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This paper studies a method for obtaining the stress with plastic deformation by finding the plastic strain on U-bent specimens of austenitic stainless steel that have been subjected to large plastic deformation using the EBSD (Electron Backscatter Diffraction) method. The Mises stress calculated on the basis of the KAM of the EBSD shows good agreement with the stress that can be geometrically calculated from the U-bent specimens. In contrast, general methods for measuring residual stress on the basis of elastic strain produce residual stress measurement results that differ specimen by specimen. Thus, for true strain not less than 0.05, stress estimation based on the EBSD method produces better results than other general methods.

The methods that have been used for evaluating residual stress on structures include the X-ray diffraction method [

Recently, based on the EBSD method, the so-called cross court method was established to enable elastic strain and stress to be calculated from deviations from the Kikuchi patterns [

In contrast, the EBSD method has been studied by many researchers as an analytical method for plastic strain [

The misorientation measured with the EBSD method is based on GND (Geometrically Necessary Dislocation) density [_{p}_{1} to ε_{p}_{2}) is larger relative to elastic strain (ε_{e}_{1} to ε_{e}_{2}), when applied stress increases from σ_{1} to σ_{2}. If misorientation caused by work hardening was large enough, stress could be evaluated by using KAM. In the course of our study on the relationship between the KAM and plastic strain, we attempted to apply a more simplified approach in this paper that estimates the stress with the plastic strain obtained through the EBSD method and a true stress-strain curve. Here, because stress and strain each have their own directional components, it is necessary to consider the misorientation obtained through the EBSD method. Ramazani et al. estimated the distribution of Mises stress with a method using a computer model called “Representative Volume Element” and found that the distribution showed good agreement with that of the KAM obtained through the EBSD method [

the true stress obtainable in uniaxial tensile tests. Thus, even for unknown samples with a complicated deformation history, the relationship between the KAM and the true strain preliminarily obtained from the samples taken out of the specimens for which tensile tests are interrupted enables the Mises stress of the uniaxial tests to be estimated in a manner that evaluates the true strain by measuring the KAM of the unknown samples through the EBSD method.

Therefore, in this paper, the stress on the U-bent specimens (prepared as samples subjected to large plastic deformation) obtained through several conventional residual stress measuring methods is compared with the Mises stress estimated by the evaluation of the plastic strain through the EBSD method and the characteristics of these methods are discussed to show the validity of the new proposal.

Tensile test specimens and U-bent specimens were taken out of large diameter type 316 NG piping that had the chemical composition shown in

C | Si | Mn | P | S | Ni | Cr | Fe | Mo | N |
---|---|---|---|---|---|---|---|---|---|

0.019 | 0.42 | 1.41 | 0.024 | 0.005 | 12.62 | 17.06 | Bal. | 2.6 | 0.105 |

Method | Specimen |
---|---|

XRD, EBSD | Specimen #1~#3 |

Strain gage | Specimen #4~#6 |

Center hole drilling | Specimen #7~#9 |

Software | TSL OIM ver.6 |
---|---|

Voltage | 20 kV |

Working Distance | 15 mm |

Pitch | 2 mm |

Area | 400 × 1000 mm |

Binning | 4 × 4 |

q step size | 0.5 degree |

Number of measure | 3 times per a specimen |

relationship was evaluated with the average KAM value (KAM_{Ave}_{, Area}) in those areas. An adjacent number can be selected in calculating the KAM. In this paper, misorientation of a specific measurement point with respect to 6 neighboring points was calculated using the adjacent number of 1. That is, the KAM value at a given measurement point can be calculated by the following equation:

where θ_{n} is misorientation between an ambient point and its surrounding points. Three U-bent specimens for each measurement method were prepared by bending specimens 2 mm in thickness with a bending jig having an 8 mm radius and the residual stress on the U-bent specimens was measured through each measurement method. The specimens were bent in a manner that keeps the distance between legs in the range of 16 + 0.1/− 0.3 mm. U-bent specimen with 2 mm thickness was used because it was easy to control to the distance. Then, the residual stress was measured with the bolts on the specimens fastened to control the distance between legs. If the bolts are loosened from a state where the distance between the legs is not more than 16 mm, the residual stress is reduced with respect to the applied plastic strain. Thus, the measurement of the residual stress was implemented with special care so as to not loosen the bolts.

The methods used for measuring the residual stress were the X-ray diffraction method, strain gauge method and center hole drilling method. In each method, a Young’s modulus of 196 GPa was used to calculate the stress. The measurement conditions are summarized in Tables 4-6. The X-ray diffraction method called the side inclination method was used for measuring stress. In this method, specimens were scanned in the direction parallel to a bending direction. The outer

Apparatus | Stress tech X3000 |
---|---|

X-ray tube | Mn-Ka |

X-ray tube voltage | 30 kV |

X-ray tube current | 6.7 mA |

Lattice plane | {3 1 1} |

Diffraction peak | 152.26 degree |

Area | f 1.0 mm |

Young’s modulus | 196 GPa |

Poisson’s ratio | 0.28 |

Depth | 20 mm |

500 mm |

Apparatus | Tokyo Measuring Instruments Laboratory |
---|---|

TDS-303 | |

Strain gage | Kyowa Electronic Instruments |

KFG-1-120-D16-16 | |

Strain gage size | 1 mm |

Strain gage type | biaxial |

Young’s modulus | 196 GPa |

Machined size | 15 × 15 mm |

10 × 10 mm |

Apparatus | SINT MTS3000 |
---|---|

Rosette gage | Tokyo Measuring Instruments Laboratory |

FRS-2-17 | |

Rosette gage type | Type A (ASTM) |

Rosette gage diameter | 5.10 mm |

Hole diameter | 1.770 mm |

Decentering | 0.005 mm Max. |

Pitch (depth) | 50 mm |

Young’s modulus | 196 GPa |

Poisson’s ratio | 0.28 |

surfaces of U-bent specimens were mirror finished by electrolytic polishing with a depth of concave of approximately 20 μm. The first measurement of the residual stress was conducted under these conditions. Then, the second measurement was conducted after finishing the surfaces of the same specimens by electrolytic polishing with a concave depth of 500 μm. In the strain gauge method, the stress was evaluated after cutting specimens into test pieces of 15 × 15 mm and then 10 × 10 mm to investigate the effect of cut size on measurement results. In the center hole drilling method, released strain was measured at 50 μm intervals and the profiles of the residual stress in the depth direction were established.

Last, in the EBSD method, the EBSD on the cross sectional surfaces at the center in the longitudinal direction of the U-bent specimens was measured in a manner that scanned each cross sectional surface along three lines in the plate thickness direction with each line made up of a series of segments having 200 μm in the plate thickness direction and 400 μm in the bending direction. Then, the plastic strain in the U-bent specimens was analyzed and the analysis results were used to estimate the stress applied by bending along with the tensile test results.

the Σ3 relation as shown in

Given that the thickness center of the U-bent specimen is the center line of bending, the true strain ε due to bending on specimens in a plane strain state is given by the following equation:

where l and l_{0} are the arc length after and before bending, δt is the thickness from the thickness center line and R is the inner bending radius with a value of 8 mm in this case. Because the plate thickness is 2 mm, the bending radius at the center line becomes 9 mm. Based on Equation (2) and the relationship between true stress and true strain in

In the following section, the calculation results in

Method | Specimen | at 20 mm | at 500 mm | ||||
---|---|---|---|---|---|---|---|

X-ray | Specimen #1 | 439 | 409 | ||||

Specimen #2 | 595 | 510 | |||||

Specimen #3 | 615 | 533 | |||||

Average | 549 | 484 | |||||

Max-min. | 176 | 124 | |||||

10 × 10 mm | 15 × 15 mm | ||||||

s_{x} | s_{y} | Mises | s_{x} | s_{y} | Mises | ||

Sectioning | Specimen #4 | 105 | 476 | 433 | 143 | 444 | 393 |

Specimen #5 | 111 | 734 | 685 | 132 | 710 | 654 | |

Specimen #6 | 116 | 639 | 590 | 148 | 612 | 553 | |

Average | 111 | 616 | 569 | 141 | 589 | 533 | |

Max-min. | 11 | 258 | 252 | 16 | 266 | 262 | |

σ_{x} | σ_{y} | Mises | |||||

Center hole drilling methods (at 500 mm) | Specimen #7 | 423 | 790 | 685 | |||

Specimen #8 | 290 | 575 | 501 | ||||

Specimen #9 | 281 | 513 | 448 | ||||

Average | 331 | 626 | 545 | ||||

Max-min. | 143 | 277 | 237 |

20 μm from the outer surface and 124 MPa at 500 μm.

As for the residual stress measurement results of the strain gauge method, the stress in the bending direction σ_{y} is larger than the stress in the plate width direction σ_{x}. The Mises stress of the 15 × 15 mm test piece and 10 × 10 mm test piece is 533 MPa and 569 MPa respectively, which means that the residual stress varies depending on the specimen sizes. Every test piece shows increases in the released elastic strain and the residual stress as the sizes get smaller. The variations in the residual stress of test pieces of respective sizes are: 262 MPa for 15 × 15 mm test pieces and 252 MPa for 10 × 10 mm test pieces. These variations are larger than those of the residual stress measurement results of the X-ray diffraction method.

In the center hole drilling method, the profiles of released strain and residual stress in the depth direction are obtained. The released strain gets larger as the drilling depth is increased as shown in

which is higher than the result in

Regarding the evaluation of the residual stress using the EBSD method, considering that the maximum true strain due to bending of the U-bent specimens, which can be calculated using Equation (2), is 0.11 as explained in 3.2, and that the relationship between bending and true strain shows good linearity until the strain ε reaches 0.18 in

Yoda et al., showed that tensile and compression have the same tendency until the strain ε reaches 0.10 approximately [_{Ave}_{. Area) in the areas at each depth from the outer surface.} The true strain values at all depths are higher than those obtained by Equation (1) but the increments are only about 0.02. Using the results in

A comparison of the residual stress estimated by Equation (2) with the measurement results of several methods is shown in

very large variation as shown in

In this paper, we compared the measurement results of residual stress on U-bent specimens of type 316 NG stainless steel to which large plastic strain was applied that were found using methods that use elastic strain and the EBSD method measuring plastic strain. The following findings were obtained:

・ The KAM values and true strain of type 316 NG stainless steel correspond well with each other until the true strain reaches 0.18;

・ The plastic strain amounts, with the bent state maintained, calculated from the correlation between the KAM values and the true strain show good agreement with the geometrically calculated plastic strain amounts and the difference in the residual stress amounts respectively converted from the two types of plastic strain amounts using the true stress-true strain curve was negligibly small at 30 MPa except in the thickness centers of the specimens;

・ The methods using elastic strain have a tendency to produce higher values of residual stress than the EBSD method using plastic strain, and large variations;

・ The EBSD method using plastic strain is suitable for calculating the residual stress when applied strain amounts are not less than 0.05;

・ In contrast, the elastic strain method is preferable in the elastic strain range because of the high work hardening ratio and little generation of misorientation; and

・ EBSD can only predict the Mises stress at the maximum strained state, but cannot predict the stress direction and whether it is tensile or compressive stress.

The authors would like to thank Dr. Mikami and Mr. Sato from IIC for residual stress measurement and tensile test.

Sakakibara, Y. and Kubushiro, K. (2017) Stress Evaluation at the Maximum Strained State by EBSD and Several Residual Stress Measurements for Plastic Deformed Austenitic Stainless Steel. World Journal of Mechanics, 7, 195-210. https://doi.org/10.4236/wjm.2017.78018