Computational Methods for Debonding in Composites

11 Prediction of Mechanical Properties of Composite Materials by AEH 229
Moreover, the deformation microstructural field is defined by
where
ε (1)
ij (x,y)=Tkl
�
ij
T kl
ij
I mn
kl
∂χmn
k −
∂yl
1 � �
= δikδ jl + δilδ jk
2
� ∂u (0)
m
∂xn
(11.20)
(11.21)
δij is the Kronecker delta. For a given point on the macroscale x, Eqs. (11.19)–
(11.20) are used to calculate approximate values of the stress and deformation fields,
respectively, within the heterogeneous material. In contrast, the homogenised stress
field, as it is the average value of the microstructural stresses σ (1)
ij in Y, is unable to
represent any microstructural fluctuations of the stress field.
11.3 Finite Element Method in AEH
11.3.1 Corrector χχχ
The solution of Eq. (11.12) is called corrector (χχχ) and contains the eigendeformations
of the representative periodic geometry [5]. The element strain and stress
matrices are ε = Bu and σ = DBu, respectively, where all the variables belong
to the microscale problem, i.e. are relative to the geometry and material of the RUC.
Therefore, the finite element approach to Eq. (11.12) results in
�
�
(11.22)
Y e BT DBdY χχχ =
Y e BT DdY = F D
where the index e denotes element quantities from the meshed unit-cell domain
(body Y) [6]. It is worthwhile to note that the corrector χχχ is a matrix, not a vector.
The second term of Eq. (11.22) consists on the columns of the matrix F D [6], which
consist on six load vectors, leading to the same number of systems of equations
to solve. The results are solutions that make up the corrector, each one defining
an eigendeformation mode. Moreover, the definition of matrix F D shows that the
force vectors appear from the integration of the gradient of elastic properties of the
material components that form the composite material.
11.3.2 Periodicity Boundary Conditions
Periodicity boundary conditions are imposed over the surface boundaries of the
RUC. For a hexahedral unit-cell in y1 ∈ [0,y 0 1 ], y2 ∈ [0,y 0 2 ] and y3 ∈ [0,y 0 3 ],the
boundary conditions can be defined as follows: