# Residual Stress Distribution in a Copper-Aluminum Multifilament Composite Fabricated by Rotary Swaging

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Cu/Al Composite

_{0}/S

_{n}), where S

_{0}and S

_{n}are the cross-sectional areas at the input and output of swaging dies, respectively, was 0.71. A special arrangement of the Al filaments was used in order to see the effect of various configurations within the sample on residual stresses. The measured cylindrical samples of approximately 37 mm in height are shown in Figure 1a,b. The initial microstructures of Cu and Al are shown in Figure 1c,d, respectively. The images were acquired via SEM-EBSD measurements on a Lyra 3 XMU scanning electron microscope (SEM, Tescan, Brno, Czech Republic). In the images, high angle grain boundaries (HAGB) defined with misorientation angles > 15° are depicted in black, low angle grain boundaries (LAGB) defined as misorientation angles 0° < θ < 15° are shown in green, and the coincidence site lattice (CSL) boundaries characterizing <111> 60° annealing twins are depicted in red. The composite matrix, i.e., Cu, was characterized by grains with an average size of 22.7 μm (measured as equivalent circle diameter), although the maximum grain size reached up to 160 μm. The structure also exhibited the notable presence of twins (Figure 1c). On the contrary, the original structure of the Al filaments did not show the presence of twins but a higher fraction of LAGB (Figure 1d). The average grain size within the Al was 38.5 μm, with the largest grains reaching up to 115 μm.

#### 2.2. Plastometric Tests

^{−1}.

#### 2.3. Neutron Diffraction Experiments

_{hkl}interplanar distance for the selected crystallographic plane. The strain was in this way determined in the processed bars using the SPN-100 neutron diffractometer of CANAM NPL infrastructure [27], installed at the LVR-15 research reactor neutron source in Řež, CZ. A sketch of the instrument is shown elsewhere [28]. This experimental facility is dedicated to the mapping of bulk-averaged residual strains in polycrystalline materials. The SPN-100 instrument is equipped with a horizontally bent Si monochromator (curvature 0.165 m

^{−1}), a radial collimator, and a position-sensitive detector for fast recording of diffraction patterns. A neutron wavelength of λ = 0.213 nm and the Cu-111 reflection were selected for the experiment, leading to a 2θ diffraction angle of approximately 61.4°.

^{2}and 5 × 5 mm

^{2}cadmium slits for the hoop/radial and for the axial strain measurements, respectively. The slits were positioned at 75 mm in front of the gauge volume center (instrumental sample axis). A larger height of the gauge volume was chosen for the hoop and radial directions due to the low diffraction intensity in these directions. This is a consequence of the preferred grain orientation (texture) formed during rotary swaging.

^{3}and ≈57 mm

^{3}for the hoop/radial and for the axial strains, respectively. To record the diffraction peaks for all three directions, three different geometrical arrangements of the examined sample with respect to the scattering vector were used. Samples were positioned with high accuracy by using a robotic arm. The lines along which the scans were carried out are indicated in Figure 3. The peaks from Cu-111 reflection were collected during the line scans through the sample cross section approximately in the middle of the length of the cylindrical sample.

#### 2.4. Strain and Stress Analysis

_{hk}

_{l}interplanar distance determination of the selected crystallographic plane with Miller indices hkl according to the following expression (see [24], p. 150):

_{hkl}and d

_{0,hkl}are the interplanar distances measured over the gauge volume for the sample and for the stress-free reference, respectively. They are determined with help of Bragg’s law [29], i.e., from the measured angular peak positions, 2θ

_{hkl}. The lower index at ε denotes the normal strain component in Voigt notation, which is parallel to the diffraction vector. Assuming an isotropic material, the residual stresses in one direction can be then determined from the lattice strains measured in three normal directions by means of the generalized Hook’s law as (see [24], p. 207)

_{1}, ε

_{2}, and ε

_{3}are the experimentally measured strains for the hkl crystallographic plane, and E and ν are the Young modulus and Poisson’s ratio, respectively, for the selected crystallographic plane. The hkl indexing is omitted for clarity of the expression. Corresponding relations for the 2nd and 3rd stress components are obtained by permutations of 1, 2, and 3 indices.

#### 2.5. FEM Analysis

_{1}and m

_{9}define the material sensitivity to T, m

_{5}is the term coupling T and strain (ε), m

_{8}is the term coupling T and $\dot{\epsilon}$, m

_{2}, m

_{4}, and m

_{7}define the material sensitivity to ε, m

_{3}depends on the material sensitivity to $\dot{\epsilon}$, and A is a multiplication factor. The dimensions of m parameters are always such that the exponent is dimensionless. The values of the used coefficients for Cu were 411.19 MPa, −0.00121, 0.21554, 0.01472, −0.00935, respectively, and m

_{5}÷ m

_{8}was 0. The values of the coefficients for Al were 151.323 MPa, −0.00253, 0.21142, 0.03177, −0.00654, respectively, and m

_{5}÷ m

_{8}was 0.

## 3. Results

#### 3.1. Sample Model, Measurement Geometry, and Simulation of Pseudo-Strain

- Irregularity of Al wires geometry.
- Variation of the scattering intensity due to texture gradients.

#### 3.2. Fitting of Intrinsic Intensities

#### 3.3. Evaluation of the Lattice Strain Distribution

#### 3.4. d_{0} Problem and Stress Differences

_{axial}−σ

_{radial}, σ

_{axial}−σ

_{hoop}, and σ

_{radial}−σ

_{hoop}through L1.1, L1.4, and in L2.1, L3.1, and L3.2 around the central Al wire for the AD sample is remarkable. In contrast, for the SD sample, stress differences are only equal in L1.3 and roughly in the L2.2 segment around the central wire, and in L2.3 at the interface Cu-Al.

_{axial}−σ

_{radial}can be considered equal to ≈0 MPa through L2.2 (except around the central Al wire) for the SD sample while it is shifted to −25 MPa in the same region for the AD sample.

_{radial}−σ

_{hoop}equals to ≈0 MPa in L2.1 for the AD specimen (leading to a coincidence of σ

_{axial}−σ

_{radial}and σ

_{axial}−σ

_{hoop}). For the SD sample, σ

_{radial}−σ

_{hoop}roughly equals to 0 MPa in L3.1 although L3.2 seems to be an extension of the first trend showing a symmetric behavior around the axis (i.e., the position of 0 mm) roughly tilted with respect to a vertical line in the diagram.

_{axial}−σ

_{radial}and σ

_{axial}−σ

_{hoop}closer to 0 MPa when compared to the SD sample.

_{0}determination formulated in the Discussion section.

#### 3.5. Full-Width at Half Maximum

#### 3.6. FEM Results

## 4. Discussion

_{0}in the Cu matrix with the methodology explained in [23,34]. The relation is based on the force equilibrium in the hoop direction [35] given by

_{3}is the total stress in the hoop direction, R is the bar radius, and r is the radial position. Since a finite number of measured points is available, the integral of Equation (7) is approximated as follows:

_{0}for the Cu-phase as follows:

_{i,k}is the interplanar distance at the measured point k in the direction i, Δr has its common meaning, and A and B are factors defined as follows:

_{0}is applied for both samples.

_{H}the hydrostatic component, σ

_{Di}the deviatoric stress in the corresponding direction i, and where σ

_{i}has its common meaning.

_{H}are roughly 0 MPa (as well as the deviatoric stresses). The SD sample exhibits a similar trend, except in (i) regions between the outer surface and the Al wires (L1.1 and L2.3) and (ii) around the central Al wire in Line 3, in which σ

_{H}are compressive.

_{D}, mainly in regions around the Al wires, are always relatively close to zero for the AD sample. The stresses have thus a hydrostatic nature in these regions, although this behavior is not always observed in the SD sample. Moreover, the σ

_{D}in the axial direction for both samples are negative or zero (except near the outer surface for the AD sample in Line 3). More specifically, the σ

_{D}are almost negligible in L1.1 and in L1.4 for the AD sample, i.e., these regions can be considered to be only under hydrostatic stress for the former and with negligible total stresses for the latter. All σ

_{D}are zero at the interface of the Al wire in L2.3 in the SD sample.

_{D}) around the central Al wire for the AD sample in Line 3 are ≈0 MPa. σ

_{D}and σ

_{H}around the Al filament, also in Line 3, for the SD sample show the same magnitude on both sides (when comparing the same stress components). It means that the hoop stresses in the Al wire can be assumed to be constant throughout this region. Therefore, Equation (8) can be used to recalculate the d

_{0}also for this sample and this region. In fact, the difference in the lattice parameter between the two determinations is ~1.5 × 10

^{−6}nm, which proves the reliability of the calculated stress-free reference.

_{D}are positive.

_{D}are close to zero for the AD sample is a consequence of both the vortex-like flow of the material and the presence of the Al filaments. When no reversal is applied between the passes, the Cu matrix flows by compressing radially (additionally to the axial flow) the Al wires. Nevertheless, a hoop component exists, which causes the stresses within the Al to be not purely hydrostatic. The significant effect of the hoop component is clearly proved by the elliptical shape of the Al wires placed near the surface (see Figure 1a,b) [39]. When the reversal is applied after the first pass, the material experiences the vortex-like flow again, although with one main difference: the hoop flow is in the other direction, with the Cu matrix recovering its previous position around the Al wires, and the hoop and, mainly, the radial stresses becoming more hydrostatic, as it would be expected if external hydrostatic forces were exerted. This effect can explain why the stresses become roughly hydrostatic around the wires in Line 1 (except in L1.3 around the outer filament), L2.1, L3.1, and L.3.2.

_{y}of the Cu matrix (130 MPa). The maximum VM stresses observed are for the SD sample in L1.4 with ≈130 MPa and in L2.3 for the AD sample with ≈120 MPa, i.e., equal to or lower than the 130 MPa yield stress. It helps to validate the reliability of the calculations and suggests that in these regions (L1.4 for the SD sample and L2.3 for the AD sample), the shear stresses can be considered negligible.

- The reversal of the bar direction contributes to lowering the deviatoric stress components in Line 1 (except in L1.3 near the outer Al wire). It seems to be advantageous in order to delay material plastification and failure;
- Within the region with a high density of Al filaments, the reversal between the passes does not change significantly the overall state for both samples;
- The line delimiting one side of the high-density Al wire region (L2.3) for the AD sample is prone to failure as well as the L1.4 for the SD;
- The filament-free region exhibits a roughly symmetric behavior but with lower VM stresses for the AD sample.

## 5. Conclusions

- Hydrostatic stresses for the AD sample tend to be tensile in regions within the Al wires or surrounding them. Only far from the Al wires do hydrostatic stresses become compressive. The SD specimen exhibits more regions with compressive hydrostatic stresses;
- Axial deviatoric stresses are zero or compressive in most of the regions for both samples;
- The reversal of the bar direction provokes a lowering of the deviatoric stresses in the regions far from the Al wires and around the low Al-wire density regions where the stresses tend to be hydrostatic;
- The reversal of the bar direction slightly changes the overall state within the region of the high density of Al wires. The stresses calculated with the von Mises relation are at a maximum when close to the outer Al wire for both samples. In regions without Al wires, the reversal of the bar seems to be advantageous to avoid a possible plastification;
- Von Mises stresses calculated with the FEM simulations are higher than those measured with neutron diffraction; among other reasons, the presence of shear stresses may be possible;
- It was found that the reversal of the bar direction seems to be advantageous for the component properties. The reversal lowers the deviatoric stress components in a significantly larger part of the volume examined, although not in all the scanned segments;
- FEM shows that there can be a large variability of residual stress along the circumference. The residual stresses calculated using the von Mises relation from the neutron diffraction data can be approaching the yield stress of the Cu matrix near the outer surface of the component. It occurs regardless of the applied deformation mode (reversal, no reversal), although this effect seems to be more pronounced for the no-reversal component. Further optimization should aim at setting the processing parameters to still lower the stresses near the surface.
- The full-width-at-half-maximum of diffraction peaks is largest in the radial direction near the central Al wire in the region with a higher density of filaments and may initially be attributed to a predominance of microstresses in the radial direction;
- A novel evaluation procedure was successfully used for pseudo-strain treatment.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Photo of the two samples used in the study: (

**a**) sample with reversal of the bar direction between the passes, denoted AD (the alternating direction) and (

**b**) sample without reversal of the bar direction between the passes, denoted SD (the same direction). The red lines indicate the sample symmetry axes on the cross-section. The initial microstructures are shown in (

**c**) for the Cu matrix and in (

**d**) for the Al filaments. The annealing twins are depicted in red, HAGB in black, and LAGB in green.

**Figure 3.**Sketch of the samples indicating the measured lines and three orthogonal strain directions: axial (1), radial (2), and hoop (3). The sketched lines were not measured at the surface but inside the sample, approximately in the middle of the length of the sample bar.

**Figure 4.**(

**a**) Detailed sketch of the sample cross section indicating the measured lines and the gauge volume used for sample scanning of hoop and radial strains. (

**b**) Zones defined between the Al wires are indicated with two numbers where the first corresponds to the scanned line and the second to the interval, both preceded by L(line).

**Figure 5.**Measurement geometry (AD sample) for Line 1, Line 2, and Line 3 for hoop, axial, and radial strains. The scan direction is shown with a red arrow and the neutron beam with black arrows. The colored spots show the distribution of the simulated scattering points and the associated pseudo-strain (see the color scale).

**Figure 6.**The measurement configuration is shown on the left (

**a**). Example of the simulated pseudo-intensity (

**b**) and pseudo-strain (

**c**), compared with the experimental data for the AD sample, scan along line 1 in radial geometry.

**Figure 7.**As in Figure 6, but accounting for the fitted distribution of the intrinsic scattering intensity. The measurement configuration is shown in (

**a**), and the fitted intrinsic intensity distribution is shown in (

**b**). The variation of simulated strain (

**c**, blue line) is due to the pseudo-strain effects only; the intrinsic strain was set to zero.

**Figure 8.**Evaluated intrinsic strain distribution by fitting the un-smeared strain distribution in the radial direction for the AD (Sample 1) and SD (Sample 2) specimens. The grey part corresponds to positions where the gauge center passes through an Al wire. Horizontal error bars indicate the size of the sampling volume along the scan direction, the vertical error bars correspond to the standard errors of the least squares fitting procedure.

**Figure 9.**Stress differences for the AD sample (

**a**,

**c**,

**e**) and SD sample (

**b**,

**d**,

**f**) in Line 1 (

**top**), Line 2 (

**center**), and Line 3 (

**bottom**).

**Figure 10.**Two-dimensional plots of the FWHM [°] for axial, radial, and hoop directions for the AD (

**a**–

**c**) and SD (

**d**–

**f**) directions. The FWHM scale is indicated on the right for each plot.

**Figure 11.**Contours of residual stresses in characteristic directions (

**a**–

**c**), and first principal stress (

**d**).

**Figure 12.**Hydrostatic and deviatoric stresses for the AD sample (

**a**,

**c**,

**e**) and SD sample (

**b**,

**d**,

**f**) in Line 1 (

**top**), Line 2 (

**center**), and Line 3 (

**bottom**).

**Figure 13.**Stresses calculated with the VM relation for the AD and SD samples in the scanned lines 1 (

**a**), 2 (

**b**), and 3 (

**c**). The VM stress distribution obtained from the FE analysis after the first pass is shown in (

**d**).

**Figure 14.**Stress states developed in the (

**a**) AD and (

**b**) SD samples. The biaxial stress states are also indicated.

Property | Unit | Cu | Al |
---|---|---|---|

Young modulus | (GPa) | 111 | 72 |

Poisson ratio | - | 0.3 | 0.3 |

Density | (g.cm^{−3}) | 8.96 | 2.80 |

Specific heat | (J.kg^{−1}.K^{−1}) | 398 | 1230 |

Emissivity | - | 0.7 | 0.03 |

Thermal expansion | (K^{−1}) | 1.7 × 10^{−5} | 2.4 × 10^{−5} |

Thermal conductivity | (W/(m.K)) | 394 | 250 |

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**MDPI and ACS Style**

Canelo-Yubero, D.; Kocich, R.; Šaroun, J.; Strunz, P.
Residual Stress Distribution in a Copper-Aluminum Multifilament Composite Fabricated by Rotary Swaging. *Materials* **2023**, *16*, 2102.
https://doi.org/10.3390/ma16052102

**AMA Style**

Canelo-Yubero D, Kocich R, Šaroun J, Strunz P.
Residual Stress Distribution in a Copper-Aluminum Multifilament Composite Fabricated by Rotary Swaging. *Materials*. 2023; 16(5):2102.
https://doi.org/10.3390/ma16052102

**Chicago/Turabian Style**

Canelo-Yubero, David, Radim Kocich, Jan Šaroun, and Pavel Strunz.
2023. "Residual Stress Distribution in a Copper-Aluminum Multifilament Composite Fabricated by Rotary Swaging" *Materials* 16, no. 5: 2102.
https://doi.org/10.3390/ma16052102