3.2 Homogenization estimates for two-phase particulate DECs with ran-
3.2.3 The partial decoupling strategy and approximation
In this section we provide a partial decoupling strategy to find the effective electro- elastic response of DECs described above. As we have seen in section2.2, the potential energy for the ideal dielectric matrix and rigid inclusions can be split into a purely mechanical contributionWme(r)(F)and an electrostatic contributionW(
r)
el (F,D). By
making use of the corresponding decompositions for W(1) and W(2) in expression (3.4) for the local energy of the composite, we obtain
W(X,F,D) =Wme(X,F)+Wel(X,F,D), (3.26) where Wme(X,F) = 2 ∑ r=1 Θ(0r)(X)Wme(r)(F) and Wel(X,F,D) = 2 ∑ r=1 Θ0(r)(X)Wel(r)(F,D). (3.27) Substituting the decomposition (3.26) into expression (3.3) for the homogenized po- tential, we obtain ˜ W(F¯,D¯) = min F∈K(F¯) min D∈D0(D¯){⟨ Wme(X,F)⟩0+⟨Wel(X,F,D)⟩0}, (3.28)
where the admissible sets K (F¯) and D0(D¯) are defined as before, and ⟨⋅⟩0 denotes the volume average in the reference configuration. It is observed that the first term on the right side of (3.28) is independent of the electric displacement field D. Therefore, we can rewrite the variational problem (3.28) as follows
˜ W(F¯,D¯) = min F∈K(F¯){⟨ Wme(X,F)⟩0+W˜el(D¯;F)}, (3.29) where ˜ Wel(D¯;F) = min D∈D0(D¯)⟨ Wel(X,F,D)⟩0 (3.30)
is the homogenized electrostatic energy for a given (fixed) deformation field F. It is important to emphasize that both terms on the right side of (3.29) depend on the
trial deformation fieldF, and therefore the mechanical and electrostatic energies are coupled together and cannot be separated, in general. Thus, to make the depen- dence of ˜Wel(D¯;F) on the deformation more transparent, it is useful to rewrite the
homogenized electrostatic energy in the current configuration, i.e., ˜
Wel(D¯;F) =J¯w˜el(d¯;F), (3.31)
where ¯J=det ¯F and
˜ wel(d¯;F) = min d∈D(¯d)⟨ Θ(1)(x)w(1) el (d)+Θ( 1)(x)w(2) el (R¯( 2),d)⟩. (3.32) In the above expression wel(1) and wel(2)are given by (2.16) and (2.22), respectively, ⟨⋅⟩ is used to denote the volume average in the current configuration, and
D (d¯) = {d∣divd=0in Ω, d⋅n=d¯⋅non∂Ω} (3.33) is the admissible set for the Eulerian electric displacement field d.
Given the assumptions of subsection 3.2.2for the evolution of the microstructure, it can be seen that the deformed characteristic functions Θ(1)(x) and Θ(2)(x) only depend on the macroscopic deformation ¯F and the average rotation of the inclusions
¯
R(2). Therefore, it is observed that by writing the homogenized electrostatic energy of the composite in its “more natural” Eulerian form the explicit dependence of
˜
wel on the local trial field F disappear. In other words, the homogenized Eulerian
electrostatic energy is seen to depend on the deformation only via the macroscopic deformation ¯F, which determines the shape of the distribution ellipses in the current configuration, and the average rotation of the inclusions ¯R(2), i.e.,
˜
wel(d¯;F) =w˜el(d¯,F¯; ¯R(2)). (3.34)
rewritten as follows ˜ W(F¯,D¯) =min ¯ R(2) {W˜me(F¯; ¯R(2))+J¯w˜ el(d¯,F¯; ¯R(2))}, (3.35) where ˜ Wme(F¯; ¯R(2)) = min F∈K′(F¯,R¯(2))⟨Wme(X,F)⟩0. (3.36) In this last expression, K′(F¯,R¯(2)) denotes the set of admissible deformations inside the matrix phase that satisfy the affine condition on the boundary of the specimen, as given by (3.1)1, and the prescribed rigid body motions at the interface of the inclusions with the matrix, as given by the rotation ¯R(2). It is seen that the variational problems (3.32) and (3.36) are decoupled from each other, for the given rotation of the the inclusions, which is in turn obtained form the outer minimization in (3.35). For this reason we refer to (3.35) as the “partial decoupling strategy”.
In summary, expression (3.35) for the effective energy ˜W(F¯,D¯) along with ex- pressions (3.32) and (3.36) for ˜wel and ˜Wme, show that for a given rotation ¯R(2), the
inner elastic and electrostatic problems can be solved independently of each other. Having solve these two decoupled variational problems, the outer minimization in (3.35) can be performed to obtain the equilibrium rotation ¯R(2) as a function of the macroscopic loading ¯F and ¯D, as well as the effective potential ˜W(F¯,D¯) for the DEC.
As it is clear from equation (3.35), performing the outer minimization, requires the knowledge of explicit expressions in terms of the prescribed rotation ¯R(2) for both
˜
Wme and ˜wel. While this is relatively simple for the electrostatic part of the effective
energy ˜wel, obtaining such explicit expressions for the effective mechanical energy,
is more difficult. For this reason, it is advantageous to make use of the “partial decoupling approximation” of Ponte Casta˜neda & Galipeau(2011), which makes use of the solution of the purely mechanical problem to obtain approximate estimates for the above general problem. Thus, letting ¯R(m2) denote the minimizer of the purely
mechanical problem ˜ Wme(F¯) =min ¯ R(2) ˜ Wme(F¯,R¯(2)), (3.37)
it follows from (3.35) that ˜
W(F¯,D¯) ≤W˜me(F¯; ¯R(m2))+J¯w˜el(d¯,F¯; ¯R(m2)). (3.38) Note that the right side of the inequality (3.38) can be treated as an estimate for the effective energy ˜W(F¯,D¯), which can be shown to become more accurate as the magnitude of the elastic interactions becomes large compared to the electrostatic in- teractions. Such conditions are expected to be met for small electric fields or for (mechanically) stiff matrix materials. A more rigorous argument for the above state- ment will be provided later on in this thesis. Finally, it is worthwhile to mention that the inequality (3.38) becomes an equality for the special case of DECs with aligned microstructure and under aligned mechanical and electrostatic loading conditions. This is because of the fact that under aligned loading conditions and when the mi- crostructure of the composite is also aligned with the loading, the average rotation of the inclusions is zero (i.e., ¯R(2)=I).