Self-Consistent Method and Quasi-Rate-Dependent Polycrystals

2009 ◽  
pp. 121-143
Author(s):  
Milan V. Mićunović
Keyword(s):  
Rate Dependent ◽  
Self Consistent ◽  
Consistent Method

Rate-dependent, finite elasto-plastic deformation of polycrystals

1986 ◽  
Vol 407 (1833) ◽  
pp. 343-375 ◽  
Keyword(s):  
Single Crystal ◽  
Single Crystals ◽  
Finite Deformation ◽  
Constitutive Relations ◽  
Slip Rate ◽  
Large Strains ◽  
Rate Dependent ◽  
Crystallographic Slip ◽  
Self Consistent ◽  
Consistent Method

Based on Hill’s method, a self-consistent averaging scheme is proposed for estimating the overall, finite deformation response of polycrystalline aggregates consisting of single crystals which undergo plastic flow by rate-dependent crystallographic slip, accompanied by elastic lattice distortion. First, constitutive relations for such single crystals are developed assuming that the slip-rate and the associated resolved shear stress are governed by: (1) a power-law relation, and (2) a viscoplastic relation. Then, Hill’s idea that the constraint imposed on a single crystal by the remaining aggregates may be represented by embedding the single crystal in a homogeneous, infinitely extended matrix having the instantaneous overall moduli, is used to formulate a completely self-consistent averaging procedure, valid for rate-dependent materials at finite strains and rotations. This method includes both the Hill and the Krӧner‒Budiansky‒Wu (K. B. W.) methods as limiting cases; when rate-effects are negligible, it reduces to Hill’s self-consistent method as formulated by Iwakuma and Nemat-Nasser for finite deformations, while it reduces to a generalized finite deformation version of the K. B. W. method for strongly rate-dependent materials. Illustrative numerical examples are presented for a plane uniaxial deformation, using a two-dimensional poly crystalline model. These examples clearly show that the rate-dependent crystallographic slip on the level of single crystals produces a more stable overall behaviour of poly crystals. This supports similar results arrived at by other investigators for single crystals and for polycrystals, by using the Taylor averaging scheme. It is shown that, while Taylor’s averaging scheme gives accurate estimates of the incremental quantities at large strains, the total overall quantities differ considerably from the ones obtained by the self-consistent method.

scholarly journals Towards B-Spline Atomic Structure Calculations

Atoms ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 50
Author(s):  
Charlotte Froese Fischer
Keyword(s):  
Atomic Structure ◽  
Bound States ◽  
Virial Theorem ◽  
B Spline ◽  
B Splines ◽  
Self Consistent ◽  
History Of ◽  
Consistent Method ◽  
Atomic Structure Calculations ◽  
Structure Calculations

The paper reviews the history of B-spline methods for atomic structure calculations for bound states. It highlights various aspects of the variational method, particularly with regard to the orthogonality requirements, the iterative self-consistent method, the eigenvalue problem, and the related sphf, dbsr-hf, and spmchf programs. B-splines facilitate the mapping of solutions from one grid to another. The following paper describes a two-stage approach where the goal of the first stage is to determine parameters of the problem, such as the range and approximate values of the orbitals, after which the level of accuracy is raised. Once convergence has been achieved the Virial Theorem, which is evaluated as a check for accuracy. For exact solutions, the V/T ratio for a non-relativistic calculation is −2.

Self-Consistent Method Using a Propagator

2004 ◽  
pp. 87-96 ◽  
Author(s):  
Motoichi Ohtsu ◽  
Kiyoshi Kobayashi
Keyword(s):  
Self Consistent ◽  
Consistent Method

Estimation of dispersion in estuaries by a self-consistent method

1985 ◽  
Vol 21 (2) ◽  
pp. 257-272
Author(s):  
D.J. Gunn ◽  
O. Yenigun
Keyword(s):  
Self Consistent ◽  
Consistent Method
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