numerical and experimental methods for the ...

Forming Technology Forum 2012

June 5 – 6, 2012, IVP, ETH Zurich, Switzerland

NUMERICAL AND EXPERIMENTAL METHODS FOR THE PREDICTION OF FAILURE IN SHEET METAL FORMING Hora P. 1*, Manopulo N. 1, Tong L. 1 1

Institute of Virtual Manufacturing, Tannenstrasse 3, ETH-Zurich, Switzerland ABSTRACT: The objective of FTF 2012 is to provide a platform for the discussion of the most recent developments concerning the experimental measurement as well as the numerical modeling of failure phenomena in sheet metal forming. The present contribution thus aims to give a short introduction to the state of the art concerning this topic. After briefly reviewing the different failure models arising in sheet metal forming, the basic problem of an adequate definition for process relevant forming limits will be discussed. The recently introduced time dependent FLC evaluation technique will be shortly exposed in this context. Chapter 3 is dedicated to the analysis of the deformation state occurring in the shear bands. It will be pointed out that Hill’s condition describing the position of the shear band is only limited to the special case of quasi-homogeneous thickness distribution and thus cannot be generalized. Chapter 4 discusses failure prediction using forming limit diagrams and briefly describes the imperfection based M-K model as well as the enhanced Modified Maximum Force Criterion (eMMFC) which does not require the definition of an imperfection. The chapter is concluded by hinting to the possibility of predicting limiting strain states by direct FE modeling. The topic of ductile fracture criteria, which is recently getting substantial attention from the scientific community, will be discussed in chapter 5. The discussion will be limited to the dependence of the plastic strain at failure on the stress triaxiality and load factor as was first proposed by Wierzbicky. KEYWORDS: Localized necking, Forming Limit Diagrams, Ductile fracture, FEM-modeling of fracture 1 INTRODUCTION The manufacturing of formed metal products is nowadays more and more planned making use of numerical simulation. An important criterion for the design of the forming operations is the need to avoid failure in form of wrinkles or ruptures. These failure models will be shortly discussed in the following. 1.1 Modelling of wrinkling The wrinkling phenomenon represents under ideal conditions a bifurcation state described by the following equation

KT  KG   0

(1)

In Eq (1) KT represents the tangent stiffness matrix, KG the geometric stiffness matrix,  the eigenvalue and  the corresponding eigenvector. The simulation of real mechanical structures, which develop a non-uniform thickness distribution, usually directly delivers the correct wrinkle structure as the only possible equilibrium state.

Fig. 1

Simulative prediction of wrinkling (AUTOFORM).

In order to correctly compute possible short-term quasi-static instable states, the stabilizing arclength methods are implemented in most implicit codes. Explicit programs on the other hand intrinsically compute the dynamic switch to the new equilibrium state correctly. The accuracy in the prediction is mostly dependent on the numerical

* Corresponding author: 8092 Zurich, phone: +41 (0)44 632 71 98, fax: +41 (0)44 632 11 65, [email protected]

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parameters such as mesh size and contact formulation. From the point of view of simulation the problem of wrinkle prediction can be considered sufficiently well solved. The problem lies more in the simulation of surface defects, which usually cannot be resolved by the finite element meshes in sufficient detail. 1.2 Failure in form of ruptures The description of the splits turns out to be both theoretically and from the point of view of numerical simulation much more complex. There exist basically three types of failure in this category:  



Cracks triggered by localized thinning of the material due to material instabilities (Type 1 localized necking) Edge cracks due to excessive hardening caused by previous cutting operations as well as possible introduction of micro cracks. (Type 2a – ductile fracture) Bending edge rupture due to large bending strains (Type 2b – ductile fracture)

Fig. 4

Even though in all three cases we can speak failure through rupture, they all occur in clearly different strain states.

Fig. 5

Fig. 2

Typical in-plane rupture induced by localized thinning effects (localized necking).

Sheet ruptures induced by bending. Al material.

Critical strains for initiation of localized necking and for initiation of rupture visualized in a forming limit diagram.

As it can be inferred from Fig. 5, localization occurs significantly earlier that the actual rupture. The former is represented in the above figure with the FLC whereas the latter by the critical strain B.

2 EXPERIMENTAL DETECTION AND EVALUATION OF FORMING LIMITS The recognition of failure, as it can be seen in Fig. 2, is usually based on the strain state at the beginning of localized necking. Fig. 3

Rupture on edges induced by crack initiation.

2.1 Time dependent evaluation of Nakajima tests The well established Nakajima test is often used to experimentally acquire the strain states where localized necking starts.

Forming Technology Forum 2012

Fig. 6

June 5 – 6, 2012, IVP, ETH Zurich, Switzerland

Time dependant evaluation method. DeFLC termination of  maj [2].

Fig. 8

Material HC220YD. Evaluation of the Nakajima tests with the time dependent method [5].

Fig. 9

Material AA 5182. Evaluation of the Nakajima tests with the time dependent method [5].

Nakajima test. Material HC 220 YD. FLC  maj has

The determination of

been until today

quite challenging. This is due to the strongly inhomogeneous strain distribution in the necking zone as well as the fact that the transit from diffuse to localized necking cannot be unambiguously recognized in the experiments. The recently proposed time dependent evaluation procedure helps overcome these difficulties [4]. To achieve an independency of the experimental setup parameters an adaptive number of pictures has been devised. The procedure starts with 4 pictures at the beginning of the forming process and with three pictures at the end. Based on this initial data the Pearson correlation coefficient rp is evaluated [6]: p

rpStart 

 t i 4

p

 t i 1

i



p

i n  p p

i 1

Start

i 1

i

Start



Start

 t .  3 i p

End

i 1



End

 

  3 

Start 2

  3 

 t  .  3 i 2

i

  3 

 t  .  3 i

 t

 t



 t .  3 i 2

i

n

rpEnd 

Fig. 7

End

  3 

(2)

 

End 2

The evaluation of the correlation coefficients is then done for continuously increasing number of picture p - by adding next pictures, until all pictures have been considered. The evaluation of the critical time step will be done for those pictures where the r-values reach the maximum. The start of localized necking is then given by the relation:

t crit 

a 2  a0 a1  a3

(3)

where the parameters a0 to a3 are the coefficients of the two intersecting lines in Fig. 7. This method has been used for the evaluation of the forming limit curves of Numisheet 2008 Benchmark 1, where a detailed description of the procedure can also be found [5]. Furthermore it has already been implemented in the ARAMIS system distributed by the company GOM GmbH.

2.2 Extension of FLC for shear fracture Many authors like Wierzbicky [25] and Stoughton [27] pointed out that shear fracture can occur in the Tension-Compression zone of the FLC i.e. having the following principle strain ratio:



 22   21  11

(4)

The extended diagrams thus feature an additional shear fracture limit as it can be seen in Fig. 10.

Forming Technology Forum 2012

June 5 – 6, 2012, IVP, ETH Zurich, Switzerland

Fig. 11

Inclination angle of the lacalized necking under the plane strain condition [7].

Fig. 10 FLC with shear induced failures. Source [21]

This kind of localized necking is often incorrectly

It is however not yet sufficiently clear whether this kind of shear fracture always occurs. This failure type has been observed in some aluminum or AHSS parts with considerable blank thickness. [21]. Thin steel blanks used in the automotive industry seem not to show this behavior.

is also in this zone larger than the corresponding shear strain ~12 . In case of the uniaxial tensile test with  = -0.5 the following values can be computed for the band:

3 DEFORMATION STATES IN LOCALIZED NECKING ZONES 3.1 Localized necking under tensile conditions The classical FLC diagram describes failure through the development of a localized necking zone. The position of the localization is derived from the assumption that having reached the maximum force, all regions except the weak spot get unloaded and return to an elastic state. The local neck is practically the only plastic zone in the remaining elastic body. Given this behavior one can derive plain strain condition for localized necking:

~   ~22  0 11

called shear band. Indeed the principal strain ~11

~



0. 0  0.67 0.47   0.47 0.0 0.0  11  0.0 0.0  0.67 

(8)

The figure below shows the generation of this zone in the DC04 material and the corresponding deformation within the localized necking.

Fig. 12 Position and local deformation inside of the localized necking.

(5)

This condition can be reached only in two cases: Case 1:

~22  0

(6.1)

Case 2:

~11 

(6.2)

Case 1 corresponds to Hill’s basic assumption for the generation of a shear band. Real deformation states however, seem to follow the pattern in Case 2. Derivation of the shear band location based on Case 1 Hill’s condition is based on the assumption that the shear band takes up the rotated position in which the strain state satisfies the condition of Case 1. Using the Mohr’s circle the rotation of the shear band with respect to the first principal stress direction can be computed as follows:   arctg   (7)

Impact of thickness inhomogenities It should be mentioned that the rotation of the shearband can only occur on thin tensile specimens which show more or less a homogeneous thinning. In cases where the local thinning is significant then the localized neck overtakes a position which has been pre announced by the thickness distribution. This is the case especially for thick sheets which develop a pronounced diffuse necking zone. Also on the Nakajima specimens with  = -0.5 the localized necking is observed due to the strong thinning on the punch pole, perpendicular to the maximal tensile strain. It thus can be said that Hill’s condition only applies under homogeneous strain distribution i.e. only in the regions where the thickness inhomogeneity is not dominant.

Forming Technology Forum 2012

June 5 – 6, 2012, IVP, ETH Zurich, Switzerland

3.2 Localized necking under biaxial stretching conditions

4 NUMERICAL PREDICTON OF LOCALIZED NECKING STRAINS

In biaxial stress states with  > 0, the second principal strain is 22 > 0 and thus the pattern proposed in Case 1 does not work. There exists no position in plane where the condition ~22  0 is satisfied. This is why many authors postulated that there can be no local necking under biaxial stretching see e.g. Backofen [7].

4.1 Hill’s criterion Numerous criteria have been developed for the numerical prediction of the FLC. A neat overview can be found in the GM symposium booklet [1], which discusses basic failure modeling. The fundamental Hill’s failure criterion for the localized necking, is based on the maximum force condition in plain strain

d 11   11d 33  0

(9)

which can be written in incremental form as follows [15].





t  t  1   (t  t )   (t )  11 33 11

(10)

Hill’s criterion delivers for materials following a hardening of the form

 y  A n the

following

relationship for the left side of the FLC

 maj  Fig. 13 Rotation of localized necking zone in dependency of the deformation state [7].

As it can be seen from the following pictures the localized necking occurs through a very local thinning of the blank. The deformation state corresponds thus to a very local diffuse thickness necking. Some materials even exhibit a fracture pattern under 45 degrees.

n 1 

(11)

Stören and Rice [6] on the other hand have taken quite a different approach. In contrast to the normally assumed smooth shape of the yield surface they proposed that an instability in the constitutive relationship, such as vertex on the yield surface can develop during the forming (Ghosh in [1], p. 291) and obtained the relation

 maj

 n   1 

 1  n     2

1  n 2 4

1

   1   2  (12)

n

4.2 Criterions with imperfection (M-K-Model) Nowadays models based on an imperfection, are widely used following the footsteps of the Marciniak-Kuczynski (M-K) failure model, which is applied in many different variants, see e.g. [14][15].

Fig. 14 Localization behaviour at biaxial stretching condition.

As it will be introduced later in conjunction to the MMFC criterion, given the strong increase in 11, the strain state also approaches a plain strain condition, this time however due to the condition described in Case 2.

Fig. 15 M-K Model with an inclined groove.

The rotation o of the shear band sketched in Fig. 15 for states with  0.5    0.0 should only be taken into consideration where it is not suppressed by the influence of the thickness inhomogeneity, as

Forming Technology Forum 2012

June 5 – 6, 2012, IVP, ETH Zurich, Switzerland

described in Chapter 3. As it can be seen there, this kind of rotation does not occur e.g. on Nakajima samples. A drawback of M-K models is the strong dependence of the FLC-position on the initial choice of the imperfection. The imperfection f = tb/ta can thus be considered a pseudo-physical parameter which needs to be calibrated to achieve the required accuracy.

FEM simulations on different Nakajima samples show that this state is effectively reached (Fig. 18).

Fig. 18 Change of stress state during the localization of the strain. Fig. 16 M-K-Models. Influence of the imperfection factors on the FLC [13].

On the other hand, strain rate effects can be investigated with M-K models in a straight forward way, Fig. 17.

The main difference to the Hill’s criterion is that the stress is now assumed to change from the initial state of =22/11 to a plain stress state during localization. The corresponding criterion can be formulated in a simplified manner as follows [16]:

 11  11

Fig. 17 M-K-Models. Influence of rate sensitivity exponent m on the FLC [13].

Hutchinson, Neale und Needleman extended the original long-wavelength M-K model, which does not specify the form of the imperfection in detail, with a short wavelength formulation. The effect of a sinusoidal geometrical imperfection on the failure behavior can be seen in [9]. The modified M-K model implemented in the program CRACH, also bases on a short wavelength definition of the imperfection [19]. 4.3 Criteria without imperfection (MMFCModel) An alternative formulation, which does not define an initial imperfection, is the Modified Maximum Force Criterion (MMFC) [16]. It is based on the assumption that during localized deformation the strain converges to a state

 0

(13)

   11  t (14) 1  2   e( E , t )      11   11

Fig. 19 Basic assumption of MMFC.

In the above equation  stands for the strain ratio,  is the curvature and t the thickness of the sheet. The term e(E,t) is a material specific calibration factor for the failure initiation. Studies have shown that this term can be determined and fixed for the different material classes e.g. metastable CrNi, Al, Cu an other alloys. Compared to the classical maximum force criterion, i. e. first term in (14), an additional stress increase due to the change in the stress state is modeled, which stabilizes the diffuse necking. The latest formulation of the MMFC criterion will be presented by Tong et al. [17] during the FTF 2012. The discontinuity discussed by Aretz [8] in the derivatives of the criterion in case of non-quadratic yield loci is only limited to the cases where the yield locus contains linear segments, which though should not occur in reality. An excellent overview on the reliability of the different failure modeling techniques can be found in the Benchmark I of Numisheet 2008 [5].

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4.4 FEM based prediction of FLC In addition to the mentioned simplified numerical procedures, necking phenomena can also be directly modelled using the Finite Element Method. This can be done on one hand by simulations with special 2½D elements [Hora-Tong, not published], which in contrast to plain strain elements are formulated with a prescribed strain condition

direct FEM simulation of the Nakajima test can be used for this purpose.

A B A  22  11 22

(15)

and thus correspond to the short-wavelength approach, see Fig. 20.

Fig. 21 FLC determination based on FEM simulation and coupled with the time-dependant evaluation method [18].

As it has been investigated by Gorji [28], explicit FE codes can also be used to very realistically simulate the actual position of the shear band up to just before the crack occurs (see Fig. 22).

Fig. 22 FE modelling of tensile behaviour with the localized neck development [28 ]. Material Ac120. Top: ARAMIS evaluation of strains. Bottom: Simulation results with LS-Dyna under consideration of initial thickness inhomogeneities.

Fig. 20 2-1/2-D-simulation of the necking for =0.5, =0.0 and =1.0 with modified plane strain elements with prescribed non-zero 22- components.

On the other hand, as proposed by Volk [18], a

4.5 Numerical Modelling of Damage Influences Although the reduction in hardening constitutes the primary effect leading to local necking, secondary influences such as generation and growth of micro voids can additionally weaken the material. This approach assumes that the carrying crosssection of the material is reduced by a factor D, due to voids. The so called effective stress is then computed as

Forming Technology Forum 2012

June 5 – 6, 2012, IVP, ETH Zurich, Switzerland

 ijeff   ij / 1  D 

(16)

The evolution of the damage factor is described by Lemaitre [26] as follows D 

DC

 Vf*

  VD

2   (17)  1     31  2  m   V2 n V  3   V  

In contrast to the FLC diagrams given in function of maj and min, fracture strongly depends on the stress triaxiality factor:



2

Damage effects are usually implemented in FE codes by using damage dependent yield locus models. A very popular yield locus model is the one proposed by Gurson-Tvergaard-Needleman (GTN-Model) [20] 2

  eq    2q1 f cosh  3 q2   1  q3 f 2  0   y   2   





(18)

m 

(21)

The so called triaxiality diagrams show that the plastic strain at fracture depends on the load parameter 



27 J 3 2 3

(22)

and assumes the maximum value under axially symmetric loading whereas reaches a minimum in plane strain conditions, see Fig. 23.

where f is the void volume fraction and describes the damage level,  is triaxiality factor and q1, q2 and q3 are material specific constants. The void volume fraction f can be computed under consideration of the shear damage extension proposed by Nahshon und Hutchinson [24] as ftot  1  f kkpl  k f ( )

sij Dijp



(19)

which is composed of the growth and nucleation of voids fgr  1  f  pl : I

(20.1)

fNucl  Ampl

(20.2)

and the additional shear term for the damage under shear deformation. The main difficulty in the application of the GTN model lies in the correct definition of the function f. The measurement of the void volume fraction is very challenging and thus the parameters are mostly inversely fitted using force-displacement curves. Examples for this kind of procedure are given in [23]. This kind of approach can be dangerous as possibly many physical phenomena are turned off by the calibration. Experience shows e.g. that the results strongly depend on the mesh size. In [29], the parameters of a modified GTN model are fitted using in-situ measurements (scanning electron microscope SEM) of the void fraction at different deformation stages.

5 PREDICTION OF DUCTILE FRACTURE Ductile fracture like the one visible in Fig. 4 cannot be modeled with the localized necking criteria discussed in chapters 3 and 4. The correct prediction of this kind of failure requires the definition of fracture criteria. An excellent introduction to ductile fracture criteria has been given by Wierzbicki in [20].

Fig. 23 Triaxiality diagram. Influence of the triaxiality stress on the critical fracture strain f [20].

The limiting surface for the plastic strain at fracture f can thus be described in dependence of the parameters  and  as in Fig. 24.

Fig. 24 Representation of the fracture locus in the space of stress triaxiality and the deviatoric state variable [20].

Wierzbicki proposes 15 different experiments for the identification of the form and position of the mentioned fracture locus.

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[2]

[3]

Fig. 25 Comparison of prediction of different failure criteria relatively to 12 plane stress test points Material Al 2024-T351 [20].

6 CONCLUSIONS The numerical and experimental failure prediction of sheet metals is since more than 50 years one of the main research interests in the field of metal forming. It nevertheless stays an open research field where conclusive solutions have not yet been found. The recently introduced time dependent evaluation methods offers at least the possibility to unite evaluation methods for the description of forming limits under a common standard procedure. However, questions about the thickness dependence of the FLC position and the influence of the strain path still remain unanswered. The methods proposed until today many times show substantial differences to the real behavior. A further not satisfactorily solved problem is the FEM modeling of failure phenomena. It is known that quasi-static implicit codes exhibit strong mesh dependence during localized necking. As a solution for this problem extensions to the classical plasticity theory (enhanced gradient plasticity) have been proposed. There nevertheless exists practically no commercial code offering the extended theory as an option. The reason lies in the fact that the elliptic equations, valid until maximum force is reached, need then to be exchanged with hyperbolic ones, leading to challenges in their numeric treatment. Another open research topic is the realistic simulation of crack initiation and propagation. Even damage based models fall short from delivering physical theories which can be easily validated in reality. The quality of the results still remains very much a function of the numerical procedures used both for identification and computation purposes.

REFERENCES [1] Koistinen D.P., Wang N-M.: Mechanics of Sheet Metal Forming – Material Behavior and Deformation Analysis. A General Motors

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[14]

Lab. Symposium. Plenum Press N.Y-London, 1978. Eberle B., Volk W., Hora P.: Automatic approach in the evaluation of the experimental FLC with a full 2D approach based on a time dependent method. In: Proceedings of the NUMISHEET 2008, pp. 279-284, Switzerland, 2008 Hora P., Eberle B., Volk W.: Numerical methods for a robust user-independent evaluation of Nakajima test for the FLC. Proceedings of the IDDRG 2009, Golden USA, 2009. Volk W., Hora P.: New algorithm for a robust user-independent evaluation of beginning instability for the experimental FLC determination, Int. J. Material Forming, 2010: 1-8. Hora P., Volk W.: Benchmark 1 – Numerical prediction of forming limits. Proceedings of the NUMISHEET 2008, Vol. 2, edited by P. Hora, ETH-Zurich, 2008. W. Stahel, W: „Statistische Datenanalyse Eine Einführung für Naturwissenschaftler“, Vieweg+Teubner Verlag, 5. Aufl., 2007. Backofen W., Deformation Processing, Addison-Wesley, 1972 Aretz H., Numerical restrictions of the modified maximum force criterion for prediction of forming limits in sheet metal forming, Modelling and Simulation in Material Science and Engineering, 12, 677-692, 2004 Hutchinson J.W., Neale K.W., Needleman A.: Sheet necking – I. Validity of plane stress assumption of the long-wavelength approximation. In: Mechanics of Sheet Metal Forming – Material Behavior and Deformation Analysis, Edited by Koistinen D.P. and Wang N-M., Plenum Press N.Y-London, 1978. Hutchinson J.W., Neale K.W.: Sheet necking – II. Time independent behavior. In: Mechanics of Sheet Metal Forming – Material Behavior and Deformation Analysis, Edited by Koistinen D.P. and Wang N-M., Plenum Press N.Y-London, 1978 Stören S., Rice J.R.: Localized Necking in thin sheets. J. Mech. Phys. Solids 23(1975) 421-441. Marciniak, Z., Kuczynski, K.: Limit strains in the processes stretch-forming sheet metal. International Journal of Mechanical Sciences 9, 609–620, 1967. Zhang C., Leotoing L., Guines D., Ragneau E.: Theoretical and numerical study of strain rate influence on AA5083 formability. J. of Materials Processing Technology 209, 3849– 3858. 2009 Aretz H.: Numerical analysis of diffuse and localized necking in orthotropic sheet metals. International Journal of Plasticity 23, 798– 840, 2007

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[15] Aretz H.: A simple isotropic-distortional hardening model and its application in elastic–plastic analysis of localized necking in orthotropic sheet metals. Int. J. of Plasticity, 24, 1457–1480, 2008 [16] Hora P., Tong L.: Theoretical prediction of the influence of curvature and thickness on the FLC by the enhanced modified maximum force criterion. In: Proceedings of the NUMISHEET 2008, pp. 205-210, Switzerland. [17] Tong L, et al., Numerical prediction of forming limit diagrams with MMFC under consideration of anisotropy and anisotropic hardening effects, FTF 12, Zurich [18] Volk W.: New experimental and numerical approach in the evaluation of the FLD with the FE-method. In: Proceedings of the FLCZurich 06, Switzerland, 2006 [19] Gese H., Dell H.: Numerical prediction of FLC with the program CRACH. Proceeding of the FLC Zurich 2006. [20] Tvergaard V., Needleman A.: Analysis of the cup-cone fracture in a round bar tensile bar. Acta Metall, 32-157, 1984. [21] Luo M, et al., Prediction of Shear-induced Crack Initiation in AHSS Deep Drawing Operation with a Phenomenological Fracture Model, Numiform, 464-472, 2010 [22] Xue Z., Pontin M.G., Zok F.W., Hutchinson J.W.: Calibration procedures for a computational model of ductile fracture. Engineering Fracture Mechanics, 77, 492–509, 2010 [23] Needleman, A.; Tvergaard, V.: An analysis of Dynamic Ductile Crack Growth in a Double Edge Cracked Specimen. Int. J. Fract. 49, 4167, 1991 [24] Nahshon K., Hutchinson JW.: Modification of the Gurson model for shear failure. Eur. J. Mech. A/Solids, 27:1–17, 2008 [25] Wierzbicki T., Bao Y., Lee Y-W., Bai Y.: Calibration and evaluation of seven fracture models. Int. J. of Mech. Sci., 47, 719-743, 2005 [26] Lemaitre J. A., A Continuous Damage Mechanics Model for Ductile Fracture, J. Eng. Mat. Tech. 107, 83-89, 1985 [27] StoughtonT., Yoon J. W., A new approach for failure criterion for sheet metals, Int. J. Plasticity, 27, 440-459, 2011 [28] Gorji M., et al., Investigation of localized necking for AL and stainless steel materials and their FEM based modeling, (to be published) [29] He M., et al., Forming Limit Stress Diagram Prediction of Aluminum Alloy 5052 Based on GTN Model Parameters Determined by In Situ Tensile Test, Chinese Journal of Aeronautics, 24, 378-386, 2011

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