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A multiscale analysis and computational method based on the Second-Order Two-Scale (SOTS) approach are proposed for the elastic quadratic eigenvalue problems in the periodic composite domain. Two typical quadratic eigenvalue problems with different damping effects are considered, and by the asymptotic expansions of both the eigenfunctions and eigenvalues, the first- and second-order cell functions, the microscale features of this heterogeneous materials are defined successively. Then, the homogenized quadratic eigenvalue problems are derived and the second-order expansions of the eigenfunctions are formed. The eigenvalues are also broadened to the second-order terms by introducing proper auxiliary elastic functions defined in the composite structure, and the nonlinear expressions of the correctors of the eigenvalues are derived. The finite element procedures are established, linearized methods are discussed for solving the quadratic eigenvalue problems and the second-order asymptotic computations are performed. Effectiveness of the asymptotic model is demonstrated by both the qualitative and quantitative comparisons between the computed SOTS approximations and the reference solutions, and the converging behavior of the eigenfunctions are numerically verified. It is also indicated that the second-order correctors are of importance to reconstruct the detailed information of the original eigenfunctions within the micro cells.

A top-down strategy is proposed for analyzing the elliptic eigenvalue problems of the hierarchically perforated materials with three-scale periodic configurations. The heterogeneous structure considered is composed of perforated cells in the mesoscopic scale and composite cells in the microscopic scale, and Neumann boundary conditions are imposed on the boundaries of the cavities. By using the classical two-scale asymptotic expansion method, the homogenized eigenfunctions and eigenvalues are obtained and the first- and second-order auxiliary cell functions are defined firstly in the mesoscale. Then, the two-scale asymptotic analysis is furtherly applied to the mesoscopic cell problems and by expanding the meso cell functions to the second-order terms, the homogenized cell functions are derived and the relations between the homogenized coefficients and the coefficients of constituent materials in the three scale levels are established. Finally, the second-order three-scale asymptotic approximations of the eigenfunctions are presented and by the idea of “corrector equation”, the three-scale expressions of the eigenvalues are obtained. The corresponding finite element algorithm is established and the successively up-scaling procedures are established. Typical two-dimensional numerical examples are performed, and both the two-scale and three-scale computed approximations of the eigenvalues are compared with the ones obtained in the classical computation. By the least squares technique, it is demonstrated that the three-scale asymptotic solutions of the eigenfunctions are good approximations of the original eigensolutions corresponding to both the simple and multiple eigenvalues. This study offers an alternative approach to describe the physical and mechanical behaviors of the hierarchically structures with more than two scales and it is indicated that the second-order terms plays an important role not only in the derivation of the expansions but also in the practical computations to capture the local oscillations within the cells.

A novel statistical second-order reduced multiscale (SSRM) approach is established for nonlinear composite materials with random distribution of grains. For these composites considered in this work, the complex microstructure of grains, including their shape, orientation, size, spatial distribution, volume fraction and so on, results in changing of the macroscopic mechanical properties. The first- and second-order unit cell functions based on two-scale asymptotic expressions are constructed at first. Then, the expected homogenized parameters are defined, and the nonlinear homogenization equation on global structure is established, successively. Further, an effective reduced model format for analyzing second-order nonlinear unit cell problem with less computation cost is introduced in detail. Finally, some numerical examples for the materials with varying distribution models are evaluated and compared with the data by theoretical models and experimental results. These examples illustrate that the proposed SSRM approaches are effective for predicting the macroscopic properties of the random composite materials and supply a potential application in actual engineering computation.

This study proposes a new elastic–plastic deformation decomposition algorithm for metal clusters to calculate micro-nanoscale elastic and plastic deformation gradients. In the macroscopic plasticity theory, the intermediate configuration is usually constructed by the dissection–unloading method. Because an atomic cluster is equivalent to a small element on a macroscopic object, our decomposition algorithm regards the unloaded configuration as the intermediate configuration for atomic clusters. This algorithm uses a new unloading method to obtain the unloaded configuration. This micro-nanoscale unloading method is constructed based on the principle of minimum potential energy and the embedded-atom method. Moreover, this method rigidly fixes atoms around dislocations during unloading. Therefore, the unloading process will not cause new plastic flow. Once the intermediate configuration is obtained, elastic and plastic deformation gradients are calculated by the interpolation method. Numerical examples of Cu nanowire stretching and bending show that the new decomposition algorithm can accurately and rapidly conduct the elastic–plastic decomposition of the total deformation. This algorithm provides a computational basis for multiscale coupling analysis of mechanical behavior of metal materials from the micro-nanoscale to the macroscale.

This paper is devoted to a study of nonlinear Robin-type boundary value problems of second-order elliptic partial differential equations. Specifically, nonlinear boundary conditions are modeled by ...

A novel high-order three-scale (HOTS) computational method for elastic behavior analysis and strength prediction of axisymmetric composite structures with multiple spatial scales is developed in this paper. The multiple heterogeneities of axisymmetric composite structures we investigated are taken into account by periodic distributions of representative unit cells on the mesoscale and microscale. First, the new micro–meso–macro coupled HOTS computational model for elastic problems of axisymmetric composite structures is established based on multiscale asymptotic analysis, which breaks through the traditional multiscale assumptions and includes three main components. Two classes of mesoscopic and microscopic auxiliary cell functions are constructed on the mesoscale and microscale, respectively. The macroscopic homogenization problems are defined on global axisymmetric structures by twice up-scaling procedures from microscale to mesoscale and then from mesoscale to macroscale. Moreover, the asymptotic HOTS solutions are constructed for approximating multiscale elastic problems of axisymmetric structures and the numerical accuracy analysis of the HOTS solutions is given in detail. Then, the strength prediction formulas for axisymmetric composite structures with multiple spatial scales are presented based on the high-accuracy elastic behavior analysis from the proposed HOTS computational model. Furthermore, the corresponding HOTS numerical algorithm based on the finite element method (FEM) is presented for analyzing the mechanical behaviors and predicting the strength of axisymmetric composite structures with multiple spatial scales in detail. Finally, some numerical examples are reported to verify the feasibility and effectiveness of the proposed HOTS computational method. In this study, a unified three-scale computational framework is offered, which enables the simulation of mechanical behaviors of axisymmetric composite structures with multiple spatial scales.

The atomic volume and surface are important geometric quantities for calculating various macroscopic physical quantities from atomistic models. This paper proposes a new analytical method to calculate the atomic volumes and surfaces of a metal cluster. This method adopts metallic radii to describe atom sizes and constructs the overall volume/surface by the molecular surface (MS) model. It divides cluster atoms into two types: interior atoms and boundary atoms. For an interior atom, the method defines a variational Voronoi cell as its volume. For a boundary atom, the method defines the intersection of the overall cluster volume and its variational Voronoi cell as its volume. The atomic surfaces are calculated along with the volume calculations. Compared with other methods, the new method considers the effect of atom sizes and does not rely on the size and location of the simulation box. Therefore, the new method can accurately calculate the overall volume of a metal cluster of arbitrary shape and the individual atomic volumes. This method provides computational support for multiscale coupled calculations from the microscale to macroscale.

This study proposes new models for thermodynamic properties and thermoelastic constitutive relation of materials with cubic crystal structures. The motion equation of atoms in cubic crystal structures is decomposed into structural deformation and thermal vibration at first. Then based on the thermo-mechanical coupling mechanism, both the thermal and mechanical aspects of structural deformation are investigated. And the thermal vibration equation is built at structural deformation positions by considering the non-harmonic approximation of interatomic potentials. Further, the improved formula of free energy is established as a function of the structural deformation and thermal vibration frequencies, which includes the thermo-mechanical coupling and non-harmonic effects. And the thermodynamic properties, including internal energy, entropy, heat capacity and thermal expansion, are derived from the free energy. Besides, based on the multiplicative decomposition of deformation gradient, the thermoelastic constitutive relation is constructed at finite deformation for cubic crystal materials. Finally, numerical results of the thermodynamic properties, thermoelastic stress–strain relations and elastic constants for face-centered cubic metals Cu, Au and Ag from 0 K to the melting points at 0–50 GPa are provided by comparing with the experimental data to demonstrate the usability of the present models.

This study presents a novel multi-scale computational method to analyze the dynamic thermo-mechanical performance of heterogeneous shell structures with orthogonal periodic configurations. The heterogeneities of heterogeneous shell structures are taken into account by periodic layouts of unit cells on the microscale in orthogonal curvilinear coordinate system. The new second-order two-scale approximate solutions for these multi-scale problems are constructed based on the multi-scale asymptotic analysis. Furthermore, the error estimates for the second-order two-scale (SOTS) solutions are obtained under some hypotheses. And then, a novel SOTS numerical algorithm based on finite element method (FEM), finite difference method (FDM) and decoupling method is brought forward in detail. Finally, some numerical examples are presented to verify the feasibility and validity of our multi-scale computational method. They also demonstrate that our multi-scale computational method can accurately capture the micro-scale dynamic thermo-mechanical responses in heterogeneous block structure, plate, cylindrical and doubly-curved shallow shells. In this paper, a unified multi-scale computational framework is established for dynamic thermo-mechanical problems of heterogeneous materials and structures with orthogonal periodic configurations. The asymptotic homogenization theory in Cartesian coordinate system and cylindrical coordinate system can be directly obtained based on the results in this paper.

In this paper, a novel high-order three-scale (HOTS) computational method is presented to analyze the dynamic thermo-mechanical performance of composite structures with multiple spatial scales. The heterogeneities of composite structures are taken into account by periodic distributions of representative unit cells on the mesoscale and microscale. By virtue of multiscale asymptotic analysis, a new unified micro-meso-macro HOTS approximate solutions are successfully constructed for these multiscale problems. Two classes of auxiliary cell functions are established on the mesoscale and microscale, respectively. Also, two kinds of equivalent material parameters are calculated by up-scaling procedure on the mesoscale and microscale, and the homogenization problems are subsequently defined on global structure. Then, the comparisons of approximate performance are obtained in the pointwise sense for the conventional two-scale solutions, low-order three-scale (LOTS) solutions and HOTS solutions, which illustrate the necessity of developing HOTS solutions for simulating the dynamic thermo-mechanical performances of composite structures with multiple spatial scales. Furthermore, the corresponding HOTS numerical algorithm based on finite element method (FEM) and Finite difference method (FDM) is brought forward in details. Finally, some numerical examples are reported to demonstrate the usability of our HOTS computational method to simulate the dynamic thermo-mechanical behaviors of composite structures with multiple spatial scales. This study offers a unified three-scale computational framework that enables the simulation and analysis of dynamic thermo-mechanical problems of composite structures with multiple spatial scales.