Mathematical and numerical correction of the DIC displacements for determination of stress field along crack front

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21st European Conference on Fracture, ECF21, 20-24 June 2016, Catania, Italy 21st European Conference on Fracture, ECF21, 20-24 June 2016, Catania, Italy

Mathematical and numerical correction of the DIC displacements for determination of stress field along crack front for determination of stress field along crack frontblade of an Thermo-mechanical modeling of a high a b pressure turbine Chernyatin A.S. *, MatvienkoYu.G. , Lopez-Crespo P.c a airplane gas turbineb, engine Chernyatin A.S. *, MatvienkoYu.G. Lopez-Crespo P.c Bauman Moscow state technical university, ul. 2-ya Baumanskaya, 5, Moscow, 105005, Russia

XV Portuguese Conference on Fracture, PCF 2016, 10-12of February 2016,displacements Paço de Arcos, Portugal Mathematical and numerical correction the DIC

a

Mechanical engineering research institute of the Russian Academy of sciences, Maly Kharitonievskiy pereulok, 4, Moscow, 101990, Russia a Bauman Moscow state technical university, ul. 2-ya Baumanskaya, 5, Moscow, a b 29071, Malaga, c 105005, Russia c University of Malaga, C/Dr Ortiz Ramos, s/n Spain b Mechanical engineering research institute of the Russian Academy of sciences, Maly Kharitonievskiy pereulok, 4, Moscow, 101990, Russia c University of Malaga, C/Dr Ortiz Ramos, s/n 29071, Malaga, Spain a Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal b IDMEC, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Abstract Portugal c Abstract CeFEMA, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, The digital image correlation method (DIC) has wide application Portugalto determine the singular (stress intensity factor) and nonb

P. Brandão , V. Infante , A.M. Deus *

singular (T-stress) components of method the stress fieldhas in the vicinity of thetocrack-tip. DIC a simple implementation and and provides The digital image correlation (DIC) wide application determine thehas singular (stress intensity factor) nonlarge arrays of experimental data. However, it requires an a priori or post processed determination of the rigid body shifting and singular (T-stress) components of the stress field in the vicinity of the crack-tip. DIC has a simple implementation and provides the crack-tip position. Abstract large arrays of experimental data. However, it requires an a priori or post processed determination of the rigid body shifting and A methodposition. for mathematical processing of the experimental displacement fields obtained by means of DIC is presenting and the crack-tip validating in this work. Themodern method provides anthe accurate and direct forfields the determination of and the During their operation, aircraft of engine components aresolution subjected to problem increasingly demanding operating conditions, A method for mathematical processing experimental displacement obtainedofbysimultaneous means of DIC is presenting especially the high pressure turbine (HPT) blades. Such conditions cause these parts to undergo different types of time-dependent body displacement and the position of the crack-tip and crack plane orientation as well as appropriate terms of Williams’s series validating in this work. The method provides an accurate and direct solution for the problem of simultaneous determination of the degradation, of on which is creep. A kinematic model using the finite method (FEM) was developed, in order to be able predict expansion. It isone basin geometric andelement involves a solution of multiparametric minimization. As atoresult, body displacement and the position and of the crack-tiprelations and crack plane orientation as well as appropriate terms of Williams’s series theprocedure creep behaviour of HPT blades. Flight data records (FDR) for a specific aircraft, provided by a commercial aviation this allows better accuracy in the estimation of fracture mechanics parameters. expansion. It is basin on geometric and kinematic relations and involves a solution of multiparametric minimization. As a result, company, were used to program obtain thermal and mechanical data for three different cycles. In of order create The the program 3D model An interactive Matlab with graphical userofinterface was developed for flight implementation the to method. this procedure allows better accuracy in the estimation fracture mechanics for the FEM analysis, a HPT blade scrap was scanned, and its parameters. chemical incomposition and material properties were hasneeded aAnwide functionality for filtration and selection of the experimental data presented the form of displacement fields as well interactive Matlab program withwas graphical user interface was and developed forsimulations implementation of thefirst method. The program obtained. The data that was gathered fed into the FEM model different were run, with a simplified as for control of the solution and it verification. Possibility of automatic accounting of real position and orientation the3D has a wide functionality forinfiltration selection ofthe themodel, experimental data presented in the the form of displacement fieldsscrap. asofwell rectangular block shape, order to and better establish and then with the real 3D mesh obtained from the blade The crack by means of definition of appropriate geometrical parameters, allows simplifying the procedure for measurement of the asoverall for control of the solution inand it verification. Possibility of automatic accounting the realedge position orientation the expected behaviour terms of displacement was observed, in particular at theoftrailing of theand blade. Thereforeofsuch a displacement fields and post processing. The method has parameters, a great potential forsimplifying application to full-scale engineering components crack by can means of definition of appropriate model be useful in the goal of predictinggeometrical turbine blade life, givenallows a set of FDR data. the procedure for measurement of the and allows automatic tracking of a fatigueThe crack with simultaneous determination of the fracture mechanics parameters. displacement fields and post processing. method has a great potential for application to full-scale engineering components The efficiency of the approach has been demonstrated on real fatigue crack on compact tension specimens with different and automatic tracking of aby fatigue crack with simultaneous determination of the fracture mechanics parameters. © allows 2016 The Authors. Published Elsevier B.V. crack lengths and loading The efficiency the conditions. approachofhas demonstrated onofreal fatigue Peer-review underofresponsibility thebeen Scientific Committee PCF 2016. crack on compact tension specimens with different It should be noted, the proposed method can be using as a basis of the determination of the fracture mechanics crack lengths and loadingthat conditions. parameters along crack front. this purpose three It should be the noted, that the For proposed method cansteps be must usingbeasperforming: a basis of the determination of the fracture mechanics Keywords: High Pressure Turbine Blade; Creep; Finite Element Method; 3D Model; Simulation.

parameters theAuthors. crack front. For this purposeB.V. three steps performing: Copyright © along 2016 The Published by Elsevier This is anmust openbe access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Scientific Committee of ECF21. * Corresponding author. Tel.: +7-499-263-6391; fax: +7-499-267-48-44. E-mail address: [email protected] * Corresponding author. Tel.: +7-499-263-6391; fax: +7-499-267-48-44. E-mail address: [email protected] 2452-3216 © 2016 The Authors. Published by Elsevier B.V. * Corresponding Tel.: +351of 218419991. Peer-review underauthor. responsibility the Scientific Committee of ECF21. 2452-3216 © 2016 The Authors. Published by Elsevier B.V. E-mail address: [email protected]

Peer-review under responsibility of the Scientific Committee of ECF21. 2452-3216 © 2016 The Authors. Published by Elsevier B.V.

Peer-review under responsibility of the Scientific Committee of PCF 2016.

Copyright © 2016 The Authors. Published by Elsevier B.V. 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 Scientific Committee of ECF21. 10.1016/j.prostr.2016.06.331

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- Calculation of dominated terms of William’s series expansion in the vicinity of the crack tip on the specimen surface, using the described method. - Implementation of previously developed experimental and numerical method of the inverse problem solution for calculation of the loading parameters act on the crack region. The corresponding boundary problem of the solid mechanics is employing with obtained information related to the crack tip location and local (near the tip) displacement fields which are restored via the expansion terms. - Solution of the direct problem for numerical calculation the stress intensity factor and the T-stress along the crack front at determined load parameters and crack geometry. © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of ECF21. Keywords: digital image correlation method; stress intensity factor; T-stress; crack tip position; fatigue crack; compact tension specimens; multiparametric minimization; finete element method

Nomenclature a W, B F E, ν X0Y U, V x0y r, θ u, v X0, Y0 α A, B φ ai KI Txx, Tzz P

crack length width and thickness of the Compact Tension specimen force applied to the specimen Young's modulus and Poisson's ratio of material global coordinate system corresponding experimental measurements displacements of the specimen points in the global coordinate system local coordinate system associated with the crack-tip polar coordinates associated with x0y displacements of the specimen points in the local coordinate system coordinates of the crack-tip in the global coordinate system before the loading angle of orientation of the crack plane before loading displacements of the crack-tip after loading angle of rotation of the crack plane after loading Williams expansion coefficients of mode I crack Stress intensity factor of mode I crack T-stresses act in the crack plane normal and along of the crack front vector of loading parameters

1. Introduction Over the past two decades, digital image correlation (DIC) has become very popular to analyse structural integrity problems. There are many works devoted to the use of DIC to determine fracture parameters. The very first work by McNeill (1987) fitted DIC full-field displacement data to the Westergaard solution (1939) and in it, Yoneyama and Murasawa (2009) determined not only mode I stress intensity factors (SIF) but also rigid body motion and other far field parameters of the truncated series type stress function. One DIC technique introduced by Riddell et al. (1999) and Sutton et al. (1999) for fatigue crack growth studies was a two-point DIC displacement gauge used to measure the local crack opening displacement. Several methods to determine the mode I SIF, the value of the T-stress and the level of crack closure were compared in works of Carroll et al. (2009). T-stress is the second term in the Williams expansion (1957) and represents the uniform stress component (Anderson (1994). Lopez-Crespo et al. (2009) and Zhang with He (2012) used DIC also to evaluate mixed-mode I and II SIFs. Thus, it can be noted that the DIC method has wide application to determine the singular and regular components of the stress field in the vicinity of the crack-tip. Knowing the stress field around the crack-tip, it is possible to apply an over-deterministic SIF calculation method as was shown by Sanford and Dally (1979), the analytical solution provided by Williams in work of Beretta (2015), the integral invariant M-theta was used by MoutouPitti (2008), Pop et al. (2011, 2013). Hellen (1975), Parks (1974), Destuynder and Djaoua (1981) used other techniques to extract the

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SIF by using interaction integrals. The main idea of a method developed by Rethore et al. (2005) is to combine both numerical and experimental techniques to determine the SIFs for modes I and II separately, using the measured displacement field from DIC and interaction integral. Two parameter fracture mechanics approach describing the near-crack-tip stress field, was applied by Erdogan and Sih (1963) to determine the SIF and the T-stress using DIC with domain integral calculation. In Q4-DIC approach of Mathieu et al. (2012) the DIC procedure consists of measuring displacement field discretized with quadratic Q4 elements. The DIC is even preferable for the investigation of the fatigue. Different least-squares regression algorithms using displacements from DIC were developed in the works of Yoneyama et al. (2006), Hamam et al. (2007), LopezCrespo et al. (2008), Abanto-Bueno and Lambros (2006), Pataky et al. (2012) to find the effective SIFs (KI and KII) and the T-stress during fatigue crack growth studies. An accurate determination of the crack-tip position during crack growth is important in the DIC analysis (Zanganeh et al. (2013). A combination of information from optical microscopy and the displacement distribution from DIC analysis was proposed by Zhu et al. (2015) and rigid body motion was considered. Otherwise, the adopted DIC algorithm of Vasco-Olmo and Díaz (2015) would consider it as a virtual displacement induced by the applied load. The precise determination of the crack-tip location has a strong influence on the amplitude of SIFs (Roux and Hild (2006). According to Lopez-Crespo et al. (2009) calculating and analysing the SIFs (modes I and II) and the plastic zone size needs firstly to locate the crack-tip. A method based on DIC can be using in order to find discontinuities even if the end of the crack is not visible, see Grégoire et al. (2009) and Zhao (2012). The strategy consists of decomposing the estimated displacement field onto the basis of test functions. In work of Delaplace and Hild the objective function is then minimizing to estimate mode I and mode II SIFs. From the absolute minimum of objective function, one locates quite precisely the crack-tip position that provides the best fit quality. This post-processing approach is using to estimate the crack-tip position in experimental part of this paper. It should be noted that there is no approach that allows accurate and direct solving of the problem of simultaneous determination of fracture mechanics parameters, the rigid body displacement in large scale and the position of the crack-tip in it. The purpose of this work is the development and validation of the method that provides a solution to this problem and can be applied to any full-scale objects (structures) and cracks of arbitrary configuration. It becomes more urgent question of determining the parameters of fracture mechanics along the crack front, not just near the crack tip on the surface. Developed and presented in the paper approach allow both directly using the displacement data measured from the object surface by DIC and post-processing such data, including the use of finite element method (Garcia-Manrique et al. (2013). It is based on geometric and kinematic relations in large scale displacements, taking into account the real localization and orientation of the tip and plane of the crack, and solution the problem of multiparametric minimization. The obtained parameters (SIF, T-stress) can be using to evaluate the admissibility of safe crack-like defects and cracks in the stability of the considered full-scale objects as was mentioned Matvienko (2013) and presented in SINTAP (1999). 2. Experimental data processing procedure 2.1. Mathematical statement of the problem Obtain a general expression with expanded number geometric and kinematic parameters to describe the problem. Fig. 1 shows a global coordinate system (GCS) XOY, which is associated to the experimental data, the horizontal (X) and vertical (Y) coordinates of arbitrary measurement points (P), the horizontal (U) and vertical (V) measured displacement data. The local coordinate system (LCS) x0y, is tied to the crack-tip and is used to describe the stress strain state in the vicinity of the crack-tip by using singular formulas for u and v displacement fields. x and y are the coordinates of the measured point P in the local coordinate system. The LCS before loading may be different from the GCS and be translated along the axes X and Y by X0 and Y0 respectively, X0 and Y0 being the crack-tip coordinates in the GCS. The LCS can also be rotated with respect to the z-axis at an angle α. α is angle of orientation of the crack plane). This discrepancy exists because in practice it is very difficult to determining the exact position of the crack-tip and therefore set the GCS of measurements at the crack-tip. The parameters A, B and φ represent the shift and rotation of the LCS after loading caused by offset of the object (structures cracked region) as a rigid body.

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The following geometric relationships can be deduced from Fig. 3: x   X  X 0  cos   Y  Y0 sin  ; . (1) y  Y  Y0  cos    X  X 0 sin  It is possible to establish the relationship between the geometric and kinematic parameters. It provide to recalculation of the fields around the crack-tip to the displacement data registered by means of DIC:

U  u cos   v sin   xcos   cos   ysin   sin    A; , V  u sin   v cos   xsin   sin    ycos   cos   B where      .

(2)

Fig. 2. The geometric and kinematic formulation of the problem of the displacement fields definition

The displacements u and v occurred around the crack-tip can be calculated by using of the singular problem equations with T-stress in the LCS:   T K r cos   M  ...; u r ,    r cos  M  1  sin 2   , (3) 2 2  4G 2 G

  T K r sin  M  2   ... r sin  M  cos 2   2 2  4G 2 G where r, θ are the polar coordinates in plane x0y attached to the crack tip, T is so called Txx-streess (the amplitudes of the second order terms in the three-dimensional series expansion of the crack-front stress field [7]) and E 2 G ,M  1  . 21    v r ,   

The equation (3) is right for plane stress state implemented at the points on the body surface. So, the determination of the K and T on the basis of experimental data can be lead in conjunction with the definition of the following parameters: X0, Y0, α, A, B, φ, which (with K and T) will be called as state parameters. Solution of the problem can be representing as a multiparameter problem of minimizing the objective function I. The objective function shows the deviation between experimentally obtained displacements U*, V* in registration points

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on the object surface with coordinates X*, Y* and displacements U, V calculated via equation (2) in conjunction with (1) and (3). Of course, the displacements U, V is computing in the registration points at the current value of the state parameters corresponding to the iteration of the minimization algorithm. The objective function is root-mean-square or maximum deviation in N number of the registration points. It should be make some remarks. Note that the local displacement model (3) can be easily supplemented by other terms of the Williams’s expansion including other crack modes, but in this case, the number of unknowns is increased and require further estimation of expansion convergence. Since experimental field U*, V* is not smooth (the objective function may not have a good form) and may contain measurement errors, It is proposed to use the mathematical programming methods of minimizing the 0-th order such as method of Nelder and Mead (1965), or modern evolutionary methods of global minimization such as particle swarm theory of Poli et al. (2007). When W component of the displacement is known from DIC, it is possible to build relationships similar (2) based on the multiplication of rotation matrices with respect to the axes. In the case, number of kinematic and geometrical unknown is increasing, but we are able to determine the spatial orientation of the crack front, as well as take into account the absolute displacement of the body, which is important for real objects. Since DIC-method gives a large arrays of the experimental data that provide obtaining reliable values of state parameters. It should be noted that the mathematical data processing approach and response bank takes into account the plasticity at the crack tip or accumulation of damage as an unknown parameter in minimization problem. 2.2. Distribution of the two-parameter fracture mechanics parameters along the crack front After determination of the Williams expansion coefficients ai* (KI* and Txx* are among of them) in the external surface of the object by means of described DIC processing method, the distribution of KI and Txx values along the crack front can be obtained for two parameters fracture mechanics. At first, using the special method that was developed by Chernyatin and Razumovskii (2009, 2011, 2013) for solution different inverse problems of solid mechanics it is possible to determine the values of the loading parameters P lead to such KI*, Txx* (and other terms of the expansion). This method basing on the finite element model of the object provides a formation so-called “response bank”. In considering problem the response bank allows to state the relations ai= ai(P,s), where s is dimensionless local coordinate along the crack front, such that s=0 corresponds to the front center and s=1 corresponds to front exit point on the free surface of the body. A minimization difference between ai* and ai(P,1) lead to real values P* of P. After implementation of direct calculation via response bank it is possible to determine the crack front distribution of KI=KI(P*,s) and Txx=Txx(P*,s). It should be noted that it can also be given the task of determining of geometry of the crack front as the DIC method provides a large amount of information. The problem of simultaneous determination of parameters of loading and front geometry settings crack was already solved by Chernyatin et al. (2015). Therefore, there are no fundamental limitations in addition to the state parameters to include as additional unknowns front geometric parameters and to use for this technique is already well established by Chernyatin and Razumovskii (2013). In this case, it is possible to stepwise implementation of the procedure for determining these parameters. 2.3. The program realization A Matlab program with graphical user interface was developed. The program conducts the following operations: • Import the results of the experiments, i.e. X and Y coordinates and U and V displacements, and the "rarefy" this data for fast pre-processing. • Delete a rectangular area around the crack line to exclude the noisy data from this region. • Select a circular region around the crack-tip. This information is U*, V* at X*, Y* and will be used subsequently to calculate the state parameters; • Build the field of u and v displacements fields. These are evaluated in GCS with equations (2). This step is very useful to compare the corrected fields to the original experimental fields and to obtain a first approximation of the state parameters.

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• Calculate the state parameters by minimizing the discrepancy between experimental U*, V* fields and the corrected U, V fields obtained with equations (2). This is doing iteratively and the values of the parameters are updating at each step of the process. Nelder-Mead method is using for the minimization of it. Previous works of Chernyatin and Razumovskii (2013) showed that Nelder-Mead method produced good results in terms of accuracy in similar type of analyses. The objective function are defining as the root mean square of the fields’ difference. • Save and load work sessions in the program. Note that to ensure stability in the process of determining the unknown state parameters, it should be, at first, the initial their assessment determine by their selection. For this purpose, the program displays the data at different stages of processing. The sequences of the state parameters calculating were determined to provide a god solution accuracy and stability at low time cost. 3. Experimental approbation of the procedure 3.1. Experiment review Compact Tension specimens of Al 2024 T351 (with E=73,4GPa, ν=0,33, yield strength is 325 MPa) were used in this work. All specimens have following geometrical parameters (see fig. 2): W=50 mm, B=12 mm, but differ from each other by crack length a and tensile loadings (Fmin and Fmax). DaVis software was used to measure experimentally the displacement fields. A 12 bit black and white video camera was used in conjunction to Navitar Precise Eye macro lenses to evaluate field of views of approximately 14×12 mm. The random pattern required by DIC algorithm was achieved by finely abrading the specimen surface with silicon carbide sand paper of grades 400, 200 and 120. This introduced a random distribution of scratches.

Fig. 2. Compact specimen and displacement fields experimentally obtained (U, V) and the theoretically expected (u, v) with eliminating of the shift and rotation of the specimen

3.2. Results and summary Fig. 2 shows the U and V displacement fields on the surface of the specimen with a=25.5 mm obtained by the DIC method between loads Fmin=0,55kN and Fmax=5,5 kN. The fields have negative values at all registration points. This may indicate a shift of the specimen due to rigid body movement during loading. Moreover, the follow inequalities v(x,0)>v(x+dx,0) and u(x,-y)
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showed existence of translation and rotation of the specimen due to rigid body movement during loading. Excluding the rigid body shifting and rotation the deviation of experimental and calculated displacement fields achieve in some point a several time. Only introduction the real position of the crack tip first of all along the x-axis (coordinate X0) can lead to the high degree of similarity between fields. It should be noted that the kinematic and geometrical parameters should be computed simultaneously since the changing one of them can lead to other values at the same deviation between experimental and computational displacements data. Table 1 shows the results of calculations of the state parameters for different crack lengths and different loads. Table 1. The results of mathematical processing of the DIC data а, mm

25,5

34,9

ΔF, kH

4,95

1,26

K, MPa*m0,5

19,5548

T, H

-

-

X0, mm

1,7178

-2,5832

Y0, mm

0,17944

0,30981

α, grad

-4,9010

-5,6679

A, mm

-0,046775

-0,033254

B, mm

-0,34749

-0,14536

φ, grad

-0,2132

-0,0337

37,1146

10,5161 22,7962

Comparison of evaluated SIFs and ones calculated according to ASTM E-399-90 (Reapproved 1997) gives a good agreement with error les then 6%. The crack tip locating coordinate X0 differences from experimentally determined is not above then 3%. Though the error of Y0 coordinate determination is much larger than X0. For the considered compact specimen with a crack length 25,5 mm, loaded with a tensile load F=1 kN (that is the loading parameter) the values of SIF and T-stresses (as Txx, and Tzz introduced by Nakamura and Parks (1992) along the crack front was calculated as described above (fig. 3).

Fig. 3. The distribution of KI and Txx, Tzz along the crack front (s=0 – front center, s=1 – front exit point)

4. Conclusion A method for mathematical processing of the experimental displacement fields obtained by DIC is presented in this work. Possibility of automatic accounting of the real position and orientation of the crack by means of defining of appropriate geometrical parameters, allows simplifying the procedure for measuring of the displacement fields and post processing. As it is sufficient to capture the interest area with a crack, without being tied exactly to the crack tip (which did not initially known) and the crack orientation. The method has a great potential for application on full-scale objects because the methodology accounts for the shifting and rotation of the region of interest studied.

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For this purpose, the geometrical and kinematic equations (10) and others should be rewrite for 3D problem. The displacement fields measured experimentally between two different loads allow extraction of singular and nonsingular terms of the stress field near the crack-tip, the crack-tip coordinates and the crack orientation. The efficacy of the approach has been demonstrated on real fatigue crack and therefore makes the approach particularly useful for fatigue crack growth studies. The methodology allows automatic tracking of a fatigue crack with simultaneous determination of the fracture mechanics parameters without additional corrections from standard DIC surface observations. Acknowledgements The authors acknowledge the support of the Russian Science Foundation (Project N 14-19-00383) References McNeill SR, Peters WH, Sutton MA., 1987. Estimation of stress intensity factor by digital image correlation. Engineering Fracture Mechanics 28(1), 101–112. Westergaard HM., 1939. Bearing pressures and cracks. Journal of Applied Mechanics 61, 49–53. Yoneyama S, Murasawa G. Digital Image Correlation, in Experimental Mechanics. In: Freire JLdF, editor. Encyclopedia of Life Support Systems (EOLSS) Oxford, UK: Eolss Publishers, 2009. Riddell WT, Piascik RS, Sutton MA, Zhao W, McNeill SR, Helm JD., 1999. Determining fatigue crack opening loads from near-crack-tip displacement measurements. In: McClung, Newman, editors. Advances in fatigue crack closure measurement and analysis, 2nd volume, ASTM STP 1343, 157–174. Sutton MA, Zhao W, McNeill SR, Helm JD, Piascik RS, Riddell WT., 1999. Local crack closure measurements: development of a measurement system using computer vision and a far-field microscope. In: McClung, Newman, editors. Advances in fatigue crack closure measurement and analysis, 2nd volume, ASTM STP 1343, 145–156. Carroll J, Efstathiou C, Lambros J, Sehitoglu H, Hauber B, Spottswood S, Chona R., 2009. Investigation of fatigue crack closure using multiscale image correlation experiments. Engineering Fracture Mechanics 76, 2384–2398. Williams ML, 1957. Stress distribution at base of stationary crack. American Society of Mechanical Engineers Transactions - Journal of Applied Mechanics 24, 109. Anderson TL. Fracture mechanics: fundamentals and applications, 2nd ed. Boca Raton: CRC Press, 1994. Lopez-Crespo P, Burguete RL, Patterson EA, Shterenlikht A, Withers PJ, Yates JR., 2009. Study of a crack at a fastener hole by digital image correlation. Experimental Mechanics 49, 551–559. Zhang R, He L., 2012. Measurement of mixed-mode stress intensity factors using digital image correlation method. Optics and Lasers in Engineering 50, 1001–1007. Sanford RJ, Dally JW, 1979. A general method for determining mixed-mode stress intensity factors from isochromatic fringe patterns. Engineering Fracture Mechanics 11, 621–633. Beretta S, Rabbolini S, Di-Bello A., 2015. Multi-scale crack closure measurements with digital image correlation on Haynes 230. Frattura ed Integrità Strutturale 33, 174–182. MoutouPitti R. Découplage des modes de rupture dans les matériaux viscoélastiques orthotropes: modélisation et expérimentation, PhD thesis. University of Limoges; 2008. Pop O, Meite M, Dubois F, Absi J., 2011. Identification algorithm for fracture parameters by combining DIC and FEM approaches. International Journal of Fracture 170, 101–114. Pop O, Dubois F, Meite M, Absi J., 2013. Mixed-mode fracture characterization by coupling digital images correlation with finite elements method. 21ème Congrès Français de Mécanique. Bordeaux, France, 101–114. Hellen T., 1975. On the method of virtual crack extensions. International Journal of Numerical Methods in Engineering 9, 187–207. Parks DM, 1974. A stiffness derivative finite element technique for determination of crack tip stress intensity factors. International Journal of Fracture 10, 487–502. Destuynder P, Djaoua M., 1981. Sur une interpretation mathematique de l’integrale de Rice en theorie de la rupture fragile. Math Meth Appl Sci. 3, 70–87. Rethore J, Gravouil A, Morestin F, Combescure A, 2005. Estimation of mixedmode stress intensity factors using digital image correlation and an interaction integral. International Journal of Fracture 132, 65–79. Erdogan F, Sih. GC, 1963. On the crack extension in plates under plane loading and transverse shear. Journal of Basic Engineering 85, 519–527. Mathieu F, Hild F, Roux S., 2012. Identication of a crack propagation law by digital image correlation. International Journal of Fatigue 36, 146– 154. Yoneyama S, Morimoto Y, Takashi M., 2006. Automatic evaluation of mixed-mode stress intensity factors utilizing digital image correlation. Strain 42, 21-29. Hamam R, Hild F, Roux S., 2007. Stress intensity factor gauging by digital image correlation: application in cyclic fatigue. Strain 43, 181–192.

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