Nonlinear elasticity theory of dislocation formation and composition change in binary alloys in three dimensions

Minami, Akihiko; Ōnuki, Akira

doi:10.1016/j.actamat.2006.11.030

Cited by 16 publications

(18 citation statements)

References 37 publications

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“…In this section, we introduce a phase field formulation to model the behaviors of intact glass under triaxial compression . As experiments showed that stress–strain follows a nonlinear relationship under triaxial compression, a nonlinear form of the elastic energy density f el is employed:

f_{e l} = \frac{1}{2} K e_{1}^{2} + Φ (e_{2}, e_{3}) + Ψ (e_{4}, e_{5}, e_{6})

where K is the material bulk modulus. The strains e i ( i = 1,…, 6) are defined as e 1 = ɛ 11 + ɛ 22 + ɛ 33 , e 2 = ɛ 11 − ɛ 22 ,

e_{3} = (2 ε_{33} - ε_{11} - ε_{22}) / \sqrt{3}

, e 4 = 2ɛ 23 , e 5 = 2ɛ 13 , e 6 = 2ɛ 12 , and

{normalε}_{i j} = (\nabla_{i} u_{j} + \nabla_{j} u_{i}) / 2

{normal∇}_{i} = normal∂ false/ \partial x_{i}

.…”

Section: Phase Field Model For Glass Systemsmentioning

confidence: 99%

“…

Φ (e_{2}, e_{3})

represents an increase in the elastic energy due to stretching or compression along the principal axes, while

Ψ (e_{4}, e_{5}, e_{6})

represents the elastic energy change due to shear deformations. They are non‐negative definite and expressed as follows:

Φ = \frac{μ {normalΓ}^{2}}{8 {normalπ}^{2}} [3 - cos (\frac{2 normalπ e_{+}}{Γ}) - cos (\frac{2 normalπ e_{-}}{Γ}) - cos (\frac{4 normalπ e_{3}}{normalΓ \sqrt{3}})]

Ψ = \frac{μ {normalΓ}^{2}}{4 {normalπ}^{2}} [3 - cos (\frac{2 normalπ e_{4}}{Γ}) - cos (\frac{2 normalπ e_{5}}{Γ}) - cos (\frac{2 normalπ e_{6}}{Γ})]

with

e_{\pm} = e_{2} \pm e_{3} false/ \sqrt{3}

, and μ is the shear modulus. The functional form of

Φ (e_{2}, e_{3})

and

Ψ (e_{4}, e_{5}, e_{6})

was determined with the following considerations: (1) compatible with linear elasticity theory when strains are small, that is,

f_{e l} = 1 false/ 2

…”

Section: Phase Field Model For Glass Systemsmentioning

confidence: 99%

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Mesoscale Phase Field Modeling of Glass Strengthening Under Triaxial Compression

Sun

2015

Int J of Appl Glass Sci

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Recent “hydraulic bomb” and “confined sleeve” tests on transparent armor glass materials such as borosilicate glass and soda‐lime glass showed that the glass strength was a function of confinement pressure. The measured stress–strain relation is not a straight line as most brittle materials behave under little or no confinement. Moreover, borosilicate glass exhibited a stronger compressive strength when compared to soda‐lime glass, even though soda‐lime has higher bulk and shear moduli as well as apparent yield strength. To better understand these experimental findings, a mesoscale phase field model is developed to simulate the nonlinear stress versus strain behaviors under confinement by considering heterogeneity formation — under triaxial compression and the energy barrier of a micro shear banding event (referred to as pseudo‐slip hereafter) in the amorphous glass. With calibrated modeling parameters, the simulation results demonstrate that the developed phase field model can quantitatively predict the pressure‐dependent strength, and it can also explain the difference between the two types of glasses from the perspective of energy barrier associated with a pseudo‐slip event.

show abstract

f_{e l} = \frac{1}{2} K e_{1}^{2} + Φ (e_{2}, e_{3}) + Ψ (e_{4}, e_{5}, e_{6})

where K is the material bulk modulus. The strains e i ( i = 1,…, 6) are defined as e 1 = ɛ 11 + ɛ 22 + ɛ 33 , e 2 = ɛ 11 − ɛ 22 ,

e_{3} = (2 ε_{33} - ε_{11} - ε_{22}) / \sqrt{3}

, e 4 = 2ɛ 23 , e 5 = 2ɛ 13 , e 6 = 2ɛ 12 , and

{normalε}_{i j} = (\nabla_{i} u_{j} + \nabla_{j} u_{i}) / 2

{normal∇}_{i} = normal∂ false/ \partial x_{i}

.…”

Section: Phase Field Model For Glass Systemsmentioning

confidence: 99%

“…

Φ (e_{2}, e_{3})

represents an increase in the elastic energy due to stretching or compression along the principal axes, while

Ψ (e_{4}, e_{5}, e_{6})

represents the elastic energy change due to shear deformations. They are non‐negative definite and expressed as follows:

Φ = \frac{μ {normalΓ}^{2}}{8 {normalπ}^{2}} [3 - cos (\frac{2 normalπ e_{+}}{Γ}) - cos (\frac{2 normalπ e_{-}}{Γ}) - cos (\frac{4 normalπ e_{3}}{normalΓ \sqrt{3}})]

Ψ = \frac{μ {normalΓ}^{2}}{4 {normalπ}^{2}} [3 - cos (\frac{2 normalπ e_{4}}{Γ}) - cos (\frac{2 normalπ e_{5}}{Γ}) - cos (\frac{2 normalπ e_{6}}{Γ})]

with

e_{\pm} = e_{2} \pm e_{3} false/ \sqrt{3}

, and μ is the shear modulus. The functional form of

Φ (e_{2}, e_{3})

and

Ψ (e_{4}, e_{5}, e_{6})

was determined with the following considerations: (1) compatible with linear elasticity theory when strains are small, that is,

f_{e l} = 1 false/ 2

…”

Section: Phase Field Model For Glass Systemsmentioning

confidence: 99%

Mesoscale Phase Field Modeling of Glass Strengthening Under Triaxial Compression

Sun

2015

Int J of Appl Glass Sci

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“…In the subsequent analysis, we focus on 2D systems; extension to 3D is straightforward. Next, we write down a local nonlinear deformation energy density, which is multivalued in the shear strain alongn ¼ ½cosðrÞ; sinðrÞ (and thus allows for slip events [31]), in terms of the displacement field uðr; tÞ:…”

mentioning

confidence: 99%

“…In the crystalline phase ÁðrÞ is constant [31], while in the glassy phase structural heterogeneity can be incorporated by assuming that ÁðrÞ is a quenched, Gaussian random variable with average hÁðrÞi ¼ Á 0 and a two-point correlation function h½ÁðrÞ À Á 0 ½Áðr 0 Þ À Á 0 i ¼ 2 expðÀjr À r 0 j=Þ, where 2 and denote the variance of the distribution and a structural correlation length, respectively. In monolithic BMGs, ÁðrÞ is used as a descriptor that distinguishes between domains of rigid and soft SRO.…”

mentioning

confidence: 99%

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Continuum Modeling of Bulk Metallic Glasses and Composites

Abdeljawad

Haataja²

2010

Phys. Rev. Lett.

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At low temperatures, monolithic bulk metallic glasses (BMGs) exhibit high strength and large elasticity limits. On the other hand, BMGs lack overall ductility due to highly localized deformation mechanisms. Recent experimental findings suggest that the problem of catastrophic failure by shear band propagation in BMGs can be mitigated by tailoring microstructural features at different length scales to promote more homogeneous plastic deformation. Herein, based on a continuum approach, we present a quantitative analysis of the effects of microstructure on the deformation behavior of monolithic BMGs and BMG composites. In particular, simulations highlight the importance of short-ranged structural correlations on ductility in monolithic BMGs and demonstrate that particle size controls the ductility of BMG composites. In broader terms, our results provide new avenues for further improvements to the mechanical properties of BMGs.

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Homogeneous Dislocation Nucleation in Landau Theory of Crystal Plasticity

Salman¹,

Baggio²

2019

Mechanics and Physics of Solids at Micro‐ and Nano‐Scales

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Nonlinear elasticity theory of dislocation formation and composition change in binary alloys in three dimensions

Cited by 16 publications

References 37 publications

Mesoscale Phase Field Modeling of Glass Strengthening Under Triaxial Compression

Mesoscale Phase Field Modeling of Glass Strengthening Under Triaxial Compression

Continuum Modeling of Bulk Metallic Glasses and Composites

Homogeneous Dislocation Nucleation in Landau Theory of Crystal Plasticity

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