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The principal parameter resonance of a ferromagnetic functionally graded (FG) cylindrical shell under the action of axial time-varying tension in magnetic and temperature fields is investigated. The temperature dependence of physical parameters for functionally graded materials (FGMs) is considered. Meanwhile, the tension bending coupling effect is eliminated by introducing the physical neutral surface. The kinetic and strain energies are gained with the Kirchhoff–Love shell theory. Based on the nonlinear magnetization characteristics of ferromagnetic materials, the electromagnetic force acting on the shell is calculated. The nonlinear vibration equations are obtained through Hamilton’s principle. Afterward, the vibration equations are discretized and solved by Galerkin and multiscale methods, respectively. The stability criterion for steady-state motion is established utilizing Lyapunov stability theory. After example analysis, the effects of magnetic field intensity, temperature and power law index on the static deflection are elucidated. Subsequently, the impacts of these parameters, as well as axial tension, on the amplitude-frequency characteristics, resonance amplitude, and multiple solution regions are discussed explicitly. Results indicate that the stiffness can be enhanced due to the generation of static deflection. The amplitude decreases with increasing magnetic field intensity, temperature, and power law index. When the magnetic field intensity surpasses a threshold, the resonance phenomenon disappears.

This paper is concerned with a finite element method (FEM) for multiscale problems in linear elasticity. We propose a method which discretizes the physical problem directly by a macroscopic FEM, coupled with a microscopic FEM resolving the micro scale on small cells or patches. The assembly process of the unknown macroscopic model is done without iterative cycles. The method allows to recover the macroscopic properties of the material in an efficient and cheap way. The microscale behavior can be reconstructed from the known micro and macro solutions. We give a fully discrete convergence analysis for the proposed method which takes into account the discretization errors at both micro and macro levels. In the case of a periodic elastic tensor, we give a priori error estimates for the displacement and for the macro and micro strains and stresses as well as an error estimate for the numerical homogenized tensor.

The subharmonic resonances of an axially moving graphene-reinforced laminated composite plate are studied based on the Galerkin and multiscale methods. Graphene nanoplatelets (GPLs) are added into matrix material which acts as the basic layer of the plate, and a graphene-reinforced nanocomposite plate is thus obtained. Different GPL distribution patterns in the laminated plate are considered. The Halpin–Tsai model is selected to predict the physical properties of the nanocomposite. Hamilton’s principle is utilized to conduct the dynamic modeling of the plate and the von Kármán deformation theory is used. The velocity is assumed to be a combination of constant and harmonically varied velocities. The natural frequencies of the linear system with constant velocity can be obtained using the eigenvalues of the coefficient matrix of the ordinary differential equations after the governing partial differential equations of motion are discretized through the Galerkin method. The instability regions of the linear system and the amplitude–frequency relations of the nonlinear system considering the harmonically varied velocity are obtained based on the multiscale analysis. The effect of GPL reinforcement on the subharmonic resonances of the linear and nonlinear systems is analyzed in detail.

By applying the multiscale method to the Möbius transformation function, we construct the multiscale analytic sampling approximation (MASA) to any function in the Hardy space $H^2(𝕋)$. The approximation error is estimated, and it is proved that the MASA is robust to sample error. We prove that the MASA can be expressed by a Hankel matrix, making use of which, a fast algorithm is established to compute the MASA. Since what we acquire in practice may well be the samples on time domain instead of the analytic ones on the unit disc of the complex plane, we establish a fast algorithm for acquiring analytic samples. Numerical experiments are carried out to demonstrate the efficiency of the MASA.

The scientific community has witnessed, lately, a tremendous progress in the fabrication and synthesis of nanomaterials. As a result, it is essential to develop new and efficient numerical techniques that are capable of modeling the behavior of materials at nanoscale with sufficient accuracy. In this work, a novel approach is presented for the multiscale analysis of brittle failure in nanostructures using the phase-field modeling. The specimen at microscale is discretized using finite elements (FEs), whose integration points lie in the representative volume elements (RVEs) at nanoscale. The displacement computed in upper scale for a microstructure that contains an evolving crack is imposed on the boundaries of the RVE in lower scale. On the other hand, the stresses and material properties obtained for the RVE in lower scale are transferred to upper scale to compute stiffness matrices and load vectors. The evolution of the phase-field variable indicates the initiation and propagation of cracks at microscale. In order to avoid time-consuming molecular dynamics (MD) simulations at nanoscale in each step of the analysis, the Mooney–Rivlin material model is used to simulate the behavior of Aluminum (AL) nanostructure at this scale. The approach that is utilized to compute the material constants and the formulation for the multiscale technique combined with the phase-field modeling in upper scale are described in detail. It is discussed how the phase-field variable in microstructure is evolved based on the properties of the RVE in nanostructure. Many numerical examples are presented to demonstrate the application of the proposed multiscale technique in the solution of engineering problems.

Given its essential nonlinearity, nonlinear energy sink (NES) has been extensively studied as a promising vibration energy harvesting device. Internal resonance, which is due to strong energy exchange between modes, also provides a valuable idea for vibration energy harvesting. Combining these two advantages, we put forward a 3:1 internal resonance system, which consists of an NES and a coupled linear oscillator, as an enhanced method for vibration energy harvesting. The multiscale method is applied to derive the relationship between amplitude and frequency response. Simulations are carried out to evaluate the performance of the proposed method. Results show that the internal resonance system can remarkably improve the vibration energy harvesting performance. The numerical solutions verify the accuracy of the analytical solutions. The results demonstrate that the internal resonance system with NES for energy harvesting has better output power and bandwidth compared with noninternal resonance system. Overall, the comprehensive design improves the performance of NES for vibration energy harvesting.

Application of porous electrode materials has sparked significant interest as a strategy to mitigate traditional electrode mechanical failure arising from its intercalation-induced large volume change. In this work, a thermal analogy method is employed for implementing the coupled chemo-mechanical model into the finite element (FE) package ABAQUS via user subroutines UMATHT and UMAT, which is used to model the lithium (Li) diffusion and the resulting deformation of the electrode during charge-discharge cycling. This work presents a Direct FE² method for modeling the chemo-mechanically coupled behavior of porous electrode materials by establishing the macro-microscopic scale transitions through concentration and displacement DOFs and the representative volume element (RVE) volume scaling relationship. The two-scale numerical simulations can be implemented in a single computational scheme. Within the present computational framework, the Li diffusion and mechanical deformation in the porous silicon electrode during charging and discharging are easily simulated in the typical FE package. Benchmarked against the traditional direct full-field numerical computational method, the Direct FE² method is validated to present significant computational efficiency improvements through two numerical examples, the constrained expansion and the pre-compression expansion of porous electrode, by 99.27% and 94.55%, respectively, while maintaining the high precision.

At nanoscale, the mechanical response of nanoparticles is largely affected by the particle size. To assess the effects of nanoparticle size (e.g., nanoparticle’s volume, cross-sectional area and length) on mechanical behaviors of bcc Fe nanoparticles under compressive loading, an atomistic field theory was introduced in current study. In the theory, atomistic definitions and continuous local density functions of fundamental physical quantities were derived. Through the atomistic potential-based method, the mechanical responses of bcc Fe nanoparticles were analyzed in different sizes. The simulation results reveal that the ultimate stress decreases as Fe nanoparticle’s volume, cross-sectional area or length increases under compressive loading. Nonetheless, the Young’s modulus increases as nanoparticle size increases. In addition, for a fixed finite volume nanoparticle, this study indicates that the ultimate stress will increase as strain rate increases, but Young’s modulus will decrease with increasing strain rate. A loading–unloading study illustrates the energy dissipation due to irreversible structure changes in Fe nanoparticles.