A note on the discrete approach for generalized continuum models

1 Introduction

The purpose of the present article is to consider the implications of a graphical representation to models of gradient elasticity [1–3], proposed to generalize the classical Hooke’s law. Starting with a discrete version of “dual” elasticity models and assuming a specific topology for the two elastic chains, we can obtain certain algebraic restrictions for the phenomenological coefficients of the continuum counterparts of the discrete models.

It is shown that, depending on the assumed topology and interaction of the two elastic chains, the sign of the new phenomenological coefficients may be positive or negative with direct implications on uniqueness and stability of the solution of related boundary value problems.

While the foundations of graph theory can be traced back to L. Euler and his “Königsberg bridge problem” [4], its growth in recent years has been explosive, covering a large number of disciplines ranging from mathematical foundations of computer science [5] to physical chemistry [6] and natural language processing [7]; only recently such developments have been applied to physics, engineering sciences and current technology problems [8], including elastic networks in manmade structures [9], and biological systems [10].

2 Graphical description of dual elasticity

We consider a one-dimensional medium composed of two interpenetrating elastic chains. The displacement of chain (or path) 1 is designated by u, the displacement of chain (or path) 2 is designated by v, and the overall average displacement by w, with all three of these quantities being measured (in general) with respect to different coordinate systems. For the representation of the average displacement w in terms of the two displacements u and v, we will employ mean square displacement relation, i.e., w²=c₁u²+c₂v² with c₁+c₂=1. When c₁=c₂ we obtain the ordinary mean square displacement. Now let x be the variable of the coordinate system of the average displacement w. Then, by taking the derivative of the mean square displacement relation with respect to x, we obtain:

where r1(x)=c1u(x)w(x) and r2(x)=c2v(x)w(x).

In the particular case that the quotients r₁ and r₂ are constants, we have

In a discretized version of such dual elasticity medium, we designate the corresponding particle displacements by u_n and v_n, n∈ℤ, and the associated spring stiffness by K_i (i=1, 2). It is also assumed that there exists an interaction force between the particles of the two superimposed lattice chains given by f_int=a(u-v), where a∈ℝ is a material coefficient. Assuming Hooke’s law, for each elastic chain, the spring-like forces acting on a particle with displacement u_n, because of the short-range interactions of particles on the same chain, are f₁=K₁(u_n-1-u_n), f₂=K₁(u_n-u_n+1) and, in combination, f₁-f₂=K₁(u_n+1-2u_n+u_n-1). Similarly the elastic forces acting on particle v_n are f₃=K₂(v_n-1-v_n), f₄=K₂(v_n-v_n+1) which gives f₃-f₄=K₂(v_n+1-2v_n+v_n-1). Equilibrium requires f₁-f₂=∂σ₁/∂x and f₃-f₄=∂σ₂/∂x, where σ₁ and σ₂ denote the stresses in each chain. Hence, for the interacting forces between particles u_n and v_n, we have ∂σ₁/∂x=f_int=a(u_n-v_n) and ∂σ₂/∂x=-f_int=a(v_n-u_n). By combining the previous equations we obtain K₁(u_n+1-2u_n+u_n-1)=a(u_n-v_n) and K₂(v_n+1-2v_n+v_n-1)=a(v_n-u_n). To continualize these equations, u_n is replaced by the displacement of the continuum u(x), whereas the displacements of the neighboring particles u_n±1 are replaced by u_n±d, and the same is done for v_n. By using a Taylor series expansions for u(x±d) and v(x±d) and neglecting terms of order d³ and higher we obtain K₁d²u_xx=a(u-v) and K2d2vxx=a(v-u)⇒u=v-K2d2avxx. By solving this system of equations we arrive at Evxx-E1E2aAdvxxxx=0 and Euxx-E1E2aAduxxxx=0, where E_i are the Young’s moduli for the two elastic chains Ei=KidA,A is the cross-sectional area, d is the particle spacing and E=E₁+E₂.

In order to derive a corresponding governing equation for the average displacement w introduced previously, and in particular for the case that the coefficients r₁ and r₂ of Eq. (1) are constants, we use the last equations to obtain E(r1uxx+r2vxx)-E1E2aAd(r1uxxxx+r2vxxxx)=0, which, in view of Eq. (2), gives Ewxx-E1E2aAdwxxxx=0. Comparing this with a corresponding equation derived for single elasticity as reviewed in [3], i.e.,

we see that the two equations become identical for a=-12K₁K₂/(K₁+K₂). In the case that we consider also terms of order d³ and d⁴, by continualization we obtain K1(d2uxx+112d4uxxxx)=a(u-v) and K2(d2vxx+112d4vxxxx)=a(v-u) and by combining the two, we have (u_2x≡u_xx, u_4x≡u_xxxx, etc)

and similarly for v. The sign of u_4x is positive if and only if (K₁+K₂)a>12K₁K₂, hence if K₁ and K₂ are linearly increasing then the value of a over which the sign of u_4x is positive is also linearly increasing. Below that value the sign is negative and when the equality a=12K₁K₂/(K₁+K₂) holds, then the higher order term u_4x vanishes. Now if we assume that the coefficients r₁ and r₂ of Eq. (1) are constants, by Eq. (4) and by neglecting terms of order d⁴ and higher we obtain

where w is the average displacement. For the coefficient of the fourth order term we have

(K1+K2)a-12K1K212a(K1+K2)d2=(K1+K2)α-12K1K2dA12α(K1+K2)d2=d212-E1E2αE

where α=aAd and Ei=KidA,E=E₁+E₂. Hence, Eq. (5) takes the form E(wxx+(d212-E1E2αE)w4x)=0. We observe that Eq. (3) is obtained from the last equation for E₁E₂=0⇒E₁=0 or E₂=0, i.e., by returning to the single elastic chain model, as also indicated by Eq. (3).

The potential energies for the two elastic springs as a result of the short-range interaction forces between particles of the same chain are Epu=12K1(un+1-un)2 and Epv=12K2(vn+1-vn)2 and the work expressions associated with the interacting forces are W1=f1un=aun2-aunvn and W2=f2vn=avn2-aunvn. By continualization we obtain Epu=12K1d2(ux)2=12K1d2(ε1)2,Epv=12K2d2(vx)2=12K2d2(ε2)2 and W=W₁+W₂=a(u-v)². Hence the total energy for the unit shell is U=12K1d2(ε1)2+12K2d2(ε2)2+a(u-v)2, where ε₁ and ε₂ denote the local individual chain strain. The energy per unit volume is

Macroscopically, if we assume an average Young’s modulus E and an average strain ε, the elastic potential energy per unit volume is given by Uv=12Eε2. Notice that for K₁=K₂ and u=v (i.e., one elastic medium), Eq. (6) becomes identical to this equation. Next, we consider the average displacement, as given by the mean square displacement (c1=c2=12) and that the average spring constant is K. Then the average energy for the unit cell is:

U=12Kwn2=12K(un+vn2)2+12K(un-vn2)2=18K(un+vn)2+18K(un-vn)2=d22Kεn2+18K(un-vn)2

where the average strain for the unit cell is taken as εn=un+vn2d. For a continuous medium the elastic potential energy per unit volume is given by

Now, in order to relate the apparent macroscopic strain with the partial strains of the two elastic chains we apply Hooke’s law for the forces F_A and F_B acting on the chains, i.e., FA=E1ε1=K1dAε1,FB=E2ε2=K2dAε2, where E_i, K_i, ε_i, d, A are as before and we also assume that the particle spacing and the cross-sectional area are the same for both elastic chains. As the two chains are assumed to be in a parallel arrangement, their combined spring stiffness is K=K₁+K₂. Hence, again by Hooke’s law, the overall force applied to both paths F_ov is F_ov=Eε=(K₁+K₂)dε/A, where E is their combined average apparent Young’s modulus and ε is their combined average apparent strain. Now the energy stored is given by the integrals of the corresponding forces ∫FAdε1=K1dAε122,∫FBdε2=K2dAε222 and ∫Fovdε=(K1+K2)dAε22 and as ∫Foνdε=∫FAdε1+∫FBdε2, we obtain the following relation between apparent and individual chain strains ε2=(K1ε12+K2ε22)/(K1+K2). In view of Eq. (6), we have

Uv=d4AK1ε12+d4AK2ε22+a2Ad(u-v)2=d(K1+K2)4AK1ε12+K2ε22K1+K2+a2Ad(u-v)2=d(K1+K2)4Aε2+a2Ad(u-v)2.

By comparing it with Eq. (7), we obtain K=K₁+K₂ and a=(K₁+K₂)/8. As a result of the above exercise, we may conclude that various relations on phenomenological coefficients of continuum macroscopic constitutive models are possible when particular assumptions are made for the local topology of their discrete counterparts. Graph theory may be a useful tool to employ in relation to three-dimensional constitutive models. The assumption of mean square displacement adopted herein is a bit unusual but it was made specifically to illustrate the various possibilities that can be explored.

3 Conclusion

An outline on the use of discrete modeling procedures, employing special assumptions on local topology, for deducing microscopic relations for the phenomenological coefficients of generalized continuum elasticity models is given. As the examples treated are of gradient type, a somewhat unusual definition of mean particle displacement, resembling that commonly utilized in discrete diffusion models, is adopted. The approach is not sensitive to that definition and corresponding results for more conventional expressions for the mean particle displacements for both deformation and diffusion problems will be given elsewhere. It is shown that the sign of the gradient coefficient in Aifantis gradient elasticity model (GRADELA) may be positive or negative, depending on the microscopic configuration and the interaction laws between particles of the same chain, as well as particles of different chains.