Unified Periodic Boundary Conditions

Consider a large sample of periodic inhomogeneous body (composite) as shown in Fig. 1. The body has two length scales, a global (macroscopic) length scale, D, which is of the order of the size of the body, and a local (microscopic) length scale, d, which is proportional to the wavelength of the variation of the microstructure. Correspondingly

8 = d / D << 1

Obviously, any function f in the body depends on two variables, global (macroscopic) coordinates Xi for the body and the local (microscopic) coordinates xt for the unit cell. For small strain elasticity, the boundary value problem for the composite body can be defined as the following, from which the unknown field quantities, stress Oj, strain £iJ and displacements ui can be solved:

, j = 0

0 ij Cijkl£kl

£kl = 2(uk, l + ul, k)

°ijnj = Ti on,

The boundary value problem has the feature that CiJtl varies very rapidly within a short wavelength (order of d) on the global length scale Xi and therefore it is difficult to find a solution that solves the global problem and accounts for the local oscillation at the same time. For example, in a FEM solution, assuming roughly each unit cell should have several hundred elements to accurately capture the large variations due to the heterogeneity of the microstructure, then for the entire composite laminate or structure, the number of elements needed will increase by several orders. Hence, there is a motivation to seek a simplified solution. Since the composite body can be envisioned as a periodical array of the RUCs, it is adequate to obtain a solution based on an individual RUC. This implies that beyond a boundary layer of the composite body, each RUC in the composite has the same deformation mode and there is no separation or overlap between the neighbouring RUCs. That is, the stress and strain fields are periodic as the microstructure (Suquet, 1987).

Since the whole body, thus each unit cell is in balance, the equilibrium equation, Eqn. (2) and Eqns.

(3) -(4) still apply in a RUC with volume V. However, the boundary conditions on the boundary of the RUC, d, should be properly determined. In the case of periodic media, the microscopic fields have to fulfill suitable periodicity conditions ensuring continuity of boundary displacements and tractions across adjacent cells. According to Suquet (1987), the displacement field for a periodic structure can be expressed as:

ut (xj, x2, x3) = £ijxj + ui(xl, x2, x3)

In the above, £j is the global strain tensor of the periodic structure and the first term on the right side

represents a linear distributed displacement field. The second term on the right side is a periodic function from one RUC to another. In addition, for a periodic RUC, the tractions on the opposite boundary surfaces should also meet the continuity condition, i. e.

Gy (P)*j (P) = – Oy (Q)nj (Q) (7)

where P and Q are two periodic points (with the same in-plane coordinates) on the two opposite boundary surfaces, n is the unit outward normal vector to the surfaces, see Fig. 1. Note that the global strain in Eqn. (6) and the corresponding global stress G tj can be defined as the averages over the RUC volume V:

where Gj(Xj, x2, x3) and Єі}(x1, x2, x3) are local (microscopic) stress and strain defined in the RUC.

Equations (6) and (7) are the periodic boundary conditions for a RUC. Together with the Eqns. (2), (3) and (4), we complete the boundary value problem for the RUC. This boundary value problem is well – posed, as shown e. g. in Suquet (1987). However, the periodic part of displacement, u*(x1, x2,x3), in Eqn. (6) is usually unknown prior to the solution, thus it is not convenient to apply Eqn. (6) directly as displacement boundary conditions. In Xia et al. (2003a), a unified form of periodic boundary conditions for any multiaxial loading and suitable for finite element analysis is developed. For the sake of brevity only the conclusions are cited here. Interested readers are encouraged to consult the reference for further details and illustrative examples.

For any RUC, its boundary surfaces must always appear in parallel pairs, the displacements of two periodic points on a pair of parallel opposite boundary surfaces can be written as

Ui (P) = ЄijXj (P) + u*(P)

(10)

Uі (Q) =єijxj (Q) + u* (Q)

(11)

In the above “P” and “Q” identify the two corresponding points on a pair of two opposite parallel boundary surfaces of the RUC, see Fig. 1.

Note that u* (x1,x2,x3) is a periodic function, i. e. its value is the same for two corresponding points at the two parallel boundaries (periodicity), therefore, the difference between the above two equations is

Ui (P) – Ui (Q) = Єу [xj (P) – xj (Q)] = ЄуAxj (12)

Since Axj = xj (P) – xj (Q) are constants for each pair of the parallel boundary surfaces, hence, for a specified Є j, the right side becomes constants. Equation (12) is a special type of displacement boundary conditions. Instead of giving known values of boundary displacements, it specifies the
displacement-differences between two periodic points at opposite boundaries. Obviously, the application of Eqn. (12) will guarantee the continuity of displacement field, i. e. the neighboring RUCs cannot separate or encroach into each other at the boundaries after the deformation.

Furthermore, it has been proved in Xia et al. (2006) that if a RUC is analyzed by using a displacement – based finite element method, the application of only Eqn. (12) can guarantee the uniqueness of the solution and thus Eqn. (7) are automatically satisfied. For the sake of brevity only the conclusions are cited here.

Theorem: In a displacement-based FEM analysis, by applying the unified displacement-difference periodic boundary conditions on a RUC, a unique solution is obtained.

Lemma 1 For a fixed periodic structure, different RUC’s may be defined, however, by applying the unified displacement-difference periodic boundary conditions, Eqn. (12), in the displacement-based FEM analysis, the solution will be independent of the choice of the RUCs.

Lemma 2 The solution obtained by applying the unified displacement-difference periodic boundary conditions, Eqn. (12), in the displacement-based FEM analysis, will also meet the traction continuity conditions, Eqn. (7).

It is noted that the unified periodic boundary conditions, Eqn. (12), can be easily applied in a FEM as the nodal displacement constraint equations. It is also noted that the proposed unified periodic boundary conditions are in the form of global strains. In the case of given global stresses, or a combination of the global stresses and strains, a proper proportion between the global strains must be applied. The required proportion can be determined without any difficulty through an iterative procedure, see the examples to follow.

One can also note that the derivation and proof procedures for the proposed unified periodic boundary conditions are not dependent on the properties of the constituent materials of the composites. Therefore, they can be applied to nonlinear micro/meso-mechanical analyses of the composites under any combination of multiaxial loads.