Microstructure evolution

which serves to characterize the “shape” and “orientation” of the random “ellipsoidal” distribution of the particle centers (see dashed lines around the ellipsoidal inclusions in Fig. 3.1). An ellipsoidal distribution implies a particular correlation between the an- gular and radial dependence of the two-point probability function. It can be thought of as an affine deformation of a random set of points from astatistically isotropic dis- tribution, which corresponds to the special case where ZD₀ =I, so that pII

0 depends

onZ−Z0 only through its magnitude|Z−Z0|. It should be emphasized that particle distributions need not be ellipsoidal and will in general exhibit independent angular and a radial dependences. However, if the microstructure can be approximated as being ellipsoidal, then simple analytical estimates may be given (see Ponte Casta˜neda and Willis (1995)) for the homogenized linear response depending only on the particle volume fraction cI₀, the particle shape and orientation, as specified by ZI₀, and the distribution shape and orientation, as specified by ZD₀ , and hence the motivation for adopting these microstructural hypotheses for the MREs of interest in this work.

3.3.2

Microstructure evolution

As illustrated in Fig. 3.1, the microstructure is expected to evolve as the deforma- tion and magnetic fields are applied: the volume fraction of the particles (assum- ing that the matrix material can accommodate non-isochoric deformations), as well as the orientation and distribution of the particles, will change because of the ap- plied mechanical and magnetic fields. However, the characterization of the evolution of this microstructure is a formidable problem which has not been resolved in full generality—even for purely mechanical constitutive behaviors. For dilute concentra- tions of deformable particles in a linearly viscous material, a theory is available from the work of Eshelby (1957). A generalization of this theory for viscoplastic compos- ites with particulate microstructures, which is valid approximately beyond the dilute range, has been given in a sequence of papers by Ponte Casta˜neda and Zaidman (1994); Kailasam et al. (1997); Kailasam and Ponte Casta˜neda (1998). In the context of finite elasticity, the problem is even more difficult and exact solutions are not avail- able even in the dilute limit. However, an approximate theory for moderate particle

concentrations has been given by Lopez-Pamies and Ponte Casta˜neda (2006a). This theory has been found to predict the microstructure evolution with good accuracy for dilute particle concentrations (Michel et al., 2010). In this subsection, anapproximate

theory is proposed for the evolution of the microstructure in MREs, building on the above-mentioned earlier works.

First of all, it is noted that, if the overall deformation includes a hydrostatic component ( ¯J 6= 1), the particle number density pI₀ and the corresponding particle volume fraction cI₀ will change with the deformation. However, it follows from the mass conservation equation, and the fact that the particles are rigid and therefore incompressible, that the particle volume fraction in the deformed configuration will be given by cI _=cI

0/J¯.

Under the applied deformation ¯F and magnetic induction field B, the particles will also rotate and change relative positions, but they will not change their shape (or size), as they are rigid. Therefore, the microstructure in the deformed configuration can be described in terms of expressions of the form

Θ(2)(x) = Z

Ω

ΘI(x−z)Ψ(z)dz, (3.33)

where ΘI _{is the characteristic function of the rotated inclusion, as defined by the}

rotated ellipsoid ΩI = x|x·ZI TZI −1 x≤1 , (3.34)

where ZI = ZI₀RI T is a (non-symmetric) second-order tensor describing the (fixed) shape and (new) orientation of the inclusion, as described by the inclusion rotation

RI induced by the deformation fieldF (which in turn will depend on both the macro- scopically applied deformation ¯F and magnetic induction fieldB). In this connection, it should be noted that all the fibers will be assumed to rotate with the same tensor

RI, which will be identified further below with the average rotation of the particles as determined by the homogenization procedure. This is clearly an approximation that neglects possible near-neighbor inclusion interactions, and would be strictly valid only in the dilute limit (when the particles do not interact). However, in the spirit of a homogenization approach it is consistent to assume that “on the average” all the particles rotate with the average deformation in the inclusion phase. In addition, it should be noted that although, in principle, distributions of orientations can be easily

considered in the context of a more general analysis, in practice, having to keep track of multiple inclusion orientations would complicate the derivations to follow, and in this first treatment of the problem, we prefer to make the simplifying assumption of perfectly aligned inclusions.

In an analogous fashion, the random positions of the particle centers is expected to evolve with the deformation, which has implications for the above-defined, two- point, probability density functions, pII₀ , for the distribution of the particles. Thus, it is clear that, at least in the dilute limit, this function will change with the macro- scopic deformation ¯F, since the particles will be convected with the deformation (see Kailasam et al. (1997) for an analogous hypotheses for particle-reinforced viscoplas- tic solids). At concentrated volume fractions, once again, neighboring particles will interact with one another, both magnetically and mechanically, and the positions of the particles will not simply be convected with the deformation. Thus, in general, it is expected that the evolution of the two-point probability density functions may depend on higher-order statistics, and that the assumed “ellipsoidal” symmetry will almost certainly be broken down. This will of course lead to significant complications in the characterization of the microstructures and the associated computation of the homogenized response, even for linear response. For this reason, we will make here the approximation that the two-point probabilities remain ellipsoidal, and that the evolution of the shape and orientation of the distributional ellipsoid will be controlled entirely by the macroscopic deformation ¯F. Again, this “closure” approximation is expected to be exact in the dilute limit, and probably not too bad for moderate particle concentrations, which is the main objective of this work in any case.

Therefore, it will be assumed here that the two-point probabilities pII_{(z −z}0₎ in the deformed configuration will also be ellipsoidal and depend on z−z0 through the combination |(ZD)−T(z−z0)|, where the distributional ellipsoid in the deformed configuration will be given by

ΩD = x|x·ZD TZD −1 x≤1 , (3.35)

with ZD = ZD₀ F¯T describing the new shape and orientation of the distributional ellipsoid in the deformed configuration.

In summary, the microstructures for the MREs have been idealized in terms of a family of initially aligned ellipsoidal inclusions, as characterized by the tensor ZI₀,

and distributed with ellipsoidal symmetry, as specified by the tensor ZD₀ . Under the applied deformation ¯F and magnetic induction field B, all the particles are assumed to rotate by identical amounts RI to new orientations specified by tensors ZI =

ZI₀RI T, and to rearrange their distribution by the applied deformation ¯F, as specified by new distributions tensors ZD = ZD₀F¯T. In this connection, it is important to emphasize that while the macroscopic deformation ¯F is prescribed (and therefore known a priori), the particle rotations RI need to be determined from the solution of the magnetoelastic problem in terms of the applied deformation ¯F and magnetic induction fieldB. This observation will play a key role in the next subsection, where we will identify certain special conditions for which the particle rotation may be determined without the need to solve the magnetoelastic problem in detail.