4.3.1. Periodic material
The homogenisation theory will be outlined here for a linear elastic material in which small strains are assumed and geometrical nonlinear effects are neglected.
Only basic features of the homogenisation method will be treated in this section.
For a more complete treatment of the subject, see for example Bensoussan [6]. It is assumed that the microstructure of the body is periodic and that the material is composed of subcells of equal shape and equal material properties. A subcell can then be chosen as a representative volume element that is repeated throughout the body as illustrated in Figure 4.2.
m
D D
f
Figure 4.2: A periodic structure composed of two different materials, Df and Dm. The representative volume element is shown in the upper right.
Each representative volume element with the volume Ω in the body has the same shape and the same material properties Dijkl = Dijkl(x, y, z) with respect to the local coordinate systems. In the undeformed and deformed configurations, adjacent base cells must always fit together at the boundaries. Moreover, for a homogeneous
stress field, cells lying far from the boundaries are subjected to the same loading conditions and will deform in the same manner. The possible shape of the cells in undeformed and deformed configurations are limited in such a way, therefore, that the boundaries of opposing sides of the base cell always have the same shape.
As an introductory example a two-dimensional case is studied in which the base cell consists basically of four corners connected by four curves. Figure 4.3 shows such a rectangular base cell in an undeformed and a deformed state with the material coordinates (x, y), displacements (u, v) and length of the sides (lx, ly).
l
Undeformed base cell
Deformed base cell
x
y,v
x,u
ly
Figure 4.3: Two-dimensional base cell in undeformed and deformed state.
The identical shapes and sizes of opposite boundary surfaces of the base cell allow relations between displacements on the boundaries to be established. Although the shape of the base cell can arbitrarily be considered periodic, for simplicity the shape is assumed here to be a right prism in an undeformed state, yielding
ui(lx, y, z) = ui(0, y, z) + Ai1lx
ui(x, ly, z) = ui(x, 0, z) + Ai2ly i = 1, 2, 3 ui(x, y, lz) = ui(x, y, 0) + Ai3lz
(4.7)
where Aij represents nine constants. In Eq.(4.7) the notations u1 = u, u2 = v and u3 = w can be used and in the same way x1 = x, x2 = y, x3 = z, l1 = lx, l2 = ly
and l3 = lz where li denotes the base cell length in direction xi. In accordance with
Eq.(4.7), the elements Aij in the matrix A are
The column j of A refers to the bounding surface normal to the xj-direction.
Since all the cells are identical and the outward normals nj to the base cell have opposite signs on opposing sides of the base cell, the tractions ti = σjinj are also opposite on opposing sides of the base cell. Accordingly, the tractions ti are termed anti-periodic and one can write
ti(lx, y, z) =−ti(0, y, z) ti(x, ly, z) =−ti(x, 0, z) ti(x, y, lz) = −ti(x, y, 0)
(4.9)
With the periodicity introduced, the equilibrium equation Eq.(4.1) for a linear elastic case involving infinitesimal displacements and no body forces can be written as
over the base cell volume Ω, with cyclic boundary conditions on the boundary surface Γ. Adopting the Galerkin method, where vi = ui, and assuming no body forces, the weak formulation of Eq.(4.10) can be written, according to Eq.(4.4) as
Z
the boundary constraints for the displacements being in accordance with Eq.(4.7).
The boundary Γ can be split up into three parts Γ = 2Γ1 + 2Γ2 + 2Γ3 where 2Γj contains the two surfaces normal to the xj-direction. This allows Eq.(4.11) to be written as
where Γj is one of the two surfaces normal to the xj-direction. It should be ob-served that ∆ui in Eq.(4.12) has different constant values for j = 1, 2, 3. According Eq.(4.8), one can write
∆ui = Aijl<j> (4.13)
where the bracketed index indicates that the summation convention is not applied
Still assuming the base cell to be a right prism in the undeformed state, the average stresses (tractions) on the boundary surfaces can be defined as
Bij = 1 Γi
Z
Γ<i>
tj dΓ (4.15)
where Γi refers to a surface normal to the xi-direction. Eq.(4.14) can then be
rewritten as Z
Ω
∂ui
∂xj
σij dΩ = ΩAijBij (4.16)
where the base cell volume Ω= lxlylz. For small macroscale deformations, the aver-age strains can be defined as
¯ij = 1
2(Aij + Aji) (4.17)
and, correspondingly, the average stresses as
¯ σij = 1
2(Bij + Bji) (4.18)
The expressions for the average strains and stresses are then substituted into Eq.(4.16) and since Bij is symmetric one obtains
Z
Ω
∂ui
∂xj
σij dΩ = Ω ¯ij¯σij (4.19) If local cavities appear in the base cell, they are treated as consisting of a material having zero stiffness.
4.3.2. Equivalent stiffness and hygroexpansion properties
A linear elastic material including hygroexpansion is assumed. The constitutive relations for such a material are
σij = Dijkl(kl− skl) (4.20) The hygroexpansion strains skl may be expressed as
skl= αkl∆w (4.21)
where αkl represents the hygroexpansion coefficients and ∆w is the change in mois-ture content. Due to the inhomogeneous material properties of the base cell, both Dijkl and αkl can vary markedly in space.
The material within the base cell is now assumed to be replaced by an equivalent fictitious material with constant stiffness and hygroexpansion properties denoted by D¯ijkland ¯αkl, respectively. With constant moisture content, the constitutive relation becomes
¯
σij = ¯Dijkl¯kl (4.22)
having the inverse relationship
¯ij = ¯Cijklσ¯kl (4.23)
A simple way of defining the constitutive parameters in Eqs.(4.22) and (4.23) is to choose six elementary cases of either stress or strain states. In the case of stress states chosen, the corresponding stress tensors are
Note that, due to the unit stress value found for each of the six cases in Eq.(4.24), one can identify that
C¯ijkl = ¯ij for k = l C¯ijkl = ¯Cijlk = 1
2¯ij for k6= l (4.25)
For the shrinkage parameters, the averaging can be performed in a similar way.
Prescribing for a seventh case that ¯σkl = 0 for all k and l and that ∆w = 1 yields the relation
¯
αkl= ¯kl (4.26)
Thus, the constitutive parameters can be identified on the basis of seven elementary cases involving either prescribed average tractions or a change in moisture content.