Hierarchical heterogeneous one-dimensional problem in linear viscoelastic media

Multi-level periodicity is a ubiquitous structural feature of natural composites. Conventional homogenisation techniques can effectively address the mechanism of uniform periodic structures, while encountering challenges in efficiently characterising the performance of multi-scale hierarchical structures. This study presents an FE2 (Finite Element Square)-nested homogenisation elastic-calculating method, integrating RVE-based homogenisation for primary periodicity and nested Direct FE2 for secondary periodicity. Harnessing micro-LPBF process, specimens with multi-level periodicity are fabricated. The high-precision manufacturing is integral for unlocking the design potential of macro–micro hierarchical material, allowing for the creation of components with intricate multi-scale features. Mechanical testing of specimens validates the accuracy of FE2-nested homogenisation method, demonstrating its effectiveness in predicting the elastic responses of multi-level periodic structures. Furthermore, the FE2-nested method is employed to elucidate the macro–micro interaction mechanisms. It underscores potential to drive innovations in hierarchical composite material, providing a method basis for mechanical customisation and enhancement of complex multi-level design.

The asymptotic homogenization method is applied to obtain formal asymptotic solution and the homogenized solution of a Dirichlet boundary-value problem for an elliptic equation with rapidly os- cillating coefficients. The proximity of the formal asymptotic solution and the homogenized solution to the exact solution is proved, which provides the mathematical justification of the homogenization pro- cess. Preservation of the symmetry and positive-definiteness of the effective coefficient in the homogenized problem is also proved. An example is presented in order to illustrate the theoretical results.

The conversion of ethanol into high-valuable chemicals and H2 by photocatalytic process provides a sustainable approach to produce carbon-chain-prolonged chemicals and hydrogen energy. In this article, Ni-MOF-74 was added to fabricate the hierarchical CdS/NiS-N composites with an elevated specific surface area during the hydrothermal synthesis of CdS microsphere, and the Ni-MOF-74 facilitate the self-assemble growth of CdS and provide a source of Ni for the formation of NiS. The as-prepared photocatalyst was subjected to photocatalytic ethanol conversion, and the hierarchical composite material CdS/NiS-N (100) formed by adding 100 mg of Ni-MOF-74 exhibits the highest photocatalytic activity and stability in an ethanol aqueous solution with a water content of 10 %. Under visible light irradiation, the conversion rate of ethanol reached 15.2 % at the photocatalytic reaction of 5 h. The selectivity of 2,3-butanediol(2,3-BDO) was 25 %, and the selectivity of acetaldehyde(AA) was 63 %. Through various characterizations, it has been proven that a large specific surface area and the coupling interface between CdS and NiS are key factors in improving photocatalytic performance. This work provides an effective strategy for constructing photocatalysts with coupled cocatalysts/semiconductors and large specific surface areas.

Hierarchical and reversible assembly of graphene oxide/polyvinyl alcohol hybrid stabilized Pickering emulsions and their templating for macroporous composite hydrogels

Bone and dentin are highly complex, hierarchical composite materials with exceptional properties due to their unique composition and structure. They are essentially the same material with varied structural organization. They are three phase composites made up of a ceramic component, hydroxyapatite (HAP), a polymeric or proteinaceous component, collagen, and fluid filled porosity. A number of macroscopic studies have shown that both dentin [1-6] and bone [7-9] undergo visco-elastic, creep deformation and stress-relaxation behaviors. The problem with these bulk experiments is that they do not give information about which phase is contributing to the macroscopic creep or how. Some of these inquiries have suggested that the collagen is not responsible [9] and that creep in hard biological materials is primarily due to dislocations in the HAP mineral. On the other hand, others have said that collagen is completely responsible for the creep [8, 10]. These uncertainties make it essential to use techniques that allow for the study of the behavior of these very different components simultaneously during loading, determining their participation in creep. One such technique is synchrotron diffraction.

For a class of nonlinear oscillation problems described by a second order ordinary differential equation containing a small nonnegative perturbation parameter $\varepsilon $ we determine asymptotic solutions which are uniformly valid as $\varepsilon \downarrow 0$ on intervals of the order of magnitude $\varepsilon ^{ - 1} $. These asymptotic solutions are obtained from formal asymptotic solutions subject to a condition in order that they approximate the exact solution asymptotically. These formal asymptotic solutions are constructed in the form of a finite generalized asymptotic power series involving functions of the periodic two variable type.

Hierarchical Dragonfly Wing: Microstructure-Biomechanical Behavior Relations

Modeling of trabecular bone as a hierarchical material

In this work, a general mathematical model for functionally graded heterogeneous equilibrium boundary value problems is considered. A methodology to find the local problems and the effective properties of functionally graded materials (FGMs) with generalized periodicity is presented, using the asymptotic homogenization method (AHM). The present models consist of the matrix metal Mo and the reinforced phase ceramic ZrC, the constituent ratios and the property gradation profiles of which can be described by the designed volume fraction. Firstly, a new threshold segmentation method is proposed to construct the gradient structure of the FGMs, which lays the groundwork for the subsequent research on the properties of materials. Further, a study of FGMs varied along a certain direction and the influence of the varied constituents and graded structures in the behavior of heterogeneous structures are investigated by the AHM. Consequently, the closed–form formulas for the effective thermo–mechanical coupling tensors are obtained, based on the solutions of local problems of FGMs with the periodic boundary conditions. These formulas provide information for the understanding of the traditional homogenized structure, and the results also be verified the correctness by the Mori–Tanaka method and AHM numerical solution. The results show that the designed structure profiles have great influence on the effective properties of the present inhomogeneous heterogeneous models. This research will be of great reference significance for the future material optimization design.

A method of investigation of the general theory of stability problems on structural elements fabricated from viscoelastic composite materials

Microfiltration of particles is modelled by the motion of particles embeddedin a Stokes flow near a porous membrane in which Darcy equations apply.Stokes flow also applies on the other side of the membrane. A pressure gradient isapplied across the membrane. Beavers and Joseph slip boundary condition applies along the membrane surfaces.This coupled Stokes-Darcy problem is solved by a perturbation method, considering that the particle size is muchlarger than the pores of the membrane. The formal asymptotic solution is developed in detail up to 3rd order.The method is applied to the example case of a spherical particle moving normal to a membrane. The solution,limited here to an impermeable slip surface (described from 3rd order expansion), uses as an intermediate step theboundary integral technique for Stokes flow near an impermeable surface with a no-slip boundary condition. Results of the perturbation solution are in good agreement with O'Neill and Bhatt analytical solution for this case.

The singularly perturbed systems for the nonlinear three-species food-chain reaction systems with time delay are considered. By using the method of the stretched variable, the formal asymptotic solution is obtained under suitable conditions.

Singular perturbation of boundary value problem for quasilinear third-order ordinary differential equations involving two small parameters

Averaging of the system of equations of motion of a viscous fluid in a porous medium

We propose an asymptotic approach for the evaluation of effective complex moduli of viscoelastic fibre‐reinforced composite materials. Our method is based on the homogenization technique. We start with a non‐trivial expansion of the input plane‐strain boundary value problem by ratios of visco‐elastic constants. This allows to simplify the governing equations to forms analogous to the complex transport problem. Then we apply the asymptotic homogenization method, coming from the original problem on multi‐connected domain to the cell problem, defined on a unit cell of the periodic structure. For the analytical solution of the cell problem we apply the boundary perturbation technique, the asymptotic expansion by a distance between two neighbouring fibres and the method of two‐point Padé approximants. As results we derive uniform analytical representations for effective complex moduli, valid for all values of the components volume fractions and properties.