Microstructure in plasticity without nonconvexity

Das, Amit Kumar; Acharya, Amit; Suquet, Pierre

doi:10.1007/s00466-015-1249-8

Cited by 9 publications

(20 citation statements)

References 26 publications

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“…For the RK1 scheme, we assume that there exists a ρ EUL independent of δt, h, v c , v (consequently, µ), S and γ such that ρ ≤ ρ EUL . Under the usual CFL criterion (15), for k = 0 (finite volume scheme), the dominant term in the error estimate for the RK1 scheme converges as O(h 1 / 2 ) [20]. Consequently, for small h i.e., h → 0, the errors incurred when using the RK1 scheme become significant.…”

Section: Cfl Conditionsmentioning

confidence: 99%

“…For this case, we have τ c = ∞. Consequently, τ * = t F = 1 sec and δt ≤ 1 sec must be respected together with the CFL condition (15) or ( 16) depending on k and the RK scheme used. Furthermore, the term α 33 v 1,1 in equation (17a) becomes equal to 0.…”

Section: Transport Of a Screw Dislocation: Parametric Studymentioning

confidence: 99%

“…In this case, we have τ c = 0.1ε sec, which leads to τ * = τ c = 10 −13 sec and δt ≤ 10 −13 sec should be respected. However, since this time step restriction is only imposed by those parts of the dislocation density field where |α h33 | < 10 −12 nm −1 , it is found that using a (7 or 8 orders of magnitude) higher time step that respects the CFL condition (15) or (16) gives satisfactory results (see sections 6.1.5 and 6.2). Note that we have used Λα h33 in equation (20).…”

Section: Transport Of a Screw Dislocation: Parametric Studymentioning

confidence: 99%

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Dislocation transport using a time-explicit Runge–Kutta discontinuous Galerkin finite element approach

Upadhyay¹,

Bleyer²

2022

Modelling Simul. Mater. Sci. Eng.

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A time-explicit Runge-Kutta discontinuous Galerkin (RKDG) finite element scheme is proposed to solve the dislocation transport initial boundary value problem in 3D. The dislocation density transport equation, which lies at the core of this problem, is a first-order unsteady-state advection-reaction-type hyperbolic partial differential equation; the DG approach is well suited to solve such equations that lack any diffusion terms. The development of the RKDG scheme follows the method of lines approach. First, a space semi-discretization is performed using the DG approach with upwinding to obtain a system of ordinary differential equations in time. Then, time discretization is performed using explicit RK schemes to solve this system. The 3D numerical implementation of the RKDG scheme is performed for the first-order (forward Euler), second-order and third-order RK methods using the strong stability preserving approach. These implementations provide (quasi-)optimal convergence rates for smooth solutions. A slope limiter is used to prevent spurious Gibbs oscillations arising from high-order space approximations (polynomial degree ≥ 1) of rough solutions. A parametric study is performed to understand the influence of key parameters of the RKDG scheme on the stability of the solution predicted during a screw dislocation transport simulation. Then, annihilation of two oppositely signed screw dislocations and the expansion of a polygonal dislocation loop are simulated. The RKDG scheme is able to resolve the shock generated during dislocation annihilation without any spurious oscillations and predict the prismatic loop expansion with very low numerical diffusion. These results demonstrate the robustness of the scheme.

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Section: Cfl Conditionsmentioning

confidence: 99%

Section: Transport Of a Screw Dislocation: Parametric Studymentioning

confidence: 99%

Section: Transport Of a Screw Dislocation: Parametric Studymentioning

confidence: 99%

See 1 more Smart Citation

Dislocation transport using a time-explicit Runge–Kutta discontinuous Galerkin finite element approach

Upadhyay¹,

Bleyer²

2022

Modelling Simul. Mater. Sci. Eng.

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“…This approach is closer to the physics since it permits to keep the memory of the discrete nature of dislocations and to access dislocations patterning. In the following, a non-convex energy density function is introduced, following the work of Zhang et al (2015); Das et al (2016). The volumic density of free (stored) energy w is assumed to be of the form…”

Section: Constitutive Law For the Dislocation's Velocitymentioning

confidence: 99%

Numerical simulation of model problems in plasticity based on field dislocation mechanics

Morin¹,

Brenner²,

Suquet³

2019

Modelling Simul. Mater. Sci. Eng.

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The aim of this paper is to investigate the numerical implementation of the Field Dislocation Mechanics (FDM) theory for the simulation of dislocation-mediated plasticity. First, the mesoscale FDM theory of Acharya and Roy (2006) is recalled which permits to express the set of equations under the form of a static problem, corresponding to the determination of the local stress field for a given dislocation density distribution, complemented by an evolution problem, corresponding to the transport of the dislocation density. The static problem is solved using FFT-based techniques (Brenner et al., 2014). The main contribution of the present study is an efficient numerical scheme based on high resolution Godunov-type solvers to solve the evolution problem. Model problems of dislocation-mediated plasticity are finally considered in a simplified layer case. First, uncoupled problems with uniform velocity are considered, which permits to reproduce annihilation of dislocations and expansion of dislocation loops. Then, the FDM theory is applied to several problems of dislocation microstructures subjected to a mechanical loading.

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“…The model in [XEA15] belongs to the same mathematical class as MFDM, being physically more involved with more state descriptors and associated coupled, nonlinear, equations of evolution. An attempt to understand the emergence of microstructure in this collection of models was made in [RA06,DAS16], in drastically simplified 1-d settings. The conclusion in [DAS16] was that in all likelihood such complexity is not essential for the emergence of dislocation microstructure in this family of models; the nature of the fundamental transport equation for Nye tensor evolution coupled to stress along with the simplest representations, from conventional plasticity theory, of the plastic strain rate due to statistical dislocations, is adequate for the stated purpose, while being faithful to representing the plastic strain rate of both resolved and unresolved dislocation populations.…”

Section: Introductionmentioning

confidence: 99%

Dislocation pattern formation in finite deformation crystal plasticity

Arora

Acharya

2020

International Journal of Solids and Structures

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Stressed dislocation pattern formation in crystal plasticity at finite deformation is demonstrated for the first time. Size effects are also demonstrated within the same mathematical model. The model involves two extra material parameters beyond the requirements of standard classical crystal plasticity theory. The dislocation microstructures shown are decoupled from deformation microstructures, and emerge without any consideration of latent hardening or constitutive assumptions related to crossslip. Crystal orientation effects on the pattern formation and mechanical response are also demonstrated. The manifest irrelevance of the necessity of a multiplicative decomposition of the deformation gradient, a plastic distortion tensor, and the choice of a reference configuration in our model to describe the micromechanics of plasticity as it arises from the existence and motion of dislocations is worthy of note.

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Microstructure in plasticity without nonconvexity

Cited by 9 publications

References 26 publications

Dislocation transport using a time-explicit Runge–Kutta discontinuous Galerkin finite element approach

Dislocation transport using a time-explicit Runge–Kutta discontinuous Galerkin finite element approach

Numerical simulation of model problems in plasticity based on field dislocation mechanics

Dislocation pattern formation in finite deformation crystal plasticity

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