Mathematical Physics - Volume II - Numerical Methods
Chapter 6. Introduction to Computational Mechanics of Discontinua
230
Stress, strain and effective stiffness The elastic strain energy of the system in the (proximity of) equilibrium can be developed in the Taylor series [34]
∂ 2 U ∂ ε αβ ∂ ε γδ
∂ U ∂ ε αβ
1 1!
1 2!
ε αβ +
ε αβ ε γδ + · · · =
U = U 0 +
(6.12)
1 2 C αβ γδ ε αβ ε γδ + · · ·
= U 0 + σ αβ ε αβ +
In the (proximity of) equilibrium state, the resultant forces acting on any atom of the system (6.2) 2 are (close to) zero, which implies that each atom lies in its potential well. Such system must be, by definition, stable in the event of an infinitesimal disturbance, such as the one imposed by the homogeneous strain tensor ε αβ acting as an external load. The linear (second) term in the Taylor series (6.12) represents the stress tensor. It is important to note that this is a general thermodynamic relation independent of the applicability of Hooke’s law [35]. With regards to the potential energy of a system, if the interatomic actions can be successfully approximated with the EAM (6.4), the stress tensor components are (6.13) where ¯ V is the average volume per atom, while ( r i j ) α and ( r i j ) β are corresponding ( α and β ) projections of the distance vectors r i j [35], [36]. Since the stress definition (6.13) is inherently related to the static equilibrium state, it is, strictly speaking, applicable only to static (or quasi-static) deformation where the resultant force acting on each atom (6.2) 2 are equal (or “close enough”) to zero. On the other hand, dynamic deformation implies wave propagation and, in order for expression (6.13) to be applicable, it must be tacitly assumed that the nonequilibrium process can be represented by a successive series of equilibrium processes. This concept is routinely used, out of necessity, in thermodynamics of nonequilibrium processes. The third term in the Taylor series (6.12) defines the elastic stiffness tensor σ αβ = 1 2 ¯ VN ∑ i , j j̸ = i d φ d r i j ( r i j ) α ( r i j ) β r i j + d Ψ d ¯ V δ αβ = − 1 2 ¯ VN ∑ i , j j̸ = i f i j ( r i j ) α ( r i j ) β r i j + d Ψ d ¯ V δ αβ
d φ d r i j !
d 2 φ d r 2
( r i j ) α ( r i j ) β ( r i j ) γ ( r i j ) δ r 2 i j
1 2 ¯ VN ∑ i , j j̸ = i
1 r i j
C αβ γδ =
i j −
−
d Ψ d ¯ V
δ αβ δ γδ ( 4 − δ αγ − δ β γ − δ αδ − δ βδ ) +
1 2
( δ αγ δ βδ + δ β γ δ αδ )( 2 − δ αβ ) − 1 2
−
d 2 Ψ d ¯ V 2
+ ¯ V
δ αβ δ γδ
(6.14) In the case of pair potentials (6.4), Ψ = 0 and only the first terms of Equations (6.13) and (6.14) remain. When the EAM (6.7) or related methods are used to model interatomic interactions, it is necessary to use the complete Equations (6.13) and (6.14) that take into account the density (the average volume per atom) dependence of the potential [24].
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