Development of Simulation System for Compliance Function and Residual Stress Measurement for Al 2124-T851 Plate

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ScienceDirect Procedia CIRP 57 (2016) 591 – 594

49th CIRP Conference on Manufacturing Systems (CIRP-CMS 2016)

Development of Simulation System for Compliance Function and Residual Stress Measurement for Al 2124-T851 Plate Xiaoming Huanga,b,c Jie Suna,* Chang’an Zhou a Jianfeng Lia a School of Mechanical Engineering, Shandong University, Jinan 250061, China Mechatronics Engineering Department, Binzhou University, Binzhou 256600, China c Ningbo Shenglong Group Co., Ltd. , Ningbo 315104, China * Corresponding author. Tel.: +86-0531-88394593; fax: +86-0531-88394593. E-mail address: [email protected] b

Abstract The crack-compliance method is an effective means of measuring the original residual stresses in pre-stretched aluminum alloy plates. The calculation of compliance functions and the choice of the order of interpolation function are important for calculation results, and the complexity of modeling the process makes this method impossible for engineering applications. In view of the problem, a metho d of crack compliance function based on the finite element parametric modeling and M atlab calculation is proposed. The calculation system for compliance functions is developed, and the automation of the analysis process is realized. The automation of optimal expansion order is obtained based on the minimization of total stress uncertainty. Then, the residual stress depth profiles in pre-stretched aluminum alloy plate 2124-T851 are determined. The results show that the method and system can accurately calculate the residual stress and prove the validity and feasibility of the proposed method. ©©2016 Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license 2015The The Authors. Published by Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of Scientific committee of the 49th CIRP Conference on M anufacturing Systems (CIRP -CM S 2016). Peer-review under responsibility of the scientific committee of the 49th CIRP Conference on Manufacturing Systems

Keywords: Crack compliance method; Compliance function; Simulation system; Residual stress; Al2124-T851

1. Introduction Aluminiu m alloy has been widely used in aircraft monolithic co mponent due to their perfect strength, high fracture toughness, and stress corrosion after supersaturated solid solution and prestretching treatment. During the mach ining process of structural components, up to 90% of the material is removed fro m the blank. For those components, it is easy to cause substantial distortion because of the initial residual stress. This has been one of the most serious problems that the aircraft manufacturing industry had to face[1-2]. The accurate measurement and quantitative description of blank init ial residual stress is prerequisite for the thin-wall mono lithic co mponent deformation prediction. Because of the large thickness and low magnitude of the alu min iu m alloy pre-stretching plate, it is difficult to measure these residual stress using common techniques such as X-ray diffraction method, neutron diffraction method, layer removal method, blind-hole method, etc. Crack co mpliance method make it almost ideal for measuring the full through thickness

stress profile. In the crack co mpliance method, strains are measured as a slot is incrementally cut through the alu miniu m alloy pre-stretching plate[3-5]. The original residual stresses were determined fro m the measured strains using series expansion approach. Crack co mpliance function calculat ion and the order of the interpolat ion function had a great influence on the accuracy of the calculation results[6-8]. Besides, the data processing is complex, which was inconvenient for engineering application. In this paper, the parametric co mpliance function calculation system was set up by using fin ite element modeling and Matlab iterat ive optimization method. Experiments were performed on aviation alu minum alloy 2124-T851 pre-stretching plate, and the through thickness residual stress profile were measured by the compliance function calculation system.

2212-8271 © 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 the 49th CIRP Conference on Manufacturing Systems doi:10.1016/j.procir.2016.11.102

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Xiaoming Huang et al. / Procedia CIRP 57 (2016) 591 – 594

2. Analysis of the testing princi ple The original residual stress was determined fro m the measured strains using the series expansion approach [9], which is tolerant enough for the noise and errors of the measured strains.Firstly, the unknown stress variation was assumed as a function of the through-thickness coordinate which can be expressed as a series expansion,

V x, y ( z)

¦

n i 1

Ai Pi ( z)

PA

(1)

Where, the Ai represents unknown coefficients to be solved. Fro m this application, Legendre polynomials expanded over the thickness of the plates were chosen for the Pi. The resulting stress distribution was guaranteed to satisfy force and mo ment equilibriu m. The strains that would be measured at the cut depth a j were calculated for each term in the series, which is called co mp liance function Cji . Using superposition, the strains given by the series expansion can be written as

H x , y (a j )

¦

n i 1

AC CA i i (a j )

removed by wire-electrode cutting fro m the p late for the rolling d irection (X d irection) and transverse direction stresses(Y d irection). According to the crack co mpliance method, the strains were obtained by the releasing of the crack stress. The cracks were introduced by the DK7725 CNC wire-cut machine. The feeding was 1.2 mm and a total of 32 cuttings were performed in the experiments. Normally, the cut is made incrementally, and the stress component normal to the cut plate is determined. In order to prevent the destruction of strain gauges by the emu lsion, the strain gauges were under insulated protection. A half bridge connection method was presented for the temperature co mpensation. BSF120-3AA-T was used to collect the strain signal and the experiment setup was shown in Fig. 1.

(2)

A least squares fit to min imize the error between the strains given by Eq.(2) and the measured strains given the Ai , and the stresses by Eq. (1), can be written as

A = (C T C )1 C T H measured

BH measured

(3)

The effect of relaxing stresses when removing the test specimen fro m a large p late was corrected by using FEM for calculating the comp liance function Cij . Each term in the series expansion for stresses, Pi in the Eq.(1), was applied to the model as an initial stress condition. One analysis step in the FEM is performed to allo w the init ial stress distribution to equilibrate with the stress free boundary condition. Subsequent FE analysis steps calculated the further stress relaxations fro m cutting the slot incrementally, and the Cij strains were taken related to the state of post-relaxing of the initial stresses. Using these compliance functions in Eq.(3) means that Eq.(1) gives the original stresses in the large plate before removing the test specimen.

Fig. 1. The crack compliance method for strain and crack.

3.2. Stress calculation Crack co mpliance function mat rix Ci (a j ) is the key point to determine the coefficients Ai in the stress calculation process. Based on the experimental values fro m the strain gauge, initial stress load was deduced by FEM iteration calculation. In the FE Modeling, the step number was set as 32, same as the number of experiment cutting. In the FE modeling, “the birth and death technology” was applied to simu late the crack generating process of the experiments. The flo w diagram of compliance function deducing was shown as Fig. 2. Start Parametric modeling of the sample

3. Experi ments and data processing 3.1. Experiments

Application of the DLOAD subroutine

2124-T851 pre -stretching blank samples were produced by the Kaiser Alu minu m & Chemical Corp., which chemical composition and mechanical properties were listed in Table 1.

Experimental strain data

Table 1. Chemical composition and mechanical properties of 2124-T851. Mechanical properties Elastic modulus

Simulation of the crack using “the birth and death technology” Crack compliance function

Chemical composition

73.1 GPa

Cu %

3.8

Mn%

0.3

Poisson’s ration

0.33

Mg %

1.2

Fe%

0.3

Shear Modulus

27.0 GPa

Zn%

0.25

Al %

81.2

A 40mm th ick 2124-T851 plate was chosen for the study. Two p iece of 150mm×150mm square specimens were

Conditions of satisfaction

No

Yes End

Fig. 2. Flow diagram of the crack compliance function in FE modeling.

Xiaoming Huang et al. / Procedia CIRP 57 (2016) 591 – 594

In each analysis step of the model, python scripts automatically judge and select the killed unit, in which the stiffness mat rix was set to zero to deactivate the element. Simu ltaneously, the quality, damp, and specific heat of the unit were all set to zero. Different orders of Legendre polynomials were applied on the FEM along the thickness direction. The A BAQUS subroutine (DLOA D) was used to introduce the Legendre polynomials and simu late the corresponding load. The graphical user interface (GUI) was developed in the form of plug-in, which includes the selection of wire cut EDM parameters, samp le coordinates parameters and sketch option. The pre-processing dialog bo x of FEM model was shown as Fig. 3.

the MaterialRS for fu rther program call; SolveRS was the main function for stress calculation. SolveRS was used to analyze the residual stress and system errors by calling the subroutine; readAll was used to read the generated data file; head was used to read the Legendre polynomial interpolation function; setUpPlot was used to set the curve parameters; LSQfit was used for least squares fit calculation; StrainPlot was used to draw the strain curve; leg was used to edit Legendre polyno mial; Splot was used to calculate and drawing stress curve. 4. Results and discussions Fig. 5 shows the strains measured by the cutting in both rolling direction and transverse direction.

Model Parameter Cutting Parameter Feed:

Feed times:

1.2

32

Coordinate Sketch

Coordinate point x1:

0.103

x2:

0.206

x3:

2.718

x4:

5.972

x5:

1.627

h:

40

L:

45

(0,h)

(x4,h)

(x2,h) (x1,h)

(L,h) (x3,h)

x1 X1is ishalf halfofofthe thecrack crackdepth depth x2=2*x1 x3=L-0.5*legendre+x1 x4=x3+legenre x5=0.5*legendre h is plate thickness L is half of plate length (x2,0) (0,0)

(x1,0)

(L,0)

(x5,0)

Fig. 5. Strain of 2124-T851 in rolling and transverse direction. cancel

OK

Fig. 3. Pre-processing dialog box of FEM model.

In addition to the random error of strain data acquisition, the stress calculation modeling error has enormous influence on the experimental results. The selection of Legendre polynomial interpolation function was crucial for reducing the stress calculation modeling error. The optimal convergence order of interpolat ion function was determined based on minimizing the total stress uncertainty. Based on the Matlab GUI, ABA QUS core solver and stress calculation program were coalesced. Fig.4 was the subroutine logic diagram of Mat lab. forrw MaterialRS rptrw

The 6~11 order Legendre polynomial order Pi(z) were applied on the p late along the plate thickness in FEM as the initial stress load. The model error, strain error and the residual stress uncertainty of different order interpolation function were shown in the Fig. 6. It is observed that the minimu m residual stress uncertainty was about 1.659 MPa when the convergence order of the Legendre polynomial was 8.

Readcomps MaterialRS

SloveRS readAll

head

setUpPlot

LSQfit

StrainPlot

leg

Splot

Fig. 4. Subroutine logic diagram of Matlab. Fig. 6. RMS of different order interpolation function in rolling direction.

Where, MaterialRS was the main function program. It was used to design the interface and extract parameter. MaterialRS includes four subfunctions: forrw was used to extract the “plate thickness” parameters and modify the “DLOAD” subroutine parameters; ReadcompsMaterialRS was used to read the data file of the MaterialRS; rptrw was used to process

The 8 order Legendre polynomial was used as the interpolation function to calculate the plate initial residual stress. The 2124-T851 alu minu m p late residual stress of 40mm thickness was shown in Fig. 7. The stresses profile

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shows a typical “M” curve in the rolling d irection, and the stresses in the transverse direction had a wave-shaped change.

method. It was observed that the stresses profile shows a typical “M” curve in the rolling direction, and the stresses in the transverse direction had a wave-shaped change. Acknowledgements The authors would like to thank the National Natural Science Foundation of China (No. 51275277) for the support. The authors also thank all the former researches that contributed to this paper. References

Fig. 7. Residual stress of Al2124-T851plate.

In the rolling direction, the max tensile stress was about 5.5MPa, which was 5mm under the surface. The compressive stress zone was in the subsurface and center zone of the plate. In the transverse direction, the tensile stress peak was in the plate center and the position was beneath the surface of 7mm, at max tensile stress value of 4.1MPa. The co mpressive stress was in the subsurfacewith max value of -5.6MPa. 5. Conclusions In the presented work, the crack co mpliance function based on the finite element parametric modeling and Matlab calculation was proposed. The calculation system with compliance functions was developed, and the automation of the optimal expansion order was obtained based on minimizing the total stress uncertainty. Polynomial stress can be loaded in the calculation model repeatedly, which improves the calculating efficiency of crack co mpliance function. Through-thickness residual stress profiles in Al 2124-T851 were measured after stress relief by the crack compliance

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