The linear theory of the nonlocal elasticity is expressed by the following set of equations for a homogeneous and an isotropic solid [10,11]:
tkl,k+ρ(fl− ¨ul)=0 (1) tkl(x)=
V
K(||x−x||,τ)σkl(x)dυ(x) (2) σkl(x)=λerr(x)δkl+2μekl(x) (3) ekl(x)= 1
2
∂uk(x)
∂xl
+∂ul(x)
∂xk
(4) whereλandμare the constants of Lamé and|| · ||is the Euclidean norm. The stress tensor, the mass density, the body force density and the displacement vectors at a reference pointxof the body at timetare denoted bytkl,ρ, flandul, respectively.
Also, the classical stress tensor atxat timetand the linear strain tensor at any point xof the body at timet are denoted byσkl(x)andekl(x), respectively. The former is associated with the latter by means of Hooke’s law. The volume integral of Eq. (2) is evaluated over the regionV of the body as well.
Equation (2) expresses the contribution of the material points of the body to the stress field at a pointxvia the attenuation function K (nonlocal modulus) [10,11].
The attenuation function is of dimension of (length)−3 as well. Ergo, it should be dependent on a characteristic length ratioτ:=e0a/, wherea is an internal char- acteristic length, i.e., a lattice parameter/bond length,is an external characteristic length, i.e., a wave length, ande0is also a dimensionless material constant.
The following properties are attributed to the nonlocal modulusK:
(i)K has to revert to the Dirac delta function (δ) asτ →0, so that the classical elasticity can be approximated when the limit of the internal characteristic length vanishes.
τ→0limK(||x−x||,τ)=δ(||x−x||) (5) (ii)K is symmetric with respect toxas well.
(iii) The maximum of K is also attained atx=xandK →0 at large distances as well.
(iv)K satisfies the normalization condition:
K(||x||,τ)d=1 (6)
wherex∈,is an infinite domain, andKcorresponds to a PDF [10,48].
2.2 A Discussion About Eringen’s Nonlocal Stress Models
2.2.1 Integral Equations, Fundamental Solutions and Green Functions The nonlocal integral constitutive equation being generally hard to handle is a first kind Fredholm. The use of either weakly singular or continuous kernels representing attenuation functions leads to ill-posed problems [30].
Eringen [10] noticed that the nonlocal integral constitutive equation could be transformed into a differential one. Provided thatK is a fundamental solution of a linear differential operatorL, it then holds in an infinite domain:
L
K(||x−x||,τ)
=δ(||x−x||) (7) Applying the differential operatorLto Eq. (2), it yields:
Ltkl=σkl (8)
Equation (8) is then called the nonlocal differential constitutive equation. Eringen attributed a physical interpretation to the nonlocal differential form [10,11].
Although this fundamental solution was adequate for a number of engineering problems [7–10,21], it is not suitable for a problem in a finite domain since it is essential to find the corresponding Green function [53].
2.2.2 Two Phase Nonlocal Integral Constitutive Model
The TPNI stress model suggested by Eringen [61] was based on Kröner’s research [2].
Eringen underlined the advantages of the above model from a mathematical point of view, yet there was some doubt if this model was suitable from a physical point of view [61].
The TPNI stress model can be regarded as the constitutive equation of a two phase elastic material, where phase 1 complies with local elasticity and phase 2 with nonlocal elasticity. The aforementioned model transforms the constitutive equation into a second kind Fredholm integral equation that is manageable and a well-posed problem. Altan [62] and subsequently Polizzotto [51] advanced the TPNI stress model which is defined as follows:
tkl(x):=k1σkl(x)+k2
V
K(||x−x||,τ)σkl(x)dυ(x) (9) wherek1+k2=1 andk1,k2>0.
The attenuation function corresponding to the TPNI stress model is defined as:
K˜(||x−x||,τ):=k1δ(||x−x||)+k2K(||x−x||,τ) (10)
2.2.3 Two Phase Constitutive Model with the Modified Kernel
An essential feature of a nonlocal kernel is associated with the normalization condi- tion. To elucidate, the normalization condition has to be such that attenuation function converges toward Dirac delta function as the nonlocal parameter tends to zero at each point of a finite body. This characteristic corroborates that the nonlocal model reverts toward the classic-local model at each point of a finite body. A PDF is defined by the attenuation function. The key point is therefore to find an appropriate transformation from an infinite domain into a finite one. Borino et al. [75] as well as Bažant and Jirásek [76] proposed an efficient transformation, which leads to a modified, sym- metric attenuation function being well-defined and satisfying all the properties of a nonlocal kernel. The model of the modified kernel is defined as follows:
Kmod(||x−x||,τ):=ξ(x,τ)δ(||x−x||)+K(||x−x||,τ) (11) whereξ(x,τ):=1−
V K(||x−ζ||,τ)dζ.
If the normalization condition is violated, the uniform straining of the finite body can not generate a uniform stress [76].
The nonlocal stresses at a reference pointxare defined by a weighted average via a nonlocal modulus (PDF) and the macroscopic (Cauchy) stresses of all the points of the body. This definition introduces the nonlocal interactions between the atoms (or the molecules) without necessitating the determination of the cohesive forces which are responsible for. The nonlocal modulus defined in an infinite domain is
symmetric with respect tox and it satisfies the normalization condition (Eq. (6)).
The normalization condition corroborates that the sum of all the weights of Cauchy stresses is equal to 1 at each pointxof the infinite body. The violation of the latter on the boundary of a finite domain is obvious, since no material points exist out of the body to contribute to the stresses [48].
To exemplify, let us consider a beam of length L with boundaries 0 and L in one-dimension. For ε>0 and ε→0, the points found in the intervals [0,ε)and (L−ε,L] have no adjacent points found in the intervals(−∞,0)and(L,+∞) to interact with each other. Ergo, the locality increases in these points and it exists together with the interactions between the adjacent points in the intervals(0,ε)and (L−ε,L).
To shed light on, the modified kernel gives rise to the locality when the interactions between the particles/points are absent. The latter consideration can be generalized in two- and three-dimensions. The modified kernel can be inserted into Eq. (2), and it can be then considered as a physical expansion of the nonlocal theory of Eq. (2) on a finite body [48].
The classic types of a kernel do not satisfy the normalization condition on the boundary of the body, because they are normalized in an infinite domain. The TPNI stress model appears both mathematical and physical inconsistencies when it is applied to a finite domain. This model additionally violates the normalization condition in a finite domain and especially on the body’s boundary.
2.3 The Correlation Between Eringen’s Nonlocal Model and Mindlin’s Gradient Model
The correlation between the nonlocal elasticity theory of Eringen and the strain gradi- ent elasticity theory of Mindlin has been an open problem. The next process following Mindlin’s rationale [29] attempts to explain the aforementioned correlation.
Consider the nonlocal constitutive equation (Eq. (2)) with respect to the strains:
ti j(xm)=
V
Ci j klK(||xm−xm||)ekl(xm)d3xm (12) whered3xm ≡d V =d xid xjd xk andCi j kl is the stiffness tensor.
Defining thatCi j kl(xm−xm):=Ci j klK(||xm−xm||), Eq. (12) yields:
ti j(xm)=
V
Ci j kl(xm−xm)ekl(xm)d3xm (13) Considering thatxm =xm+xm, Taylor series of the strainekl(xm)gives:
ekl(xm)=ekl(xm)+∂ekl(xm)
∂xq
xq+1 2
∂2ekl(xm)
∂xq∂xn
xqxn+. . . (14)
Inserting Eq. (14) into Eq. (13), it yields:
ti j(xm)=
V
Ci j kl(xm−xm)
ekl(xm)+∂ekl(xm)
∂xq
xq
+ 1 2
∂2ekl(xm)
∂xq∂xn
xqxn+. . . d3xm (15) The stresses are ergo defined by selecting the first three terms of Taylor series of Eq. (15):
¯
ti j(xm)=ekl(xm)
V
Ci j kl(xm−xm)d3xm +∂ekl(xm)
∂xq
V
Ci j kl(xm−xm)xqd3xm +1
2
∂2ekl(xm)
∂xq∂xn
V
Ci j kl(xm−xm)xqxnd3xm (16) We also consider the mathematical expressions:
Di j kl :=Di j kl(xm)=
V
Ci j kl(xm−xm)d3xm =K0, Bi j klq :=Bi j klq(xm)=
V
Ci j kl(xm−xm)xqd3xm, Ei j klqn :=Ei j klqn(xm)=
V
Ci j kl(xm−xm)xqxnd3xm (17) Thereby, Eq. (16) can be written as:
¯
ti j(xm)=Di j klekl(xm)+Bi j klqekl,q(xm)+Ei j klqnekl,qn(xm) (18) It will hold Bi j klq =0, provided that the material is centroid and symmetric.
Equation (18) will therefore give:
t¯i j(xm)=Di j klekl(xm)+Ei j klqnekl,qn(xm) (19) As regards the strain gradient elasticity theory, the equilibrium equation can be expressed as:
σpq =τpq−μr pq,r (20) whereσpq are the total stresses,τpq are the Cauchy stresses according to Mindlin andμr pq,rare the double stresses.
Based on the simplified theory (i.e., the form II of the strain gradient elasticity theory of Mindlin), we have:
μr pq =2τpq,r (21) where is a length scale which represents a material length associated with the surface elastic strain energy.
We thus obtain by combining Eq. (20) with Eq. (21):
σpq=τpq−2τpq,rr (22) We next consider the simplified form:
Ei j klqn= −2δqnDi j kl (23)
whereδqnis Kronecker delta.
Equation (19) can be then written by inserting Eq. (23):
¯
ti j(xm)=Di j klekl(xm)−2δqnDi j klekl,qn(xm) (24)
or t¯i j(xm)=Di j klekl(xm)−2Di j klekl,rr(xm) (25)
Equation (25) will be identical to Eq. (22) provided thatτkl =Di j klekl(xm).