Multi-scale FE computation for the structures of composite materials with small periodic configuration under condition of coupled thermoelasticity

In this paper, the multi-scale computational method for a structure of composite materials with a small periodic configuration under the coupled thermoelasticity condition is presented. The two-scale asymptotic (TSA) expression of the displacement and the increment of temperature for composite materials with a small periodic configuration under the condition of thermoelasticity are briefly shown at first, then the multi-scale finite element algorithms based on TSA are discussed. Finally the numerical results evaluated by the multi-scale computational method are shown. It demonstrates that the basic configuration and the increment of temperature strongly influence the local strains and local stresses inside a basic cell. The project supported by the National Natural Science Foundation of China (19932030) and Special Funds for Major State Basic Research Projects     

Two-scale finite element method for piezoelectric problem in periodic structure

The prediction of the mechanical and electric properties of piezoelectric fibre composites has become an active research area in recent years. By means of introducing a boundary layer problem, some new kinds of two-scale finite element methods for solutions to the electric potential and the displacement for composite material in periodic structure under the coupled piezoelectricity are derived. The coupled two-scale relation of the electric potential and the displacement is set up, and some finite element approximate estimates and numerical examples which show the effectiveness of the method are presented.     

Fully implicit formulation of elastoplastic homogenization problem for two-scale analysis

In this study, using a virtual work equation, a micro-/macro-kinematic relation and a linearized constitutive relation, a boundary value problem is fully implicitly formulated to determine perturbed displacement increment fields in elastoplastic unit cells for two-scale analysis. It is shown that this implicit homogenization problem can be iteratively solved with quadratic convergences by successively updating strain increment fields in unit cells, and that the boundary value problem formulated provides a computational algorithm which is versatile for initial setting of strain increment fields. The computational algorithm developed is then examined by performing a two-scale analysis of a holed plate with an elastoplastic micro-structure, subjected to tensile loading. This demonstrates that the convergence in iteratively solving the implicit homogenization problem strongly depends on the initial setting of strain increment fields in unit cells.     

Two-scale finite element method for piezoelectric problem in periodicstructure

The prediction of the mechanical and electric properties of piezoelectric fibre composites has become an active research area in recent years. By means of introducing a boundary layer problem, some new kinds of two-scale finite element methods for solutions to the electric potential and the displacement for composite material in periodic struc- ture under the coupled piezoelectricity are derived. The coupled two-scale relation of the electric potential and the displacement is set up, and some finite element...     

NonLinearly Elastic Membrane Model For Heterogeneous Shells by Using a New Double Scale Variational Formulation: A Formal Asymptotic Approach

This paper is concerned with the asymptotic analysis of shells with periodically rapidly varying heterogeneities. The asymptotic analysis is performed when both the periods of changes of the material properties and the thickness of the shell are of the same orders of magnitude. We consider a shell made of Saint Venant–Kirchhoff type materials for which we justify a new two-scale variational formulation. We assume that both the data and the displacement field admit a formal asymptotic expansion with a negative order of the leading term. We prove that the lowest order term of the displacement field must be of order zero. When the space of nonlinear inextensional displacement is reduced to , this displacement field is a solution of a two-dimensional membrane model which is obtained by solving two coupled problems. The first, posed on the middle surface of the shell is two-dimensional and global and the second, posed on the periodicity cell, is three-dimensional and local.     

Bloch-wave homogenization for a spectral problem in fluid-solid structures

This paper is concerned with the study of the vibrations of a coupled fluid-solid periodic structure. As the period goes to zero, an asymptotic analysis of the spectrum (i.e., the set of eigenfrequencies) is performed with the help of a new method, the so-called Bloch-wave homogenization method (which is a blend of two-scale convergence and Bloch-wave decomposition). The limit spectrum is made of three parts: the macroscopic or homogenized spectrum, the microscopic or Bloch spectrum, and the boundary-layer spectrum. The two first parts are completely characterized: The homogenized and the Bloch spectra are purely essential, and have a band structure. The boundary-layer spectrum is shown to be empty in the special case of periodic boundary condition.     

圆板在物体撞击下的非线性动力响应

本文在Von Kármán大位移的意义上,利用虚位移原理伽辽金方法建立了圆板在物体撞击下的非线性动力响应的控制微分方程,在研究响应问题时,考虑了冲击载荷与圆板位移响应之间的耦合影响,文中使用时间增量法和奇异摄动理论求解问题的控制方程,获得了固支圆板非线性动力响应的近似解,并且求解了具体算例,绘出了圆板位移、应力响应曲线以及冲击力随时间的变化曲线。     

A two-scale non-local model of swelling porous media incorporating ion size correlation effects

A new two-scale model is proposed for derivation of the macroscopic modified effective stress principle for swelling porous media saturated by an electrolyte solution containing finite size ions. A non-local pore-scale model is developed within the framework of Statistical Mechanics in conjunction with the thermodynamic approach based on Density Functional Theory leading to a nonlinear integral Fredholm equation of second kind for the ion/nanopore correlation function coupled with Poisson problem for the electric double layer potential. When combined with the fluid equilibrium condition such non-local electrochemical problem gives rise to a constitutive law for the fluid stress tensor in terms of the disjoining pressure which is decomposed into several components of different nature. The homogenization procedure based on formal asymptotic expansions is applied to up-scale the model to the macroscale leading to a two-scale constitutive law for the swelling pressure appearing in the modified effective stress principle with improved accuracy incorporating the deviations from the Gouy–Chapman Poisson–Boltzmann-based theory due to the finite size short-range ion–ion correlation effects. The integro-differential problem posed in a periodic cell is discretized by collocation schemes. Numerical results are obtained for a stratified arrangement of parallel macromolecules showing that the effects of ion–ion correlation forces give rise to anomalous attraction patterns between the particles for divalent ions.     

物性与温度相关材料广义热弹性模型及渐近分析

计及材料物性与温度的相关性,基于Clausius不等式和L-S广义热弹性理论,通过对自由能公式的高阶展开,构建了具有变物性特征的广义耦合热弹性动力学模型。推导了各向同性材料表面受热冲击问题的线性化控制方程组,利用热冲击的瞬时特征,借助于Laplace正、逆变换技术及其极限性质,给出了变物性条件下一维热冲击问题的温度场、位移场和应力场的渐近表达式。通过算例,得到了热冲击作用下各物理场的分布规律以及材料物性与温度相关性对于热弹性响应的影响规律。结果表明：材料物性与温度相关性对于各物理场的阶跃位置、阶跃间隔以及阶跃峰值均产生影响,但值得注意的是,相比于位移场和应力场的显著影响,其对温度场的影响效果并不明显。     

Consistent incremental approximation of dissipation pseudo-potentials in the variational formulation of thermo-mechanical constitutive updates

In this paper, we detail a consistent approximate expression for incremental dissipation pseudo-potentials which appear in the variational formulation of coupled thermo-mechanical boundary-value problems. We explain why the most intuitive expression does not work in the case of an explicit temperature dependence in the dissipation, and propose an alternative expression ensuring consistent results when reducing the time increment towards zero.     

圆板在物体撞击下的非线性动力响应

本文在Von Kármán大位移的意义上,利用虚位移原理伽辽金方法建立了圆板在物体撞击下的非线性动力响应的控制微分方程,在研究响应问题时,考虑了冲击载荷与圆板位移响应之间的耦合影响,文中使用时间增量法和奇异摄动理论求解问题的控制方程,获得了固支圆板非线性动力响应的近似解,并且求解了具体算例,绘出了圆板位移、应力响应曲线以及冲击力随时间的变化曲线。     

A Two-Scale Model for Coupled Electro-Chemo-Mechanical Phenomena and Onsager’s Reciprocity Relations in Expansive Clays: II Computational Validation

In Part I Moyne and Murad [Transport in Porous Media 62, (2006), 333–380] a two-scale model of coupled electro-chemo-mechanical phenomena in swelling porous media was derived by a formal asymptotic homogenization analysis. The microscopic portrait of the model consists of a two-phase system composed of an electrolyte solution and colloidal clay particles. The movement of the liquid at the microscale is ruled by the modified Stokes problem; the advection, diffusion and electro-migration of monovalent ions Na⁺ and Cl⁻ are governed by the Nernst–Planck equations and the local electric potential distribution is dictated by the Poisson problem. The microscopic governing equations in the fluid domain are coupled with the elasticity problem for the clay particles through boundary conditions on the solid–fluid interface. The up-scaling procedure led to a macroscopic model based on Onsager’s reciprocity relations coupled with a modified form of Terzaghi’s effective stress principle including an additional swelling stress component. A notable consequence of the two-scale framework are the new closure problems derived for the macroscopic electro-chemo-mechanical parameters. Such local representation bridge the gap between the macroscopic Thermodynamics of Irreversible Processes and microscopic Electro-Hydrodynamics by establishing a direct correlation between the magnitude of the effective properties and the electrical double layer potential, whose local distribution is governed by a microscale Poisson–Boltzmann equation. The purpose of this paper is to validate computationally the two-scale model and to introduce new concepts inherent to the problem considering a particular form of microstructure wherein the clay fabric is composed of parallel particles of face-to-face contact. By discretizing the local Poisson–Boltzmann equation and solving numerically the closure problems, the constitutive behavior of the diffusion coefficients of cations and anions, chemico-osmotic and electro-osmotic conductivities in Darcy’s law, Onsager’s parameters, swelling pressure, electro-chemical compressibility, surface tension, primary/secondary electroviscous effects and the reflection coefficient are computed for a range particle distances and sat concentrations.     

Non-axisymmetric Homann stagnation-point flow of Walter's B nanofluid over a cylindrical disk

The study of non-axisymmetric Homann stagnation-point flow of Walter's B nanofluid along with magnetohydrodynamic(MHD) and non-linear Rosseland thermal radiation over a cylindrical disk in the existence of the time-independent free stream is considered. Moreover, the notable impacts of thermophoresis and Brownian motion are analyzed by Buongiorno's model. The momentum, energy, and concentration equations are converted into the dimensionless coupled ordinary differential equations via similarity transformations, which are later numerically solved by altering the values of the pertinent parameters. The numerical and asymptotic solutions for the large shear-to-strain rate ratio γ =a/bfor the parameters of the displacement thicknesses and the wall-shear stress are computed by perturbative expansion and analyzed. Furthermore, the technique bvp4c in MATLAB is deployed as an efficient method to analyze the calculations for the non-dimensional velocities, temperature, displacement thickness, and concentration profiles. It is observed that the two-dimensional displacement thickness parameters α andβ are reduced due to the viscoelasticity and magnetic field effects. Moreover, when the shear-to-strain rate ratio approaches infinity, α is closer to its asymptotic value, while βand the three-dimensional displacement thickness parameter δ_1 show the opposite trend.The outcomes of the viscoelasticity and the magnetic field on the skin friction are also determined. It is concluded that ■ reaches its asymptotic behavior when the shearto-strain rate ratio approaches infinity. Meanwhile, ■ shows different results.     

Analysis of thermal responses in a two-dimensional porous medium caused by pulse heat flux

In this article, the generalized model for thermoelastic waves with two relaxation times is utilized to compute the increment of temperature, the displacement components, the stress components, and the changes in the volume fraction field in a two-dimensional porous medium. By using the Fourier-Laplace transform and the eigenvalue method, the considered variables are obtained analytically. The derived approach is estimated with numerical outcomes which are applied to the porous media with a geometrical simplification. The numerical results for the considered variables are performed and presented graphically. Finally, the outcomes are represented graphically to display the difference among the classical dynamical(CD) coupled, the Lord-Shulman(LS), and the Green-Lindsay(GL) models.     

Energy release rate and path-independent integral in dynamic fracture of magneto-electro-thermo-elastic solids

The energy release rate and associated energy flux integral in dynamic fracture of magneto-electro-thermo-elastic solids are formulated with the inclusion of multi-field fully coupled effects based on fundamental principles of thermodynamics. The difference between the global and local dynamic contour integrals is caused by unsteady state, mechanical body force, electricity conduction and thermal effect as the closed contour including crack faces is chosen. This formulation successfully captures the crack-tip singularity of coupled fields, offers the right expression for the crack driving force, and resolves the controversial issue on magneto-electro-thermo-elastic fracture criterion. Especially, for steady-state crack propagation in a magneto-electro-elastic solid, the path-independent dynamic contour integral is determined from the asymptotic near-tip field solution based on the Stroh-type formalism and the resulting dynamic energy release rate has an odd dependence on the dynamic magnetic induction intensity factor and the dynamic electric displacement intensity factor.     

考虑微结构连接性的双尺度结构自然频率拓扑优化

多孔材料因具有轻量化、高孔隙率和减振/散热等优良多物理特性,在航空航天等领域具有广阔应用前景。采用拓扑优化方法对含多种多孔材料的结构进行结构与材料微结构构型一体化设计,有助于获得具有优良力学性能的结构设计。然而,传统逆均匀化微结构设计方法无法确保不同多孔材料微结构之间的连接性,设计结果不具备可制造性。本文面向含多种多孔材料的双尺度结构基频最大化设计问题,考虑不同微结构之间的连接性,协同设计多孔材料的微结构构型及其在宏观尺度下的布局。采用均匀化方法计算多孔材料的宏观等效力学性能,通过对不同多孔材料微结构单胞的边界区域采用相同的拓扑描述确保双尺度优化过程中任意空间排布下不同微结构的连接性,并通过优化算法确定微结构间的连接形式及微结构拓扑。在宏观尺度,提出结合离散材料插值模型和RAMP插值模型RAMP (Rational Approximation of Material Properties)的多孔材料各向异性宏观等效刚度及质量插值模型,获得清晰的多孔材料宏观尺度布局并减轻优化过程中伪振动模态的影响。建立以双尺度结构基频最大化为目标,以材料用量为约束的优化列式,推导灵敏度表达式,并基于梯度优化算法求解双尺度结构拓扑优化问题。数值算例表明,采用本文优化方法能够有效确保基频最大化双尺度结构设计中不同多孔材料微结构之间的连接性,增强优化设计结果的可制造性。     

Effect of the thermal relaxation and magnetic field on generalized micropolar thermoelasticity

The model of generalized micropolar magneto-thermoelasticity for a thermally and perfectly conducting half-space is studied. The initial magnetic field is parallel to the boundary of the half-space. The formulation is applied to the generalized thermo-elasticity theories of Lord and Shulman, Green and Lindsay, as well as to the coupled dynamic theory. The normal mode analysis is used to obtain expressions for the temperature increment, the displacement, and the stress components of the model at the interface. By using potential functions, the governing equations are reduced to two fourth-order differential equations. By numerical calculation, the variation of the considered variables is given and illustrated graphically for a magnesium crystal micropolar elastic material. Comparisons are performed with the results predicted by the three theories in the presence of a magnetic field.     

Evaluation of continuous modelings for the modulated vibration modes of long repetitive structures

The aim of this paper is to evaluate a recently proposed continuous modeling for the modulated vibrations modes of long periodic structures. We describe an asymptotic two-scale method to study free vibrations of this type of structures in two cases: beam bending theory and 2D elasticity. A validation of the two proposed continuous modelings is presented and a comparison between 1D and 2D approach is given.     

A very complicated structure of resonances in a nonlinear system with cyclic symmetry: Nonlinear forced localization

The fundamental and subharmonic resonances of a nonlinear cyclic assembly are examined using the asymptotic method of multiple-scales. The system consists of a number of identical cantilever beams coupled by means of weak linear stiffnesses. Assuming beam inextensionality, geometric nonlinearities arise due to longitudinal inertia and the nonlinear relation between beam curvature and transverse displacement. The governing nonlinear partial differential equations are discretized by a Galerkin procedure and the resulting set of coupled ordinary differential equations is solved using an asymptotic analysis. The unforced assembly is known to possess localized nonlinear normal modes, which give rise to a very complicated topological structure of fundamental and subharmonic response curves. In contrast to the linear system which exhibits as many forced resonances as its number of degrees of freedom, the nonlinear system is found to possess a number of additional resonance branches which have no counterparts in linear theory. Some of the additional resonances are spatially localized, corresponding to motions of only a small subset of periodic elements. The analytical results are verified by numerical Poincaré maps, and the forced localization features of the nonlinear assembly are demonstrated by considering its response to impulsive excitations.     

A long wave low frequency motion model for a finitely sheared elastic layer

We consider two-dimensional long wave low frequency motion in a pre-stressed layer composed of neo-Hookean material. Specifically, the pre-stress is a simple shear deformation. Derivation of the dispersion relation associated with traction-free boundary conditions is briefly reviewed. Appropriate approximations are established for the two associated long wave modes. From these approximations it is clear that there may be either two, one or no real long wave limiting phase speeds. These approximations are also used to establish the relative asymptotic orders of the displacement components and pressure increment. Using these relative orders to motivate the introduction of appropriate a scales, an asymptotically consistent model long wave low frequency motion is established. It is shown that in the presence of shear there is neither bending nor extension, or analogues of their previously established pre-stressed counterparts. In fact, both the in-plane and normal displacement components have the same asymptotic orders and the derived governing equation is of vector form.