with a part-through defect has its peculiarities. The plane strain condition has been assumed to exist for the crack front area close to the deepest point b, whereas near the internal free surface of cylinder it is generally believed that plane stress might prevail there. Evidently, it is necessary to take into account along with the crack front of growing defect, the change of factors such as stress asymptotic, constrain effect, local fracture stress, physical–mechanical material properties and the stress redistribution due to the plastic strain.
2. Crack growth model In this paper, we shall examine the growth of initially semi-elliptical surface cracks in a pressurized cylinder. The model describing the growth of such flaws is employed here, based on the following assumptions: • the defect is assumed to have, in the general case, an elliptical-arc shape; • the modeling of the flaw growth conditions is simplified by considering these as small scale yielding states in an area close to the crack front. When conditions of small scale plastic deformation take place (or are fulfilled), crack growth will be characterized in terms of elastic– plastic SIF, Kp ; • the change of the local fracture stress along the partthrough crack front is proportional to the distribution of local plastic strain close to the curvilinear crack front. To describe the elastic SIF distribution at any point along the crack front in a pressurized hollow cylinder (Fig. 1), we have used a numerical solution obtained from the 3D finite element analysis [8] PR p·b 1=2 b b t ; ; ;f Fe t Q a t R q p s n pbY Q ⫺1=2 ; Fe ; Q 1 ⫹ 1:464 b=a1:65
K
1
where P is the internal pressure, PR=t is the mean hoop nominal stress. The complete expression for a functional Fe is adduced in Ref. [8]. It is proposed to set the position of any points of the semi-elliptical crack front by two coordinates re and fe : These are the distances to the crack
V.N. Shlyannikov / International Journal of Pressure Vessels and Piping 77 (2000) 227–234
front and the parametrical angle, respectively (Fig. 1)
fe arctg
( for
b tg f a
2
points of the crack front coinciding with the semi-axes tips a and b and lying within the idealized plane stress and plane strain conditions, respectively, it is accepted that the dominant singularity governing the asymptotic
q e0 ⱕ 1; r0 b0 ; y0 r0 sin f; xe y0 =tg fe ; re x2e ⫹ y20 q e0 ⬎ 1; r0 a0 ; x0 r0 cos f; ye x0 =tg fe ; re x20 ⫹ y2e
Under cyclic loading in an area close to the crack front, the interconnected processes of accumulation and development of microdamages take place. The macroscopic effect of these processes of nucleation, growth and coalescence of voids is a new free surface creation or increment of the crack length that is apparent in various degrees in all materials from the quasi-brittle to the quasi-plastic. The magnitude of this increment is determined by the realization of the leading micromechanism of the crack growth in the elementary act of fracture, depending on stress–strain-state near the crack tip and the elastic–plastic material properties. In works [10– 12] for both static and cyclic loading conditions, the concept of the fracture damage zone (FDZ) is proposed, according to which the realization of the leading fracture micromechanism, the so-called decohesive (brittle) or coalescent (ductile) one, depends on the ratio between the FDZ-size and that of the crack tip plastic zone. The FDZ-size has been considered as a fundamental characteristic setting interrelation between the processes occurring on both micro- and macroscale level with respect to the material structure. In the present paper the FDZ ahead of the crack front is assumed to be located where the stress strain state in the element reaches a certain critical value that can be measured from standard test data. A relative FDZ-size has been introduced by Ref. [12] as ( ) 2 ^ S 22 ⫺ 4 W ⴱc ⫺ S 3 S 1 ⫹ S p 2 S d c ; 2 W ⴱc ⫺ S 3 4 2 s0 1 2 an s ⫹ s n⫹1 W ⴱc sn 2 f n⫹1 f where s n is the nominal hoop stress, s f is the local fracture stress, a and n are the hardening parameters of the Ramberg–Osgood power-law, S i i 1; 2; 3 and S p are the elastic and plastic coefficients that will be given below. According to model (4), the crack front with semi-axes a and b growth, after one cyclic loading step to the new configuration can be obtained provided that the coordinates of points A and B (Fig. 1) deduced from the FDZ equation satisfy Eqs. (2) and (3). The proposed model (4) assumes that the growth variation can be modeled by assuming different growth rate properties in the depth and surface directions. For boundary
229
3
behavior of the stresses at the crack tip (for the strain hardening material known as the HRR-singular field) has the form [13–15]
s ij
s ij s K p r ⫺1= n⫹1 s~ ij u; n; s e e s0 s0
K p r ⫺1= n⫹1 s~ e u; n
5
aIn 1= n⫹1 is the plastic SIF that is where K p J= related to the J-integral and a numerically computed quantity In ; s e the effective stress, r and u are the polar coordinates centered at the crack tip. In small scale yielding conditions, the plastic SIF is related to the elastic SIF as [13] JE 0 K2 as 02 12 I K n⫹1 2 E0 n p s0 s0 or for the plane strain (3D) 1= n⫹1 " 2 #1= n⫹1 2 J 1 ⫺ n K 1 3D ; K p s0 aIn3D aIn3D
6
7
E E0 1⫺n2 and the plane stress (2D) 1= n⫹1 " 2 #1= n⫹1 1 K J 1 2D ; K p aIn2D aIn2D s 0
E0 E 8
For formulae (7) and (8) containing the elastic SIF it is assumed that their distribution along the semi-elliptical crack front is described by Eq. (1), therefore it is possible to present the following complete expressions for coefficients S i i 1; 2; 3 and Sp from formula (4) S1 0:25 1 ⫹ n k ⫹ 1Y Q; Fe 2 ; S3 0:125 1 ⫹ n k ⫹ 1Y Q; Fe 2 p S2 0:5 1 ⫹ n 1 ⫺ kY Q; Fe 2 = 2; Sp
nl p n⫹1 s~ Y Q; Fe 2 n ⫹ 1 In e
9
230
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the plastic zone rp
Table 1 Variable parameters of cylinder and crack geometry t=R
R=R0
0.10 0.25 0.50 0.75 1.00
0.91 0.80 0.67 0.57 0.50
R (mm)
150 60 30 20 15
R0 (mm)
165 75 45 35 30
s s n =s 0
e0
b0 =t
b0 (mm)
a0 (mm)
0.10
0.01 0.10 0.30 0.01 0.10 0.30 0.01 0.10 0.30 0.01 0.10 0.30
0.15 1.50 4.50 0.15 1.50 4.50 0.15 1.50 4.50 0.15 1.50 4.50
1.50 15.0 45.0 0.60 6.00 18.0 0.30 3.00 9.00 0.15 1.50 4.50
0.25 0.2 0.4 0.5 0.7
0.50
1.00
where we have for the appropriate two points A and B of the crack front B-plane strain 3D—fe
p ⫺ l l3D 1 ⫺ n2 ; 2
A-plane strain 2D—fe 0 ⫺ l l2D 1;
k k2D 3 ⫺ n= 1 ⫹ n; In In2D ; s~ e s~ e2D : Approach by Eq. (4) enables us to predict the crack growth, when at the crack tip both elastic and plastic SIFs are known. It is often used in plane analysis to estimate the growth of advance of crack tip, making no distinction between the variation of stress state along the crack front, but it is here applied in a pointwise sense, so that the local crack growth is assumed to be related to the local fracture stress. In the present problem this relationship is applied in a finite sense. Let us assume that all material properties, the stress– strain-state and the fracture characteristics continuously vary along the crack front from the plane strain conditions in the cylinder wall to the plane stress on its interior surface. Then the local fracture stress s f that affects the FDZ-size will change appropriately from s f3D up to s f2D : For the plane strain case it may be tentatively assumed that s f3D ⬇ s utrue or
su 1 ⫺ c s0
10
s utrue is the true ultimate stress, c is the reduction of area. The second degree of freedom for the distribution s f fe at s f fe 0 s f2D is indefinite. Then for the purpose of the present analysis let us assume that this distribution s f fe of local fracture stress s f along the crack front in qualitative terms repeats the behavior of
11
The size of the plastic zone in the crack growth direction can be obtained as the distance from the crack tip to the boundary between the elastic and plastic areas taking into account that s e 1 on this boundary or ~ e3D 1= n⫹1 ; rp3D K 3D p s
2D ~ e2D 1= n⫹1 r2D p K p s
12
Then taking into consideration Eqs. (1), (7) and (8) we shall obtain for small scale yielding !1= n⫹1 2D r3D s~ e3D Y fe p=2 2 p 2 In 1 ⫺ n 3D 13 Y fe 0 r2D In s~ e2D p and the distribution of s f fe along the crack front of a semi-elliptical surface flaw by Eq. (11) can be rewritten in the following form
s f
k k3D 3 ⫺ 4n; In In3D ; s~ e s~ e3D
s f3D ⬇ s utrue
rp2D s f2D : rp3D s f3D
sf s f3D v !2 u 3D s0 u rp t cos fe ⫹sin f2e r2D p
14
where s f3D s f3D =s 0 and rp3D =r2D p are given by Eqs. (10) and (13), respectively. Thus, the function defining the distribution of the FDZ-sizes or increment of crack length along its curvilinear front is found. Strictly speaking, the use of formula (14) within Eq. (4) allows one to calculate the increment of crack length not only for two degrees of freedoms (two extreme points of front A and B), but for an infinite number of degrees of freedom assigned to a value of parametrical angle f e. Application of Eq. (4) together with Eq. (14) leads us to the following relation for fatigue crack growth rate prediction !1=m dl s n2 K 2f ⫺ s th2 DK 2th 2d l 15 dN 4s fⴱ eⴱf Ed p where l re fe ; d d=l; K f S1 ⫹ S p ⫹ S 2 d ⫹ S3 d ; DK th ; s fⴱ ; eⴱf ; m are the conventional constants of cyclic deformation and fracture curves. The growth of surface semi-elliptical defects is simulated by using a numerical solution for the crack front SIF. From these results the crack growth rate (Eq. (15)) at all points is inferred from the fracture damage zone model (Eq. (4)), and the crack front is advanced. The SIFs are re-evaluated, and a new crack front shape implied by the FDZ-model found. This process, which allows the crack front shape to evolve freely, is repeated until a terminal value of crack size is reached. The elastic–plastic singular solution (5) governing the asymptotic behavior of the stresses and strains at the crack tip, used in the present work, is correct only for small scale yielding conditions, that is when the plastic zone size does not exceed 10%
V.N. Shlyannikov / International Journal of Pressure Vessels and Piping 77 (2000) 227–234
231
Table 2 Chemical composition of steel Material
C
Si
Mn
S
P
Cr
Ni
Mo
V
Cu
Steel A
0.13 0.18
0.20 0.22
0.37 0.46
0.012
0.008 0.011
2.38 2.63
0.19 0.27
0.62
0.26 0.28
0.10 0.14
Steel B
0.17
0.28
0.42
0.004
–
2.40
1.02
0.50
0.15
–
of a characteristic body size. Such characteristic size for the hollow cylinder is the remaining ligament between the deepest point of the flaw and the opposite wall of the cylinder. The similar condition can be extended as restriction of existence for the idealized plane strain condition because of constraint effect in the deepest point of the crack front. Thus present approach can be used until then while the plastic zone size at the crack tip will not exceed 10%, the remaining ligament between the deepest point of the flaw and the opposite wall of the hollow cylinder. The size of plastic zone in the crack growth direction can be obtained first by Eq. (12).
3. Results and discussion In the present paper, initially semi-elliptical internal surface flaws in a steel cylinder under constant amplitude cyclic pressure are considered. The general regularities of crack growth for the variation of the geometry from a thinwalled t=R 0:1 to a thick-walled t=R 1:0 cylinder, an initial crack shape from oblong semi-elliptical up to true half-circular, the depth of the initial crack from shallow surface up to deep part-through have been established. The initial aspect ratio ranges from e0 0:1 to e0 1:0; whereas the relative initial crack depth is made to vary from b0 =t 0:01 to b0 =t 0:3: For each cylinder geometry variant t=R 0:1; 0.25; 0.5; 0.75; 1.0), 12 combinations of initial forms and sizes of surface flaws have been calculated (Table 1). The total number of calculation variants is 60 for each type of steel. The cylinders were endowed with the properties of two
steels, whose chemical composition is shown in Table 2. From Refs. [16,17] basic mechanical steel properties have been obtained for tests performed at low and room temperatures. As can be seen from Table 3, for steel A the state from brittle up to plastic was thus reached with the same chemical composition. As a matter of fact, each line of Table 3 may be considered as a separate material with peculiar properties. Moreover, such selection of data from Table 3 enables one to estimate the effect of test temperature on the development of cracks in the cylinder. The aspect ratio during the crack growth process was predicted using the present simulation approach. To substantiate the proposed model (Eqs. (4)–(15)), a comparison between the numerical and experimental results has been made for aspect ratio change. The experimental data reported in Ref. [2] for a pressure vessel of cylindrical type made of steel B and in Ref. [7] for a bending plate made of steel A has been compared with the simulation results obtained with the present model, which is shown in Fig. 2. It can be seen that the agreement between them is fairly good from a part-through crack growth to external surface of cylinder and plate respectively. The flaw propagation paths determined for both t=R 0:1 and t=R 1:0 under cyclic loading of the steel B thin- and thick-walled cylinders are displayed in Fig. 3. The initial crack configurations examined have a relative crack depth b0 =t equal to 0.01, 0.03, 0.1, 0.3 and a crack aspect ratio e0 equal to 0.1 (semi-elliptical flaw), 1.0 (half-circular flaw) or the intermediate values 0.25, 0.5 and 0.75. The diagrams in Fig. 3 show the advance of the crack front during the early stages up to the point of breakthrough. The crack behavior in thin-walled cylinder (Fig. 3a) and thick-walled cylinder
Table 3 Mechanical and fracture properties of type A and B steel Material heat treatment
Reference
Steel A quenching 1000⬚, 4 h., oil hardening, tempering 620⬚C 4 h., air cooling
[18]
Steel B quenching 920⬚C, 15 h,water cooling tempering 600⬚C 20 h., tempering 650⬚C air cooling
[2]
s 0;2 (MPa)
su (MPa)
sf (MPa)
20
1100
1157
2246
⫺196
1440
1590
1655
20
606
698
1377
Test temperature T (⬚C)
DKth p (MPa m 9.23
12.7 5.57
e (%)
c (%)
ef
n
16.6
67.2
1.115
7.034
3.1
2.9
0.029
8.511
18.5
69.5
1.187
6.674
232
V.N. Shlyannikov / International Journal of Pressure Vessels and Piping 77 (2000) 227–234
Fig. 2. Comparison of aspect ratio changes predicted by the present model with experimental data.
(Fig. 3b) is different. The most remarkable feature of the early stage of growth is that the crack shape change is strongly dependent on the initial shape, but a common characteristic can be found. Thus, for b0 =t ⱕ 0:01 these initial cracks with different e0 asymptotically tend towards a preferred profile as they propagate to the external surface of the cylinder, reaching a very similar shape. The crack aspect ratio tends to converge to some asymptote rapidly, that is, the front of such a flaw becomes nearly circular-arcshape for low values of b0 =t. On the contrary, for b0 =t ⬎ 0:01 the flaw aspect ratio changes slowly during the fatigue crack
growth. In other words, in the case internal surface flaw in a cylinder, the fatigue behavior for b0 =t ⱕ 0:01 is different from that for b0 =t ⬎ 0:01: Several authors have shown that the surface defects in pipes tend to follow preferred fatigue propagation paths, that is, the flaw aspect ratio is a function of the relative crack depth [2,3], and an analogous conclusion has also been drawn for edge flaws in flat plates [18]. Fig. 4 illustrates the fatigue crack shape developments for a range applied nominal stress level s n 0:2 up to s n 0:7 beginning with two different initially semi-elliptical cracks in thin-walled (a) and thick-walled (b) cylinders
Fig. 3. Propagation paths under cyclic pressure for different initial flaw configurations.
V.N. Shlyannikov / International Journal of Pressure Vessels and Piping 77 (2000) 227–234
233
Fig. 4. Variation of crack aspect ratio with nominal stress level for: (a) thin-walled cylinders; and (b) thick-walled cylinders.
made from both steels A and B with four different b0 =t ratios. For an initial surface flaw that depth b0 =t is greater than 0.1, the influence of s n s n =s 0 is really not excessively great. Similar experimentally observable behavior of the aspect ratio for initial deep cracks is noticed in the literature [5–7]. It should be noted that experimental data for more small-sized cracks b0 =t ⬍ 0:1 are absent in the literature because there are difficulties of exact measurement and development observation of part-through cracks. In these cases, when b0 =t ⬍ 0:1 the influence of an applied
nominal stress level is significant, which is more noticeable in a thick-walled cylinder (Fig. 4b). Distinctions in the cracks behavior for steels A and B also depends on an applied nominal stress level. For moderate low stresses s n 0:2 the influence of a material properties of cylinders is noticeable. For stresses s n 0:7 these differences decrease and almost disappear for thick-walled cylinders made from steels A and B. These calculated data agreed well with experiments of the authors of Ref. [5] who have shown that for s n 0:6–0:8; behavior of aspect ratio as a
Fig. 5. Predicted aspect ratio changes with test temperature for: (a) thin-walled cylinders; and (b) surface crack behavior at room temperature for steel A.
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function of depth ratio for several steel types and titanium alloys is described in one common curve. It can be remarked that the results computed by the present study differ for room (Fig. 5a) and low (Fig. 5b) temperatures. An especially strong influence on surface crack growth is rendered by low temperature (Fig. 5a). The lowering of test temperature to ⫺196⬚C reduces (see Table 3) almost entirely the restriction of plastic properties for steel A. In this brittle state, steel A ceases to respond to a change in the relative wall-thickness of a cylinder. At the same time, at room temperature (Fig. 5b) the scale effect in the cylinder of this material is exhibited. By comparing the crack profiles adopted during the fatigue growth process in Fig. 5b, the effect of the dimensionless wall-thickness ratio, t/R, on the fatigue shape development is seen to exist. The aspect ratio change is determined for b0 =t 0:01; 0.05 and different values of t=R; in particular, the two upper curves are related to semicircular surface flaws with e0 1:0; while the lower curves represent the semi-elliptical defects with e0 0:1; : 0.25; 0.5. Note that the influence of the parameter t/R for initial semi-elliptical flaw with e0 0:1 is not significant for low values of b=t; that is, the different curves for a given value of t=R are very close to one another. On the contrary, for relative depth b=t ⬍ 0:05 the influence of the scale factor t/R on the aspect ratio changes is significant for full range of initial semi-elliptical-arc-shaped flaw. As it follows from our numerical results, FDZ-size is a sensitive parameter affected by numerous factors. It can be expected that the aspect ratio change is also affected by a number of principal mechanical and fracture properties of a material. It will be remembered that the basic equations of our model contain elastic constants, static or cyclic strain hardening exponent, constants of cyclic deformation and fracture curves. 4. Conclusions The numerical analysis has been carried out to calculate the aspect ratio changes for different values of the geometrical parameters for both cylinder and surface flaw. Thickand thin-walled cylinders containing initial semi-elliptical internal surface flaws have been analyzed. Crack propagation paths in the diagram of flaw aspect ratio against relative crack depth have been determined through a fracture damage zone model. The influence of the main steel properties stipulated by various aspects of thermal treatment and test temperature on the aspect ratio change is significant, especially for low temperature. It has been found that
surface crack behavior is sensitive to the change of initial flaw configuration, nominal stress level and dimensionless wall-thickness. The comparison between the numerical predictions and experimental data shows that the agreement is good for the aspect ratio change, demonstrating that the modeling crack growth by fracture damage zone model is reliable.
References [1] Nair PK. Fatigue crack growth model for part-through flaws in plates and pipes. Trans ASME 1979;101:54–60. [2] Burak MI, Kaidalov VB, Pichkov SN, et al. Experimental study of cyclic crack resistance for pressure vessels in tests on large-scale models. Applied problems of strength & plasticity. Methods for solution 1988. p. 115 (p. 115–22, in Russian). [3] Carpinteri A, Brighenti R. Circumferential surface flaws in pipes under cyclic axial loading. Engng Fract Mech 1998;60:383–96. [4] Letunov VI, Shulginov BS, Plungrova I, et al. Regularities in development of surface cracks in low-alloy steel under asymmetric cyclic bending. Communication I. Strength Mater 1985;11:41–6. [5] Lukiyanov VF, Naprasnikov VV, Korobtsov AS. Kinetics of surface crack front variation under axisymmetrical bending. Strength Mater 1988;7:8–13. [6] Varfolomeev IV, Vainshtok VA, Krasovsky AYa. Criteria and stable growth forms of part-through cracks under cyclic loading. Communication I. Strength Mater 1990;8:3–10. [7] Varfolomeev IV, Vainshtok VA, Krasovsky AYa. Criteria and stable growth forms of part-through cracks under cyclic loading. Communication 2. Strength Mater 1990;9:11–7. [8] Murakami Y. Stress intensity factors handbook. Oxford: Pergamon Press, 1987. [9] Rooke DR, Cartwrigth DJ. Compendium of stress intensity factors. London: HMSO, 1976. [10] Shlyannikov VN. Strain energy density and fracture process zone. Communication I. Theoretical assumptions. Strength Mater 1995;10:3–17. [11] Shlyannikov VN. Strain energy density and fracture process zone. Communication 2. Experimental background. Strength Mater 1995;11/12:3–21. [12] Shlyannikov VN. Modelling of crack growth by fracture damage zone. Theor Appl Fract Mech 1996;25:187–201. [13] Hutchinson JW. Singular behaviour at the end of a tensile crack in a hardening material. J Mech Phys Solids 1968;16:13–31. [14] Hutchinson JW. Plastic stress and strain fields at a crack tip. J Mech Phys Solids 1968;16:337–47. [15] Rice JR, Rosengren GF. Plane strain deformation near a crack tip in a power-law hardening material. J Mech Phys Solids 1968;16:1–12. [16] Makhutov NA. Deformation criteria of fracture and calculation elements structure strength. Mashinistroenie Press, 1981 (in Russian). [17] Troshchenko VT, Pokrovsky VV, Kaplunenko VG. Prediction of crack resistance for thermostable steels with taking into account specimen sizes. Communication I. Experimental results, Strength Mater 1997;1:5–25. [18] Lin XB, Smith RA. Fatigue shape analysis for corner cracks at fastener holes. Engng Fract Mech 1998;59:73–87.
Fatigue shape analysis for internal surface flaw in a pressurized hollow cylinder V.N. Shlyannikov* Kazan State Power Engineering Institute, 51 Krasnoselskaya Street, 420066 Kazan, Russia Received 2 August 1999; received in revised form 20 October 1999; accepted 3 February 2000
Abstract Fatigue crack shape is simulated through a theoretical model for part-through cracks in a hollow cylinder under cyclic pressure. The defect is assumed to have an elliptical-arc shape. The propagation path of the surface flaw is obtained as a diagram of aspect ratio e against relative depth b=t: The numerical procedure calculates the local growth increments at a set of points defining a crack front by employing a fracture damage zone size model, that can directly predict the shape development of propagating cracks. The aspect ratio changes during crack growth are obtained and compared with the limited experimental data found in the literature. Thick- and thin-walled cylinders, and also semielliptical internal surface flaws are considered. The influence of the elasto-plastic characteristics of the steels’ different properties, related to test temperature, on the aspect ratio variation is studied. 䉷 2000 Elsevier Science Ltd. All rights reserved. Keywords: Pressurized cylinder; Surface flaw; Fatigue growth simulation; Aspect ratio; Fracture process zone; Elasto-plastic behavior and properties
1. Introduction The existence of crack-like flaws cannot be excluded in pressured vessels and pipings. Therefore, the interdisciplinary technology related to pressurized components is of considerable importance in many branches of industry, such as power engineering, petrochemicals, process plants, transport and space communications. In order to provide service in a safe condition, it is important to perform fracture mechanics assessment. During the manufacture and industrial applications of metallic cylinders there often appear some part-through surface defects. The defects may appear in different ways. Thus, defects are approximately considered as internal semi-elliptical cracks. In most cases, part-through cracks appear in the center of the cylinder on the inner surface. The assessment of both the form and size changes of the surface crack during propagation is an essential element for structural integrity prediction of the pressured vessels and pipelines in the presence of initial and accumulated operation damages. This problem has been studied by several investigators [1–3]. However, these results have no systematic or generalized character, concerning the influence of the geometric parameters of * Tel.: ⫹ 7-8432-438-634; fax: ⫹ 7-8432-438-634. E-mail address: [email protected] (V.N. Shlyannikov).
cylinder, elasto-plastic behavior and properties, changes of local fracture characteristics along the curvilinear crack front. The objectives of this paper are to study and present the behavior of internal surface flaws in steel cylinders subjected to fluctuating pressures, due to refilling operations. As shown in Fig. 1, a long tube of internal radius R and external radius R0 contains a radial crack of length 2a and depth b. In the present work, semi-elliptical internal surface flaws in a metallic cylinder subjected to the cyclic pressure P are considered. The elliptical-arc part-through defect is described by the flaw aspect ratio e b=a (a, b are the ellipse semi-axis), and the relative depth b=t of the deepest point B (Fig. 1) on the front is equal to the ratio between the maximum crack depth b, and the cylinder wallthickness t. The initial value of the parameter e0 ranges from 0.1 to 1.0, whereas the relative crack depth b0 =t is made to vary from 0.01 to 0.3. The dimensionless wall-thickness t=R ranges from 0.1 (thin-walled cylinder) to 1.0 (thick-walled cylinder). In each individual case of loading these steel cylinders are subjected to constant amplitude cyclic pressure. Hereby in all the considered cases the total variation of the maximum hoop stress s u s u =s 0 s u is normalized by the yielding stress of the material s 0 that is perpendicular to the crack growth direction, ranges from 0.2 to 0.7.
0308-0161/00/$ - see front matter 䉷 2000 Elsevier Science Ltd. All rights reserved. PII: S0308-016 1(00)00014-4
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Fig. 1. Basic geometry of problem. Simulating fatigue crack growth of crack front with varying defect sizes.
In accordance with the experimental observations on crack shape development in Refs. [4–7], the possibility of approximation by semi-elliptical shape for the front of growing part-through cracks during the fatigue process is proved. The marginal crack shape deviation from the canonical semi-elliptical form allows the use of well-known numerical solutions for stress intensity factors (SIFs) for the hollow cylinder loaded by internal pressure. For most practical purposes, the SIF’s of surface defects are sufficiently well described by solutions for idealized geometries and load configurations as found in handbooks such as Refs. [8,9]. Following the fundamentals of fracture mechanics, it may be assumed that the position of the front of a growing part-through defect will be determined by the magnitude of the SIF at each point of the curvilinear crack contour. Hence, it will be described by an infinite number of degrees of freedom. However, the authors [6] by means of numerical investigations have shown for semi-elliptical crack a possibility of the reduction of the number of degrees of freedom from 21 to 2 without a major loss of accuracy. Thus, the prediction of shape development of propagating cracks may be described by a model with two degrees of freedom, each of which determines the position of semi-axes tips a and b of an ellipse at any instant. The stress–strain-state of the pressurized cylinder with a part-through defect has its peculiarities. The plane strain condition has been assumed to exist for the crack front area close to the deepest point b, whereas near the internal free surface of cylinder it is generally believed that plane stress might prevail there. Evidently, it is necessary to take into account along with the crack front of growing defect, the change of factors such as stress asymptotic, constrain effect, local fracture stress, physical–mechanical material properties and the stress redistribution due to the plastic strain.
2. Crack growth model In this paper, we shall examine the growth of initially semi-elliptical surface cracks in a pressurized cylinder. The model describing the growth of such flaws is employed here, based on the following assumptions: • the defect is assumed to have, in the general case, an elliptical-arc shape; • the modeling of the flaw growth conditions is simplified by considering these as small scale yielding states in an area close to the crack front. When conditions of small scale plastic deformation take place (or are fulfilled), crack growth will be characterized in terms of elastic– plastic SIF, Kp ; • the change of the local fracture stress along the partthrough crack front is proportional to the distribution of local plastic strain close to the curvilinear crack front. To describe the elastic SIF distribution at any point along the crack front in a pressurized hollow cylinder (Fig. 1), we have used a numerical solution obtained from the 3D finite element analysis [8] PR p·b 1=2 b b t ; ; ;f Fe t Q a t R q p s n pbY Q ⫺1=2 ; Fe ; Q 1 ⫹ 1:464 b=a1:65
K
1
where P is the internal pressure, PR=t is the mean hoop nominal stress. The complete expression for a functional Fe is adduced in Ref. [8]. It is proposed to set the position of any points of the semi-elliptical crack front by two coordinates re and fe : These are the distances to the crack
V.N. Shlyannikov / International Journal of Pressure Vessels and Piping 77 (2000) 227–234
front and the parametrical angle, respectively (Fig. 1)
fe arctg
( for
b tg f a
2
points of the crack front coinciding with the semi-axes tips a and b and lying within the idealized plane stress and plane strain conditions, respectively, it is accepted that the dominant singularity governing the asymptotic
q e0 ⱕ 1; r0 b0 ; y0 r0 sin f; xe y0 =tg fe ; re x2e ⫹ y20 q e0 ⬎ 1; r0 a0 ; x0 r0 cos f; ye x0 =tg fe ; re x20 ⫹ y2e
Under cyclic loading in an area close to the crack front, the interconnected processes of accumulation and development of microdamages take place. The macroscopic effect of these processes of nucleation, growth and coalescence of voids is a new free surface creation or increment of the crack length that is apparent in various degrees in all materials from the quasi-brittle to the quasi-plastic. The magnitude of this increment is determined by the realization of the leading micromechanism of the crack growth in the elementary act of fracture, depending on stress–strain-state near the crack tip and the elastic–plastic material properties. In works [10– 12] for both static and cyclic loading conditions, the concept of the fracture damage zone (FDZ) is proposed, according to which the realization of the leading fracture micromechanism, the so-called decohesive (brittle) or coalescent (ductile) one, depends on the ratio between the FDZ-size and that of the crack tip plastic zone. The FDZ-size has been considered as a fundamental characteristic setting interrelation between the processes occurring on both micro- and macroscale level with respect to the material structure. In the present paper the FDZ ahead of the crack front is assumed to be located where the stress strain state in the element reaches a certain critical value that can be measured from standard test data. A relative FDZ-size has been introduced by Ref. [12] as ( ) 2 ^ S 22 ⫺ 4 W ⴱc ⫺ S 3 S 1 ⫹ S p 2 S d c ; 2 W ⴱc ⫺ S 3 4 2 s0 1 2 an s ⫹ s n⫹1 W ⴱc sn 2 f n⫹1 f where s n is the nominal hoop stress, s f is the local fracture stress, a and n are the hardening parameters of the Ramberg–Osgood power-law, S i i 1; 2; 3 and S p are the elastic and plastic coefficients that will be given below. According to model (4), the crack front with semi-axes a and b growth, after one cyclic loading step to the new configuration can be obtained provided that the coordinates of points A and B (Fig. 1) deduced from the FDZ equation satisfy Eqs. (2) and (3). The proposed model (4) assumes that the growth variation can be modeled by assuming different growth rate properties in the depth and surface directions. For boundary
229
3
behavior of the stresses at the crack tip (for the strain hardening material known as the HRR-singular field) has the form [13–15]
s ij
s ij s K p r ⫺1= n⫹1 s~ ij u; n; s e e s0 s0
K p r ⫺1= n⫹1 s~ e u; n
5
aIn 1= n⫹1 is the plastic SIF that is where K p J= related to the J-integral and a numerically computed quantity In ; s e the effective stress, r and u are the polar coordinates centered at the crack tip. In small scale yielding conditions, the plastic SIF is related to the elastic SIF as [13] JE 0 K2 as 02 12 I K n⫹1 2 E0 n p s0 s0 or for the plane strain (3D) 1= n⫹1 " 2 #1= n⫹1 2 J 1 ⫺ n K 1 3D ; K p s0 aIn3D aIn3D
6
7
E E0 1⫺n2 and the plane stress (2D) 1= n⫹1 " 2 #1= n⫹1 1 K J 1 2D ; K p aIn2D aIn2D s 0
E0 E 8
For formulae (7) and (8) containing the elastic SIF it is assumed that their distribution along the semi-elliptical crack front is described by Eq. (1), therefore it is possible to present the following complete expressions for coefficients S i i 1; 2; 3 and Sp from formula (4) S1 0:25 1 ⫹ n k ⫹ 1Y Q; Fe 2 ; S3 0:125 1 ⫹ n k ⫹ 1Y Q; Fe 2 p S2 0:5 1 ⫹ n 1 ⫺ kY Q; Fe 2 = 2; Sp
nl p n⫹1 s~ Y Q; Fe 2 n ⫹ 1 In e
9
230
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the plastic zone rp
Table 1 Variable parameters of cylinder and crack geometry t=R
R=R0
0.10 0.25 0.50 0.75 1.00
0.91 0.80 0.67 0.57 0.50
R (mm)
150 60 30 20 15
R0 (mm)
165 75 45 35 30
s s n =s 0
e0
b0 =t
b0 (mm)
a0 (mm)
0.10
0.01 0.10 0.30 0.01 0.10 0.30 0.01 0.10 0.30 0.01 0.10 0.30
0.15 1.50 4.50 0.15 1.50 4.50 0.15 1.50 4.50 0.15 1.50 4.50
1.50 15.0 45.0 0.60 6.00 18.0 0.30 3.00 9.00 0.15 1.50 4.50
0.25 0.2 0.4 0.5 0.7
0.50
1.00
where we have for the appropriate two points A and B of the crack front B-plane strain 3D—fe
p ⫺ l l3D 1 ⫺ n2 ; 2
A-plane strain 2D—fe 0 ⫺ l l2D 1;
k k2D 3 ⫺ n= 1 ⫹ n; In In2D ; s~ e s~ e2D : Approach by Eq. (4) enables us to predict the crack growth, when at the crack tip both elastic and plastic SIFs are known. It is often used in plane analysis to estimate the growth of advance of crack tip, making no distinction between the variation of stress state along the crack front, but it is here applied in a pointwise sense, so that the local crack growth is assumed to be related to the local fracture stress. In the present problem this relationship is applied in a finite sense. Let us assume that all material properties, the stress– strain-state and the fracture characteristics continuously vary along the crack front from the plane strain conditions in the cylinder wall to the plane stress on its interior surface. Then the local fracture stress s f that affects the FDZ-size will change appropriately from s f3D up to s f2D : For the plane strain case it may be tentatively assumed that s f3D ⬇ s utrue or
su 1 ⫺ c s0
10
s utrue is the true ultimate stress, c is the reduction of area. The second degree of freedom for the distribution s f fe at s f fe 0 s f2D is indefinite. Then for the purpose of the present analysis let us assume that this distribution s f fe of local fracture stress s f along the crack front in qualitative terms repeats the behavior of
11
The size of the plastic zone in the crack growth direction can be obtained as the distance from the crack tip to the boundary between the elastic and plastic areas taking into account that s e 1 on this boundary or ~ e3D 1= n⫹1 ; rp3D K 3D p s
2D ~ e2D 1= n⫹1 r2D p K p s
12
Then taking into consideration Eqs. (1), (7) and (8) we shall obtain for small scale yielding !1= n⫹1 2D r3D s~ e3D Y fe p=2 2 p 2 In 1 ⫺ n 3D 13 Y fe 0 r2D In s~ e2D p and the distribution of s f fe along the crack front of a semi-elliptical surface flaw by Eq. (11) can be rewritten in the following form
s f
k k3D 3 ⫺ 4n; In In3D ; s~ e s~ e3D
s f3D ⬇ s utrue
rp2D s f2D : rp3D s f3D
sf s f3D v !2 u 3D s0 u rp t cos fe ⫹sin f2e r2D p
14
where s f3D s f3D =s 0 and rp3D =r2D p are given by Eqs. (10) and (13), respectively. Thus, the function defining the distribution of the FDZ-sizes or increment of crack length along its curvilinear front is found. Strictly speaking, the use of formula (14) within Eq. (4) allows one to calculate the increment of crack length not only for two degrees of freedoms (two extreme points of front A and B), but for an infinite number of degrees of freedom assigned to a value of parametrical angle f e. Application of Eq. (4) together with Eq. (14) leads us to the following relation for fatigue crack growth rate prediction !1=m dl s n2 K 2f ⫺ s th2 DK 2th 2d l 15 dN 4s fⴱ eⴱf Ed p where l re fe ; d d=l; K f S1 ⫹ S p ⫹ S 2 d ⫹ S3 d ; DK th ; s fⴱ ; eⴱf ; m are the conventional constants of cyclic deformation and fracture curves. The growth of surface semi-elliptical defects is simulated by using a numerical solution for the crack front SIF. From these results the crack growth rate (Eq. (15)) at all points is inferred from the fracture damage zone model (Eq. (4)), and the crack front is advanced. The SIFs are re-evaluated, and a new crack front shape implied by the FDZ-model found. This process, which allows the crack front shape to evolve freely, is repeated until a terminal value of crack size is reached. The elastic–plastic singular solution (5) governing the asymptotic behavior of the stresses and strains at the crack tip, used in the present work, is correct only for small scale yielding conditions, that is when the plastic zone size does not exceed 10%
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231
Table 2 Chemical composition of steel Material
C
Si
Mn
S
P
Cr
Ni
Mo
V
Cu
Steel A
0.13 0.18
0.20 0.22
0.37 0.46
0.012
0.008 0.011
2.38 2.63
0.19 0.27
0.62
0.26 0.28
0.10 0.14
Steel B
0.17
0.28
0.42
0.004
–
2.40
1.02
0.50
0.15
–
of a characteristic body size. Such characteristic size for the hollow cylinder is the remaining ligament between the deepest point of the flaw and the opposite wall of the cylinder. The similar condition can be extended as restriction of existence for the idealized plane strain condition because of constraint effect in the deepest point of the crack front. Thus present approach can be used until then while the plastic zone size at the crack tip will not exceed 10%, the remaining ligament between the deepest point of the flaw and the opposite wall of the hollow cylinder. The size of plastic zone in the crack growth direction can be obtained first by Eq. (12).
3. Results and discussion In the present paper, initially semi-elliptical internal surface flaws in a steel cylinder under constant amplitude cyclic pressure are considered. The general regularities of crack growth for the variation of the geometry from a thinwalled t=R 0:1 to a thick-walled t=R 1:0 cylinder, an initial crack shape from oblong semi-elliptical up to true half-circular, the depth of the initial crack from shallow surface up to deep part-through have been established. The initial aspect ratio ranges from e0 0:1 to e0 1:0; whereas the relative initial crack depth is made to vary from b0 =t 0:01 to b0 =t 0:3: For each cylinder geometry variant t=R 0:1; 0.25; 0.5; 0.75; 1.0), 12 combinations of initial forms and sizes of surface flaws have been calculated (Table 1). The total number of calculation variants is 60 for each type of steel. The cylinders were endowed with the properties of two
steels, whose chemical composition is shown in Table 2. From Refs. [16,17] basic mechanical steel properties have been obtained for tests performed at low and room temperatures. As can be seen from Table 3, for steel A the state from brittle up to plastic was thus reached with the same chemical composition. As a matter of fact, each line of Table 3 may be considered as a separate material with peculiar properties. Moreover, such selection of data from Table 3 enables one to estimate the effect of test temperature on the development of cracks in the cylinder. The aspect ratio during the crack growth process was predicted using the present simulation approach. To substantiate the proposed model (Eqs. (4)–(15)), a comparison between the numerical and experimental results has been made for aspect ratio change. The experimental data reported in Ref. [2] for a pressure vessel of cylindrical type made of steel B and in Ref. [7] for a bending plate made of steel A has been compared with the simulation results obtained with the present model, which is shown in Fig. 2. It can be seen that the agreement between them is fairly good from a part-through crack growth to external surface of cylinder and plate respectively. The flaw propagation paths determined for both t=R 0:1 and t=R 1:0 under cyclic loading of the steel B thin- and thick-walled cylinders are displayed in Fig. 3. The initial crack configurations examined have a relative crack depth b0 =t equal to 0.01, 0.03, 0.1, 0.3 and a crack aspect ratio e0 equal to 0.1 (semi-elliptical flaw), 1.0 (half-circular flaw) or the intermediate values 0.25, 0.5 and 0.75. The diagrams in Fig. 3 show the advance of the crack front during the early stages up to the point of breakthrough. The crack behavior in thin-walled cylinder (Fig. 3a) and thick-walled cylinder
Table 3 Mechanical and fracture properties of type A and B steel Material heat treatment
Reference
Steel A quenching 1000⬚, 4 h., oil hardening, tempering 620⬚C 4 h., air cooling
[18]
Steel B quenching 920⬚C, 15 h,water cooling tempering 600⬚C 20 h., tempering 650⬚C air cooling
[2]
s 0;2 (MPa)
su (MPa)
sf (MPa)
20
1100
1157
2246
⫺196
1440
1590
1655
20
606
698
1377
Test temperature T (⬚C)
DKth p (MPa m 9.23
12.7 5.57
e (%)
c (%)
ef
n
16.6
67.2
1.115
7.034
3.1
2.9
0.029
8.511
18.5
69.5
1.187
6.674
232
V.N. Shlyannikov / International Journal of Pressure Vessels and Piping 77 (2000) 227–234
Fig. 2. Comparison of aspect ratio changes predicted by the present model with experimental data.
(Fig. 3b) is different. The most remarkable feature of the early stage of growth is that the crack shape change is strongly dependent on the initial shape, but a common characteristic can be found. Thus, for b0 =t ⱕ 0:01 these initial cracks with different e0 asymptotically tend towards a preferred profile as they propagate to the external surface of the cylinder, reaching a very similar shape. The crack aspect ratio tends to converge to some asymptote rapidly, that is, the front of such a flaw becomes nearly circular-arcshape for low values of b0 =t. On the contrary, for b0 =t ⬎ 0:01 the flaw aspect ratio changes slowly during the fatigue crack
growth. In other words, in the case internal surface flaw in a cylinder, the fatigue behavior for b0 =t ⱕ 0:01 is different from that for b0 =t ⬎ 0:01: Several authors have shown that the surface defects in pipes tend to follow preferred fatigue propagation paths, that is, the flaw aspect ratio is a function of the relative crack depth [2,3], and an analogous conclusion has also been drawn for edge flaws in flat plates [18]. Fig. 4 illustrates the fatigue crack shape developments for a range applied nominal stress level s n 0:2 up to s n 0:7 beginning with two different initially semi-elliptical cracks in thin-walled (a) and thick-walled (b) cylinders
Fig. 3. Propagation paths under cyclic pressure for different initial flaw configurations.
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233
Fig. 4. Variation of crack aspect ratio with nominal stress level for: (a) thin-walled cylinders; and (b) thick-walled cylinders.
made from both steels A and B with four different b0 =t ratios. For an initial surface flaw that depth b0 =t is greater than 0.1, the influence of s n s n =s 0 is really not excessively great. Similar experimentally observable behavior of the aspect ratio for initial deep cracks is noticed in the literature [5–7]. It should be noted that experimental data for more small-sized cracks b0 =t ⬍ 0:1 are absent in the literature because there are difficulties of exact measurement and development observation of part-through cracks. In these cases, when b0 =t ⬍ 0:1 the influence of an applied
nominal stress level is significant, which is more noticeable in a thick-walled cylinder (Fig. 4b). Distinctions in the cracks behavior for steels A and B also depends on an applied nominal stress level. For moderate low stresses s n 0:2 the influence of a material properties of cylinders is noticeable. For stresses s n 0:7 these differences decrease and almost disappear for thick-walled cylinders made from steels A and B. These calculated data agreed well with experiments of the authors of Ref. [5] who have shown that for s n 0:6–0:8; behavior of aspect ratio as a
Fig. 5. Predicted aspect ratio changes with test temperature for: (a) thin-walled cylinders; and (b) surface crack behavior at room temperature for steel A.
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function of depth ratio for several steel types and titanium alloys is described in one common curve. It can be remarked that the results computed by the present study differ for room (Fig. 5a) and low (Fig. 5b) temperatures. An especially strong influence on surface crack growth is rendered by low temperature (Fig. 5a). The lowering of test temperature to ⫺196⬚C reduces (see Table 3) almost entirely the restriction of plastic properties for steel A. In this brittle state, steel A ceases to respond to a change in the relative wall-thickness of a cylinder. At the same time, at room temperature (Fig. 5b) the scale effect in the cylinder of this material is exhibited. By comparing the crack profiles adopted during the fatigue growth process in Fig. 5b, the effect of the dimensionless wall-thickness ratio, t/R, on the fatigue shape development is seen to exist. The aspect ratio change is determined for b0 =t 0:01; 0.05 and different values of t=R; in particular, the two upper curves are related to semicircular surface flaws with e0 1:0; while the lower curves represent the semi-elliptical defects with e0 0:1; : 0.25; 0.5. Note that the influence of the parameter t/R for initial semi-elliptical flaw with e0 0:1 is not significant for low values of b=t; that is, the different curves for a given value of t=R are very close to one another. On the contrary, for relative depth b=t ⬍ 0:05 the influence of the scale factor t/R on the aspect ratio changes is significant for full range of initial semi-elliptical-arc-shaped flaw. As it follows from our numerical results, FDZ-size is a sensitive parameter affected by numerous factors. It can be expected that the aspect ratio change is also affected by a number of principal mechanical and fracture properties of a material. It will be remembered that the basic equations of our model contain elastic constants, static or cyclic strain hardening exponent, constants of cyclic deformation and fracture curves. 4. Conclusions The numerical analysis has been carried out to calculate the aspect ratio changes for different values of the geometrical parameters for both cylinder and surface flaw. Thickand thin-walled cylinders containing initial semi-elliptical internal surface flaws have been analyzed. Crack propagation paths in the diagram of flaw aspect ratio against relative crack depth have been determined through a fracture damage zone model. The influence of the main steel properties stipulated by various aspects of thermal treatment and test temperature on the aspect ratio change is significant, especially for low temperature. It has been found that
surface crack behavior is sensitive to the change of initial flaw configuration, nominal stress level and dimensionless wall-thickness. The comparison between the numerical predictions and experimental data shows that the agreement is good for the aspect ratio change, demonstrating that the modeling crack growth by fracture damage zone model is reliable.
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