### Constitutive model

#### Liang Wan, Akio Ishii, Jun-Ping Du, Wei-Zhong Han, Qingsong Mei, and Shigenobu Ogata, “Atomistic modeling study of a strain-free stress driven grain boundary migration mechanism”, Scripta Materialia, 134 (2017) 52-56.

A recent experiment (Scripta Mater., 65:990, 2011) shows that the Σ7 {132}/{132} grain boundary in Al can migrate under external stress but produces no strain. Here, based on a bi-crystallographic analysis, an atomic shuffling path was identified as the feasible mechanism for this grain boundary migration. By a density functional theory calculation, it reveals that the enthalpy barrier of this atomic shuffling path increases by external shear stress applied with shear of the grain boundary along the tilt axis〈111〉, which is in good agreement with experimentally measured shear-direction-dependence of activation enthalpy for this grain boundary migration.

#### Xu-Sheng Yang, Yun-Jiang Wang, Hui-Ru Zhai, Guo-Yong Wang, Yan-Jing Su, L. H. Dai, Shigenobu Ogata, and Tong-Yi Zhang, “Time-, stress-, and temperature-dependent deformation in nanostructured copper: Creep tests and simulations”, Journal of the Mechanics and Physics of Solids, 94 (2016) 191-206.

In the present work, we performed experiments, atomistic simulations, and high-resolution electron microscopy (HREM) to study the creep behaviors of the nanotwinned (nt) and nanograined (ng) copper at temperatures of 22 °C (RT), 40 °C, 50 °C, 60 °C, and 70 °C. The experimental data at various temperatures and different sustained stress levels provide sufficient information, which allows one to extract the deformation parameters reliably. The determined activation parameters and microscopic observations indicate transition of creep mechanisms with variation in stress level in the nt-Cu, i.e., from the Coble creep to the twin boundary (TB) migration and eventually to the perfect dislocation nucleation and activities. The experimental and simulation results imply that nanotwinning could be an effective approach to enhance the creep resistance of twin-free ng-Cu. The experimental creep results further verify the newly developed formula (Yang et al., 2016) that describes the time-, stress-, and temperature-dependent plastic deformation in polycrystalline copper.

#### Wei-Zhong Han, Ling Huang, Shigenobu Ogata, Hajime Kimizuka, Zhao-Chun Yang, Christopher Weinberger, Qing-Jie Li, Bo-Yu Liu, Xi-Xiang Zhang, Ju Li, Evan Ma, Zhi-Wei Shan, “From “smaller is stronger” to “size-independent strength plateau”: towards measuring the ideal strength of iron”, Advanced Materials, published online 17 APR 2015.

The trend from “smaller is stronger” to “size-independent strength plateau” is observed in the compression of spherical iron nanoparticles. When the diameter of iron nanospheres is less than a critical value, the maximum contact pressure saturates at 10.7 GPa, corresponding to a local shear stress of ≈9.4 GPa, which is comparable to the theoretical shear strength of iron.

#### Yun-Jinag Wang, Guo-Jie Jason Gao, and Shigenobu Ogata, "Atomistic understanding of diffusion kinetics in nanocrystals from molecular dynamics simulations", Physical Review B, 88-11 (2013) 115413-1-7.

Understanding the grain size effect on diffusion in nanocrystals has been hampered by the difficulty of measuring diffusion directly in experiments. Here large-scale atomistic modeling is applied to understand the diffusion kinetics in nanocrystals. Enhanced short-circuit diffusivity is revealed to be controlled by the rule of mixtures for grain-boundary diffusion and lattice diffusion, which can be accurately described by the Maxwell-Garnett equation instead of the commonly thought Hart equation, and the thermodynamics of pure grain-boundary self-diffusion is not remarkably affected by varying grain size. Experimentally comparable Arrhenius parameters with atomic detail validate our results. We also propose a free-volume diffusion mechanism considering negative activation entropy and small activation volume. These help provide a fundamental understanding of how the activation parameters depend on size and the structure-property relationship of nanostructured materials from a physical viewpoint.

#### Guo-Jie J.Gao, Yun-Jiang Wang, and Shigenobu Ogata, "Studying the elastic properties of nanocrystalline copper using a model of randomly packed uniform grains", Computational Materials Science, 79 (2013) 56-62.

We develop a new Voronoi protocol, which is a space tessellation method, to generate a fully dense (containing no voids) model of nanocrystalline copper with precise grain size control; we also perform uniaxial tensile tests using molecular dynamical (MD) simulations to measure the elastic moduli of the grain boundary and the grain interior components at 300 K. We find that the grain boundary deforms more locally compared with the grain core region under thermal vibrations and is elastically less stiff than the core component at finite temperature. The elastic modulus of the grain boundary is lower than 30% of that of the grain interior. Our results will aid in the development of more accurate continuum models of nanocrystalline metals.

#### Yun-jiang Wang, Akio Ishii, and Shigenobu Ogata, "Entropic effect on creep in nanocrystalline metals", Acta Materialia, 61-10 (2013) 3866-3871.

We report a significant entropic effect on creep of nanocrystalline metal using molecular dynamics. Our simulations reveal that the activation entropy may contribute a multiplicative factor of many orders of magnitude to the steady-state creep rate. The relationship between activation entropy and enthalpy obeys an empirical Meyer-Neldel compensation rule. The activation volume is found to decrease with increasing temperature for dislocation nucleation creep, which agrees well with experimental results. The study opens up an avenue for quantitatively discussing the entropic effects on various thermally activated deformations in nanocrystals.

#### Yun-Jiang Wang, Guo-Jie Jason Gao, and Shigenobu Ogata, "Size-dependent transition of deformation mechanism, and nonlinear elasticity in Ni3Al nanowires", Applied Physics Letters, 102 (2013) 041902-1-5 .

A size-dependent transition of deformation mechanism is revealed in Ni3Al nanowire under atomistic uniaxial tension. Deformation twinning is replaced by phase transformation when the diameter of Ni3Al nanowire reduces to a critical value near 4?nm. Enhanced size-dependent nonlinear elasticity is observed in the nanowires, in comparison to their bulk counterpart which is benchmarked by combined density functional and atomistic study. This study provide fundamental understanding on the size-dependent deformation mechanisms of nanostructured alloys.

#### Yun-jiang Wang, Akio Ishii, and Shigenobu Ogata, "Grain Size Dependence of Creep in Nanocrystalline Copper by Molecular Dynamics", Materials Transactions, 53-1 (2012) 156-160.

The grain size dependence of creep is critical to understand the plastic deformation mechanism of nanoscale metals. Here we used molecular dynamics to study the stress-induced grain size exponent transition in creep of nanocrystalline copper. The grain size exponent was found to initially increase with increasing stress, then decrease after some critical stress. The derived grain size exponents are in agreement with experimental results for diffusional and grain boundary sliding creep at low stress. While, the founded decreasing grain size exponent with increasing stress for dislocation nucleation creep in nanocrystal is in contrast with conventional materials. We propose a constitutive equation for dislocation nucleation governed creep in nanocrystal to explain its grain size dependence transition with stress.

#### Yun-Jiang Wang, Akio Ishii, and Shigenobu Ogata, "Transition of creep mechanism in nanocrystalline metals", Physical Review B, 84-22 (2011) 224102-1-7.

Understanding creep mechanisms with atomistic details is of great importance to achieve the mechanical and thermodynamical stabilities of nanocrystalline (NC) metals over a wide temperature range. Here we report a molecular dynamics analysis of creep in NC copper dominated by competing deformation mechanisms. We found the dominating creep mechanism transits from grain boundary (GB) diffusion to GB sliding, and then dislocation nucleation with increasing stress. The derived stress exponent, small activation volume of 0.1?10b3, and grain size exponent all agree quantitatively with experimental values. We proposed a stress-temperature deformation map in NC metals accommodated by the competition among different stress-driven, thermally activated processes. The model is general to answer the question why deformation mechanism transits with stress in NC metals.

#### Naohiro Toda, Hajime Kimizuka, and Shigenobu Ogata, "DFT-Based FEM Analysis of Nonlinear Effects on Indentation Process in Diamond Crystal", International Journal of Mechanical Sciences, 52-2 (2010) 303-308.

We apply a new framework of a finite-element method (FEM) analysis with constitutive relations based on density functional theory (DFT), as an efficient method to characterize the nonlinear and anisotropic elastic deformation of single-crystal diamond. In our scheme, the stress?strain relations are obtained during FEM analysis on the fly based on the plane-wave-based DFT total-energy calculations and their numerical database is simultaneously constructed, which enables us to obtain high-precision stress without any empirical parameters even under finite strained conditions. To check its validity and accuracy, the shear deformation behavior of diamond crystal is analyzed under the strained condition. Then we examine the nonlinear effects on the indentation deformation of diamond single crystal, by comparing the results from the DFT-based constitutive relations with those from the linear elastic ones.

#### Naohiro Toda, Hajime Kimizuka, and Shigenobu Ogata, "Density-Functional-Theory-Based Finite-Element Analysis of Diamond Single Crystal", Journal of Solid Mechanics and Materials Engineering, 3-3 (2009) 541-551.

We apply a finite-element analysis method based on first-principles density functional theory, to evaluate the nonlinear large elastic deformation of single-crystal diamond. The stress-strain relations are obtained during finite-element analysis on the fly based on the first-principles calculations and their numerical database is simultaneously constructed, which enables us to obtain high-precision stress without any empirical parameters even under finite strained conditions. The shear strength and mechanical behavior of diamond crystal are analyzed under various stress conditions, and then the uniaxial deformation of a diamond-crystal pillar model is examined through the present analysis method.

#### Masato Wakeda, Yoji Shibutani, Shigenobu Ogata, and Junyoung Park, "Multiple Shear Banding in a Computational Amorphous Alloy Model", Applied Physics A, 91-2 (2008) 281-285.

The strain localized phenomenon, so called shear bands (SBs), in an amorphous alloy have received a lot of attention in recent years. In this study, we microscopically investigated the nature and dynamics of multiple SBs using molecular dynamics model. In the SB region, intense shear-induced structural change occurred, typified by the annihilation of pentagonal short-range order, and significant localized heating accompanied with the SB propagation was observed. Moreover, a large number of fine SBs operated simultaneously at a high strain rate, whereas, only a few SBs appeared and propagated abruptly at a low strain rate. These results were discussed with respect to brittle/ductile deformation of bulk metallic glasses.

#### Hajime Kimizuka, Shigenobu Ogata, and Ju Li, "Hydrostatic Compression and High-Pressure Elastic Constants of Coesite, SiO2", Jounal of Applied Physics, 103-5 (2008) 053506-1-4.

Using density-functional theory, we computed all the independent elastic constants of coesite, a high-pressure polymorph of silica, as functions of pressure up to 15 GPa. The results are in good agreement with experimental measurements under ambient conditions. Also, the predicted pressure-dependent elastic properties are consistent with x-ray data in the literature concerning lattice strains at high pressures. We find that coesite, like quartz, exhibits a gradual softening of a shear modulus B44 with increasing pressure, in contrast to the rising bulk modulus.

#### Naohiro Toda, Hajime Kimizuka, and Shigenobu Ogata, "Database-accelerated parallelized local quasicontinuum method based on first-principles density-functional theory for crystals possessing internal degrees of freedom", Transactions of the Japan Society for Computational Methods in Engineering, 7-2 (2008) 273-278. [in Japanese].

We proposed a new finite-element analysis method based on a numerical stress-strain database to calculate the non-linear large elastic deformation for single crystals possessing internal degrees of freedom. The database is constructed during finite-element analysis on-the-fly based on the first-principles density functional theory, which is able to obtain high-precision stress without any empirical parameters even under finite strained conditions. The database significantly improves the total computational efficiency without loss of accuracy. We also carried out parallel-computational method for the database construction process to realize further improvement of computational efficiency. The effectiveness of the proposed inverse analysis method is demonstrated through numerical simulation for several example problems

#### Masato Wakeda, Shibutani Yoji, and Shigenobu Ogata, "Atomistic Formation Mechanism of Multiple Shear Bands in Amorphous Metals", Journal of the Society of Materials Science, Japan, 57-2 (2008) 119-125. [in Japanese].

At room temperature, plastic deformation of amorphous metals is localized in narrow bands, so called shear bands (SBs), which is a key issue of their brittle/ductile deformations. The extremely disordered structure and the abrupt fracture obtained in the experiments hinder us from making clear the nature and dynamics of SBs. In this study, atomistic modeling to produce the multiple shear bands (MSBs), not single shear band, was proposed, and the evolution process of MSBs including their coalescence and stationary was investigated using molecular dynamics simulations. We prepared a plate of Cu-Zr binary amorphous alloy model by the melt-quench process, and then performed uniaxial tensile and compressive tests under plane stress condition. During the loading, the deformation was suddenly localized in narrow bands shortly after the onset of yielding. The propagation of SBs was accompanied with drastic stress drop and significant local heating caused by the friction of atoms. Also, critical stresses of SB nucleation considerably differed between under tension and compression. This result indicates that Tresca or von Mises criterion, commonly used as a yield condition in the crystalline metals, is not appropriate to describe the yielding of the amorphous metals. The SB angles to the loading axis are observed to be 45?57° under tension, while 40?46° under compression. These angles agree well with the fracture angles observed in the experiments with multi-component metallic glasses. It is concluded that the critical stress state of SB nucleation is dependent on not only the shear stress but also the stress normal to the SB, and it can be described by Mohr-Coulomb criterion.

#### Futoshi Shimizu, Shigenobu Ogata, and Ju Li, "Theory of Yield Point in Metallic Glass and Molecular Dynamics Calculations", Materials Transactions, 48-11 (2007) 2923-2927.

The aged-rejuvenation-glue-liquid (ARGL) shear band model has been proposed for metallic glasses (Acta Mater. 54 (2006) 4293), based on small-scale molecular dynamics simulations up to 20,000 atoms and thermomechanical analysis. The model predicts the existence of a critical lengthscale ?10 nm, above which melting could occur in shear-alienated glass. Large-scale molecular dynamics simulations with up to 5 million atoms have directly verified this prediction. When the applied stress exceeds the glue traction (computed separately before in a shear cohesive zone, or an amorphous-amorphous ``generalized stacking fault energy'' calculation), we indeed observe maturation of the shear band embryo into bona fide shear crack, accompanied by melting. In contrast, when the applied stress is below the glue traction, the shear band embryo does not propagate, becomes diffuse, and eventually dies. Thus this all-important quantity, the glue traction which is a property of shear-alienated glass, controls the macroscopic yield point of well-aged glass. We further suggest that the disruption of chemical short-range order (``chemical softening'') governs the glue traction microscopically. Catastrophic thermal softening occurs only after chemical alienation and softening in our simulation, after the shear band embryo has already run a critical length.

#### Hideki Mori, Shigenobu Ogata, Ju Li, Seiji Akita, and Yoshikazu Nakayama, "Plastic Bending and Shape Memory Effect of Double-Wall Carbon Nanotubes", Physical Review B, 76-16 (2007) 165405.

Plastic bending of (5,5)@(10,10) double-wall carbon nanotube is analyzed using nudged elastic band minimum energy path calculations. At lower applied bending curvature, only the outer tube deforms plastically. However, at higher bending curvature, both the inner and outer tubes deform plastically. We find that the plastic deformation of the outer tube is more difficult than that of isolated single-wall carbon nanotube of the same diameter due to tube-tube interactions. In contrast, the plastic deformation of the inner tube is not strongly affected by the presence of the outer tube. We also analyze the shape-memory effect (SME) discovered experimentally, which is a thermal recovery process from the plastically bent state to the straight defect-free state, which can be repeated multiple times. We analyze the physics behind SME of carbon nanotubes, which is quite different from that of traditional shape-memory alloys.

#### Junyoung Park, Yoji Shibutani, Masato Wakeda, and Shigenobu Ogata, "Influence of Size and Number of Nanocrystals on Shear Band Formation in Amorphous Alloys", Materials Transactions, 48-5 (2007) 1001-1006.

In this study, binary (copper and zirconium) amorphous metals with embedded nanosized crystal structures are subjected to uniaxial tension using molecular dynamics simulations to reveal the mechanism of shear band structure formation. The number and the size of the nanocrystals are chosen as the study parameters. The number of nanocrystals affects the stress-strain curve and shear band formation while the size of the nanocrystals does not significantly affect the results. As reported in the experimental work published so far, under tension coalescent voids are found in the shear bands or at the interface between crystalline and amorphous materials. The simulation results show that the number of shear bands under compressive loading is much larger than that under tensile loading. We also found that, even under compressive loading, the shear bands started from regions with enough free volume.