Anisotropic concurrent coupled atomistic and discrete dislocation for partial dislocations in FCC materials

doi:10.1007/s10483-025-3275-6

摘要/Abstract

Abstract:

Spurious forces are a significant challenge for multi-scale methods, e.g., the coupled atomistic/discrete dislocation (CADD) method. The assumption of isotropic matter in the continuum domain is a critical factor leading to such forces. This study aims to minimize spurious forces, ensuring that atomic dislocations experience more precise forces from the continuum domain. The authors have already implemented this idea using a simplified and unrealistic slipping system. To create a comprehensive and realistic model, this paper considers all possible slip systems in the face center cubic (FCC) lattice structure, and derives the required relationships for the displacement fields. An anisotropic version of the three-dimensional CADD (CADD3D) method is presented, which generates the anisotropic displacement fields for the partial dislocations in all the twelve slip systems of the FCC lattice structure. These displacement fields are tested for the most probable slip systems of aluminum, nickel, and copper with different anisotropic levels. Implementing these anisotropic displacement fields significantly reduces the spurious forces on the slip systems of FCC materials. This improvement is particularly pronounced at greater distances from the interface and in more anisotropic materials. Furthermore, the anisotropic CADD3D method enhances the spurious stress difference between the slip systems, particularly for materials with higher anisotropy.

Key words: multi-scale method, anisotropic coupled atomistic/discrete dislocation (CADD), spurious force, partial dislocation, face center cubic (FCC) material

中图分类号:

O302

. [J]. Applied Mathematics and Mechanics (English Edition), 2025, 46(7): 1365-1382.

S. FORGHANI, N. KHAJI. Anisotropic concurrent coupled atomistic and discrete dislocation for partial dislocations in FCC materials[J]. Applied Mathematics and Mechanics (English Edition), 2025, 46(7): 1365-1382.

图/表 19

Number	Step		MD	DDD	Linear elasticity
1	Predict in MD and update segments in DDD		Predict	Update segments
		Input	$f a k, r a k, v a k, ∀ a ∈ Ω A$	$f d k, ∀ d ∈ Ω C$ $s e k, ∀ e ∈ γ c$
		Output	$r a k + 1, ∀ a ∈ Ω A$ $v a k + 1 / 2, ∀ a ∈ Ω A$ $u n k + 1, ∀ n ∈ ∂ Ω I$	$s e k + 1, ∀ e ∈ γ c$
2	Detect atomic dislocations in MD		Detect atomic dislocations
		Input	$r a k + 1, ∀ a ∈ Ω A$
		Output	$s e k + 1, ∀ a ∈ γ A D$
3	Compute $* ˜$ contributions by DDD			Compute $* ˜$ contributions
		Input		$s e k, ∀ e$ $σ ˜ n k + 1, ∀ n ∈ ∂ Ω t$
		Output		$u ˜ n k + 1, ∀ n ∈ ∂ Ω I ∪ ∂ Ω u$ $u ˜ a k + 1, ∀ a ∈ ∂ Ω P$
4	Compute boundary corrections $*^$				Compute boundary corrections $*^$
		Input			$σ ˜ n k + 1, ∀ n ∈ ∂ Ω t$ $u ˜ n k + 1, u n k + 1, ∀ n ∈ ∂ Ω I ∪ ∂ Ω u$
		Output			$t^n k + 1, ∀ n ∈ ∂ Ω t$ $u^n k + 1, ∀ n ∈ ∂ Ω I ∪ ∂ Ω u$
5	Solve elasticity by linear elasticity				Solve elasticity
		Input			$t^n k + 1, ∀ n ∈ ∂ Ω t$ $u^n k + 1, ∀ n ∈ ∂ Ω I ∪ ∂ Ω u$
		Output			$σ^d k + 1, ∀ d ∈ N DD$ $u^n k + 1, ∀ n ∈ Ω C$ $u^a k + 1, ∀ a ∈ Ω P$
6	Compute forces by MD and DDD		Compute forces	Compute forces
		Input	$r a k + 1, ∀ a ∈ Ω A$ $u^a k + 1, u ˜ a k + 1, ∀ a ∈ Ω P$	$s e k, ∀ e$ $σ^d k + 1, ∀ d ∈ N DD$
		Output	$f a k + 1, ∀ a ∈ Ω A$	$f d k + 1, ∀ d ∈ N DD$
7	Correct by MD		Correct
		Input	$v a k + 1 / 2, f a k + 1, ∀ a ∈ Ω A$
		Output	$v a k + 1, ∀ a ∈ Ω A$

Material	Elastic constant/GPa			References
Material	$c 11$	$c 12$	$c 44$	References
Aluminum	113.0	61.6	32.0	Ercolessi and Adams (1994)^[44]
Nickel	263.0	154.0	127.0	Sheng et al. (2011)^[45]
Copper	175.0	124.0	79.0	Sheng et al. (2011)^[45]

System	Slip plane	Burgers vector $(b)$
1	$(1 1 1)$	$12 [1 1 ¯ 0]$
2	$(1 ¯ 1 1)$	$12 [1 1 0]$
3	$(1 1 ¯ 1)$	$12 [1 1 0]$
4	$(1 1 1 ¯)$	$12 [1 1 ¯ 0]$

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Method	The onset distance of spurious stress/nm
Method	Aluminum	Nickel	Copper
CADD1	2.31	2.80	4.22
CADD2	2.00	2.41	3.31
CADD3	1.85	1.79	2.70

Comparison of methods	The reduction amount of the onset distance
Comparison of methods	Aluminum	Nickel	Copper
CADD2 compared with CADD1	13%	14%	22%
CADD3 compared with CADD1	20%	36%	36%
CADD3 compared with CADD2	7.5%	25%	18%