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•Unstable mode for crack in hcp-Mg is studied with the atomic elastic stiffness.•The mode is specified with the principal strain axis for negative eigenvalue atoms.•Basal plane crack ...has symmetric mode and actually shows unstable propagation.•Unstable mode for dislocation emission is also captured in the prismatic crack.•Crack tip singularity deviates from the linear facture mechanics by unstable atoms.
As a series study discussing local deformation by the atomic elastic stiffness, Bijα=Δσiα/Δεj, various molecular dynamics simulations are performed on the basal, prismatic and pyramidal cracks in hcp Mg; and their different deformation behaviors are discussed by the 1st eigenvalue of BijαΔεj=ηαΔεi and their principal axis of the eigenvector {Δεi} at each atom point. The basal and pyramidal cracks show brittle cracking and the crack tip stress coincides with the magnitude of the σyy=KIC/2πr in the linear fracture mechanics, despite of the different conditions of single crack vs. periodic crack array. In the basal crack, the principal axes of ηα(1)<0 atoms clearly show symmetrical deformation mode against crack plane. The pyramidal crack propagates in the 30° tilted adjacent pyramidal plane, and ηα(1)<0 atoms emerge around the crack tip and the corresponding principal axes are often normal to the new crack surface. On the other hand, the prismatic crack shows dislocation emission and tip blunting. ηα(1)<0 atoms widely emerge in butterfly shape around crack tip before the stress–strain peak or the onset of dislocation emission. There is no remarkable structural change in these ηα(1)<0 region; however, the material property or local stiffness is clearly different from the surrounding media so that the stress distribution on the crack plane remarkably deviates from the continuum theory. The principal axes of these atoms are not in the crystal orientation but in 45° against loading axis; this difference possibly prevents the structural change. The trigger of dislocation emission is captured by the principal axis of ηα(1)<0 atoms, which is located on the tip surface and oriented in the slip direction.
In order to clarify the physical meaning of the eigenvector of the atomic elastic stiffness matrix, Bija=Δσia/Δεj, static calculations of uniaxial tension are performed on various fcc, bcc, and hcp ...metals with four different embedded atom method (EAM) potentials. Many fcc metals show instability for the constant volume mode, or the eigenvector of (Δεxx, Δεyy, Δεzz) = (±1, ∓1, 0), under the 001 tension. Bcc also loses resistance against other constant volume mode, (Δεxx, Δεyy, Δεzz) = (±1, ±1, ∓2), in the 001 tension. Hcp shows shear modes Δγyz and Δγzx under the 0001 tension, which correspond to atom migration by dislocation on the slip plane. Similar shear modes appear in the 111 tension of fcc and 110 tension of bcc. Hcp also changes the mode to constant volume and shear in the 1¯010 tension, which imply the deformation in the pyramidal and prismatic planes.
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•We discuss the onset of crack propagation by the atomic elastic stiffness in Si Tersoff potential.•We find the precursor instability in the 1st eigenvalue of the stiffness ...matrix.•The eigenvector indicates mode I, II and III cracking for (010), (110) and (111) cracks.•We also find the stress concentrated area has higher stiffness than that in perfect lattice.•The area may prevent dislocation emission and give brittle character to Si.
We performed molecular dynamics simulations on the 001(010), 001(110) and 1¯1¯2(111) mode I through cracks in Si, and discussed the unstable crack propagation based on the determinant and eigenvalue of the atomic elastic stiffness (AES), Bijα, under the finite temperature of 300K. Two significant findings are obtained; the one is the existence of extremely high detBijα atoms, or “highly stable” zone, at the crack tip although the tip surface actually has negative detBijα atoms. The other finding is the precursor instability of the 1st eigenvalue of Bijα{Δεj}=ηα{Δεj}. Although there is a little difference in the responses of the 001(010), 001(110) and 1¯1¯2(111) cracks, extremely negative eigenvalue of ηα(1) is observed before the unstable crack propagation. The corresponding eigenvector shows large positive Δε2 (crack opening or mode I) in the 001(010) crack, large negative Δε1 (intrusion in the x-direction or mode II) in the 001(110) crack and large Δε5 (shear in the zx or mode III) in the 1¯1¯2(111) crack. However, the fracture process in the later stage is rather complicated so that the final fracture mode is not correspond to these instability mode.
Basic characteristics of 6 × 6 matrix of atomic elastic stiffness (AES), Bija=Δσia/Δεj, or the deformation resistance at each atom point, are discussed first in static analyses of generalized ...stacking fault (GSF) energy surface for 8 fcc, 4 bcc and 4 hcp metals with Zhou’s EAM potential. For hcp metals, the stress–strain peak along the GSF path exactly coincides with the point where the AES loses the resistance showing negative 1st eigenvalue ηa(1), or the solution of BijaΔεj=ηaΔεi=Δσia; however, all fcc and 2 bcc (Mo and W) never have negative ηa(1) along the GSF path. Fe and Ta transiently show ηa(1) < 0 while they also have positive ηa(1) at the GSF energy peak. Then we performed MD simulations of edge and screw dislocation dipoles in a periodic slab cell of typical elements of fcc, bcc and hcp; and discussed the eigenvalue and the corresponding eigenvector {Δεxx, Δεyy, Δεzz, Δγyz, Δγzx, Δγxy} of the dislocation cores. As expected from the results of the GSF analyses, dislocation cores in fcc Ni have no ηa(1) < 0 atoms, even in their glide process under external shear loading. Bcc Fe and hcp Co definitely have ηa(1) < 0 atoms in the dislocation cores and their migration direction can be visualized by the maximum shear direction of the strain tensor of the corresponding eigenvector.
•A calculation scheme for ferroelastic phase transformation is newly developed.•The phase-field model incorporating the elastic strain energy is used for this computation.•Nucleations and nucleus ...growths of the rhombohedral phase in the cubic matrix are numerically reproduced.•This simulation predicts that the ferroelastic structure in from a single matrix and polycrystalline matrices.•This scheme well expresses the qualitative behavior of ferroelastic phase transformation.
In order to predict the formation of ferroelastic phases in crystal grains of La0.6Sr0.4Co0.2Fe0.8O3−δ (LSCF), which is a common material used for solid oxide fuel cells (SOFCs), we propose an analysis method based on a phase-field model. The phase-field model equipped with the elastic energy is introduced to realize the ferroelastic phase formation in a crystal grain. The finite element method (FEM) is employed to solve the phase transformation, and strain distributions are also calculated by FEM. On the basis of the developed analysis method, some numerical examples of analysis are performed to reproduce the deformation-induced nucleation and growth of ferroelastic phases of LSCF. Various microstructures are obtained from the simulations in accordance with the initial microstructure and their tendencies are discussed. These microstructures are reproduced by the proposed analysis method based on finite element method, which enables us to evaluate the deformation field in terms of changes of shape and stress.
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•Twinning in Fe is discussed with the positiveness of atomic elastic stiffness (AES)•Negative AES layers or unstable bands appear just before the stress drop•The first twin initiates ...at the point where the negative AES appears first•Multiple twins are formed at the unstable bands after the stress drop
As a series study that discusses local instability by atomic elastic stiffness, Bija=Δσia/Δεj, molecular dynamics simulations of simple shear on Fe perfect lattice are performed to discuss twin deformation. In the simulation of extremely low temperature of 0.1K, many twin boundaries are nucleated in a periodic slab cell of stacked (1¯1¯2) planes just after the stress drop or elastic limit under shear. The 1st eigenvalue ηa(1) of the 6×6 matrix Bija shows fluctuation in each (1¯1¯2) planes just before the elastic limit. At the stress-strain peak, ηa(1)<0 or unstable layers emerge and twin deformation occurs at these unstable “band” in the simulation cell. The period of the ηa(1) fluctuation is almost constant even if the cell size is expanded in the 1¯1¯2 and 11¯0 direction, while the unstable band arrays tilt normal to the diagonal line of the simulation cell, under the cell expansion in the 111 direction. At the elevated temperature of 300K, the simultaneous periodic band never appears but a few unstable bands appear locally in time sequence and bring twinning there.
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