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The effect of microstructure parameters on the residual stresses in the ultrafine-grained sheets
Journal Pre-proof The effect of microstructure parameters on the residual stresses in the ultrafine-grained sheets Farshad Nazari, Mohammad Honarpisheh, Haiyan Zhao
PII:
S0968-4328(19)30346-4
DOI:
https://doi.org/10.1016/j.micron.2020.102843
Reference:
JMIC 102843
To appear in:
Micron
Received Date:
20 October 2019
Revised Date:
2 February 2020
Accepted Date:
2 February 2020
Please cite this article as: Nazari F, Honarpisheh M, Zhao H, The effect of microstructure parameters on the residual stresses in the ultrafine-grained sheets, Micron (2020), doi: https://doi.org/10.1016/j.micron.2020.102843
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The effect of microstructure parameters on the residual stresses in the ultrafine-grained sheets
Farshad Nazari a,b, Mohammad Honarpisheh a,*, Haiyan Zhao b a b
Department of Mechanical Engineering, University of Kashan, Kashan, Iran
Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China
Corresponding author: [email protected], Tel ++98-31-55913404, Fax ++98-31-55913444
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Highlights
Investigation of microstructure parameters on the residual stresses of ultrafine-grained sheets.
Microstructure parameters are crystallites size, dislocations density, and lattice strain.
Microstructure parameters have a significant effect on the macro-residual stresses, and
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strain is the most effective parameter.
In the ultrafine-grained sheets, micro-parameters, with an undeniable effect, have a
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contribution as same as macro-parameters on the macro-residual stresses.
Abstract In this research, the influence of microstructure parameters on the residual stresses of ultrafine-
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grained sheets was investigated. For this purpose, the constrained groove pressing (CGP) process was carried out on the copper sheets with 3 mm thickness, and residual stresses of the CGPed sheets was measured using the contour method. Microstructure of the CGPed
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specimens was evaluated by the optical microscopy, micro x-ray diffraction (micro-XRD), and transmission electron microscopy (TEM) experiments. Microstructure parameters including
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crystallites size, dislocations density, and lattice strain were calculated using Williamson-Hall and Williamson-Smallman equations, and the calculated results were validated by the TEM images. The influence of these parameters on the residual stresses was investigated by analysis of variance (ANOVA) method, and two approaches were considered in this way. According to the results, the CGP process can create nanostructures in the CGPed sheets, and with increasing number of CGP passes, grains size, crystallites size, lattice strain, and residual stresses decrease, and density of dislocations increases. Microstructure parameters have a significant effect on the macro-residual stresses, and strain is the most effective parameter. 1
Also, in the ultrafine-grained sheets, micro-parameters have an undeniable contribution, which is the same as that of macro-parameters on the macro-residual stresses. Keywords: crystallite size, dislocations density, lattice strain, residual stress, CGP. 1- Introduction The development of ultrafine-grained structures is one of the methods for improving materials properties. Constrained groove pressing (CGP) process is one of the severe plastic deformation (SPD) methods which can produce ultrafine-grained or nanostructures sheets [1]. Evaluation of the effect of the CGP process on the mechanical properties and microstructure of the CGPed copper sheets illustrated that CGP process decreases grain size, and increases
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hardness and strength of the sheets [2]. Similar results have been presented for other materials such as aluminum [3], and steel [4]. In the previous studies of the authors [5, 6] deformation behavior and residual stresses in the CGP process were investigated, and demonstrated compressive residual stress was created near the surfaces of the CGPed sheet, and with moving
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along the thickness, it changes to tension residual stress at the middle of the sheet.
Residual stresses are classified in the three categories as the types I, II and III [7]. Type I is
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macro-residual stress which is created within the specimens in the rage of much larger than grain size. Types II and III are micro-residual stresses which they are operated at the grain-size
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and atomic level, respectively. Micro-residual stresses are generated due to heterogeneity and interaction between parameters of the microstructure of materials [8]. Salvati and Korsunsky [9] studied the different types of residual stress using FIB-DIC ring core technique and multi-
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scale FEM simulation on the aluminum alloy, and presented the majority of residual stresses (about 2/3) in the plastic deformation is type I. Pouraliakbar et al. [10] measured microstructure parameters of the CGPed sheet of Al-Mn-Si alloy. They utilized the X-ray diffraction (XRD)
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method for measuring the parameters, including crystallite size, dislocation density, and lattice strain, and revealed that microstructure parameters extremely depend on the plastic
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deformation and temperature of the process. They presented, with an increasing number of CGP passes, effective plastic strain increased which causes to increase dislocation density and reduce crystallite size in the microstructure. Moradpour et al. [11, 12] investigated the crystallite size and dislocation density in the Al-Mg CGPed sheets by the XRD analysis and transmission electron microscopy (TEM) imaging. They studied two passes of the CGP process and showed this process causes to grain refinement and reduces crystallites size based on the severe plastic deformation, and calculation of the dislocations density revealed that the average of dislocation density in the second pass is 4×1014 m-2 which is two times higher than the related 2
value of the initial sheet. Sergo et al. [13] investigated the influence of grains size on the residual stresses in the alumina/zirconia composite and showed that a large number of grains boundaries reduce the residual stresses due to spreading relaxation, and local microstructural heterogeneity causes to increase residual stresses. Weglewski et al. [14] studied the effects of grain size on the thermal residual stresses in the sintered Cr/Al2O3 composite and presented that a composite with the fine grain size has lower residual stresses than the composite with the coarse grain size. Also, Cao et al. [15] with simulating the grains size and dislocations on a nanocrystalline thin film, and evaluating the effect of them on the residual stresses revealed that residual stresses are reduced due to grains size reduction, and increasing homogeneity.
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Roy et al. [16] evaluated the relationship of dislocations density and residual stresses and indicated that increasing dislocations density in the cold working processes causes to enhance residual stresses due to increasing micro-defects, and inhomogeneity in the microstructure. Sato et al. [17] studied the relationship between dislocation density and residual stress in the wire drawing process. They carried out the process in the room temperature, and utilized
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energy-dispersive X-ray diffraction method for investigation, and showed that higher residual stress and dislocation density induced in the center of the wire, and those are decreased at the
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surface of the wire. Meyer et al. [18], with investigating a single track deep rolled steel by XRD analysis, illustrated that residual stress and dislocation density have an almost linear-relation
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together.
According to the fundamentals of X-ray diffraction in residual stress measurement, and experimental results, residual stress increases with increasing lattice strain [19-21]. The
equation,
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relationship between lattice strain and residual stress is shown in equation (1) [21, 22]. In this is residual stress, E is elastic modulus, is Poisson’s ratio,
E
(1 ) s in 2
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spacing of diffracting plane, and
(
d d 0
d
0
is the lattice spacing which
d d d
0
d
is the measured
is lattice strain.
0
(1)
)
d0
According to the literature, microstructure parameters have effects on the residual stresses, but the contribution and effective curve for each parameter have not been evaluated. In this study, the effects of lattice strain, dislocations density, and crystallites size on the residual stresses were investigated. Experimental and computational study and statistical analysis were utilized, and the research was carried out on the CGPed sheets using optical microscopy, X-ray diffraction methods, TEM imaging, and a contour method. 3
2- Materials and methods 2.1- CGP process
For performing the CGP process, some samples were prepared from a commercial pure copper (99.93%) sheet with a 3 mm thickness and dimensions of 72×50 mm. Before carrying out the process, the samples were fully annealed at a temperature of 650 ºC for 2 hours [23] to eliminate the rolling effects and initial residual stresses, to form a uniform microstructure. The CGP process applies a continuous severe plastic deformation on the specimen that it is performed by using two pairs of symmetrical grooved and flattened dies. Each pass of the process includes four steps. During the first and second steps, the specimen is grooved and
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flattened, then in the following the sheet is rotated through 180º, and grooving and flattening are conducted again on the sheet in the third and fourth steps, respectively. The CGP process was carried out up to three passes, and in this way, a 60-ton hydraulic press machine was used. All of the experiments were performed at room temperature (25℃) with the forming speed of
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0.4 mm/s, and lubrication with oil was utilized. Figure 1 shows the steps of the CGP process.
Fig. 1 Steps of the CGP process
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2.2- Residual stress
A 2D map of residual stresses was measured using the contour method at the first, second, and
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third passes of CGP. In the contour method, cutting the samples and releasing residual stress cause to deform the cutting surface, so by measuring the displacements on the cutting planes and processing them, and then simulating processed displacements using a finite element model, the residual stresses are calculated [24-28]. With considering the aging effect can be expressed that duration time between CGP process and residual stress measurement (about 12 hours) is not important, because with forming at the room temperature and using commercial pure copper alloy with high purity of Cu (99.93%) and lack of alloying components (<0.07%) such as Mg, Al, Ni, etc., the aging ability of the sheet is not significant. In this research, residual 4
stresses were measured on the middle cross-section of the samples, and figure 2 illustrates the
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dimensions of the specimens, cutting line, and situation of the experiments.
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Fig. 2 Dimensions of the CGPed specimens, cutting line, and situation of the experiments
Cutting the samples was performed using Charmilles ROBOFIL 2000 wire electro-discharge
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machine, and the cutting process was accomplished by a brass wire (Cu-Zn37), and with the 0.6 mm/min cutting speed in a full immersion condition. For measuring the surface
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displacements which resulted from releasing residual stresses, the cutting surfaces (both sides of the cutting plane) were measured by POLI-SKY coordinate-measurement machine using a RENISHAW PH6 touch-trigger probe with 2 mm diameter. Sampling was carried out with a
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distance of 0.2 mm in the Y-direction, and 0.5 mm in the X-direction, that on each cross-section 1485 points were measured. The specimens were completely constrained to prevent any displacement during the measurement. After measuring, an average of the data was calculated
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at the corresponding points on both surfaces, and the resulted contours were smoothed using a bivariate spline for eliminating the probability of measurement error, and the roughness which
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resulted from the cutting process. According to fundamentals of the contour method, for calculating the residual stresses, processed surface displacements must be simulated inverted as a boundary condition in a finite element model. Finite element simulation and residual stress computation were developed in the ABAQUS commercial package using a 3D linear elastic model. In this model, the element of C3D8R was used, and for increasing the accuracy of the simulation, finer elements were utilized near the cutting surface than the other areas. Figure 3 shows the finite element model, and contour of the surface displacements after averaging, smoothing, and inverting at the first pass of the CGP. 5
Fig. 3 Finite element model, and contour of the surface displacements after averaging, and inverting at the first pass of the CGP
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2.3- Microstructure
To evaluate the effect of microstructure on the residual stresses, microstructure of the samples was first investigated by optical microscopy, and the average of grains size was determined by vertical and horizontal intercept lines method according to the ASTM E112-96 [29] standard.
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In the following, the microstructure parameters were measured by micro x-ray diffraction (micro-XRD) on the surface of the cross-section. Micro-XRD was used to study and calculate
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the crystallites size, lattice strain, and dislocation density, and the tests were carried out on a Rigaku High-Resolution SmartLab x-ray diffractometer by Cu Kα radiation (λ=0.154183 nm).
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Radiance diameter was 400 µm, and the operation was performed in the range of 20-100º (2θ), step width of 0.02º, count time of 0.5 s per step, and potential of 40 KV with 200 mA. The results were analyzed using Williamson-Hall and Williamson-Smallman methods, and for
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verifying the micro-XRD results, transmission electron microscopy imaging was performed on the specimens. Nine points on each cross-section were investigated by the experiments that for the first, second, and third passes of CGP, 27 points were measured. The points map of the
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XRD test is illustrated in figure 2.
To investigate the effect of microstructure parameters, and interaction of them on the residual
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stresses, two approaches have been considered. In the first approach, the effects of microparameters on the residual stresses were evaluated to determine the significance of each parameter and the interaction of them. Then, in the second approach, the microstructure parameters besides the effective strain as a macro-parameter were investigated to determine that have micro-parameters a significant effect on the macro-residual stresses against macrofactors, or they have only a local effect and are limited to the micro-areas. The effect of the parameters and their interaction on the residual stresses were investigated using the response
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surface method and analysis of variance (ANOVA). The values of parameters were considered as magnitudes and the analyses were computed by Design-Expert commercial software.
3- Calculation of crystallite size, lattice strain, and dislocations density Crystallites size and lattice strain were calculated using the Williamson-Hall method. Based on this method, the intensity and broadening of the diffraction peaks are dependent on the crystallite size, and lattice strain [30-32]. Relation (2) shows the Williamson-Hall equation. C os
K
2 S in
(2)
D
In this Equation, β is the peak broadening measured as the full width at half maximum
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(FWHM), θ is the Bragg angle, K is Scherer constant which generally is 0.9, λ is the X-ray wavelength, D is the crystallite size, and ε is the lattice strain. According to the WilliamsonHall equation, if βCosθ is plotted against Sinθ, and a linear equation as Y=mX+c is fitted to the scatter plot, the slope and intercept of the equation give the lattice strain (ε=m/2) and the
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crystallite size (D=Kλ/c), respectively. Figure 4 illustrates the XRD pattern of CGPed copper
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sheets at the center of the surface (point 5).
Fig. 4 XRD pattern of CGPed copper sheets at the center of the surface (point 5)
Dislocation density is one of the important parameters on the residual stress and mechanical behavior of the materials, and evaluation of the dislocations density is possible by using the XRD test and analyzing the results. Williamson and Smallman [33] showed the density of dislocations depends on the crystallite size and lattice strain and presented an equation to calculate the dislocation density based on them. Equation (3) shows the relation of dislocation 7
density and crystallite size, that in this equation, ρp is density of dislocations per unit cell resulted from crystallites, and n is the number of dislocations per face of unit cell, which for annealed or severe plastic deformed metals, n is equal to 1 (minimum value) [33]. p
3n D
(3)
2
Study the relationship of dislocation density and lattice strain indicated that the density of dislocations is equal to the ratio of lattice strain energy to dislocation energy into the structure. In this way, Williamson and Smallman [33] presented equation (4) to calculate the density of dislocation based on the lattice strain. k F
b
2
(4)
2
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s
In this equation, ρs is the density of dislocations resulted from lattice strain, b is Burgers vector, and k is a function that it is dependent on the microstructure and mechanical properties of the materials which can be variable between 2 to 25. According to studies, the value of k for metals
-p
with FCC structure and [1 1 0] Burgers vector is 16.1, and for metals with BCC structure and [1 1 1] Burgers vector is 14.4 [33]. Also, F is a constant which is called F factor and based on
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the calorimetric experiments, it is considered to one (F=1) [33]. According to the equations (3) and (4), and considering the expressed parameters, the total density of dislocations is calculated
p s (
D
2
b
2 2
1
)2
(5)
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3k
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by equation (5) as follows [33-35]:
4- Results and discussion 4.1- Residual stress
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The CGP process creates residual stress in the CGPed sheet because of non-uniform deformation and forming ultrafine-grain structure. Investigation of residual stresses results
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showed, at the first and second passes of CGP, residual stresses are compressive near the surface, and with moving along the thickness, they gradually change to tension at the middle of the thickness. But at the third pass, residual stress is tension near the surface and converts to compression by moving toward the thickness. The reason for this distribution of residual stresses at the first, and second passes is deformation behavior of the material, and changing pattern of residual stress at the third pass is because of micro-cracks on the surface that causes the residual stress at the third pass be very low [5]. Figure 5 illustrates the results of residual stresses at the first, second, and third passes of CGP. 8
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Fig. 5 Contour of residual stresses at the (a) first, (b) second, and (c) third passes of CGP
Study the effect of pass number on the residual stresses showed that, the residual stresses are
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decreased with increasing number of CGP passes that it is because of extending the plastic deformed area, and increasing homogeneity due to the increasing number of CGP passes. According to figure 5, maximum tension and compression residual stresses, with the 66 MPa
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and -74.75 MPa, were occurred at the first pass of the CGP, and minimum tension and compression residual stresses, with the 9.3 MPa and -8.3 MPa, were occurred at the third pass
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of the CGP.
4.2- Microstructure
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Investigation of the effect of CGP process on the microstructure and evaluation of the grain size showed that this process causes a reduction in grain size, and repetition of the process with an increasing number of passes causes more grain refinement and creates the ultrafine-grain or nano structures. According to the results, at the first pass of CGP, the average of grains size reduced from 60 µm to 39.5 µm, and at the second and third passes, the average of grains size decreased to 34.5 µm and 28.3 µm, respectively. Figure 6 shows optical microscopy images of the CGPed sheets.
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Fig. 6 Microstructures of (a) annealed sheet, and CGPed sheets at the (b) first, (c) second, and (d) third
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passes of CGP
Grain refinement during the CGP process is based on the application of high plastic strain. The
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plastic strain causes to generate dislocations into the structure, and density of dislocations increases due to increasing strain and interacting the dislocations together. At first, the dislocations aggregate at the grain-boundaries, and with increasing the dislocations density,
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they transfer into the grains, and after rearranging, sub-boundaries are created. With following this procedure, sub-grains are formed, and by transferring more dislocations into the grains, sub-grains convert to main grains with the large-angle boundaries, causing a refined
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microstructure [36, 37]. Figure 7 shows TEM images of aggregation of dislocations, and formation of sub-grain and main-grain in the CGP process, and table 1 presents the value of
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calculated parameters on the measured points (Fig. 2).
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Fig. 7 Aggregation of dislocations, and formation of sub-grain and main-grain at the (a) first, and (b)
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third passes of CGP
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Table 1 Crystallites size, lattice strain, dislocations density, and residual stress of the measured points at the different CGP passes
Crystallite size* (nm)
Dislocation density* (1014×1/m2)
Residual stress (MPa)
1
0.0016
30.14
14.41
-25.5
2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
0.00225 0.0021 0.0025 0.00235 0.00225 0.00085 -0.00235 0.0008 0.00195 0.001 0.0008 0.0021 0.0029 0.0023 0.00115 0.002 0.0011 -0.007 -0.00045 -0.000225 0.00085 -0.0019 0.00235 -0.00095 -0.00155 0.00315
33.81 30.81 30.14 33.01 34.66 25.67 14.90 20.69 30.81 25.67 24.75 30.81 32.24 27.73 22.36 29.50 25.21 17.11 18.48 13.33 22.73 14.29 29.50 16.70 15.23 35.55
18.06 18.5 22.51 19.32 19.97 8.98 42.79 10.49 17.18 10.57 8.77 22.47 24.41 22.51 13.96 18.40 11.84 11.10 6.60 45.81 10.15 36.08 21.62 15.43 27.61 24.05
59.8 -33.4 -21.5 49.5 -37.6 -22.11 46.7 -22.4 -0.4 19.4 -13.8 -11.7 14.8 -11.7 -13.7 19.4 -0.4 3.6 -6.7 1.2 4.1 -5.1 4.1 1.2 -6.7 3.6
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Third pass
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Second pass
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Lattice strain*
First pass
Point number
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* Micro-parameters were calculated based on the measured micro-XRD results by Rigaku HighResolution SmarlLab machine, and have a 6% allowable tolerance.
Evaluation of crystallites size by using XRD results and Williamson-Hall equation showed,
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with increasing the number of CGP passes, the size of crystallites decreases in accordance with the changing microstructure and average grain size. According to the results, average of the crystallites size was 28.2 nm at the first pass of the CGP, and it reduces to 27.6 nm and 20.3 nm at the second and third passes, respectively. To validate the results of the XRD and observe the crystallites, TEM imaging was utilized, and the investigation of TEM images shows that crystallite sizes have a positive agreement with the calculated results of XRD. Figure 8 shows the TEM images of crystallites at the different passes of CGP. 12
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Fig. 8 TEM images of crystallites at the (a) first, and (b) third passes of the CGP
According to the results, the CGP process can create nanostructures in the CGPed sheets and
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with reducing the size of grains and crystallites, the residual stresses are decreased due to increasing homogeneity and spreading relaxation in the structure. Other research shows, the
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same results have been presented for the other processes [13, 14]. Calculation of dislocation density using Williamson-Smallman equation showed, that the average density of dislocations
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at the first and second passes of the CGP are 19.45×1014 and 16.68×1014 1/m2, and at the third pass of the CGP, dislocations are increased to maximum density with 22.05×1014 1/m2. Figure
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9 shows the TEM images of dislocations density at the first, second and third passes of CGP.
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Fig. 9 TEM images of dislocations density at the (a) first, (b) second, and (c) third passes of CGP
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According to the results, the size of grains and crystallites monotonically decreases due to increasing the number of CGP passes. Because of a low density of dislocations and coarse grains in the initial sheet, during the first pass, dislocation density grew up and grain refinement
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occurred, but during the second pass, more grains refinement and increasing sub-grains boundaries with consuming dislocations cause to reducing dislocations density at the second
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pass. At the third pass of the CGP, applying deformation and plastic strain to the microstructure lead to producing and increasing the dislocations density, and increasing the dislocations due to rising the collision of them together and with the grains boundaries causes to intensification and increasing dislocations production and forming a refined microstructure. Plastic deformation, and stress distribution in the microstructure cause to create elastic-plastic deformation of grains proportional to the crystal directions, value of stress, and heterogeneity of the materials. Elastic strain of the grains is named lattice strain [38], and according to the
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results, average of lattice strain at the first, second, and third CGP passes is 19×10-2, 17×10-2, and 15×10-2, respectively which it reveals that with increasing CGP passes the lattice strain reduces due to increasing plastic strain in the structure. Based on the results, with decreasing lattice strain, residual stress is reduced that this relationship is observable in equation (1). Figure 10 demonstrates a line plot of the average of crystallite size, dislocation density, and lattice strain in term of CGP passes. Crystallite size (nm)
Dislocation density (1/m2)*e14
Lattice strain *e-2
30 28 26
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24 22 20 18 16
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14 12
Second pass
Third pass
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First pass
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Fig. 10 The average of crystallite size, dislocation density, and lattice strain in term of CGP passes
4.3- Influence of micro-parameters on the residual stresses
Residual stresses, based on the affecting length and balancing, are classified in the two
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categories of macro, and micro residual stresses. Macro-residual stresses are under the influence of parameters with the widespread affecting area, but micro-residual stresses are affected by micro-parameters and interaction of them. In the first approach, to investigate the
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influence of micro parameters such as lattice strain, crystallites size, and dislocations density on the macro-residual stresses, a multiple regression analysis was applied on the experimental
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data, and a predictive quadratic model was constructed as follow: S tre s s ( M P a ) 6 4 .3 7 A 1 3 0 .2 9 B 2 4 5 .8 8 C 5 2 7 .9 8 A B 6 8 0 .5 9 A C 1 5 1 8 .8 9 B C 8 4 .9 0 A
2
4 9 1 .0 3 B
2
1 0 2 9 .7 8 C
2
(6)
In this model, the determination coefficient, R2 (0.8388) indicated that predicted values have an acceptable agreement with the experimental results, and the residuals plot of residual stresses with a linear pattern shows that there are no abnormalities in the data. Figure 11 illustrates the normal probability plot of residuals, and table 2 presents the ANOVA results of the model. 15
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Fig. 11 Normal probability plot of residuals of the residual stresses
Table 2 ANOVA results of the effect of microstructure parameters on the residual stress
Mean Square 1368.12 717.77 1467.99 1664.51 481.11 356.96 877.63 95.26 785.50 788.27
F-value 10.41 5.46 11.17 12.66 3.66 2.72 6.68 0.7246 5.97 6.00
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df 9 1 1 1 1 1 1 1 1 1
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Sum of Squares 12313.12 717.77 1467.99 1664.51 481.11 356.96 877.63 95.26 785.50 788.27
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Source Model A-crystallite size B-dislocation C-lattice strain AB AC BC A² B² C²
p-value < 0.0001 0.0312 0.0036 0.0022 0.0718 0.1167 0.0187 0.4058 0.0250 0.0248
significant
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According to the results, three factors of lattice strain, crystallites size, and dislocations density, with the P-value less than 0.05, are the significant factors on the residual stresses. Lattice strain with the minimum P-value is the most effective parameter on the residual stresses, and
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dislocations density, the interaction of lattice strain-dislocations density, and crystallites size are in the lower levels of importance, respectively. Figures 12 and 13 show the affective curves
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and the interaction contours of the parameters. As be seen in the figures, all of the parameters have a nonlinear behavior, and the effect of each parameter on the residual stresses is dependent on the interaction of them.
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Fig. 12 Perturbation plot of the effective parameters in the first approach
Fig. 13 Interaction contours of (a) lattice strain-dislocations density, and (b) crystallites size-dislocations
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density on the residual stresses
Calculation of percentage of the parameters influence shows that lattice strain (11.3% effect)
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has the most effect on the residual stresses, whereas dislocation density and interaction of dislocations-lattice strain (with contributions of 10% and 6%) are in the next ranking. Figure 14 illustrates the effect percentage of the effective parameters on the residual stresses.
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Affecting percentage
12 10 8 6 4 2 0
C-lattice strain
AB
BC
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A-crystallite B-dislocations size density
Fig. 14 Effect percentage of the effective micro-parameters on the residual stresses
Severe plastic deformation in the CGP process causes to form residual stress in the CGPed sheet, and effective strain is the most important factor in creating macro-residual stresses. In
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the second approach, to determine the effect of the micro-parameters in presence of the macrofactor, effective strain as the macro-factor was considered. The effective strain is 1.16 at the
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first pass of the CGP, and at the second and third passes, it increases to 2.32 and 3.48, respectively [5, 39]. In this analysis, a modified quadratic model was utilized that it is presented
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in the relation (7). The R-squared (0.9176), and a linear pattern of residuals plot showed that the predicted and experimental data have a positive agreement together, and there aren’t any abnormalities in the results. Figure 15 illustrates the normal probability plot of residuals, and
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the results of ANOVA are shown in table 3.
S tre s s ( M P a ) 1 6 .3 5 A 2 6 .1 2 B 5 6 .4 9 C 5 6 .1 1 D 6 .1 9 A C
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1 0 6 .1 0 B C 9 6 .6 9 B D 1 9 9 .7 7 C D 9 3 .9 9 C
18
2
9 8 .9 7 D
2
(7)
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Fig. 15 Normal probability plot of residuals of the residual stresses
Table 3 ANOVA results of the effect of micro, and macro factors on the residual stresses
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Source Sum of Squares df Mean Square F-value p-value 13469.68 10 1346.97 18.93 < 0.0001 significant Model A-effective strain 1651.65 1 1651.65 23.21 0.0002 B-crystallite size 408.90 1 408.90 5.75 0.0283 C-dislocation 361.70 1 361.70 5.08 0.0377 D-lattice strain 136.35 1 136.35 1.92 0.1842 AC 81.43 1 81.43 1.14 0.2997 BC 420.88 1 420.88 5.91 0.0264 BD 407.39 1 407.39 5.72 0.0286 CD 125.09 1 125.09 1.76 0.2024 C² 110.91 1 110.91 1.56 0.2288 D² 174.86 1 174.86 2.46 0.1354
According to the obtained results, effective strain with the minimum P-value is the most influential factor on the residual stresses. Also, lattice strain that it was the most effective factor
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in the first approach, it is not an important parameter solely in the second approach that it is because of lattice strain shows its effect in the effective strain. Based on the results of both
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approaches strain is the most influential factor on the residual stresses. After effective strain, the interaction of crystallite size-dislocations density is in the second rating, and other factors including crystallite size, the interaction of dislocations-lattice strain, and dislocations density are in the next ratings, respectively. Investigation of the affective curve of the parameters showed that effective strain has a linear behavior, and with increasing effective strain the residual stresses decrease, but microstructure parameters have a nonlinear behavior which it shows the effect of each micro-factor is dependent on the interaction of them together. Figures
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16 and 17 illustrate the affective curves and interaction contours of the parameters on the
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residual stresses.
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Fig. 16 Perturbation plot of the effective parameters in the second approach
Fig. 17 Interaction contours of (a) crystallites size-dislocations density, and (b) crystallites size-lattice
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strain on the residual stresses
Calculation of percentage of the effective parameters shows that effective strain with 11.25% has the most effects on the residual stresses, and the interaction of crystallite size-dislocations density and crystallites size with the contribution of 2.86% and 2.78% are in the next rankings, respectively. Figure 18 demonstrates the contribution of the effective factors.
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Affecting percentage
12 10 8 6 4 2 0
B-crystallite C-dislocations size density
BC
BD
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A-effective strain
Fig. 18 Effect percentage of the effective factors
Comparison of the results shows, effective strain has a high difference with the other factors, but summation of microstructure parameters effects with the 10.9% has a close effect to the
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effective strain that it shows in the ultra-fine grained or nanostructure sheets, the effects of micro-parameters on the macro-residual stresses are undeniable, and contribution of micro-
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parameters is as same as macro-parameters on the macro-residual stresses.
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5- Conclusion
In this study, the effects of micro-parameters on the macro-residual stresses were investigated, and the contribution of each parameter on the residual stresses was determined. The main
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results are summarized as follows:
The CGP process can create nanostructures in the CGPed sheets, and with increasing CGP passes, the size of grains and crystallites is decreased, causing a reduction residual stresses
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due to spreading relaxation and homogeneity in the structure. Increasing the number of CGP passes causes an increase in the density of dislocations and
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reduces lattice strain due to enhancing plastic strain in the structure.
According to the results of both approaches, lattice strain, crystallites size, and dislocation density have a significant effect on the macro-residual stresses, and strain is the most effective parameter.
There is a direct relationship between effective strain and residual stresses, while microstructure parameters have a nonlinear behavior, which it shows the effect of each micro-parameter on the residual stresses is dependent on the interaction between them.
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In the ultrafine-grained or nanostructure sheets, micro-parameters have an undeniable effect on the macro-residual stresses, and they have a contribution similar to that of macroparameters on the macro-residual stresses.
Conflict of interest
The authors declared no potential conflicts of interest with respect to the research, authorship
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and/or publication of this article.
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PII:
S0968-4328(19)30346-4
DOI:
https://doi.org/10.1016/j.micron.2020.102843
Reference:
JMIC 102843
To appear in:
Micron
Received Date:
20 October 2019
Revised Date:
2 February 2020
Accepted Date:
2 February 2020
Please cite this article as: Nazari F, Honarpisheh M, Zhao H, The effect of microstructure parameters on the residual stresses in the ultrafine-grained sheets, Micron (2020), doi: https://doi.org/10.1016/j.micron.2020.102843
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The effect of microstructure parameters on the residual stresses in the ultrafine-grained sheets
Farshad Nazari a,b, Mohammad Honarpisheh a,*, Haiyan Zhao b a b
Department of Mechanical Engineering, University of Kashan, Kashan, Iran
Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China
Corresponding author: [email protected], Tel ++98-31-55913404, Fax ++98-31-55913444
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Highlights
Investigation of microstructure parameters on the residual stresses of ultrafine-grained sheets.
Microstructure parameters are crystallites size, dislocations density, and lattice strain.
Microstructure parameters have a significant effect on the macro-residual stresses, and
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strain is the most effective parameter.
In the ultrafine-grained sheets, micro-parameters, with an undeniable effect, have a
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contribution as same as macro-parameters on the macro-residual stresses.
Abstract In this research, the influence of microstructure parameters on the residual stresses of ultrafine-
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grained sheets was investigated. For this purpose, the constrained groove pressing (CGP) process was carried out on the copper sheets with 3 mm thickness, and residual stresses of the CGPed sheets was measured using the contour method. Microstructure of the CGPed
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specimens was evaluated by the optical microscopy, micro x-ray diffraction (micro-XRD), and transmission electron microscopy (TEM) experiments. Microstructure parameters including
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crystallites size, dislocations density, and lattice strain were calculated using Williamson-Hall and Williamson-Smallman equations, and the calculated results were validated by the TEM images. The influence of these parameters on the residual stresses was investigated by analysis of variance (ANOVA) method, and two approaches were considered in this way. According to the results, the CGP process can create nanostructures in the CGPed sheets, and with increasing number of CGP passes, grains size, crystallites size, lattice strain, and residual stresses decrease, and density of dislocations increases. Microstructure parameters have a significant effect on the macro-residual stresses, and strain is the most effective parameter. 1
Also, in the ultrafine-grained sheets, micro-parameters have an undeniable contribution, which is the same as that of macro-parameters on the macro-residual stresses. Keywords: crystallite size, dislocations density, lattice strain, residual stress, CGP. 1- Introduction The development of ultrafine-grained structures is one of the methods for improving materials properties. Constrained groove pressing (CGP) process is one of the severe plastic deformation (SPD) methods which can produce ultrafine-grained or nanostructures sheets [1]. Evaluation of the effect of the CGP process on the mechanical properties and microstructure of the CGPed copper sheets illustrated that CGP process decreases grain size, and increases
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hardness and strength of the sheets [2]. Similar results have been presented for other materials such as aluminum [3], and steel [4]. In the previous studies of the authors [5, 6] deformation behavior and residual stresses in the CGP process were investigated, and demonstrated compressive residual stress was created near the surfaces of the CGPed sheet, and with moving
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along the thickness, it changes to tension residual stress at the middle of the sheet.
Residual stresses are classified in the three categories as the types I, II and III [7]. Type I is
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macro-residual stress which is created within the specimens in the rage of much larger than grain size. Types II and III are micro-residual stresses which they are operated at the grain-size
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and atomic level, respectively. Micro-residual stresses are generated due to heterogeneity and interaction between parameters of the microstructure of materials [8]. Salvati and Korsunsky [9] studied the different types of residual stress using FIB-DIC ring core technique and multi-
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scale FEM simulation on the aluminum alloy, and presented the majority of residual stresses (about 2/3) in the plastic deformation is type I. Pouraliakbar et al. [10] measured microstructure parameters of the CGPed sheet of Al-Mn-Si alloy. They utilized the X-ray diffraction (XRD)
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method for measuring the parameters, including crystallite size, dislocation density, and lattice strain, and revealed that microstructure parameters extremely depend on the plastic
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deformation and temperature of the process. They presented, with an increasing number of CGP passes, effective plastic strain increased which causes to increase dislocation density and reduce crystallite size in the microstructure. Moradpour et al. [11, 12] investigated the crystallite size and dislocation density in the Al-Mg CGPed sheets by the XRD analysis and transmission electron microscopy (TEM) imaging. They studied two passes of the CGP process and showed this process causes to grain refinement and reduces crystallites size based on the severe plastic deformation, and calculation of the dislocations density revealed that the average of dislocation density in the second pass is 4×1014 m-2 which is two times higher than the related 2
value of the initial sheet. Sergo et al. [13] investigated the influence of grains size on the residual stresses in the alumina/zirconia composite and showed that a large number of grains boundaries reduce the residual stresses due to spreading relaxation, and local microstructural heterogeneity causes to increase residual stresses. Weglewski et al. [14] studied the effects of grain size on the thermal residual stresses in the sintered Cr/Al2O3 composite and presented that a composite with the fine grain size has lower residual stresses than the composite with the coarse grain size. Also, Cao et al. [15] with simulating the grains size and dislocations on a nanocrystalline thin film, and evaluating the effect of them on the residual stresses revealed that residual stresses are reduced due to grains size reduction, and increasing homogeneity.
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Roy et al. [16] evaluated the relationship of dislocations density and residual stresses and indicated that increasing dislocations density in the cold working processes causes to enhance residual stresses due to increasing micro-defects, and inhomogeneity in the microstructure. Sato et al. [17] studied the relationship between dislocation density and residual stress in the wire drawing process. They carried out the process in the room temperature, and utilized
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energy-dispersive X-ray diffraction method for investigation, and showed that higher residual stress and dislocation density induced in the center of the wire, and those are decreased at the
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surface of the wire. Meyer et al. [18], with investigating a single track deep rolled steel by XRD analysis, illustrated that residual stress and dislocation density have an almost linear-relation
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together.
According to the fundamentals of X-ray diffraction in residual stress measurement, and experimental results, residual stress increases with increasing lattice strain [19-21]. The
equation,
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relationship between lattice strain and residual stress is shown in equation (1) [21, 22]. In this is residual stress, E is elastic modulus, is Poisson’s ratio,
E
(1 ) s in 2
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spacing of diffracting plane, and
(
d d 0
d
0
is the lattice spacing which
d d d
0
d
is the measured
is lattice strain.
0
(1)
)
d0
According to the literature, microstructure parameters have effects on the residual stresses, but the contribution and effective curve for each parameter have not been evaluated. In this study, the effects of lattice strain, dislocations density, and crystallites size on the residual stresses were investigated. Experimental and computational study and statistical analysis were utilized, and the research was carried out on the CGPed sheets using optical microscopy, X-ray diffraction methods, TEM imaging, and a contour method. 3
2- Materials and methods 2.1- CGP process
For performing the CGP process, some samples were prepared from a commercial pure copper (99.93%) sheet with a 3 mm thickness and dimensions of 72×50 mm. Before carrying out the process, the samples were fully annealed at a temperature of 650 ºC for 2 hours [23] to eliminate the rolling effects and initial residual stresses, to form a uniform microstructure. The CGP process applies a continuous severe plastic deformation on the specimen that it is performed by using two pairs of symmetrical grooved and flattened dies. Each pass of the process includes four steps. During the first and second steps, the specimen is grooved and
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flattened, then in the following the sheet is rotated through 180º, and grooving and flattening are conducted again on the sheet in the third and fourth steps, respectively. The CGP process was carried out up to three passes, and in this way, a 60-ton hydraulic press machine was used. All of the experiments were performed at room temperature (25℃) with the forming speed of
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0.4 mm/s, and lubrication with oil was utilized. Figure 1 shows the steps of the CGP process.
Fig. 1 Steps of the CGP process
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2.2- Residual stress
A 2D map of residual stresses was measured using the contour method at the first, second, and
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third passes of CGP. In the contour method, cutting the samples and releasing residual stress cause to deform the cutting surface, so by measuring the displacements on the cutting planes and processing them, and then simulating processed displacements using a finite element model, the residual stresses are calculated [24-28]. With considering the aging effect can be expressed that duration time between CGP process and residual stress measurement (about 12 hours) is not important, because with forming at the room temperature and using commercial pure copper alloy with high purity of Cu (99.93%) and lack of alloying components (<0.07%) such as Mg, Al, Ni, etc., the aging ability of the sheet is not significant. In this research, residual 4
stresses were measured on the middle cross-section of the samples, and figure 2 illustrates the
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dimensions of the specimens, cutting line, and situation of the experiments.
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Fig. 2 Dimensions of the CGPed specimens, cutting line, and situation of the experiments
Cutting the samples was performed using Charmilles ROBOFIL 2000 wire electro-discharge
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machine, and the cutting process was accomplished by a brass wire (Cu-Zn37), and with the 0.6 mm/min cutting speed in a full immersion condition. For measuring the surface
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displacements which resulted from releasing residual stresses, the cutting surfaces (both sides of the cutting plane) were measured by POLI-SKY coordinate-measurement machine using a RENISHAW PH6 touch-trigger probe with 2 mm diameter. Sampling was carried out with a
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distance of 0.2 mm in the Y-direction, and 0.5 mm in the X-direction, that on each cross-section 1485 points were measured. The specimens were completely constrained to prevent any displacement during the measurement. After measuring, an average of the data was calculated
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at the corresponding points on both surfaces, and the resulted contours were smoothed using a bivariate spline for eliminating the probability of measurement error, and the roughness which
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resulted from the cutting process. According to fundamentals of the contour method, for calculating the residual stresses, processed surface displacements must be simulated inverted as a boundary condition in a finite element model. Finite element simulation and residual stress computation were developed in the ABAQUS commercial package using a 3D linear elastic model. In this model, the element of C3D8R was used, and for increasing the accuracy of the simulation, finer elements were utilized near the cutting surface than the other areas. Figure 3 shows the finite element model, and contour of the surface displacements after averaging, smoothing, and inverting at the first pass of the CGP. 5
Fig. 3 Finite element model, and contour of the surface displacements after averaging, and inverting at the first pass of the CGP
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2.3- Microstructure
To evaluate the effect of microstructure on the residual stresses, microstructure of the samples was first investigated by optical microscopy, and the average of grains size was determined by vertical and horizontal intercept lines method according to the ASTM E112-96 [29] standard.
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In the following, the microstructure parameters were measured by micro x-ray diffraction (micro-XRD) on the surface of the cross-section. Micro-XRD was used to study and calculate
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the crystallites size, lattice strain, and dislocation density, and the tests were carried out on a Rigaku High-Resolution SmartLab x-ray diffractometer by Cu Kα radiation (λ=0.154183 nm).
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Radiance diameter was 400 µm, and the operation was performed in the range of 20-100º (2θ), step width of 0.02º, count time of 0.5 s per step, and potential of 40 KV with 200 mA. The results were analyzed using Williamson-Hall and Williamson-Smallman methods, and for
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verifying the micro-XRD results, transmission electron microscopy imaging was performed on the specimens. Nine points on each cross-section were investigated by the experiments that for the first, second, and third passes of CGP, 27 points were measured. The points map of the
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XRD test is illustrated in figure 2.
To investigate the effect of microstructure parameters, and interaction of them on the residual
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stresses, two approaches have been considered. In the first approach, the effects of microparameters on the residual stresses were evaluated to determine the significance of each parameter and the interaction of them. Then, in the second approach, the microstructure parameters besides the effective strain as a macro-parameter were investigated to determine that have micro-parameters a significant effect on the macro-residual stresses against macrofactors, or they have only a local effect and are limited to the micro-areas. The effect of the parameters and their interaction on the residual stresses were investigated using the response
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surface method and analysis of variance (ANOVA). The values of parameters were considered as magnitudes and the analyses were computed by Design-Expert commercial software.
3- Calculation of crystallite size, lattice strain, and dislocations density Crystallites size and lattice strain were calculated using the Williamson-Hall method. Based on this method, the intensity and broadening of the diffraction peaks are dependent on the crystallite size, and lattice strain [30-32]. Relation (2) shows the Williamson-Hall equation. C os
K
2 S in
(2)
D
In this Equation, β is the peak broadening measured as the full width at half maximum
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(FWHM), θ is the Bragg angle, K is Scherer constant which generally is 0.9, λ is the X-ray wavelength, D is the crystallite size, and ε is the lattice strain. According to the WilliamsonHall equation, if βCosθ is plotted against Sinθ, and a linear equation as Y=mX+c is fitted to the scatter plot, the slope and intercept of the equation give the lattice strain (ε=m/2) and the
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crystallite size (D=Kλ/c), respectively. Figure 4 illustrates the XRD pattern of CGPed copper
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sheets at the center of the surface (point 5).
Fig. 4 XRD pattern of CGPed copper sheets at the center of the surface (point 5)
Dislocation density is one of the important parameters on the residual stress and mechanical behavior of the materials, and evaluation of the dislocations density is possible by using the XRD test and analyzing the results. Williamson and Smallman [33] showed the density of dislocations depends on the crystallite size and lattice strain and presented an equation to calculate the dislocation density based on them. Equation (3) shows the relation of dislocation 7
density and crystallite size, that in this equation, ρp is density of dislocations per unit cell resulted from crystallites, and n is the number of dislocations per face of unit cell, which for annealed or severe plastic deformed metals, n is equal to 1 (minimum value) [33]. p
3n D
(3)
2
Study the relationship of dislocation density and lattice strain indicated that the density of dislocations is equal to the ratio of lattice strain energy to dislocation energy into the structure. In this way, Williamson and Smallman [33] presented equation (4) to calculate the density of dislocation based on the lattice strain. k F
b
2
(4)
2
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s
In this equation, ρs is the density of dislocations resulted from lattice strain, b is Burgers vector, and k is a function that it is dependent on the microstructure and mechanical properties of the materials which can be variable between 2 to 25. According to studies, the value of k for metals
-p
with FCC structure and [1 1 0] Burgers vector is 16.1, and for metals with BCC structure and [1 1 1] Burgers vector is 14.4 [33]. Also, F is a constant which is called F factor and based on
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the calorimetric experiments, it is considered to one (F=1) [33]. According to the equations (3) and (4), and considering the expressed parameters, the total density of dislocations is calculated
p s (
D
2
b
2 2
1
)2
(5)
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3k
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by equation (5) as follows [33-35]:
4- Results and discussion 4.1- Residual stress
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The CGP process creates residual stress in the CGPed sheet because of non-uniform deformation and forming ultrafine-grain structure. Investigation of residual stresses results
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showed, at the first and second passes of CGP, residual stresses are compressive near the surface, and with moving along the thickness, they gradually change to tension at the middle of the thickness. But at the third pass, residual stress is tension near the surface and converts to compression by moving toward the thickness. The reason for this distribution of residual stresses at the first, and second passes is deformation behavior of the material, and changing pattern of residual stress at the third pass is because of micro-cracks on the surface that causes the residual stress at the third pass be very low [5]. Figure 5 illustrates the results of residual stresses at the first, second, and third passes of CGP. 8
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Fig. 5 Contour of residual stresses at the (a) first, (b) second, and (c) third passes of CGP
Study the effect of pass number on the residual stresses showed that, the residual stresses are
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decreased with increasing number of CGP passes that it is because of extending the plastic deformed area, and increasing homogeneity due to the increasing number of CGP passes. According to figure 5, maximum tension and compression residual stresses, with the 66 MPa
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and -74.75 MPa, were occurred at the first pass of the CGP, and minimum tension and compression residual stresses, with the 9.3 MPa and -8.3 MPa, were occurred at the third pass
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of the CGP.
4.2- Microstructure
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Investigation of the effect of CGP process on the microstructure and evaluation of the grain size showed that this process causes a reduction in grain size, and repetition of the process with an increasing number of passes causes more grain refinement and creates the ultrafine-grain or nano structures. According to the results, at the first pass of CGP, the average of grains size reduced from 60 µm to 39.5 µm, and at the second and third passes, the average of grains size decreased to 34.5 µm and 28.3 µm, respectively. Figure 6 shows optical microscopy images of the CGPed sheets.
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Fig. 6 Microstructures of (a) annealed sheet, and CGPed sheets at the (b) first, (c) second, and (d) third
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passes of CGP
Grain refinement during the CGP process is based on the application of high plastic strain. The
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plastic strain causes to generate dislocations into the structure, and density of dislocations increases due to increasing strain and interacting the dislocations together. At first, the dislocations aggregate at the grain-boundaries, and with increasing the dislocations density,
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they transfer into the grains, and after rearranging, sub-boundaries are created. With following this procedure, sub-grains are formed, and by transferring more dislocations into the grains, sub-grains convert to main grains with the large-angle boundaries, causing a refined
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microstructure [36, 37]. Figure 7 shows TEM images of aggregation of dislocations, and formation of sub-grain and main-grain in the CGP process, and table 1 presents the value of
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calculated parameters on the measured points (Fig. 2).
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Fig. 7 Aggregation of dislocations, and formation of sub-grain and main-grain at the (a) first, and (b)
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third passes of CGP
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Table 1 Crystallites size, lattice strain, dislocations density, and residual stress of the measured points at the different CGP passes
Crystallite size* (nm)
Dislocation density* (1014×1/m2)
Residual stress (MPa)
1
0.0016
30.14
14.41
-25.5
2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
0.00225 0.0021 0.0025 0.00235 0.00225 0.00085 -0.00235 0.0008 0.00195 0.001 0.0008 0.0021 0.0029 0.0023 0.00115 0.002 0.0011 -0.007 -0.00045 -0.000225 0.00085 -0.0019 0.00235 -0.00095 -0.00155 0.00315
33.81 30.81 30.14 33.01 34.66 25.67 14.90 20.69 30.81 25.67 24.75 30.81 32.24 27.73 22.36 29.50 25.21 17.11 18.48 13.33 22.73 14.29 29.50 16.70 15.23 35.55
18.06 18.5 22.51 19.32 19.97 8.98 42.79 10.49 17.18 10.57 8.77 22.47 24.41 22.51 13.96 18.40 11.84 11.10 6.60 45.81 10.15 36.08 21.62 15.43 27.61 24.05
59.8 -33.4 -21.5 49.5 -37.6 -22.11 46.7 -22.4 -0.4 19.4 -13.8 -11.7 14.8 -11.7 -13.7 19.4 -0.4 3.6 -6.7 1.2 4.1 -5.1 4.1 1.2 -6.7 3.6
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Third pass
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Second pass
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Lattice strain*
First pass
Point number
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* Micro-parameters were calculated based on the measured micro-XRD results by Rigaku HighResolution SmarlLab machine, and have a 6% allowable tolerance.
Evaluation of crystallites size by using XRD results and Williamson-Hall equation showed,
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with increasing the number of CGP passes, the size of crystallites decreases in accordance with the changing microstructure and average grain size. According to the results, average of the crystallites size was 28.2 nm at the first pass of the CGP, and it reduces to 27.6 nm and 20.3 nm at the second and third passes, respectively. To validate the results of the XRD and observe the crystallites, TEM imaging was utilized, and the investigation of TEM images shows that crystallite sizes have a positive agreement with the calculated results of XRD. Figure 8 shows the TEM images of crystallites at the different passes of CGP. 12
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Fig. 8 TEM images of crystallites at the (a) first, and (b) third passes of the CGP
According to the results, the CGP process can create nanostructures in the CGPed sheets and
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with reducing the size of grains and crystallites, the residual stresses are decreased due to increasing homogeneity and spreading relaxation in the structure. Other research shows, the
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same results have been presented for the other processes [13, 14]. Calculation of dislocation density using Williamson-Smallman equation showed, that the average density of dislocations
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at the first and second passes of the CGP are 19.45×1014 and 16.68×1014 1/m2, and at the third pass of the CGP, dislocations are increased to maximum density with 22.05×1014 1/m2. Figure
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9 shows the TEM images of dislocations density at the first, second and third passes of CGP.
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Fig. 9 TEM images of dislocations density at the (a) first, (b) second, and (c) third passes of CGP
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According to the results, the size of grains and crystallites monotonically decreases due to increasing the number of CGP passes. Because of a low density of dislocations and coarse grains in the initial sheet, during the first pass, dislocation density grew up and grain refinement
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occurred, but during the second pass, more grains refinement and increasing sub-grains boundaries with consuming dislocations cause to reducing dislocations density at the second
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pass. At the third pass of the CGP, applying deformation and plastic strain to the microstructure lead to producing and increasing the dislocations density, and increasing the dislocations due to rising the collision of them together and with the grains boundaries causes to intensification and increasing dislocations production and forming a refined microstructure. Plastic deformation, and stress distribution in the microstructure cause to create elastic-plastic deformation of grains proportional to the crystal directions, value of stress, and heterogeneity of the materials. Elastic strain of the grains is named lattice strain [38], and according to the
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results, average of lattice strain at the first, second, and third CGP passes is 19×10-2, 17×10-2, and 15×10-2, respectively which it reveals that with increasing CGP passes the lattice strain reduces due to increasing plastic strain in the structure. Based on the results, with decreasing lattice strain, residual stress is reduced that this relationship is observable in equation (1). Figure 10 demonstrates a line plot of the average of crystallite size, dislocation density, and lattice strain in term of CGP passes. Crystallite size (nm)
Dislocation density (1/m2)*e14
Lattice strain *e-2
30 28 26
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24 22 20 18 16
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14 12
Second pass
Third pass
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First pass
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Fig. 10 The average of crystallite size, dislocation density, and lattice strain in term of CGP passes
4.3- Influence of micro-parameters on the residual stresses
Residual stresses, based on the affecting length and balancing, are classified in the two
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categories of macro, and micro residual stresses. Macro-residual stresses are under the influence of parameters with the widespread affecting area, but micro-residual stresses are affected by micro-parameters and interaction of them. In the first approach, to investigate the
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influence of micro parameters such as lattice strain, crystallites size, and dislocations density on the macro-residual stresses, a multiple regression analysis was applied on the experimental
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data, and a predictive quadratic model was constructed as follow: S tre s s ( M P a ) 6 4 .3 7 A 1 3 0 .2 9 B 2 4 5 .8 8 C 5 2 7 .9 8 A B 6 8 0 .5 9 A C 1 5 1 8 .8 9 B C 8 4 .9 0 A
2
4 9 1 .0 3 B
2
1 0 2 9 .7 8 C
2
(6)
In this model, the determination coefficient, R2 (0.8388) indicated that predicted values have an acceptable agreement with the experimental results, and the residuals plot of residual stresses with a linear pattern shows that there are no abnormalities in the data. Figure 11 illustrates the normal probability plot of residuals, and table 2 presents the ANOVA results of the model. 15
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Fig. 11 Normal probability plot of residuals of the residual stresses
Table 2 ANOVA results of the effect of microstructure parameters on the residual stress
Mean Square 1368.12 717.77 1467.99 1664.51 481.11 356.96 877.63 95.26 785.50 788.27
F-value 10.41 5.46 11.17 12.66 3.66 2.72 6.68 0.7246 5.97 6.00
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df 9 1 1 1 1 1 1 1 1 1
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Sum of Squares 12313.12 717.77 1467.99 1664.51 481.11 356.96 877.63 95.26 785.50 788.27
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Source Model A-crystallite size B-dislocation C-lattice strain AB AC BC A² B² C²
p-value < 0.0001 0.0312 0.0036 0.0022 0.0718 0.1167 0.0187 0.4058 0.0250 0.0248
significant
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According to the results, three factors of lattice strain, crystallites size, and dislocations density, with the P-value less than 0.05, are the significant factors on the residual stresses. Lattice strain with the minimum P-value is the most effective parameter on the residual stresses, and
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dislocations density, the interaction of lattice strain-dislocations density, and crystallites size are in the lower levels of importance, respectively. Figures 12 and 13 show the affective curves
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and the interaction contours of the parameters. As be seen in the figures, all of the parameters have a nonlinear behavior, and the effect of each parameter on the residual stresses is dependent on the interaction of them.
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Fig. 12 Perturbation plot of the effective parameters in the first approach
Fig. 13 Interaction contours of (a) lattice strain-dislocations density, and (b) crystallites size-dislocations
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density on the residual stresses
Calculation of percentage of the parameters influence shows that lattice strain (11.3% effect)
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has the most effect on the residual stresses, whereas dislocation density and interaction of dislocations-lattice strain (with contributions of 10% and 6%) are in the next ranking. Figure 14 illustrates the effect percentage of the effective parameters on the residual stresses.
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Affecting percentage
12 10 8 6 4 2 0
C-lattice strain
AB
BC
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A-crystallite B-dislocations size density
Fig. 14 Effect percentage of the effective micro-parameters on the residual stresses
Severe plastic deformation in the CGP process causes to form residual stress in the CGPed sheet, and effective strain is the most important factor in creating macro-residual stresses. In
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the second approach, to determine the effect of the micro-parameters in presence of the macrofactor, effective strain as the macro-factor was considered. The effective strain is 1.16 at the
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first pass of the CGP, and at the second and third passes, it increases to 2.32 and 3.48, respectively [5, 39]. In this analysis, a modified quadratic model was utilized that it is presented
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in the relation (7). The R-squared (0.9176), and a linear pattern of residuals plot showed that the predicted and experimental data have a positive agreement together, and there aren’t any abnormalities in the results. Figure 15 illustrates the normal probability plot of residuals, and
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the results of ANOVA are shown in table 3.
S tre s s ( M P a ) 1 6 .3 5 A 2 6 .1 2 B 5 6 .4 9 C 5 6 .1 1 D 6 .1 9 A C
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1 0 6 .1 0 B C 9 6 .6 9 B D 1 9 9 .7 7 C D 9 3 .9 9 C
18
2
9 8 .9 7 D
2
(7)
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Fig. 15 Normal probability plot of residuals of the residual stresses
Table 3 ANOVA results of the effect of micro, and macro factors on the residual stresses
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Source Sum of Squares df Mean Square F-value p-value 13469.68 10 1346.97 18.93 < 0.0001 significant Model A-effective strain 1651.65 1 1651.65 23.21 0.0002 B-crystallite size 408.90 1 408.90 5.75 0.0283 C-dislocation 361.70 1 361.70 5.08 0.0377 D-lattice strain 136.35 1 136.35 1.92 0.1842 AC 81.43 1 81.43 1.14 0.2997 BC 420.88 1 420.88 5.91 0.0264 BD 407.39 1 407.39 5.72 0.0286 CD 125.09 1 125.09 1.76 0.2024 C² 110.91 1 110.91 1.56 0.2288 D² 174.86 1 174.86 2.46 0.1354
According to the obtained results, effective strain with the minimum P-value is the most influential factor on the residual stresses. Also, lattice strain that it was the most effective factor
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in the first approach, it is not an important parameter solely in the second approach that it is because of lattice strain shows its effect in the effective strain. Based on the results of both
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approaches strain is the most influential factor on the residual stresses. After effective strain, the interaction of crystallite size-dislocations density is in the second rating, and other factors including crystallite size, the interaction of dislocations-lattice strain, and dislocations density are in the next ratings, respectively. Investigation of the affective curve of the parameters showed that effective strain has a linear behavior, and with increasing effective strain the residual stresses decrease, but microstructure parameters have a nonlinear behavior which it shows the effect of each micro-factor is dependent on the interaction of them together. Figures
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16 and 17 illustrate the affective curves and interaction contours of the parameters on the
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residual stresses.
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Fig. 16 Perturbation plot of the effective parameters in the second approach
Fig. 17 Interaction contours of (a) crystallites size-dislocations density, and (b) crystallites size-lattice
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strain on the residual stresses
Calculation of percentage of the effective parameters shows that effective strain with 11.25% has the most effects on the residual stresses, and the interaction of crystallite size-dislocations density and crystallites size with the contribution of 2.86% and 2.78% are in the next rankings, respectively. Figure 18 demonstrates the contribution of the effective factors.
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Affecting percentage
12 10 8 6 4 2 0
B-crystallite C-dislocations size density
BC
BD
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A-effective strain
Fig. 18 Effect percentage of the effective factors
Comparison of the results shows, effective strain has a high difference with the other factors, but summation of microstructure parameters effects with the 10.9% has a close effect to the
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effective strain that it shows in the ultra-fine grained or nanostructure sheets, the effects of micro-parameters on the macro-residual stresses are undeniable, and contribution of micro-
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parameters is as same as macro-parameters on the macro-residual stresses.
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5- Conclusion
In this study, the effects of micro-parameters on the macro-residual stresses were investigated, and the contribution of each parameter on the residual stresses was determined. The main
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results are summarized as follows:
The CGP process can create nanostructures in the CGPed sheets, and with increasing CGP passes, the size of grains and crystallites is decreased, causing a reduction residual stresses
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due to spreading relaxation and homogeneity in the structure. Increasing the number of CGP passes causes an increase in the density of dislocations and
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reduces lattice strain due to enhancing plastic strain in the structure.
According to the results of both approaches, lattice strain, crystallites size, and dislocation density have a significant effect on the macro-residual stresses, and strain is the most effective parameter.
There is a direct relationship between effective strain and residual stresses, while microstructure parameters have a nonlinear behavior, which it shows the effect of each micro-parameter on the residual stresses is dependent on the interaction between them.
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In the ultrafine-grained or nanostructure sheets, micro-parameters have an undeniable effect on the macro-residual stresses, and they have a contribution similar to that of macroparameters on the macro-residual stresses.
Conflict of interest
The authors declared no potential conflicts of interest with respect to the research, authorship
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and/or publication of this article.
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