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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 ...

Convergence of Dziuk's fully discrete linearly implicit scheme which was proposed in [G. Dziuk, Math. Models Methods Appl. Sci., 4 (1994), pp. 589--606] for curve shortening flow still remains open...

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.

The second-order coupled piezoelectric models based on the multi-scale asymptotic expansion method are developed for the axially and spherically symmetric composites. The governing piezoelectric equations are compactly formulated in cylindrical and spherical coordinates, and the composite domains are assumed to be periodically occupied by the representative cells. The multi-scale asymptotic expansions for the displacement and the electric potential are formally defined and the effective elastic, piezoelectric, and dielectric coefficients are expressed in terms of the microscopic functions defined on the cell domain. Particularly, the cell solutions and the homogenized solutions for the plane axisymmetric and spherically symmetric problem are derived analytically. The corresponding finite element procedure is proposed, in which the Newmark algorithm is applied to construct the computational scheme in the temporal domain. Numerical experiments are carried out to simulate both the static and dynamic asymptotic behavior of the space axisymmetric and the one-dimensional plane axisymmetric structures. It is validated from the numerical examples that the asymptotic models proposed in the current work are effective to capture the macroscopic performance of the piezoelectric structures and the second-order expansions of the solutions is essential for obtaining the correct distributions of the stress and electric displacement.

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.

A novel three-scale asymptotic expansion used to evaluate viscoelastic analysis of composites with multiple configuration is systematically studied. The heterogeneities of composite structures are described by periodic layout of unit cells on the microscale and mesoscale. Firstly, the three-scale asymptotic formulations based on homogenization approach and Laplace transform are established, and the local cell solutions in microscale and mesoscale are also defined. Further, two kinds of homogenized coefficients are derived by upscaling methods, and the homogenized equations are obtained on whole macroscopic domain. Also, the strain and stress fields are constructed as three-scale asymptotic solutions by assembling the unit cell solutions and homogenized solutions. Then, the finite element algorithms based on inverse Laplace transform and the three-scale asymptotic homogenization are proposed. Finally, some typical numerical results are employed to validate the presented approaches. They illustrate that the three-scale asymptotic approaches proposed in this paper are efficient and accurate for analyzing the viscoelastic problems of the composites with multiple configuration.

In this paper, a novel multiscale computational method is presented for transient heat conduction problems of periodic porous materials with diverse periodic configurations in different subdomains. In these porous materials, heat transfer at microscale has an important impact on the macroscopic temperature field. Firstly, the second-order two-scale (SOTS) solutions for these multiscale problems are successfully obtained based on asymptotic homogenization method. Then, the error analysis in the pointwise sense is given to illustrate the importance of developing SOTS solutions. Furthermore, the error estimate for the SOTS approximate solutions in the integral sense is presented. In addition, a SOTS numerical algorithm is proposed to effectively solve these problems based on finite element method (FEM) and finite difference method (FDM). Finally, some numerical examples are shown, which demonstrate the feasibility and effectiveness of the SOTS numerical algorithm we proposed. In this paper, a unified two-scale computational framework is established for transient heat conduction problems of periodic porous materials with diverse periodic configurations in different subdomains.

In this paper, a novel multiscale computational method is presented to simulate and analyze dynamic thermo-mechanical problems of composite structures with diverse periodic configurations in different subdomains. In these composite structures, thermo-mechanical coupled behaviors at microscale have an important impact on the macroscopic displacement and temperature fields. Firstly, the novel second-order two-scale (SOTS) solutions for these multiscale problems are successfully obtained based on multiscale asymptotic analysis. Then, the error analysis in the pointwise sense is given to illustrate the importance of developing the SOTS solutions. Furthermore, the error estimate for the SOTS approximate solutions in the integral sense is presented. In addition, a SOTS numerical algorithm is proposed to effectively solve these problems based on finite element method, finite difference method and decoupling method. Finally, some numerical examples are shown, which demonstrate the feasibility and effectiveness of the SOTS numerical algorithm we proposed. In this paper, a unified two-scale computational framework is established for dynamic thermo-mechanical problems of composite structures with diverse periodic configurations in different subdomains.

This study develops a novel multiscale computational method for heat conduction problems of composite structures with diverse periodic configurations in different subdomains. Firstly, the second-order two-scale (SOTS) solutions for these multiscale problems are successfully obtained based on multiscale asymptotic analysis. Then, the error analysis of SOTS solutions in the pointwise sense is given to illustrate the importance of developing the SOTS solutions. Furthermore, the error estimate for the SOTS approximate solutions in the integral sense is presented. In addition, a SOTS numerical algorithm is proposed to effectively solve these problems based on finite element method (FEM). Finally, some numerical examples verify the feasibility and effectiveness of the SOTS numerical algorithm we proposed. In this paper, a unified two-scale computational framework is established for heat conduction problems of composite structures with diverse periodic configurations in different subdomains.

In this paper, a novel second-order two-scale (SOTS) analysis method and related numerical algorithm are developed for damped dynamic thermo-mechanical problems of quasi-periodic composite materials. The formal SOTS solutions for these problems are constructed by the multiscale asymptotic analysis. Then we theoretically explain the importance of the SOTS solutions by the error analysis in the pointwise sense. Furthermore, the convergence result with an explicit rate for the SOTS solutions is obtained in the integral sense. In addition, a SOTS numerical algorithm is proposed to solve these problems effectively. Finally, some numerical examples verify the feasibility and effectiveness of the SOTS numerical algorithm we proposed.

Mechanical responses of nanoporous aluminum samples under shock in different crystallographic orientations (, , , and ) are investigated by molecular dynamics simulations. The shape evolution of void during collapse is found to have no relationship with the shock orientation. Void collapse rate and dislocation activities at the void surface are found to strongly dependent on the shock orientation. For a relatively weaker shock, void collapses fastest when shocked along the orientation; while for a relatively stronger shock, void collapses fastest in the orientation. The dislocation nucleation position is strongly depended on the impacting crystallographic orientation. A theory based on resolved shear stress is used to explain which slip planes the earliest-appearing dislocations prefer to nucleate on under different shock orientations.

A novel second-order two-scale asymptotic method is presented for the eigenvalue problems of the second-order elliptic operator in the general composite domain. The eigenvalue equation is firstly reformulated in curvilinear coordinates with periodic configuration using proper coordinate transformation, and by applying the asymptotic expansion technique, the eigenfunctions of the system are expanded to the second-order terms. Using the argument of the so-called “corrector equations”, the eigenvalues are expressed in terms of the homogenized eigenfunctions and the cell functions are defined in the representative cell domain. The feature of the proposed model is that some homogenized material coefficients and all the microscopic cell functions are dependent on the macroscopic coordinates. Various reduced expressions of the eigenfunctions and eigenvalues are discussed under specific coordinate transformations, and the conditions that the cell functions and homogenized coefficients are decoupled from the macroscopic coordinates are elaborated. The finite element algorithm is developed and three numerical experiments are carried out, which demonstrate the effectiveness of our proposed method in simulating and predicting the vibration behavior of the composite structures. It is also indicated that the second-order correctors are of necessity to capture the locally oscillating behavior within a periodicity of the eigenfunctions. By the coordinate transformation, the asymptotic analysis method can be generalized to more general composite domain with quasi-periodic and non-periodic configurations.

This work establishes a novel three-scale homogenization algorithm to predict heat transfer performance of porous materials with multiple periodic configurations. The heterogeneities of porous structures are considered by periodic distributions of unit cells on the microscale and mesoscale. A new micro-meso-macro formula based on homogenization methods and multiscale asymptotic expansions is given at first. Two types of unit cell solutions in microscale and mesoscale are obtained by solving the distinct multiscale cell functions. Also, two kinds of homogenization coefficients are calculated by up-scaling procedure, and the homogenization equations are defined on global structure. Further, the temperature and heat flux fields are established as three-scale approximate solutions by assembling the various local cell solutions and homogenization solutions. Then, the associated finite element algorithm based on the three-scale homogenization methods is proposed in detail. Finally, some numerical examples are reported to validate the methods. They illustrate that the three-scale homogenization methods presented in this work are effective and accurate for calculating the heat transfer performance of the porous materials with multiple configurations.

This paper develops a novel second-order two-scale (SOTS) method to predict the mechanical performance of statistically inhomogeneous core–shell materials. As for these kinds of materials, the sophisticated microscopic information of inclusions, including their shape, size, orientation, spatial distribution, volume fraction, and so on, leads to changes of the macroscopic mechanical properties. Firstly, the microscopic configuration for the core–shell structure with random distribution is briefly characterized. Secondly, the SOTS formulae for solving the mechanical problems are proposed, and the importance of the two-scale approximate solutions in pointwise sense is given. Then, the associated prediction algorithm based on the two-scale model is brought forward in detail. Finally, some numerical results for the effective properties of the materials with varying probability distribution models are calculated and compared with the data by analytical and experimental results. The comparison shows that the statistical second-order two-scale model is validated for determination of the mechanical properties of inhomogeneous core–shell materials and demonstrates its potential applications in actual engineering applications.

This study develops a novel multiscale computational method for thermoelastic problems of composite materials with orthogonal periodic configurations. Firstly, the multiscale asymptotic analysis for these multiscale problems is given successfully, and the formal second-order two-scale approximate solutions for these multiscale problems are constructed based on the above-mentioned analysis. Then, the error estimates for the second-order two-scale (SOTS) solutions are obtained. Furthermore, the corresponding SOTS numerical algorithm based on finite element method (FEM) is brought forward in details. Finally, some numerical examples are presented to verify the feasibility and effectiveness of our multiscale computational method. Moreover, our multiscale computational method can accurately capture the local thermoelastic responses in composite block structure, plate, cylindrical and doubly-curved shallow shells.

This study develops a novel high-order three-scale (HOTS) computational method for heat conduction problems of axisymmetric composite structures with multiple spatial scales. The heterogeneities of the composites are taken into account by periodic distributions of unit cells on the mesoscale and microscale. Firstly, the multiscale asymptotic analysis for these multiscale problems is given detailedly. Based on the above-mentioned analysis, the 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. Then, the error analyses for the conventional two-scale solutions, low-order three-scale (LOTS) solutions and HOTS solutions are obtained in the pointwise sense, which illustrate the necessity of developing HOTS solutions for simulating the heat conduction behaviors of composite structures with multiple periodic configurations. Furthermore, the corresponding HOTS numerical algorithm based on finite element method (FEM) is brought forward in details. Finally, some numerical examples are presented to verify the feasibility and validity of our HOTS computational method. In this paper, a unified three-scale computational framework is established for heat conduction problems of axisymmetric composite structures with multiple spatial scales.

In this paper the macroscopic damping model for dynamical behavior of the structures with random polycrystalline configurations at micro–nano scales is established. First, the global motion equation of a crystal is decomposed into a set of motion equations with independent single degree of freedom (SDOF) along normal discrete modes, and then damping behavior is introduced into each SDOF motion. Through the interpolation of discrete modes, the continuous representation of damping effects for the crystal is obtained. Second, from energy conservation law the expression of the damping coefficient is derived, and the approximate formula of damping coefficient is given. Next, the continuous damping coefficient for polycrystalline cluster is expressed, the continuous dynamical equation with damping term is obtained, and then the concrete damping coefficients for a polycrystalline Cu sample are shown. Finally, by using statistical two-scale homogenization method, the macroscopic homogenized dynamical equation containing damping term for the structures with random polycrystalline configurations at micro–nano scales is set up.

A novel second-order two-scale method is developed to predict thermo-mechanical coupling performance of statistically inhomogeneous porous materials. For these materials, the sophisticated microstructure information of pores, including their shapes, orientations, sizes, spatial distributions, volume fractions and so on, leads to changes of the macroscopic thermo-mechanical properties. Also, radiation effect at microscale is considered in this study which has an important influence on the macroscopic temperature fields. Firstly, the microscopic configuration for the structure with random distribution is briefly characterized by a periodic layout of unit cells. Secondly, the second-order two-scale formulations for computing the thermo-mechanical coupling problems which consider the interaction between temperature and displacement fields are given in detail. Then, the statistical prediction algorithm based on the two-scale model is brought forward. Finally, some numerical examples for porous materials with varying probability distribution models are calculated by the algorithm proposed, and compared with the data by the theoretical method and experimental results. The comparison shows that the statistical two-scale model is validated for determination of the thermo-mechanical coupling properties of porous materials and demonstrates its potential applications in actual engineering computation.