Some open problems in the theory of composites
Abstract: A selection of open problems in the theory of composites is presented. Particular attention is drawn to the question of whether two-dimensional, two-phase composites with general geometries have the same set of possible effective tensors as those of hierarchical laminates. Other questions involve the conductivity and elasticity of composites. Finally, some future directions for wave and other equations are mentioned. This article is part of the theme issue ‘Topics in mathematical design of complex ma… Show more
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Cited by 7 publications
References 106 publications
The continuing miniaturization of optoelectronic devices, alongside the rise of electromagnetic metamaterials, poses an ongoing challenge to nanofabrication. With the increasing impracticality of quality control at a single-feature (-device) resolution, there is an increasing demand for array-based metrologies, where compliance to specifications can be monitored via signals arising from a multitude of features (devices). To this end, a square grid with quadratic sub-features is amongst the more common designs in nanotechnology (e.g. nanofishnets, nanoholes, nanopyramids, μLED arrays etc). The electrical resistivity of such a quadratic grid may be essential to its functionality; it can also be used to characterize the critical dimensions of the periodic features. While the problem of the effective electrical resistivity ρ eff of a thin sheet with resistivity ρ 1, hosting a doubly-periodic array of oriented square inclusions with resistivity ρ 2, has been treated before (Obnosov 1999 SIAM J. Appl. Math. 59 1267–87), a closed-form solution has been found for only one case, where the inclusion occupies c = 1/4 of the unit cell. Here we combine first-principle approximations, numerical modeling, and mathematical analysis to generalize ρ eff for an arbitrary inclusion size (0 < c < 1). We find that in the range 0.01 ≤ c ≤ 0.99, ρ eff may be approximated (to within <0.3% error with respect to finite element simulations) by: ρ e f f = ρ 1 α ( c ) − 1 − ρ 2 / ρ 1 1 + ρ 2 / ρ 1 α ( c ) + 1 − ρ 2 / ρ 1 1 + ρ 2 / ρ 1 c · α ( c ) , α ( c ) = 1 + 0.9707 0.9193 + c 1 − c 2.1261 0.4671 . whereby at the limiting cases of c → 0 and c → 1, α approaches asymptotic values of α = 2.039 and α = 1/c − 1, respectively. The applicability of the approximation to considerably more complex structures, such as recursively-nested inclusions and/or nonplanar topologies, is demonstrated and discussed. While certainly not limited to, the theory is examined from within the scope of micro four-point probe (M4PP) metrology, which currently lacks data reduction schemes for periodic materials whose cell is smaller than the typical μm-scale M4PP footprint.
The continuing miniaturization of optoelectronic devices, alongside the rise of electromagnetic metamaterials, poses an ongoing challenge to nanofabrication. With the increasing impracticality of quality control at a single-feature (-device) resolution, there is an increasing demand for array-based metrologies, where compliance to specifications can be monitored via signals arising from a multitude of features (devices). To this end, a square grid with quadratic sub-features is amongst the more common designs in nanotechnology (e.g. nanofishnets, nanoholes, nanopyramids, μLED arrays etc). The electrical resistivity of such a quadratic grid may be essential to its functionality; it can also be used to characterize the critical dimensions of the periodic features. While the problem of the effective electrical resistivity ρ eff of a thin sheet with resistivity ρ 1, hosting a doubly-periodic array of oriented square inclusions with resistivity ρ 2, has been treated before (Obnosov 1999 SIAM J. Appl. Math. 59 1267–87), a closed-form solution has been found for only one case, where the inclusion occupies c = 1/4 of the unit cell. Here we combine first-principle approximations, numerical modeling, and mathematical analysis to generalize ρ eff for an arbitrary inclusion size (0 < c < 1). We find that in the range 0.01 ≤ c ≤ 0.99, ρ eff may be approximated (to within <0.3% error with respect to finite element simulations) by: ρ e f f = ρ 1 α ( c ) − 1 − ρ 2 / ρ 1 1 + ρ 2 / ρ 1 α ( c ) + 1 − ρ 2 / ρ 1 1 + ρ 2 / ρ 1 c · α ( c ) , α ( c ) = 1 + 0.9707 0.9193 + c 1 − c 2.1261 0.4671 . whereby at the limiting cases of c → 0 and c → 1, α approaches asymptotic values of α = 2.039 and α = 1/c − 1, respectively. The applicability of the approximation to considerably more complex structures, such as recursively-nested inclusions and/or nonplanar topologies, is demonstrated and discussed. While certainly not limited to, the theory is examined from within the scope of micro four-point probe (M4PP) metrology, which currently lacks data reduction schemes for periodic materials whose cell is smaller than the typical μm-scale M4PP footprint.
An FE-DMN method for the multiscale analysis of fiber reinforced plastic components
In this work, we propose a fully coupled multiscale strategy for components made from short fiber reinforced composites, where each Gauss point of the macroscopic finite element model is equipped with a deep material network (DMN) which covers the different fiber orientation states varying within the component. These DMNs need to be identified by linear elastic precomputations on representative volume elements, and serve as high-fidelity surrogates for full-field simulations on microstructures with inelastic constituents. We discuss how to extend direct DMNs to account for varying fiber orientation, and propose a simplified sampling strategy which significantly speeds up the training process. To enable concurrent multiscale simulations, evaluating the DMNs efficiently is crucial. We discuss dedicated techniques for exploiting sparsity and high-performance linear algebra modules, and demonstrate the power of the proposed approach on an industrial-scale three-dimensional component. Indeed, the DMN is capable of accelerating two-scale simulations significantly, providing possible speed-ups of several magnitudes.
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