Crack Growth Resistance Characteristics Determination ... - Core

Available online at www.sciencedirect.com

ScienceDirect Procedia Engineering 160 (2016) 29 – 36

XVIII International Colloquium on Mechanical Fatigue of Metals (ICMFM XVIII)

Crack growth resistance characteristics determination algorithm for creep-fatigue interaction V.N. Shlyannikova,b*, A.V. Tumanova, N.V. Boychenkoa a

Kazan scientific Center of Russian Academy of Sciences, Experimental and Numerical Mechanics of Deformation and Fracture Laboratory, Lobachevsky str, 2/31,Kazan, Russia; b Kazan National Research Technical University, Kazan, 420111, Russia

Abstract A creep-fatigue crack growth rate interpretation method and algorithm and test results are described. The algorithm allows to obtain a crack growth rate diagram in terms of creep stress intensity factor. The proposed algorithm is realized on steel compact tension specimens (12Х1MF). Crack growth rate was determined on standard compact specimens under temperature 550°C. The waveforms for the loading and unloading portions were trapezoidal, and the loading/unloading times were held constant – 5 s. A hold time of predetermined duration, 60 s, was superimposed on the trapezoidal waveforms at maximum load. The potential drop and the unloading compliance methods were used to monitor crack length during the creep–fatigue tests. For the experimental crack paths in tested specimens the governing parameter for the 3D-fields of the stresses and strains at the crack tip in the form of In-integral was calculated by finite element analysis along crack front. The governing parameter of the stress-strain fields in form of In-integral was used as the foundation of the creep stress intensity factor. The creep stress intensity factor approach was applied to the fatigue crack growth. The numerical and experimental results demonstrated that Kcr is the most effective crack tip parameter in correlating the creep–fatigue crack growth rates in power plant materials and can be used reliably for practical purposes. © Published by Elsevier Ltd. This ©2016 2016The TheAuthors. Authors. Published by Elsevier Ltd.is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the University of Oviedo. Peer-review under responsibility of the University of Oviedo Keywords: Creep-fatigue interaction, creep stress intensity factor, In-integral, creep–fatigue crack growth rate

* Corresponding author. Tel.: +7-843-236-31-02; Fax: +7-843-236-31-02. E-mail address: [email protected]

1877-7058 © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the University of Oviedo

doi:10.1016/j.proeng.2016.08.859

30

V.N. Shlyannikov et al. / Procedia Engineering 160 (2016) 29 – 36

1. Introduction The majority of power engineering objects work in high-temperature conditions. It causes a demand to consider creep properties of materials. Residual life of loaded details of power engineering objects usually depended on the amount of launchings. In durability studies a single loading cycle is often taken from a start moment to reaching of a nominal output mode. High temperature and cyclic loading conditions lead to cracks initiation in stress concentration points. In literature a lot of attention is paid to the processes which are taking place during the cracks’ growth due to action of fatigue loads. Also, a lot of attention is paid to studies of creep properties under monotonic static loading. Recently, special attention is paid to creep-fatigue interaction [1]. Procedure of the crack growth characteristics determination at the creep-fatigue interaction in the international standards can be found [2]. However, the postprocessing methods are described in the above standard [2], are not sufficient to describe such phenomena as difference of a crack growth rate along curvilinear front. In addition, some of equations presented in these standard lead to erroneous results. In the present study, the creep–fatigue crack growth rate methodology is extended for the curvilinear through-thickness crack case, and an analysis method is proposed that does not require simplifying assumptions regarding the crack geometry and the value for the governing parameter of creep stress–strain field in the specified test specimen configuration. 2. Algorithm for crack growth resistance characteristics determination under creep-fatigue interaction Our primary interest is to obtain an accurate description for the creep–fatigue crack growth rate on the base of nonlinear fracture mechanics parameters. In the past characterization of growth rates have been attempted in terms of stress intensity factor (SIF), the reference stress, crack opening displacement and creep fracture mechanics parameter C-integral. Various aspects of the characterization of creep crack growth have been reviewed in [1]. It is well known that at high temperature can be realized several mechanisms of deformation and failure in polycrystalline materials including diffusion, coupled diffusion and power-law creep and separate power-law creep. According to Hutchinson’s opinion [11], grain boundary cavitation is one of the primary mechanisms which is accompanied by voids nucleation, coalescence and growth. In the case of uniform the density of cavitating facets in the region of concern near the crack tip, the material meets the conditions required for application of the C-integral to the macroscopic crack problem. Fig.1 shows the algorithm of the creep-fatigue crack growth resistance characteristics determination for materials with secondary creep dominates and deformation behavior is described by the Norton elastic-nonlinear viscous constitutive relation. The proposed algorithm consists of three main parts: experiment, numerical calculations and postprocessing. The subject for both the experimental studies and the numerical analyses are compact tension specimens, which are most frequently used for characterizing creep crack growth rate. The proposed method is based on the concept of a creep stress intensity factor (CSIF) introduced by authors [3]. According to the authors [3], CSIF can be used as a generalized parameter for fracture resistance characteristic determination for creep and creep-fatigue interaction. This algorithm allows to determine the characteristics of a crack growth rate equation in terms of the creep stress intensity factor. In a simplest case these are is exponential equation constants which describe the linear segment of the crack growth rate diagram in log-log scale. In addition, the proposed CSIF is free from the limitation of two-parameter fracture criterions and it takes into account the in-plane and out-of-plane constraint effects near the crack tip. 3. Theoretical background The Hoff analogy [4] is used for CSIF determination method described in [3]. A substitution of strains and displacements to their rates gives us a possibility to signify energy supplied to a crack tip in terms of C(t)-integral which is similar to Hutchinson-Rice-Rosengren solution [5-9].

31


C (t )

wu

§

·

³ ¨©W * (t )dy T wx ds ¸¹

(1)

Г

Fig.1. Creep-fatigue crack growth resistance characteristics determination algorithm where W * (t ) is a rate of energy per unit of volume, Г is a closed contour at the crack tip, ds is a increment along the contour path, Т is a normal vector to the contour, u is a displacements rate, x, y, z are Cartesian coordinates. For the second creep stage the behavior of the material is described by Norton equation:

H

n

§V · H0 ¨¨ ¸¸ , H E ©V0 ¹

V

where

E

V E

n

BV , B

is a Young's modulus,

V0

H0

(2)

n

V0

is a reference stress,

H0

is a strain rate at the reference stress,

B

is an

amplitude factor, n is a exponent factor. In this case, the first term of the asymptotic expansion for creep conditions can be present as:

32


A1 (t )

§ C (t ) · ¨¨ ¸¸ © V 0H0 I n L ¹

1 n 1

(3)

where I n is a governing parameter of stress-strain fields, which depends on material properties, loading conditions and geometry of a cracked body, L is a crack characteristic length. For steady creep the equation (3), with taking into account (2) can be presented as [3]: 1 n 1

§ C (t ) · ¨¨ ¸¸ © BI n L ¹

K cr

(4)

In this equation the governing parameter of stress-strain fields can be obtained by numerical methods:

I n T , T *, n

S

³ :(T ,T *, n)dT

(5)

S

:T , T *, n

ª § du~ ·º du~ · n ~ n 1 § V e cosT «V~rr ¨ u~T r ¸ V~rT ¨ u~ r T ¸»sin T n 1 dT ¹ dT ¹¼ © ¬ ©

1 ~ ~ V rr u r V~rT u~T cos T n 1

(6)

where V~e is a dimensionless equivalent stress intensity,

r ,T

are coordinates in polar coordinate system

u~i is a dimensionless displacement rates. C (t ) -integral for the standard compact tension specimens:

centered at a crack tip, According [2], the

C (t )

PVc bW

'P('Vc ) § f ' · ¨ ¸ bWt h ¨© f ¸¹

§ f '· ¨¨ ¸¸ © f ¹

(7)

where

f

º ª « 2 a » 2 3 4 « W »§¨ 0.866 4.64§¨ a ·¸ 13.32§¨ a ·¸ 14.72§¨ a ·¸ 5.6§¨ a ·¸ ·¸ « 3 »¨ ©W ¹ ©W ¹ ©W ¹ © W ¹ ¸¹ «§ a · 2 »© « ¨1 ¸ » ¬© W ¹ ¼

f'

df , d (a / W )

P

is a applied load,

(8)

Vc

is a crack opening displacement rate at the load line, b is a specimen thickness, W is a

length of the working area of the specimen, each load cycle.

a is a crack length, f is a correction function, t h is a hold time on

33


It should be noted that the equation for f ' / f determination presented in [2] leads to erroneous results. Therefore, an approximate method to calculate derivatives was used. The final equation for the creep stress intensity factor can be expressed as [3]: 1 n1

K cr (t )

§ Vc § f ' · · P ¸ ¨ ¨ V BI (t ) L bWL ¨¨ f ¸¸ ¸ © ¹¹ © 0 n

(9)

Thus, it’s necessary to know the dependencies between a crack opening displacement rate crack length

a

f (t h )

Vc

f (t h )

and a

via creep time to calculate all the required parameters. These dependencies give a

possibility to determinate a governing parameter in the form of In-integral by numerical methods. It allows to obtain a crack growth rate diagram in terms of creep stress intensity factor. 4. Experimental procedure and numerical calculations Verification of the proposed algorithm was realized on the UTS-110MH-5-0U test system with a hightemperature oven and a high-precision crack opening displacement extensometer (Fig.2).

Fig.2. Test equipment The specimen configuration chosen for the creep–fatigue test is a standard compact type, C(T) specimen. In general, the dimensions chosen are as recommended by the ASTM test standard (Fig.3a). The size of the working area is W 50 mm, specimen thickness is b 12.5 mm. The waveforms for the loading and unloading portions were trapezoidal, and the loading/unloading times were held constant (5 s rise and decay times). A hold time of predetermined duration, 60 s, was superimposed on the trapezoidal waveforms at maximum load, as shown in Fig. 3b. The tests were carried out at 5500C with a load ratio, R, of 0.1. Two times during the total creep–fatigue life was the test was stopped, and the C(T) specimen cooled to room temperature; then the specimen was again subjected to creep–fatigue loading at 5500C. This practice helped to determine the intermediate crack front positions during the creep–fatigue loading between the initial (pre-crack) and the final crack fronts. After total failure of the specimen, measurements of the crack sizes were taken for four positions of the crack front by means of an optical microscope. In addition to the unloading compliance method a potential drop method were used to monitor crack length during the creep-fatigue tests. It was assumed that the relationship between the potential drop and the crack length is linear. The average crack length was calculated according to:

34


ª U (t ) U 0 º «a f a0 U f U 0 »¼» a0 ¬«

ars (t ) where

a0

is an initial crack length,

U0

(10)

is an initial potential drop,

a f , U f is a final crack length and potential

drop respectively. Dependences between a crack opening displacement rate and a crack length via number of cycles are experimentally obtained. The next stage of the proposed algorithm is an In-parameter determination for all crack front points depending of creep time. To confront the theoretical aspects relevant to the experiments, finite element analysis was used to determine a governing parameter In for a power-law creep material constitutive relationship. The FE modeling and stress-strain state analysis were realized using the commercial code ANSYS [10]. For stress-strain fields and governing parameter I n calculation a finite element models of compact tension specimen was created (Fig.4). The twentynode quadrilateral brick isoparametric three-dimensional solid elements were used for the specimen model. In order to ensure the convergence of numerical results the element model has a significant mesh concentration near a crack front. Each of the FE-meshes contains about 240,000 nodes. To ensure the convergence results a minimum distance between nodes is lmin 1.6 105 mm.

a)

b)

Fig.3. Specimen geometry (a) and creep–fatigue cycle shape (b)

Fig.4. Finite element model As was noted earlier the crack front markers were applied to every experimental specimen. These markers were used for determine a relationship between In-integral and creep time. For displacement rates field calculation three numerical solutions is conducted for each point of dependencies InIntegral via relative crack length. For each of the experimental crack front positions numerical calculations for experimentally determined creep time, and two additional were performed. Two additional solutions with time


incriments (± 5 hours) to obtain a displacement rates is needed. For each point of the specified contour near a crack tip the dependencies of displacements via creep time were obtained from numerical calculations. Further, these dependencies were approximated by polynomials of the second order. The displacement rates were defined as a first derivative of the obtained curves. It’s gives an opportunity to establish a history of In-integral changing along crack front during the experiment for each specimen. 5. Results and discussions The proposed algorithm is implemented on compact specimens cut from a steel (12Х1MF) pipe bend. Test results for different cutting directions and different values of the applied load are presented in this section. Experimental load for the specimens cuted out in a longitudinal direction is 8 kN, in a transverse direction is 7 kN. Dependences of the potential drop and crack opening displacements during the hold time on each test cycle via summary time are presented at Fig.5. On the basis of crack front markers the In-integral distributions were obtained numerically (Fig. 6).

a) b) Fig.5. Potential drop on crack edges (a) and force-line displacement rate vs creep time (b)

Fig.6. In-factor behavior as a function of relative crack length (1 – L-direction, specimen surface; 2 – L-direction, mid-section; 3 – T-direction, specimen surface; 4 – T-direction, mid-section)

Knowing trends of changing of all parameters in the equation (9) we can create a crack growth rate diagram in terms of a creep stress intensity factor (Fig.7). It is seen that the experimental data fall within a relatively narrow scatter band, and the dependence of the crack growth rate, da/dt, on Kcr follows a near linear trend on a log–log scale. Fig. 7 shows the creep–fatigue fracture diagrams of the compact tension specimens. Constraint effects is usually occurred through the variability of the crack growth rate along crack front. As seen from the figure 7 the creep stress intensity factor is sensitive to a crack growth rate variability along crack front. This cannot be achieved using methods known in the literature.

35

36


a) b) Fig.7. Crack growth rate as a function of creep stress intensity factor (a - Longitudinal direction, b - Transverse direction) Thus the proposed method is free from the limitations of suppositions of state, plane strain or plane stress. Equally important, this method accounting for the in-plane and out-of-plane constraint effects is applicable for experimental specimens and real structural elements with through-the-thickness or part-through surface curved cracks. 6. Conclusions Testing and results interpretation algorithm allowing to create a crack growth rate diagram in terms of the creep stress intensity factor was proposed. The numerical and experimental results demonstrated that Kcr is the most effective crack tip parameter in correlating the creep–fatigue crack growth rates in power plant materials and can be used reliably for practical purposes. The application of the introduced parameters through experimental study of the creep–fatigue crack growth rate confirms effectiveness of the proposed approach. Equally important, this method accounting for the in-plane and out-of-plane constraint effects is applicable for experimental specimens and real structural elements with through-the-thickness or part-through surface curved cracks. 7. Acknowledgment The authors gratefully acknowledge the financial support of the Russian Foundation for Basic Research under the Project 15-38-20169. References [1]. Saxena A. Nonlinear fracture mechanics for engineers. CRC Press LCC; 1998. 472p. [2]. ASTM E2760-10e2, Standard Test Method for Creep-Fatigue Crack Growth Testing, ASTM International, West Conshohocken, PA, 2010 [3]. Shlyannikov V.N., Tumanov A.V., Boychenko N.V. A creep stress intensity factor approach to creep–fatigue crack growth. Eng. Fract. Mech 2015. V142, p.201-219. [4]. Hoff NJ. Approximate analysis of structures in the presence of moderately large creep deformation. Quarterly Appl Mech 1954;12:49–55. [5]. Hutchinson J.W. Plastic stress and strain fields at a crack tip. J Mech Phys Solids. 1968. 16:337–347 [6]. Rice J.R., Rosengren G.F. Plane strain deformation near a crack tip in a power-law hardening material. J Mech Phys Solids 1968. 16:1–12 [7]. Shlyannikov V.N., Tumanov A.V. Characterization of crack tip stress fields in test specimens using mode mixity parameters. Int. J. Fract. 2014, 185:49-76. [8]. Rice J.R. A path independent integral and the approximate analysis of strain concentration by notches and cracks. J Appl Mech. 1968. 35:379–386 [9]. Hilton P.D., Hutchinson J.W. Plastic intensity factors for cracked plates. Eng. Fract. Mech. 1971, vol.3, p. 435-451. [10]. ANSYS. Theory Reference. 001242. Eleventh Edition. SAS IP, Inc., 1999 [11] Hutchinson JW. Constitutive behavior and crack tip fields for materials undergoing creep-constrained grain boundary cavitation. Acta Metall 1983;31:1079–88.