**1. Introduction**

The concept of LRIS in metals refers to the deviation of local stress from the applied stress that occurs over relatively long length scales such as that of the spacing of dislocation heterogeneities (e.g. labyrinth microstructure [1], persistent slip bands (PSB) walls, cell walls, subgrain boundaries, etc). LRIS may have been initially discussed in connection with the Bauschinger effect [2]. Understanding LRIS is essential for characterizing the Bauschinger effect and plastic deformation in cyclically deformed metals [3,4]. Metals strain harden during plastic deformation and upon reversal of the applied stress, yielding occurs at a much lower (absolute value) stress than if the material continued monotonic

deformation. This effect is referred to as the Bauschinger effect and is contrary to what is expected based on isotropic hardening. When a metal is cyclically deformed, the lost strength due to the Bauschinger effect occurs with each reversal of the applied stress. This results in low hardening rates and saturation stresses compared to the (monotonic) fracture stress [3]. Different theories have been proposed for rationalizing the Bauschinger effect [5–8]. The two prominent theories are Mughrabi's composite model [5], which proposed relatively high values of LRIS, and the Orowan-type mechanism [6], which involves no internal stresses.

The Mughrabi composite model presented the heterogenous dislocation microstructure as hard (high dislocation density walls) and soft (low dislocation density channels or cell interiors) sections. In the forward direction of deformation (tension), the stress is positive in both the walls and channels (hard and soft regions). However, in the reverse direction of deformation (compression), while the stress in the walls are still positive, these regions can place the cell interiors, PSB channels, etc. in compression. Thus, the Bauschinger effect, which is the occurrence of yielding at lower stresses, is observed in this simplified model [2].

The Orowan-type mechanism, which was discussed in more details by Brown in [9,10], suggests that mobile dislocations in the forward direction of straining encounter an increasing lineal density of obstacles (forest dislocations or dislocation walls). However, in the reverse direction, there is a lower lineal density of obstacles (channels or cells interiors). Therefore, plastic strains are accumulated at a much lower stresses on reversal [6,7].

Historic LRIS assessment studies on high dislocation density heterogeneities of cyclically deformed single crystals oriented for single slip suggest internal stresses in the so-called hard phase varying from a factor of 1.0 (no LRIS) to 3, or more (larger than the applied stress). These studies were based on measurements of dislocation dipole height, dislocation loop radii, asymmetry in X-ray peaks, and lattice parameter measurements through CBED analysis [5,11–19]. In an earlier recent study by the authors [1], the maximum dipole heights were discovered to be approximately independent of location, being almost identical in the walls and channels of the labyrinth dislocation microstructure in <001> Cu single crystals oriented for multiple slip. Since the maximum dipole height strength values may be indicative of the local stresses, nearly equal maximum dipole heights in the walls and channels support a uniform stress state and low LRIS. However, the maximum value for dipole heights suggest dipole strengths that were about a factor of 2.4 higher than the applied stress based on the usual athermal equation to separate the dislocations of a dipole. Extra stress at the dipoles may be provided by tripoles or small dislocation pile-ups [20]. A nearly homogenous stress distribution with only small internal stresses were suggested by the authors in an earlier study [1] to be present based on the maximum dipole separation stress values. This is consistent with the observation of uniform dipole height across the heterogeneous dislocation microstructure. Other studies reported similar behaviors, observing homogenous dipole heights and higher dipole separation stresses for cyclically deformed metal single crystals oriented in single slip (except aluminum, which has a similar dipole separation stress to the applied stress but homogenous dipole heights as usual) [11–13]. It should be noted that accounting on either the anisotropy of cubic elasticity or the finite elongation ratios of dipoles did not allow the authors to explain the aluminum specificity. As stated earlier, since the maximum dipole heights are the upper limit of stable dipoles under the imposed local stress, they can predict the local stresses in cyclically deformed materials. The local stresses may be more accurately measured by determining the lattice parameters using convergent beam electron diffraction. This method involves using a small convergent electron probe to generate a diffraction pattern containing the higher order Laue zone (HOLZ) lines that are very sensitive to small elastic distortions in the lattice.

There have been few studies on internal stress assessment using the CBED technique in creep deformed and fatigued polycrystals and single crystals oriented in single slip. Such studies on cyclically deformed single crystals oriented in multiple slip are missing in the literature. Straub et al. [21] and Maier et al. [22] examined internal stresses using CBED analysis in polycrystalline copper specimens experiencing either creep or cyclic deformation. They did not quantify the internal stresses but

suggested that internal stresses exist. It should be noted that both of these studies used kinematical simulations for deriving the position of the HOLZ lines, but dynamical effects may be important. Kassner et al. observed a homogenous stress distribution with no internal stresses in an unloaded monotonically (creep deformed) polycrystalline copper using CBED analysis [23]. In another CBED study by Kassner et al., an absence of internal stresses in creep deformed aluminum single crystal was reported [24]. In the most recent study by Kassner et al. on a cyclically deformed copper single crystal oriented for single slip at ambient temperature, lattice parameter measurements in the channels and close to the vein bundles showed no evidence of LRIS. The uncertainty of these measurements was ±30 MPa, which is 80% of their applied stress [25]. Furthermore, it was not determined whether relaxation occurred leading to a reduction in the LRIS. Legros et al. [26] assessed the internal stresses in a cyclically deformed silicon single crystal oriented for single slip at 1078 K using chi-squared analysis on CBED patterns. Chi-squared is the typical refinement method for producing the best match between the simulated and experimental CBED patterns [27]. Legros et al. suggested small internal stresses closer to the dislocation wall (7 MPa or about 14% of their applied stress) and negligible internal stresses within the cell interior exist in the cyclically deformed silicon single crystal [26]. This is basically consistent with the earlier work by the authors of this paper on the structures without PSBs. Again, all of the studies in the literature that use CBED for strain measurements were performed on unloaded material, and, of course, examined on the thin regions of the foil. Thus, LRIS relaxation is possible.

In this study, we evaluated the lattice parameters using CBED in the channels and close to the walls of the labyrinth microstructure of a cyclically deformed copper <001> single crystal oriented in multiple slip which complements the dipole study of our earlier work [1].
