Asymptotic analysis of perforated plates and membranes. Part 2: Static and dynamic problems for large holes

Static and dynamic problems for the elastic plates and membranes periodically perforated by holes of different shapes are solved using the combination of the singular perturbation technique and the multi-scale asymptotic homogenization method. The problems of bending and vibration of perforated plates are considered. Using the asymptotic homogenization method the original boundary-value problems are reduced to the combination of two types of problems. First one is a recurrent system of unit cell problems with the conditions of periodic continuation. And the second problem is a homogenized boundary-value problem for the entire domain, characterized by the constant effective coefficients obtained from the solution of the unit cell problems. In the present paper the perforated plates with large holes are considered, and the singular perturbation method is used to solve the pertinent unit cell problems. Matching of limiting solutions for small and large holes using the two-point Padé approximants is also accomplished, and the analytical expressions for the effective stiffnesses of perforated plates with holes of arbitrary sizes are obtained.     

On the analysis of a plate with a local shape perturbation

The asymptotic behavior of the solution of the bending problem of plates with local shape perturbations (connections, ribs, holes comparable in size with the plate thickness) is studied in a three-dimensional formulation using the local perturbation method. The problem is completely decomposed into a two-dimensional problem of plate theory and local problems describing the threedimensional stress-strain state in the perturbation region. The local problems are solved using numerical methods.     

Asymptotic analysis of boundary-value problems in thin perforated domains with rapidly varying thickness

For a second-order symmetric uniformly elliptic differential operator with rapidly oscillating coefficients, we study the asymptotic behavior of solutions of a mixed inhomogeneous boundary-value problem and a spectral Neumann problem in a thin perforated domain with rapidly varying thickness. We obtain asymptotic estimates for the differences between solutions of the original problems and the corresponding homogenized problems. These results were announced in Dopovidi Akademii Nauk Ukrainy, No. 10, 15–19 (1991). The new results obtained in the present paper are related to the construction of an asymptotic expansion of a solution of a mixed homogeneous boundary-value problem under additional assumptions of symmetry for the coefficients of the operator and for the thin perforated domain.     

On singular perturbation boundary-value problem of coupling type system of convection-diffusion equations

In this paper we consider the singular perturbation boundary-value problem of thefollowing coupling type system of convection-diffusion equationsWe advance two methods:the first one is the initial value solving method,by which theoriginal boundary-value problem is changed into a series of unperturbed initial-valueproblems of the first order ordinary differential equation or system so that an asymptoticexpansion is obtained;the second one is the boundary-value solving method,by which theoriginal problem is changed into a few boundary-value problems having no phenomenon ofboundary-layer so that the exact solution can be obtained and any classical numericalmethods can be used to obtain the numerical solution of consismethods can be used to obtainthe numerical solution of consistant high accuracy with respect to the perturbationparameterε     

Solving the problem of linear viscoelasticity for multiply connected anisotropic plates

This paper describes the method for solving the problems of linear viscoelasticity for thin plates under the influence of bending moments and transverse forces. The small parameter method was used to reduce the original problem to a sequence of boundary-value problems solved via complex potentials of the bending theory of multiply connected anisotropic plates. The general representations of complex potentials and boundary conditions for their determination are obtained. The method for determining the stress state of the plate at any time with respect to complex approximation potentials is developed by replacing the powers of the small parameter by the Rabotnov operators. The problem of a plate with elliptical holes is solved. The numerical calculation results in the case of a plate with one or two holes are given. The variation of bending moments in time until stationary condition is reached is studied, and the influence of geometric characteristics of the plate on these variable is described.     

Second-order homogenization of periodic materials based on asymptotic approximation of the strain energy: formulation and validity limits

In this paper a second-order homogenization approach for periodic material is derived from an appropriate representation of the down-scaling that correlates the micro-displacement field to the macro-displacement field and the macro-strain tensors involving unknown perturbation functions. These functions take into account of the effects of the heterogeneities and are obtained by the solution of properly defined recursive cell problems. Moreover, the perturbation functions and therefore the micro-displacement fields result to be sufficiently regular to guarantee the anti-periodicity of the traction on the periodic unit cell. A generalization of the macro-homogeneity condition is obtained through an asymptotic expansion of the mean strain energy at the micro-scale in terms of the microstructural characteristic size ?; the obtained overall elastic moduli result to be not affected by the choice of periodic cell. The coupling between the macro- and micro-stress tensor in the periodic cell is deduced from an application of the generalised macro-homogeneity condition applied to a representative portion of the heterogeneous material (cluster of periodic cell). The correlation between the proposed asymptotic homogenization approach and the computational second-order homogenization methods (which are based on the so called quadratic ansätze) is obtained through an approximation of the macro-displacement field based on a second-order Taylor expansion. The form of the overall elastic moduli obtained through the two homogenization approaches, here proposed, is analyzed and the differences are highlighted. An evaluation of the developed method in comparison with other recently proposed in literature is carried out in the example where a three-phase orthotropic material is considered. The characteristic lengths of the second-order equivalent continuum are obtained by both the asymptotic and the computational procedures here analyzed. The reliability of the proposed approach is evaluated for the case of shear and extensional deformation of the considered two-dimensional infinite elastic medium subjected to periodic body forces; the results from the second-order model are compared with those of the heterogeneous continuum.     

On the first order change of bifurcation buckling loads due to structural geometry perturbations

It is shown that a simple method may be applied in order to determine the first order change of eigenmodes and critical loads at bifurcation buckling of slender structures when cavities or reinforcements are introduced. The perturbation procedure developed is applicable whenever a characteristic diameter of the structural geometry perturbation is one order less than the wave length of the unperturbed eigenmode and necessary computations involve at most the solution of two linear boundary value problems. Situations when the procedure may be further simplified are discussed and statements are made regarding requisite conditions for the buckling load definitely to increase or decrease due to geometry perturbations. Application of the method is illustrated by means of four cases of engineering interest, which involve perforated and reinforced beams and plates with holes and cracks.     

The Solution Asymptotics of Classical and Nonclassical, Static and Dynamic Boundary-Value Problems for Thin Bodies

An asymptotic method for solution of classical and nonclassical boundary-value problems of the theory of elasticity for thin bodies (beams, rods, plates, and shells) is expounded. Studies on the asymptotic theory of thin bodies are reviewed. Asymptotic results are compared with those obtained by other applied theories. The asymptotic approach has been found out to be related to Saint Venant's principle. The correctness of this principle is mathematically proved for one class of problems. A fundamentally new asymptotics in the components of the stress tensor and the displacement vector is revealed in considering new classes of problems. On their basis, the applicability domains are outlined for various models of understructures. Solutions are obtained to certain classes of dynamic problems for thin bodies, particularly, those simulating seismic effects. The resonance conditions are established and ways of preventing them are pointed out.     

Asymptotic analysis of laminated plates and shallow shells

It was noted long ago [1] that the material strength theory develops both by improving computational methods and by widening the physical foundations. In the present paper, we develop a computational technique based on asymptotic methods, first of all, on the homogenization method [2, 3]. A modification of the homogenization method for plates periodic in the horizontal projection was proposed in [4], where the bending of a homogeneous plate with periodically repeating inhomogeneities on its surface was studied. A more detailed asymptotic analysis of elastic plates periodic in the horizontal projection can be found, e.g., in [5, 6]. In [6], three asymptotic approximations were considered, local problems on the periodicity cell were obtained for them, and the solvability of these problems was proved. In [7], it was shown that the techniques developed for plates periodic in the horizontal projection can also be used for laminated plates. In [7], this was illustrated by an example of asymptotic analysis of an isotropic plate symmetric with respect to the midplane.     

Homogenization of viscoelastic-matrix fibrous composites with square-lattice reinforcement

The present paper provides details on the application of asymptotic homogenization techniques to the analysis of viscoelastic-matrix fibrous composites with square-lattice reinforcement and their effective properties. The correspondence principle allows transformation of the governing boundary value problems to quasi-static ones. Thereafter, the homogenization procedure is used. To solve the cell problem, modified boundary shape perturbation procedure is proposed. The resulting Laplace transforms are inverted by the effective and accurate Gaver algorithm. The proposed approach, however, yields a computationally intense solution.     

Dynamics stress concentrations in Ambartsumian's plate with a cutout

Based on the motion equations of flexural wave in Ambartsumian's plates including the effects of transverse shear deformations, by using perturbation method of small parameter, the scattering of flexural waves and dynamic stress concentrations in the plate with a cutout have been studied. The asymptotic solution of the dynamic stress problem is obtained. Numerical results for the dynamic stress concentration factor in Ambartsumian's plates with a circular cutour are graphically presented and discussed. The project supported by the National Natural Science Foundation of China.     

基于均匀化理论的多孔板弯曲问题新解

由于订型分布孔的存在,通常的有限元法不能有效地分析多孔板的弯曲问题。该文基于均匀化理论建立了该类问题的新解法。对含密集型分布的阶梯型圆孔板的分析结果,说明了该文方法是有效的。     

Reiterated homogenization of a laminate with imperfect contact: gain-enhancement of effective properties

A family of one-dimensional (1D) elliptic boundary-value problems with periodic and rapidly-oscillating piecewise-smooth coefficients is considered. The coefficients depend on the local or fast variables corresponding to two different structural scales. A finite number of imperfect contact conditions are analyzed at each of the scales. The reiterated homogenization method (RHM) is used to construct a formal asymptotic solution. The homogenized problem, the local problems, and the corresponding effective coefficients are obtained. A variational formulation is derived to obtain an estimate to prove the proximity between the solutions of the original problem and the homogenized problem. Numerical computations are used to illustrate both the convergence of the solutions and the gain of the effective properties of a three-scale heterogeneous 1D laminate with respect to their two-scale counterparts. The theoretical and practical ideas exposed here could be used to mathematically model multidimensional problems involving multiscale composite materials with imperfect contact at the interfaces.     

Vibrations of highly prestressed anisotropic plates via a numerical-perturbation technique

The method of matched asymptotic expansions is used to reduce the problem of the transverse vibrations of a highly prestressed anisotropic plate into the simpler problem of the vibration of an anisotropic membrane with modified boundary conditions that account for the bending effects. In the absence of an exact solution the membrane problem can be solved by any well-known numerical technique. The numerical-perturbation results for a clamped circular plate with rectangular orthotropy and a uniform tensile stress applied on its boundary show an excellent correlation with finite-element solutions for the original problem. Furthermore, the solutions obtained for annular plates form the basis for solutions to problems involving near-annular plates.     

Novel numerical implementation of asymptotic homogenization method for periodic plate structures

The present paper develops and implements finite element formulation for the asymptotic homogenization theory for periodic composite plate and shell structures, earlier developed in  and , and thus adopts this analytical method for the analysis of periodic inhomogeneous plates and shells with more complicated periodic microstructures. It provides a benchmark test platform for evaluating various methods such as representative volume approaches to calculate effective properties. Furthermore, the new numerical implementation (Cheng et al., 2013) of asymptotic homogenization method of 2D and 3D materials with periodic microstructure is shown to be directly applicable to predict effective properties of periodic plates without any complicated mathematical derivation. The new numerical implementation is based on the rigorous mathematical foundation of the asymptotic homogenization method, and also simplicity similar to the representative volume method. It can be applied easily using commercial software as a black box. Different kinds of elements and modeling techniques available in commercial software can be used to discretize the unit cell. Several numerical examples are given to demonstrate the validity of the proposed methods.     

孔结构金属复合夹芯板抗爆炸冲击响应及吸能特性研究

采用有限元方法研究爆炸载荷下四边固支孔结构金属复合夹芯板的动力响应及吸能特性，给出了孔结构金属复合夹芯板的动力响应过程，得到夹芯板的变形模式，比较了孔结构金属复合夹芯板与非孔结构金属复合夹芯板的抗爆炸冲击性能，同时讨论了孔大小、间距、排布方式和面板质量分布等因素对孔结构金属复合夹芯板抗爆炸冲击性能的影响。研究结果表明，迎爆面外面板的孔设计使爆炸冲击波穿过孔洞直接作用在芯材上，增强了芯材的压缩，从而提高了夹芯板的能量吸收能力。同等面密度情况下，内外面板厚度比大于1的孔结构金属复合夹芯板变形挠度小于内外面板厚度比小于1的孔结构金属复合夹芯板。进一步研究发现，通过合理设计内外面板的质量分布，可以使孔结构金属复合夹芯板的抗爆炸冲击性能最优。     

Homogenization-based method for bending analysis of perforated plate

Owing to the existence of distributed holes, it is difficult to solve the bending problem of perforated plates by the conventional finite element method. A homogenization-based method for this problem is presented in this paper. As an example, the bending analysis of a circular perforated plate with distributed step-wise cylindrical holes is made. The deflection and the fundamental frequency obtained by present method are in good agreement with experimental data, this implies that the method is effective. This project is supported by the National Natural Science Foundation (19602007) and National Outstanding Youth Foundation (19525206).     

Multiscale asymptotic homogenization for multiphysics problems with multiple spatial and temporal scales: a coupled thermo-viscoelastic example problem

A systematic approach for analyzing multiple physical processes interacting at multiple spatial and temporal scales is developed. The proposed computational framework is applied to the coupled thermo-viscoelastic composites with microscopically periodic mechanical and thermal properties. A rapidly varying spatial and temporal scales are introduced to capture the effects of spatial and temporal fluctuations induced by spatial heterogeneities at diverse time scales. The initial-boundary value problem on the macroscale is derived by using the double scale asymptotic analysis in space and time. It is shown that an extra history-dependent long-term memory term introduced by the homogenization process in space and time can be obtained by solving a first order initial value problem. This is in contrast to the long-term memory term obtained by the classical spatial homogenization, which requires solutions of the initial-boundary value problem in the unit cell domain. The validity limits of the proposed spatial–temporal homogenized solution are established. Numerical example shows a good agreement between the proposed model and the reference solution obtained by using a finite element mesh with element size comparable to that of material heterogeneity.     

Combination of the homogenization and effective medium methods for flow through porous media in a nonperiodic external perturbation field

Models described by parabolic equations with a rapidly oscillating nonperiodic right side are investigated by means of averaging theory methods. For a nonperiodic perturbation field a combined homogenization and effective medium method is developed. This method makes it possible to obtain the solution of the cell problems in a finite form correct to the second order in the inhomogeneity parameter. The method is applied to problems of single-phase and two-phase flow through porous media. The technique of the method is outlined and explicit solutions of cell problems are constructed.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 108–115, July–August, 1995.     

Homogenization of nonlinear problems in the mechanics of composites

A further development of the homogenization method is proposed to solve the physically nonlinear equilibrium problems for the laminated plates or the plates made of functionally graded materials. In the linear case, according to this method, the corresponding solution is a superposition of the solution to the global problem in the entire domain and the solution to the local problem in a representative domain, e.g., in a periodicity cell. In the nonlinear case, such a superposition is not valid, which complicates the application of the homogenization method. In order to eliminate this difficulty, it is possible to combine the homogenization method and the linearization method when solving a boundary value problem or a variational problem. In the mechanics of deformable solids, the constitutive relations can be considered as equations with respect to velocities or the stress and strain differentials in time or in the loading parameter. When these equations are linear with respect to velocities, it is possible to use the homogenization method. In this paper such an approach is illustrated by the example of a symmetric laminated plate bent under a uniformly distributed time-dependent load.