Modeling of spatially periodic dielectric sensors in the presence of a top ground plane bounding the test dielectric

Effective balance equations for elastic composites subject to inhomogeneous potentials

“…Remark 4 (Properties of the cell problems) Nontrivial solutions of the cell problems (52-55) (which correspond to six elastic-type cell problems for every fixed (k, l), k, l = 1, 2, 3, k ≥ l, as M is minor symmetric, see [43,44]) are driven by local variations of the elastic constants within the composite, which formally appear as body forces on the right hand side of (52)(53), as well as the interface loads in the stress discontinuity conditions (54), see also [43,44]. In the latter case, the contribution is directly related to the difference in the elastic constants between the matrix and the inclusion/fiber, and to the interface geometry.…”

“…Therefore, our formulation directly applies to the investigation of electrostriction and magnetostriction driven by electric and magnetic fields that are spatially periodic (see, e.g. [53] for a semianalytical and experimental analysis of periodic electric fields in the context of dielectric sensors) and characterized by a wavelength of the same order of the microscale d.…”

“…The effective elastic coefficients and body force that appear in (63) are to be computed by solving (in general, numerically) the six elastic-type cell problems (52)(53)(54)(55) as well as the problem (58-61). Although we are assuming macroscopic uniformity of the geometry (see Remark 2), the cell problems still retain a parametric dependence on the macroscale variable x, which is encoded in the elastic moduli C(x, y) and in the local volume load f y .…”

“…This way, the contribution due to C M (x) would only appear in the homogenized balance equations (63), whereas the cell problems would be analogous to (52)(53)(54)(55) with C replaced by C L , and therefore would depend on the microscale y only. However, our model is intrinsically more complex than the standard one, as macroscale heterogeneities in the cell problem (58)(59)(60)(61) are also encoded in the local volume load f y (x, y).…”

“…In fact, a multiplicative decomposition of the potentials leads to a complete decoupling between macroscale and microscale problems. Given the geometry of the cell Ω and the macroscopically uniform constituents' elastic moduli C I and C II , it is just sufficient to solve the six cell elastictype cell problems (52)(53)(54)(55) (124) to recover the effective elasticity tensorC and volume loadf that are needed to completely specify the homogenized balance equations (63).…”