Because the general nonlocal theory satisfies, both the force and the moment global balance conditions, the nonlocal stress tensor is a symmetric tensor with six indepen- dent components. The skew-symmetric part of the nonlocal stress tensor vanishes upon achieving the moment balance globally. However, if the moment balance is considered locally and is violated globally, the skew-symmetric part of the nonlocal stress tensor shall be non-vanishing. Given this observation, the balance equations of the general nonlocal theory can be written in three different forms, as follows:
Form I:
tj i,j+ fi =ρu¨i
ei j ktj k=0 (40)
Form II:
σj i,j+ fi =ρu¨i
ei j ktj k=0 (41)
Form II:
tj i,j+ fi =ρu¨i
ei j kσj k=0 (42)
whereσi jandti jare the local and the nonlocal stress tensors, respectively. The stress tensors, σi j andti j, in Eqs. (40)–(42) are general tensors each of which has nine independent components. These tensors are defined by:
σi j(x)= ¯Ci j kluk,l(x) (43)
ti j(x)=
V
Ci j klx−x, uk,l
x d V
x
(44)
where C¯i j kl andCi j kl are local and nonlocal stiffnesses, respectively. Form I (Eq.
(40)) is a general form of the nonlocal theory, which achieves the force and the moment global balance requirements. This form is an equivalent form of the balance Eq. (9). According to the first equation of Eq. (40), the resultant force at a point is calculated as the sum of the local interaction of this point with its neighbors and the nonlocal interactions with all other points of the elastic domain. On the other hand, the second equation of Eq. (40) indicates that the resultant moment at a point is calculated as the sum of the moments of the local interaction of the point with its neighbors and the moments of the nonlocal interactions with all other points of the elastic domain.
Forms II and III are reduced versions of the general nonlocal theory (Form I).
Form II achieves the force balance locally but violates it globally. In Form III, the moment balance is achieved locally, but it is violated globally. These different forms would lead to different versions of the nonlocal theory.
In the following, we show that Form I of the nonlocal theory can be reduced to the strain gradient theory. In addition, we show that Form II can be reduced to the couple stress theory.
4.1 Strain Gradient Theory
For weak nonlocal fields (i.e., weak nonlocality), the general nonlocal theory can be reduced to a strain gradient theory in which the deformation energy is a function of the strain and its gradients. The simplification of the general nonlocal theory to a strain gradient theory is achieved by considering Form I of the nonlocal model and expanding the strain field,ε
x
, about pointxas follows [18]:
ε x
=ε (x)+ x−x
.∇ε (x)+1 2
x−x .∇2
ε (x) + 1
6
x−x .∇3
ε (x)+ 1 24
x−x .∇4
ε (x)+. . . (45) By the substitution of Eq. (45) into Eq. (22), the nonlocal stress is formed depend- ing on the strain and its gradients, as follows:
tj i =tj i =
λεrr +a3εrr,ll+a5εrr,llmm+. . . δi j
+
2μεi j +b3εi j,ll+b5εi j,llmm+. . .
(46) where
an= λ (n−1)
V
x−xn−1βλx−x, λ d V
x for n=3,5, ...
bn= 2μ (n−1)
V
x−xn−1βμx−x, μ d V
x
for n=3,5, ... (47) Substituting Eq. (46) into Eq. (40) gives the equation of motion in the form:
λεrr,j+a3εrr,ll j+a5εrr,llmm j+. . . δi j
+
2μεi j,j+b3εi j,ll j+b5εi j,llmm j+. . .
+ fi =ρu¨i (48) The strain can be expressed in terms of the displacement vector,u, as follows:
ε= ∇u−1
2∇ ×u (49)
which gives the field equation with the form:
⎡
⎣λ+2μ+ ∞ n=3,5,...
(an+bn)∇(n−1)
⎤
⎦(∇∇.u)
−1 2
⎡
⎣2μ+ ∞
n=3,5, ...
bn∇(n−1)
⎤
⎦(∇ × ∇ ×u)+F=ρu¨ (50)
4.2 Couple Stress Theory
Form II of the nonlocal theory can be reduced to a couple stress theory. Form II indicates that the force resultant at a point is locally balanced while the moment resultant at a point is globally balanced. According to Eqs. (41) and (43), the skew- symmetric part of the nonlocal stress tensorti jis obtained for isotropic-linear elastic materials with the form:
ei j ktj k(x)=
V
2μβμx−x, μ θmn
x d V
x
(51)
whereθmn is the rotation tensor that expresses the rigid rotation of the elastic body.
For a weak nonlocal moment field (i.e., weak nonlocality), Form II of the nonlocal theory can be reduced to the couple stress theory by expanding the rigid-rotation fieldθ
x
about pointxas follows [18]:
θ x
=θ (x)+ x−x
.∇θ (x)+1 2
x−x .∇2
θ (x) + 1
6
x−x .∇3
θ (x)+ 1 24
x−x .∇4
θ (x)+. . . (52) The substitution of Eq. (52) into Eq. (51) yields:
ei j ktj k(x)=2μθj k(x)+b3θj k,ll(x)+b5θj k,llmm(x)+. . . (53) wherebnare defined in Eq. (47). Equation (53) can be rewritten in terms of a rotation gradient tensor, Ri j, i.e.,
ei j ktj k(x)=ej ki
2μθi(x)+b3Rli,l(x)+b5Rli,lmm(x)+. . .
(54) where
Rj i=θi,j (55)
whereθi =(ei j kθk j)/2 andθi j =ei j kθk.
According to Eq. (41), the skew-symmetric part of the general nonlocal stress is zero, i.e.,ei j k tj k(x)=0. This gives the skew-symmetric part of the local stress tensor,σ[i j]= −ej i k(μθk), with the form:
σ[i j]= 1 2ej i k
b3Rlk,l(x)+b5Rlk,lmm(x)+. . .
(56) In light of the preceding equations, the balance equation of the couple stress theory is obtained in the following form by the substitution of Eq. (56) into the first equation of Eq. (41):
σ(j i),j+1 2ej i k
b3Rlk,l j+b5Rlk,lmm j+. . .
+ fi =ρu¨i (57) whereσ(j i)=λεrrδi j+2μεi jis the classical symmetric-stress tensor. The field equa- tion can be obtained by substitutingR=(∇∇ ×u) /2 into Eq. (57):
(λ+2μ) (∇∇.u)−1 2
⎡
⎣2μ+1 2
∞ n=3,5,...
bn∇(n−1)
⎤
⎦(∇ × ∇ ×u)+F=ρu¨(58)
4.3 Relation to Mindlin’s Strain Gradient Theories
In the framework of Mindlin’s strain gradient theory, the total strain energy is a function of the strain and its first and second gradients [2,5,43]. According to this theory, the field equation is expressed with the form [2]:
(λ+2μ)
1−l12∇2+l24∇4 (∇∇.u)
−μ
1−l32∇2+l44∇4
(∇ × ∇ ×u)+F=ρu¨ (59) wherel1,l2,l3, andl4 are different length scales. By comparing Eqs. (59) to (50), the length scales,l1,l2,l3, andl4, can be related to the material coefficients,λ,an, μ, andbn, as follows:
l12= −a3+b3
λ+2μ,l24=a5+b5
λ+2μ, l32= −b3
2μ,l44= b5
2μ (60)
The preceding discussion indicates that Mindlin’s strain gradient is a nonlocal theory but for weak nonlocality. The strain gradient theory can model long-range interatomic/intermolecular interactions up to a certain number of neighbors depend- ing on the gradients of the strain considered.
4.4 Relation to Couple Stress Theory
The couple stress theory [7,8] was originally developed on the basis of the first-order gradient of the rotation tensor. This theory was then extended to the second rotation gradient theory [9] in which the internal energy depends on the strain and the first and the second gradients of the rotation. According to the couple stress theory, the field equation is expressed as follows [9]:
(λ+2μ) (∇∇.u)−μ
1−l52∇2+l46∇4
(∇ × ∇ ×u)+F=ρu¨ (61) wherel5andl6are length scales. According to Eqs. (61) and (58), the length scales, l5andl6, can be related to the material coefficientsμ, andbn, as follows:
l52= −b3
4μ, l64= b5
4μ (62)
It is clear that the couple stress theory, like the strain gradient theory, is a nonlocal theory but with weak nonlocality. The couple stress theory can model long-range interatomic/intermolecular interactions up to a certain number of neighbors depend- ing on the considered gradients of the rotation tensor.
4.5 Wave Propagation
To derive the dispersion relations along [100] direction in the context of the strain gradient and couple stress theories, Eq. (14) is used. Equation (14) is substituted into the field Eq. (50), which gives the dispersion relations according to the strain gradient theory with the form:
ω2L =
⎛
⎝(λ+2μ)k2+ ∞
n=3,5,....
−(i)n+1(an+bn)kn+1
⎞
⎠/ρ (63)
ω2T =
⎛
⎝μk2+1 2
∞ n=3,5,...
−(i)n+1bnkn+1
⎞
⎠/ρ (64)
Similarly, the dispersion relations according to the couple stress theory are derived in the following form by substituting Eq. (14) into Eq. (58):
ω2L =
(λ+2μ)k2
/ρ (65)
ω2T =
⎛
⎝μk2+1 4
∞ n=3,5,...
−(i)n+1bnkn+1
⎞
⎠/ρ (66)