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Using a single crystal of a Cu-6 mass% Ag alloy aged at 723 K for various times, nano-sized Ag precipitates in a Cu matrix were observed by conventional and ultra high-voltage transmission electron microscopes. The nano-sized Ag precipitates with a radius of about 5 nm had a nearly spherical shape. The shape change from the nearly spherical shape to the {111} octahedral shape occurred with increasing the size of the Ag precipitates to 11 nm. Elastic states of the Ag precipitates were evaluated by observing moiré fringes formed between the Ag precipitates and the Cu matrix. The origin of the shape change of the nano-sized Ag precipitates was discussed by considering the sum of elastic strain energy and interfacial energy. [ABSTRACT FROM AUTHOR]

Small crystalline particles or precipitates are often formed comprising near polyhedral shapes with round edges. Using the anisotropy of the surface energy given by a simple broken-bond model for fcc crystals, a geometrical analysis is performed to consider the particle-shape dependence of surface energy. Polyhedral and nearly polyhedral particles composed of {100} and {111} planes are treated as examples. The effect of round edges on the variation of surface energy of the nearly polyhedral particles is discussed. [ABSTRACT FROM AUTHOR]

Abstract: The dependence of precipitate shape on the elastic strain energy caused by anisotropy of the elastic moduli is discussed for polyhedral precipitates with purely dilatational misfit strains. By considering cubic structures for the precipitates and matrix phase, the elastic strain energy is calculated for the precipitate shapes of a {100} cube, a {110} rhombic-dodecahedron, a {111} octahedron, and the truncated shapes of all three polyhedra. The anisotropy of the elastic moduli of Cu is used for the numerical calculations. The polyhedral shapes of the {111} octahedron and {100} cube result in the highest and lowest values of the elastic strain energy per unit volume of precipitate. The calculated values of the elastic strain energy obtained for the truncated polyhedra are compared with values estimated using a geometrical method that considers the area changes of the {100}, {110} and {111} interfaces. [Copyright &y& Elsevier]

Inclusion problems for elongated toroidal inclusions are solved to discuss the strain fields caused by eigenstrained doughnut-like and tubular inclusions. For an infinitely long tubular inclusion, uniform eigenstrains do not generate elastic strains in the matrix region inside the tube. Effects of tube length on the elastic strains are shown for the purely dilatational eigenstrain #x03B5;*. For the tubular inclusion with the length 2b, the elastic strain at the center of the matrix region inside the tube becomes the maximum as much as ≈(a/b)#x03B5;* when the tubular inclusion has the inner diameter ≈2(b-a) and the outer diameter ≈2(b+a). [Copyright &y& Elsevier]

Abstract: The two-dimensional inclusion problem of elasticity has been extended to treat general non-elliptic shapes described as , p≥2. By using the Green function, the average Eshelby tensors are evaluated and elastic strain energy of the superelliptic inclusion is calculated for a material with cubic elastic anisotropy. In addition, by introducing the isotropic interface energy, equilibrium inclusion shapes to minimize the sum of elastic strain energy and interface energy are discussed as a function of the size and shape of the two-dimensional inclusion. It is found that an intermediate shape between a circle and a square or between an ellipse and a rectangle can be a minimum-energy shape under certain sets of given conditions. [Copyright &y& Elsevier]

When elastic moduli are anisotropic, the elastic strain energy of materials containing precipitates with purely dilatational misfit strains depends on the precipitate shape. This dependence is discussed for the octahedral to spherical shape transition. Precipitates modelled as superspherical shapes described by (x12/a2)p/2+(x22/a2)p/2+(x32/a2)p/2⩽1 with various values of p (1⩽p⩽2) and embedded in matrices with cubic structure are treated. The micromechanics inclusion problem is solved with the assumption of the same elastic moduli for the precipitates and matrices to calculate the elastic strain energy. When the anisotropy ratio of the elastic moduli is larger than unity, octahedral precipitates cause larger elastic strain energy than spherical precipitates. However, the precipitate-shape dependence of the elastic strain energy is not monotonic and the elastic strain energy peaks for an intermediate shape between octahedral and spherical. The stability of misfit precipitates of octahedral shape is discussed. [Copyright &y& Elsevier]

Energy consideration is made to discuss the spherical to cubical shape transitions of coherent precipitates in cubic materials. A superspherical shape approximation is adopted to describe intermediate shapes of the precipitates between a sphere and a cube. The elastic strain energy due to the misfit precipitates is evaluated by solving three-dimensional inclusion problems of cubic materials for superspherical inclusions with uniform eigenstrains. One of the advantages of the present analysis is that the shape transitions of the precipitates from spherical to cubical can be discussed with simple analysis using a parameter describing the cubic character of precipitates. The present theoretical results well explain the previous experimental results on the shape transitions of precipitates for Ni–Ni3Al and Cu–Co–Cr alloys. [Copyright &y& Elsevier]

The shape transitions of coherent γ′ precipitates from spherical to cuboidal observed in Ni-base alloys with γ matrix are discussed with simplified energy consideration. Possible shapes of the γ′ precipitates are approximated with superspheres that are described by x12/a2p/2+x22/a2p/2+x32/a2p/2⩽1 with appropriate values of p p⩾2, where the xi axes are parallel with 100 of the γ′ precipitates. The supersphere becomes a sphere when p=2 and a cuboid when p→∞. The elastic strain energy and interface energy due to the misfit γ′ precipitates are evaluated approximately. The equilibrium shape of the γ′ precipitates is obtained as the shape that causes the minimum of sum total of the elastic strain energy and interface energy. It is shown that the present analysis reasonably explains the previous experimental results on the shape change of the γ′ precipitates in the Ni–Ni3Al alloy. [ABSTRACT FROM AUTHOR]

In Ni-base alloys, the shape of γ′ precipitates with purely dilatational misfit strains often becomes an intermediate shape between the sphere and cuboid. The present study provides a new two-dimensional model to understand the elastic strain energy caused by the intermediate precipitate shape. A two-dimensional inclusion problem is solved for an inclusion shape described by (x12/a2)p/2+(x22/a2)p/2⩽1 (p⩾2). This shape that may be called a supercircle becomes a circle and a square when p=2 and p→∞, respectively. Intermediate shapes between the circle and square can be expressed by choosing appropriate values of p. Using the two-dimensional Green function of elasticity, the elastic states are examined for an infinitely extended material containing a uniformly eigenstrained supercircular inclusion. Anisotropic elastic moduli of cubic crystals are considered as those of the material containing the supercircular inclusion. The averaged Eshelby tensors of the supercircular inclusions are calculated as a function of p and the anisotropic elastic moduli. Using the averaged Eshelby tensors calculated, the elastic strain energy due to the supercircular inclusion with purely dilatational eigenstrains and its variation with the shape change from circular to square are discussed. [Copyright &y& Elsevier]

A copper alloy containing amorphous B2O3 dispersed particles is made by internal oxidation of a dilute Cu-B solid-solution alloy. Although the B2O3 particles behave as plastically non-deformable particles at room temperature, these become liquid particles in the solid Cu matrix at high temperatures. Cu single crystals containing the B2O3 particles are tensile tested at high temperatures. The shape change of the liquid particles due to the plastic deformation of the Cu matrix is observed with electron imcroscopy. The occurrence of relaxation by the diffusion of the Cu atoms along the Cu/B2O3 interface affects the shape change. The kinetics of the diffusional relaxation is discussed on the basis of the results of observations of the particle shape. [ABSTRACT FROM AUTHOR]

Steady-state sliding of grain boundaries is discussed by taking into account the change in the potential energy of an applied stress, when the sliding occurs by diffusional accommodation. The present analysis enables us to derive the sliding rates at boundaries of various shapes in a simple manner without knowing local stress fields. Sliding rate of a boundary containing second phase particles is also derived by similar calculations. The results are compared with previous calculations by other investigators, and it is found that the resultant expressions for the sliding rates are essentially identical to those derived from more complex analyses. [ABSTRACT FROM AUTHOR]

Ultra-fine-grained (UFG) Cu shows little total elongation in tensile tests because simple shear deformation is concentrated in narrow regions during the initial stage of plastic deformation. Here, we attempted to improve the total elongation of UFG Cu obtained by equal-channel angular pressing. By making shallow dents on the side surfaces of the plate-like specimens, this induced pure shear deformation and increased their total elongation. During the tensile tests, we observed the overall and local deformation of the dented and undented UFG Cu specimens. Using three-dimensional digital image correlation, we found that the dented specimens showed suppression of thickness reduction and delay in fracture by enhancement of pure shear deformation. However, the dented and undented specimens had the same ultimate tensile strength. These results provide us a new concept to increase total elongation of UFG materials. [ABSTRACT FROM AUTHOR]

To further understand the internal residual stress that is microscopically generated via martensitic transformation in steels, the origin of internal strain attributed to Bain correspondence between face-centered cubic (fcc) and body-centered cubic (bcc) was evaluated from macro- and micro-viewpoints and its effect on hardness was investigated in an interstitial free Fe-16%Ni martensite. Neutron diffractometry and electron backscatter diffraction analysis showed that body-centered cubic (bcc) crystal structure of as-quenched martensite contained elastic distortions leading to small tetragonality, even in martensitic steels without solute carbon, and that the extended [001] bcc of martensite tended to be parallel to <001> fcc of prior fcc austenite. In addition, the combination of micro-scale focused ion beam (FIB) and high-precision digital image correlation techniques revealed that a micropillar fabricated by FIB processing within a martensite block was anisotropically deformed by the release of the residual strain distributed in as-quenched martensite in correspondence with the tetragonal distortions of the bcc crystal structure. These results prove that a small part of the Bain strain remained as an internal elastic residual strain and was microscopically distributed among Bain groups in lath martensite after martensitic transformation. Furthermore, the residual strain generated a hydrostatic internal stress, and therefore, the nanohardness decreased considerably by the micropillar fabrication accompanied by the release of the internal stress. This means that the internal residual stress in martensite among Bain groups influences the mechanical properties of martensitic steel. Image, graphical abstract [ABSTRACT FROM AUTHOR]

Slip transmission between neighbouring grains is needed for the occurrence of plastic deformation in polycrystals. A geometric factor defined for a pair of grains with a certain orientation relationship is used to determine the slip transmission. Two slip transmission factors are calculated for randomly oriented grains. One is the slip transmission factor averaged for randomly oriented grains and the other is the minimum of the transmission factors among all possible orientation relationships. The possible number of slip systems in a grain varies for different metals and alloys. The effects of the number of slip systems on the transmission factors are discussed. [ABSTRACT FROM AUTHOR]

Using scanning electron microscopy, we investigated how the microstructure of a Cu single crystal with a {15 12 9}〈9 10 3〉 orientation evolved from cold rolling. The first 50 % rolling caused its crystal orientation to rotate to {211}〈111〉. Although orientation splitting occurred near the surface of the single crystal, band-like regions with near-{211}〈111〉 orientations were still present after the fourth 50 % rolling. We measured the spread of crystal orientation in the near-{211}〈111〉 regions as a function of the total plastic equivalent strain induced by all rolling steps. When 50 % rolling was performed less than twice, the spread of crystal orientation was proportional to the square root of the plastic equivalent strain. Based on our results, we discussed the relationship between the spread of crystal orientation and the plastic equivalent strain generated by rolling. [ABSTRACT FROM AUTHOR]

Orientation relationships of grains in tricrystals are discussed. Disorientation angles among grains are evaluated and a constrained condition of these around a triple junction is shown. Probability densities of the disorientation angles are obtained for randomly oriented grains by numerical calculations. A disorientation-angle diagram for tricrystal is proposed to characterize the orientation relationships of grains around the triple junctions. [ABSTRACT FROM AUTHOR]

Fully reversed tension–compression fatigue tests were performed on pure aluminum single crystals with the [1 1 2], [1 1 4] and [1 1 2 0] double-slip orientations under constant plastic strain amplitudes in the range of 5 × 10-4 and 1 × 10-2. No stress saturation was obtained at any strain level in the cyclic hardening curves (CHCs). The CHCs were characterized as: initial hardening, intermediate softening and secondary hardening. The degree of the softening was found to be strongly dependent on the stress-axis: the softening stages were significant for the [1 1 2 0] and [1 1 4] specimens, and conversely, indefinable at any plastic strain amplitude for the [1 1 2] specimens. The changes in surface topography were also found to be dependent on the crystal orientation. Characteristic macrobands in the order of several hundred micrometers were formed in the [1 1 2 0] specimens. The appearance of cyclic softening during fatigue deformation in aluminum single crystals is explained in terms of both the activity of cross-slip and the strain localization in the macrobands. [Copyright &y& Elsevier]

Low-cycle fatigue behavior of [1 1 3] double-slip oriented Cu–Fe alloy single crystals with and without precipitated f.c.c. γ-Fe particles were examined under plastic-strain-controlled fatigue tests and compared with that of single-slip oriented crystals such as [4 1 9]. Although the γ-Fe particles are coherent and shearable, no cyclic softening was observed in all the particle bearing specimens. The cyclic stress–strain curves of the [1 1 3] crystals show a plateau region similar to those in the [4 1 9] crystals. On the other hand, shear stress amplitude increased monotonically with increase in shear strain amplitude on solution treated specimens. Dislocation microstructures characteristic of cyclic deformation were more clearly developed in the [1 1 3] crystals. Magnetic measurement of fatigued specimens revealed that more particles had transformed martensitically into b.c.c. α-Fe in the [1 1 3] crystals than in the [4 1 9] crystals. The above results can be understood reasonably if orientation-dependent deformation and transformation characteristics are taken into account. [Copyright &y& Elsevier]

The dislocation density of pure copper fabricated by two severe plastic deformation (SPD) processes, i.e., accumulative roll bonding and equal-channel angular pressing, was evaluated using scanning transmission electron microscopy/transmission electron microscopy observations. The dislocation density drastically increased from ~ 10 13 m − 2 to about 5 × 10 14 m − 2 , and then saturated, for both SPD processes. [ABSTRACT FROM AUTHOR]

Abstract: Pure Cu, Cu–0.41at.% Si and Cu–1.64at.% Si alloys were severely deformed by an accumulative roll bonding (ARB) method and the resultant microstructure was observed using high-voltage electron microscopy and scanning transmission electron microscopy. Although the mean boundary separation along the ARB normal direction was dependent on the Si concentration, the dislocation density within grains and cells depended very weakly on the Si concentration and stayed around 5×1014 m−2. Uniaxial tensile tests were also performed at room temperature. The contribution of dislocations and grain boundaries to the strength was estimated using two models: one based on the Taylor-type and Hall–Petch-type strengthening mechanisms, the other based on thermally activated dislocation depinning from grain boundaries. Both model provide reasonable values for yield strength compared with 0.2% proof stress values, and therefore the grain size region used in this study may be the transition region between these two models. [Copyright &y& Elsevier]

Abstract: A new severe plastic deformation (SPD) process to obtain nanocrystalline (NC) structure by wear is reported. An Al–Al3Ti functionally graded material fabricated by a centrifugal method is wear tested to investigate the microstructure of the wear-induced layer. When the sliding distance of the wear exceeds l =100m, the wear-induced layer is obtained just below the worn surface. The microstructure of the wear-induced layer consists of fine Al3Ti fibrous particles and an NC solid-solution matrix containing a partly amorphized phase. The NC matrix has fine grains with a mean diameter of 16nm and is a supersaturated solid-solution in which Ti is dissolved in the Al matrix. From the microstructure, it is estimated that the wear-induced layer is formed at a nominal shear strain of more than 90 and an effective shear strain of more than 52. An NC structured wear-induced layer is generated by the SPD process. [Copyright &y& Elsevier]

Abstract: We propose a method of mechanical test for measuring strength of electrodeposited Ni film using micro-sized cantilever. Two cantilevers were fabricated for each beam direction parallel and vertical to grain growth direction. Bending tests were carried out with a test machine designed for micro-sized specimens. Bending tests of the two cantilever specimens showed that strength of the cantilever which beam direction is parallel to grain growth direction is higher than that of vertical to grain growth direction. EBSD analysis of the specimens after the bending test implies that difference in strength was caused by deformation twin. [Copyright &y& Elsevier]

The Boerdijk-Coxeter helix (BC helix or tetrahelix) is a linear stacking of regular tetrahedra. Although the BC helix exhibits an aperiodic nature, structures resembling the BC helix with periodicity are found in materials. To understand such structures, we considered a modification of the BC helix to introduce periodicity. By adjusting the relative rotation of adjacent tetrahedra, we demonstrated that periodic arrangements consisting of 8, 11, and 14 tetrahedra have appearances similar to that of the BC helix., (© 2024. The Author(s).)

The logarithm ln⁡ R of rotation matrix R is a skew symmetric tensor consisting of three independent elements of real numbers. In addition to the Euler angles and the axis/angle pair, the elements of ln⁡ R called the log angles are also the set of three parameters of R . In this paper, we will show that the concept of the log angles is also useful to discuss changes in crystal orientations. The changes in R as a function of the position are given by the changes in the log angles. As an example, orientation changes caused by arrays of dislocations in a plastically deformed Cu single crystal are discussed., Competing Interests: The authors declare that there is no conflict of interests regarding the publication of this paper.

Crystalline nanoparticles or nanoprecipitates with a cubic structure often have near polyhedral shapes composed of low-index planes with {100}, {111} and {110}. To consider such near polyhedral shapes, algebraic formulas of extended superspheres that can express intermediate shapes between spheres and various polyhedra have been presented. Four extended superspheres, (i) {100} regular-hexahedral; (ii) {111} regular-octahedral (iii) {110} rhombic-dodecahedral and (iv) {100}-{111}-{110} rhombicuboctahedral superspheres are treated in this study. A measure ∏ to indicate the degree of polyhedrality is presented to discuss shape transitions of the extended superspheres. As an application of ∏ superspherical coherent precipitate is shown.