3.2.2. Computational Homogenization of Mechanical Properties
In order to predict and complement the stiffness increase which has been experimentally observed of the reinforced matrix, a method that combines the image analysis presented above with computational homogenization was developed to assess the effective elastic response of the graphene enhanced polymer.
As alluded to in
Figure 1f,g, the nano-structures of the images were embedded 1D elements, cf. in [
20], whereas the neat PE resin represents the bulk of the image area. In order to predict the theoretical increase in the stiffness of the overall composite, a finite element model-based method was developed. The effective elastic representation of the 2-D nano-structure-enhanced composite was extracted from computational homogenization, cf. in [
2,
23] in this method. In this homogenization, the 2-D GnP embedded in the neat resin were considered as 1-D reinforcements resolved via the image analysis method presented above.
Consider the 1-D resolved embedded GnP from the image in a Representative Area Element (RAE), as shown in
Figure 2. The square region of the RAE is 𝛺□, of width and breadth 𝑎 = 30 μm, with embedded GnP considered as 1-D elements is shown in
Figure 1b. The embedded 1-D elements have individual length (2–15 μm), thickness, orientation, and discrete locations given by the image analysis in
Figure 1f. From the image analysis, the average GnP length and thickness are
l and
t, respectively. Considering the 1-D element in detail, which can be viewed in
Figure 2b, the natural coordinate
s is introduced, running along the 1-D structure with tangent vector 𝒏 in the integration interval 𝐼. The “stress” response is governed by the axial force 𝑁. It is assumed that the RAE plate has thickness
l and that plane strain conditions holds.
For the homogenization, we consider the classical scheme of Suquet [
24] stating the virtual work equivalence between the macroscopic and the microscopic fields in the RAE formulated as
where
is the area of 𝛺□ and
is the number of 1D elements identified from the image analysis. It is further assumed that there is perfect bonding between the GnPs and the polymer, whereby the normal GnP strain
is affine with the straining of the polymer, i.e.,
. Moreover, in Equation (1) the sub-index
m relates to quantities at the macro-level, whereby
is the homogenized macro stress and
is the virtual strain at the macro level. On the micro-level,
σ is the microscopic stress of the polymer,
N is axial force of the 1-D resolved GnP, cf.
Figure 2, and
ε is the strain tensor at the micro level.
In order to link the macro- and micro-stress responses, the strain at the micro-level
is subdivided in terms of a subscale strain
defined by
, where
is the fluctuating displacement field of the subscale. Upon combining this with Equation (1), the homogenized macro-stress of the GnP enhanced polymer is obtained as
corresponding to self-balancing of the micro-field
In addition, the condition for kinematic macro-homogeneity can be written as
where
is the boundary of the region
as shown in
Figure 1. Upon assuming Dirichlet boundary conditions
along
, indeed the kinematic macro-homogeneity is fulfilled.
The balance relations in Equation (3) is solved using finite elements, approximating the fluctuation field
so that the polymer bulk is discretized using standard 2-D bilinear elements. To kinematically couple the 1-D elements to the bulk, the GnP structures are located at the common face between two adjacent bulk elements, cf.
Figure 1g. In this development, elastic response is assumed for the bulk and the 1-D elements so that
where
and
are the modulus of elasticity and Poisson’s ratio of the bulk and
is the axial stiffness of the 1-D elements. Please note that
is the given macroscopic strain and
is the FE-resolved microscale displacement field from the balance relations in Equation (3). As alluded to in Equation (4), Dirichlet boundary conditions are selected for
.
The elastic modulus of the GnP (
) is used to calculate the axial stiffness
whereby
is defined in terms of the average depth
l through the RAE plane, cf.
Figure 2, and the average thickness
t of the GnPs. In order to validate the procedure problem, the stiffness properties
,
, and
are compared with the experimental measured data. Current results are in the form of basic stiffness response of components
and
. It should be noted that
has another dimension as compared to
.
For the homogenization, the mesh was auto-created with the extracted GnP from the images mapped onto 2-D surfaces using the Abaqus CAE [
25] surface meshing tool (
Figure 1g). The mesh for the matrix, GnPs, and boundaries were then assigned to sets using scripts that were developed to be suitable for running in batch format for large data sets. From parameter studies using the computational homogenization, it was noted that GnPs with lengths less than 2 µm had a negligible effect on resultant stiffness during the homogenization procedure and were therefore omitted. Unit displacement was applied in the vertical direction, the horizontal direction, and in shear in order to extract the corresponding homogenized material response. An example can be seen in
Figure 1h of the deformation due to the applied extension in the vertical direction. The effective elastic stiffness of the matrix can then be used for component design of e.g., carbon fiber reinforced polymers (CFRP)s, via upscaling the matrix to the ply level response using standard methods.