4.1. Constitutive Law
We choose a very simple constitutive law for our solid material, which is the de Saint-Venant strain energy function given by:
where
is described by different parameters depending on the solid domain that it is describing and
is the Green-Lagrangian strain tensor for each solid domain, that is,
We adopt the notation that:
and the subscript
, where
is the subphase and
is the matrix throughout this section. We have that the expansion of
is given by:
and we note that
is the fourth rank elasticity tensor with major and minor symmetries, namely:
We can now determine
. We have:
where the subscript
denotes the subphase and the matrix. We now apply the asymptotic homogenization technique to (175). Using the transformation of the gradient operator (70):
we can write
as:
Then, using (177), we are able to write the expansion of the strain energy function
as:
Multiplying (178) by
, we obtain:
We can then equate the coefficients of
,
and
in (179). For
, we have:
Equating the coefficients of
gives:
and finally, equating the coefficients of
gives:
where we have that:
Therefore, we have that the leading order term of the constitutive law in the subphase and the matrix respectively are:
Now that we have an expression for
, we can use this to find the leading order term of the second Piola–Kirchoff stress, that is we take the derivatives of (184) with respect to
and
, respectively. Therefore, we have:
where we have a different second Piola–Kirchoff stress for both the subphase and the matrix, respectively, that is,
We now wish to linearise the second Piola–Kirchoff stress
. To do this, we can use the expression
where:
Therefore, carrying out the linearisation, we have that:
Ignoring the nonlinear terms, we have:
We are now able to use the second Piola–Kirchoff stress
to find the first Piola–Kirchoff stress:
where we can define the first Piola stress
in both of the solid constituents, the subphase and matrix, as:
We can also linearise the first Piola–Kirchoff stress
. Therefore, we have:
and since we have that
is major and minor symmetric, then we can write this as:
We therefore can define the solid stresses in the subphase and matrix as:
respectively, for our chosen constitutive law. We can then use these expressions in the problem for
and
given by Equations (164)–(169), that is,
The problem given by (197)–(202) admits a unique solution up to a
-constant function. Exploiting linearity, the solution is given as:
where
and
are
-constant functions. The third rank tensors
and
are the solutions of the pore-scale problems given by:
and the vectors
and
are the solution to this pore-scale problem:
Both (205)–(210) and (211)–(216) are to be solved on the periodic cell. To ensure the uniqueness of the solution, we also require a further condition on
,
,
and
, for example:
Here, we wish to discuss in detail the cell problems (143)–(145), (205)–(210) and (211)–(216) and how they can potentially be solved. These pore-scale periodic cell problems are to be solved to determine the model coefficients of the final macroscale model. It is through these model coefficients that the complexity of the materials microstructure is encoded in the final model.
In general, with the asymptotic homogenization technique, these cell problems would only depend on the pore-scale and therefore can be solved in a straight-forward way. For example, solving the pore-scale asymptotic homogenization cell problems for linear elastic composites was carried out in [
44], and for linear poroelasticity, the problems were solved in [
40]. Similarly, it would be possible to solve the cell problems arising from linear poroelastic composites by combining the techniques used in both of these previous works. In the linear case, we have the problems (143)–(145), (205)–(210) and (211)–(216) with the simplification that
approaches the identity. This simplification means that the two scales are fully decoupled, and we can solve the fluid and the elastic-type cell problems.
However, due to the nonlinearity of the system we consider here, the two scales are coupled, meaning that the pore-scale periodic cell problems have a dependence on the macroscale and therefore cannot be easily solved. This dependence is through the quantity appearing in (143)–(145) and (215)–(216). This quantity is the Piola transform, which involves the leading order deformation gradient . This depends on both the pore-scale and the macroscale, as can be seen in Equations (120) and (121). This means that the two scales are not fully decoupled, and therefore, this dramatically increases the computational complexity.
It is however crucial for a realistic analysis of the scenarios of interest (such as biological tissues) to be able to solve problems of this type. Despite the complexity, there are some potential emerging techniques that may mean it would be possible to solve this model numerically in the future. A recent example of a proposed method that could be potentially used to solve the types of problems arising in this work is found in [
45]. This work investigated the potential of using Artificial Neural Networks (ANNs) for quick, accurate upscaling and localisation of problems. The method involves an incremental numerical approach where there is a rearrangement of the cell properties relating to the current deformation, and this means that there is a remodelling of the macroscopic model after each incremental time step. This method is applicable to finite strain and large deformation problems, whilst there will only be infinitesimal deformation within each incremental time step. Reference [
45] investigated the full effects of the coupling between the macroscale and microscale for the first time in the analysis of fluid-saturated porous media. We believe that by following an approach similar to the one set out in [
45], we could obtain a solution to our model numerically.
We can use our expressions (203) and (204) for
and
to rewrite
and
. We have:
and:
where we define the pore-scale gradients of the auxiliary variables as:
Then, we can return to (163) and use our linearised solid stresses to find the effective stress:
As mentioned in
Section 3.4, we return to the expression (157), restated here for convenience,
We obtain expressions for
and
by taking the time derivative of (203) and (204), and we then substitute these expressions into (222) to obtain:
Expanding the LHS in (223) and using (220), we obtain:
Expanding the second term on the LHS further and rearranging, we obtain:
Rearranging and collecting
terms gives:
We define:
and:
and we can then use (227) and (228) to rewrite (226) as:
Finally, we can divide by
M to obtain:
We therefore have now derived the effective macroscale governing equations for a nonlin