基于宏观三角形分区平板壳单元的非线性有限元分析

针对剪切闭锁效应,本文研究了一种基于假设自然应变方法的宏观三角形分区平板壳单元。利用通用有限元软件ABAQUS所提供的用户自定义单元(UEL)和用户自定义材料(UMAT)子程序,本文将宏观三角形分区平板壳单元和基于损伤能释放率的混凝土弹塑性损伤本构模型成功嵌入了ABAQUS的主分析模块。经典试验McNeice双向混凝土板的数值模拟结果表明:宏观三角形分区平板壳单元对于描述板壳结构的非线性损伤行为是行之有效的。     

Exact mesh shape design of large cable-network antenna reflectors with flexible ring truss supports

An exact-designed mesh shape with favorable surface accuracy is of practical significance to the performance of large cable-network antenna reflectors. In this study, a novel design approach that could guide the generation of exact spatial parabolic mesh configurations of such reflector was proposed. By incorporating the traditional force density method with the standard finite element method, this proposed approach had taken the deformation effects of flexible ring truss supports into consideration, and searched for the desired mesh shapes that can satisfy the requirement that all the free nodes are exactly located on the objective paraboloid. Compared with the conventional design method,a remarkable improvement of surface accuracy in the obtained mesh shapes had been demonstrated by numerical examples. The present work would provide a helpful technical reference for the mesh shape design of such cable-network antenna reflector in engineering practice.     

Shear deformable finite beam elements for composite box beams

The shear deformable thin-walled composite beams with closed cross-sections have been developed for coupled flexural, torsional, and buckling analyses. A theoretical model applicable to the thin-walled laminated composite box beams is presented by taking into account all the structural couplings coming from the material anisotropy and the shear deformation effects. The current composite beam includes the transverse shear and the restrained warping induced shear deformation by using the first-order shear deformation beam theory. Seven governing equations are derived for the coupled axial-flexural-torsional-shearing buckling based on the principle of minimum total potential energy. Based on the present analytical model, three different types of finite composite beam elements, namely, linear, quadratic and cubic elements are developed to analyze the flexural, torsional, and buckling problems. In order to demonstrate the accuracy and superiority of the beam theory and the finite beam elements developed by this study,numerical solutions are presented and compared with the results obtained by other researchers and the detailed threedimensional analysis results using the shell elements of ABAQUS. Especially, the influences of the modulus ratio and the simplified assumptions in stress–strain relations on the deflection, twisting angle, and critical buckling loads of composite box beams are investigated.     

Coupled Electrokinetic–Hydromechanic Model for \text{ CO}_{2} Sequestration in Porous Media

In this paper, a computational model for the simulation of coupled electrokinetic and hydromechanical flow in a multiphase domain is introduced. Particular emphasis is placed on modeling $\text{ CO}_{2}$ flow in a deformed, unsaturated geologic formation and its associated streaming potential. The governing field equations are derived based on the averaging theory and solved numerically based on a mixed discretization scheme. The standard Galerkin finite element method is utilized to discretize the deformation and the diffusive dominant field equations, and the extended finite element method, together with the level-set method, is utilized to discretize the advective dominant field equations. The level-set method is employed to trace the $\text{ CO}_{2}$ plume front, and the extended finite element method is employed to model the high gradient in the saturation field front. This mixed discretization scheme leads to a highly convergent system, giving a stable and effectively mesh-independent model; furthermore, it minimizes the number of degrees of freedom, making the numerical scheme computationally efficient. The capability of the proposed model is evaluated by verification and numerical examples. Effects of the formation stiffness on the $\text{ CO}_{2}$ flow and the salinity content on the streaming potential are discussed.     

New nonconforming finite element method for solving transient Naiver-Stokes equations

For transient Naiver-Stokes problems, a stabilized nonconforming finite element method is presented, focusing on two pairs inf-sup unstable finite element spaces, i.e., pNC/pNC triangular and pNQ/pNQ quadrilateral finite element spaces. The semi- and full-discrete schemes of the stabilized method are studied based on the pressure projection and a variational multi-scale method. It has some attractive features： avoiding higher-order derivatives and edge-based data structures, adding a discrete velocity term only on the fine scale, being effective for high Reynolds number fluid flows, and avoiding increased computation cost. For the full-discrete scheme, it has second-order estimations of time and is unconditionally stable. The presented numerical results agree well with the theoretical results.     

A New Linear Shell Model for Shells with Little Regularity

In this paper, a new model of linearly elastic shell is formulated. The model is of Koiter’s type capturing the membrane and bending effects. However, the model is well formulated for shells with little regularity, namely the shells whose middle surface is parameterized by a W ^1,∞ function. Unknowns in the model are the displacement \(\tilde{\boldsymbol {u}}\) of the middle surface of the shell and the infinitesimal rotation \(\tilde{\boldsymbol {\omega}}\) of the shell cross-section. The existence and uniqueness of the solution of the model has been proved for \((\tilde{\boldsymbol {u}}, \tilde{\boldsymbol {\omega}}) \in H^{1} \times H^{1}\) with the help of the Lax-Milgram lemma. The model is analyzed asymptotically with respect to the small thickness of the shell. It is shown that the model asymptotically behaves just as the membrane model, the flexural model, and the generalized membrane model in each regime. In this way the model is justified. Since the model is well formulated for shells whose middle surface is with corners, we compare the model with Le Dret’s model of folded plates. It turns out that in the regime of the flexural shells the models are the same. The differential equations of the model are derived. They imply that the model is as a special case of the Cosserat shell model with a single director for a particular constitutive law.     

A Fast Algorithm for Invasion Percolation

We present a computationally fast Invasion Percolation (IP) algorithm. IP is a numerical approach for generating realistic fluid distributions for quasi-static (i.e., slow) immiscible fluid invasion in porous media. The algorithm proposed here uses a binary-tree data structure to identify the site (pore) connected to the invasion cluster that is the next to be invaded. Gravity is included. Trapping is not explicitly treated in the numerical examples but can be added, for example, using a Hoshen–Kopelman algorithm. Computation time to percolation for a 3D system having $N$ total sites and $M$ invaded sites at percolation goes as $O(M \log M)$ for the proposed binary-tree algorithm and as $O(M N)$ for a standard implementation of IP that searches through all of the uninvaded sites at each step. The relation between $M$ and $N$ is $M = N^{D/E}$ , where $D$ is the fractal dimension of an infinite cluster and $E$ is Euclidean space dimension. In numerical practice, on finite-sized cubic lattices with invasion structures influenced by the injection boundary and boundary conditions lateral to the flow direction, we observe the scaling $M = N^{0.852}$ in 3D (valid through the second decimal place) instead of $M= N^{0.843}$ based on the infinite cluster fractal dimension $D=2.53$ .     

Leray-α Regularization of the Smagorinsky-Closed Filtered Equations for Turbulent Jets at High Reynolds Numbers

The article reports on blending of the Leray-α regularization with the conventional Smagorinsky subgrid-scale closure as an option for large-eddy-simulation of turbulent flows at very high Reynolds number on coarse meshes. The model has been tested in the self-similar far-field region of a jet at a range of Reynolds numbers spanning over two decades (4×10³, 4×10⁴ and 4×10⁵) on two very coarse meshes of 2×10⁵ and 3×10⁴ mesh cells. The results are compared with the well-resolved DNS for $Re_D=4\times 10^3$ on 15 million cells and experimental data for higher Re numbers. While the pure Leray-α can fail badly at high Re numbers on very coarse meshes, a blending of the two strategies by adding a small amount of extra-dissipation performs well even at a huge jet Reynolds number of $Re_D=4\times 10^5$ on a very coarse mesh (2×10⁵ cells), despite the ratio of the typical mesh spacing to the Kolmogorov length exceeding 300. It is found that the main prerequisite for successful LES, both for the classic Smagorinsky and the blended Leray-α/Smagorinsky model, is to resolve the shear-length $L_s=\sqrt{\varepsilon/{\cal S}^3}$ (where ${\cal S}$ is the shear-rate modulus), defined by the constraint Δ/L _s ?<?1, where Δ is the typical mesh-cell size. For the mixed Leray-α/Smagorinsky model the regularization parameter should also be related to the shear-length rather than the local mesh size or Reynolds number, for which we propose a guide criterion α?=?0.15÷0.3 L _s .     

Estimability Analysis for Optimization of Hysteretic Soil Hydraulic Parameters Using Data of a Field Irrigation Experiment

To improve the quality of parameter optimization, estimability analysis has been proposed as the first step before inverse modeling. When using field data of irrigation experiments for the determination of soil hydraulic parameters, wetting and drying processes may complicate optimization. The objectives of this study were to compare estimability analysis and inverse optimization of the soil hydraulic parameters in the models with and without considering hysteresis of the soil water retention function. Soil water pressure head data of a field irrigation experiment were used. The one-dimensional vertical water movement in variably saturated soil was described with the Richards equation using the HYDRUS-1D code. Estimability of the unimodal van Genuchten–Mualem hydraulic model parameters as well as of the hysteretic parameter model of Parker and Lenhard was classified according to a sensitivity coefficient matrix. The matrix was obtained by sequentially calculating effects of initial parameter variations on changes in the simulated pressure head values. Optimization was carried out by means of the Levenberg-Marquardt method implemented in the HYDRUS-1D code. The parameters \(\alpha , K_{s}, \theta _{s}\) , and \(n\) in the nonhysteretic model were found sensitive and parameter \(\theta _{s}\) strongly correlated with parameter \(n\) . When assuming hysteresis, the estimability was decreased with soil depth for \(K_{s}\) and \(\alpha ^{d}\) , and increased for \(\theta _{s}\) and n. Among the shape parameters, \(\alpha ^{w}\) was the most estimable. The hysteretic model could approximate the pressure heads in the soil by considering parameters from wetting and drying periods separately as initial estimates. The inverse optimization could be carried out more efficiently with most estimable parameters. Despite the remaining weaknesses of the local optimization algorithm and the inflexibility of the unimodal van Genuchten model, the results suggested that estimability analysis could be considered as a guidance to better define the optimization scenarios and then improved the determination of soil hydraulic parameters.     

Dynamic modeling thin skeletonal shallow shells as 2D structures with nonuniform microstructures

The subject of this consideration is a thin skeletonal elastic shallow shell with an orthogonal beam-grid microstructures. The important feature of the considered shells is that a dimension of the microstructure is of an order of the shell thickness. The formulation of 2D-macroscopic mathematical model of these shells, based on a tolerance averaging approximation (Wo?niak et?al., 2008), is the aim of the paper. During the modeling procedure, the shell under consideration is treated as a structure with a nonuniform microstructure. The general results of the contribution will be illustrated by the analysis of natural vibrations of a cylindrical thin skeletonal shallow shell.     

A piecewise beam element based on absolute nodal coordinate formulation

The element created in this investigation is based on the it absolute nodal coordinate formulation (ANCF) which has been successfully used in flexible multibody system dynamic and integration of computer aid design and analysis (ICADA). When modeling a B-spline curve with ANCF beam element, it is the common manner to convert this curve into a series of Bézier curves because the systematical conversion between ANCF beam element and a Bézier curve has already been built. In order to avoid the constrain equation produced in this method and to express a B-spline curve using only one element, an alternative approach is developed, leading to the piecewise ANCF (PANCF) beam element. It is demonstrated that when two ANCF beam elements are connected according to a particular continuity, they can constitute a PANCF element. Besides, a new PANCF element will be produced when an ANCF element is connected to an existing PANCF element. The continuity condition can be automatically ensured by the selection of nodal coordinates and the calculation of the piecewise continuous shape functions. The algorithm for converting a B-spline curve to a PANCF beam element is then given. There also are discussions on the features of PANCF element. When two neighboring segments of PANCF element have the same assumed length, the position vector at the interface cannot be expressed by the other coordinates so the position vector is preserved in the \(C^{2}\) continuous situation. Two examples are given to conclude the interpolation and continuity properties of the shape function and to demonstrate the feasibility of this PANCF in the ICADA.     

Multi-component Reynolds-averaged Navier–Stokes simulations of Richtmyer–Meshkov instability and mixing induced by reshock at different times

Turbulent mixing generated by shock-driven acceleration of a perturbed interface is simulated using a new multi-component Reynolds-averaged Navier–Stokes (RANS) model closed with a two-equation $K$ – $\epsilon $ model. The model is implemented in a hydrodynamics code using a third-order weighted essentially non-oscillatory finite-difference method for the advection terms and a second-order central difference method for the gradients in the source and diffusion terms. In the present reshocked Richtmyer–Meshkov instability and mixing study, an incident shock with Mach number $M\!a_{\mathrm{s}}=1.20$ is generated in air and progresses into a sulfur hexafluoride test section. The time evolution of the predicted mixing layer widths corresponding to six shock tube test section lengths are compared with experimental measurements and three-dimensional multi-mode numerical simulations. The mixing layer widths are also compared with the analytical self-similar power-law solution of the simplified model equations prior to reshock. A set of model coefficients and initial conditions specific to these six experiments is established, for which the widths before and after reshock agree very well with experimental and numerical simulation data. A second set of general coefficients that accommodates a broader range of incident shock Mach numbers, Atwood numbers, and test section lengths is also established by incorporating additional experimental data for $M\!a_{\mathrm{s}}=1.24$ , $1.50$ , and $1.98$ with $At=0.67$ and $M\!a_{\mathrm{s}}=1.45$ with $At=-0.67$ and previous RANS modeling. Terms in the budgets of the turbulent kinetic energy and dissipation rate equations are examined to evaluate the relative importance of turbulence production, dissipation and diffusion mechanisms during mixing. Convergence results for the mixing layer widths, mean fields, and turbulent fields under grid refinement are presented for each of the $M\!a_{\mathrm{s}}=1.20$ cases.     

Double-Diffusive Natural Convection in Anisotropic Porous Medium Bounded by Finite Thickness Walls: Validity of Local Thermal Equilibrium Assumption

Double-diffusive natural convection in fluid-saturated porous medium inside a vertical enclosure bounded by finite thickness walls with opposing temperature, concentration gradients on vertical walls as well as adiabatic and impermeable horizontal ones has been performed numerically. The Darcy model was used to predict fluid flow inside the porous material, while thermal fields are simulated based on two-energy equations for fluid and solid phases on the basis of a local thermal non-equilibrium model. Computations have been performed for different controlling parameters such as the buoyancy ratio $N$ , the Lewis number Le, the anisotropic permeability ratio $R_\mathrm{p}$ , the fluid-to-solid thermal conductivity ratio $R_\mathrm{c}$ , the interphase heat transfer coefficient $\mathcal{H}$ , the ratio of the wall thickness to its height $D$ , the wall-to-porous medium thermal diffusivity ratio $R_\mathrm{w}$ , and the solid-to-fluid heat capacity ratio $\gamma $ . Thus, the effects of the controlling parameters on heat and mass transfer characteristics are discussed in detail. Moreover, the validity domain of the local thermal equilibrium (LTE) assumption has been delimited for different set of the governing parameters. It has been shown that Le has a noticeable significant effect on fluid temperature profiles and that higher $N$ values lead to a significant enhancement in heat and mass transfer rates. Moreover, for higher $\mathcal{H}, R_\mathrm{c}$ , $R_\mathrm{p}, R_\mathrm{w}$ , or $D$ values and/or lower $\gamma $ values, the solid and fluid phases tend toward LTE.     

Gas Sorption and the Consequent Volumetric and Permeability Change of Coal II: Numerical Modeling

The permeability of coalbed methane reservoirs may evolve during the recovery of methane and injection of gas, due to the change of effective stress and gas adsorption and desorption. Experimental and numerical studies were conducted to investigate the sorption-induced permeability change of coal. This paper presents the numerical modeling part of the work. It was found that adsorption of pure gases on coal was well represented by parametric adsorption isotherm models in the literature. Based on the experimental data of this study, adsorption of pure \(\hbox {N}_2\) was modeled using the Langmuir equation, and adsorption of pure \(\hbox {CO}_2\) was well represented by the N-Layer BET equation. For the modeling of CO \(_2\) & N \(_2\) binary mixture adsorption, the ideal adsorbed solution (IAS) model and the real adsorbed solution (RAS) model were used. The IAS model estimated the total amount of mixture adsorption and the composition of the adsorbed phase based on the pure adsorption isotherms. The estimated total adsorption and adsorbed-phase composition were very different from the experimental results, indicating nonideality of the CO \(_2\) –N \(_2\) –Coal-adsorption system. The measured sorption-induced strain was linearly proportional to the total amount of adsorption despite the species of the adsorbed gas. Permeability reduction followed a linear correlation with the volumetric strain with the adsorption of pure \(\hbox {N}_2\) and the tested CO \(_2\) & N \(_2\) binary mixtures, and an exponential correlation with the adsorption of pure \(\hbox {CO}_2\) .     

Saint-Venant End Effects in Anti-plane Shear for Classes of Linear Piezoelectric Materials

This paper is concerned with further investigation of the effect of mechanical/electrical coupling on the decay of Saint-Venant end effects in linear piezoelectricity. Saint-Venant's principle and related results for elasticity theory have received considerable attention in the literature but relatively little is known about analogous issues in piezoelectricity. The current rapidly developing smart structures technology provides motivation for the investigation of such problems. The decay of Saint-Venant end effects is investigated in the context of anti-plane shear deformations for linear homogeneous piezoelectric solids. For a rather general class of anisotropic piezoelectric materials, the governing partial differential equations of equilibrium are a coupled system of second-order partial differential equations for the mechanical displacement u and electric potential ?. The traction boundary-value problem with prescribed surface charge can be formulated as an oblique derivative boundary-value problem for this elliptic system. Energy-decay estimates using differential inequality methods are used to study the axial decay of solutions on a semi-infinite strip subjected to non-zero boundary conditions only at the near end. This analysis is carried out for a rather general class of materials (the tetragonal ${\bar 4}$ crystal class). The boundary-value problem involves a full coupling of mechanical and electrical effects. There are four independent material constants appearing in the problem. An explicit estimated decay rate (a lower bound for the actual decay rate) is obtained in terms of two dimensionless piezoelectric parameters d ₀,r, the first of which provides a measure of the degree of piezoelectric coupling. The estimated decay rate is shown to be monotone decreasing with increasing values of the coupling parameter d ₀. In the limit as d ₀→0, we recover the exact decay rate for the purely mechanical case. Thus, for the tetragonal ${\bar 4}$ class of materials, piezoelectric end effects are predicted to penetrate further into the strip than their elastic counterparts, confirming recent results obtained in other contexts in linear piezoelectricity.     

Spherical combustion clouds in explosions

This study explores the properties of spherical combustion clouds in explosions. Two cases are investigated: (1) detonation of a TNT charge and combustion of its detonation products with air, and (2) shock dispersion of aluminum powder and its combustion with air. The evolution of the blast wave and ensuing combustion cloud dynamics are studied via numerical simulations with our adaptive mesh refinement combustion code. The code solves the multi-phase conservation laws for a dilute heterogeneous continuum as formulated by Nigmatulin. Single-phase combustion (e.g., TNT with air) is modeled in the fast-chemistry limit. Two-phase combustion (e.g., Al powder with air) uses an induction time model based on Arrhenius fits to Boiko’s shock tube data, along with an ignition temperature criterion based on fits to Gurevich’s data, and an ignition probability model that accounts for multi-particle effects on cloud ignition. Equations of state are based on polynomial fits to thermodynamic calculations with the Cheetah code, assuming frozen reactants and equilibrium products. Adaptive mesh refinement is used to resolve thin reaction zones and capture the energy-bearing scales of turbulence on the computational mesh (ILES approach). Taking advantage of the symmetry of the problem, azimuthal averaging was used to extract the mean and rms fluctuations from the numerical solution, including: thermodynamic profiles, kinematic profiles, and reaction-zone profiles across the combustion cloud. Fuel consumption was limited to $\sim $ 60–70 %, due to the limited amount of air a spherical combustion cloud can entrain before the turbulent velocity field decays away. Turbulent kinetic energy spectra of the solution were found to have both rotational and dilatational components, due to compressibility effects. The dilatational component was typically about 1 % of the rotational component; both seemed to preserve their spectra as they decayed. Kinetic energy of the blast wave decayed due to the pressure field. Turbulent kinetic energy of the combustion cloud decayed due to enstrophy $\overline{\omega ^{2}} $ and dilatation $\overline{\Delta ^{2}} $ .     

Estimability Analysis and Optimisation of Soil Hydraulic Parameters from Field Lysimeter Data

Modelling the water-flow in the vadose zone requires accurate hydraulic parameters to be obtained at the relevant scale. Weighable lysimeters enable us to monitor hydraulic data at an intermediate scale between lab and field scales and they can be used to optimise these parameters. Parameter optimisation using inverse methods may be limited by the non-uniqueness of the solution. In this contribution, an estimability method has been used to assess the estimability of the van Genuchten–Mualem parameters, to evaluate the information content of the data collected from a bare field lysimeter and to optimise the estimable model parameters. Daily data were monitored from a 2  $\text{ m }^{3}$ lysimeter, filled with the soil of a former coking plant: pressure heads and water contents were measured at three depths (50, 100, 150 cm), cumulative boundary water fluxes. Water-flow was represented using the one-dimensional single-porosity model implemented in HYDRUS-1D code. The estimability of the 5 van Genuchten–Mualem hydraulic parameters and the information content of different data were evaluated by sequentially calculating a sensitivity coefficient matrix. Optimisation was achieved by the Levenberg-Marquardt algorithm. The estimability analysis revealed that estimability of the soil hydraulic parameters, based on the combination of daily pressure heads and water contents, was higher than those based on these data separately. In case of 2.4 being considered as a cut-off criterion for this study, all the parameters were considered estimable from daily data in the decreasing order: $\theta _\mathrm{s},\, \textit{n}, K_\mathrm{s},\, \alpha ,\, \theta _\mathrm{r}$ . Hydraulic parameters were optimised in four scenarios: $\theta _\mathrm{s}$ and n were estimated with reliability while $\alpha ,\, K_\mathrm{s}$ and $\theta _\mathrm{r}$ were uncertain. However, the narrow variations in measured data restricted parameter optimisation.     

The evolution and homogeneity of EU economies (with an econometric approach)

We consider a model for the evolutions of \(m>2\) economies \(y_{i}(t),\) where we assume that their interaction is based on the differences of the values of their evolution status. Since the economies have structures which cause delays, we introduce in the equations a mathematical memory formalism represented by a derivative of fractional order which leads to a system of integro-differential equations. The solution is obtained from a set of \(m\) linear equations in the Laplace Transform (LT) of the \(y_{i}(t)\) . Differently from a previous note Caputo (Non Linear Dyn Econ 16:2–22, 2012), in the present one each economy is affected by a different memory. Is found that the asymptotic values of the state of evolutions of the economies are all equal to the initial value of the economy which has the memory represented by a fractional derivative of order smaller than the others. A method for the definition and estimate of the measure of economies and their comparison is also presented and applied to study the homogeneities of 5 EU economies with an estimate of fields, where they are inhomogeneous.     

Flow temporal reconstruction from non-time-resolved data part I: mathematic fundamentals

At least two circumstances point to the need of postprocessing techniques to recover lost time information from non-time-resolved data: the increasing interest in identifying and tracking coherent structures in flows of industrial interest and the high data throughput of global measuring techniques, such as PIV, for the validation of computational fluid dynamics (CFD) codes. This paper offers the mathematic fundamentals of a space--time reconstruction technique from non-time-resolved, statistically independent data. An algorithm has been developed to identify and track traveling coherent structures in periodic flows. Phase-averaged flow fields are reconstructed with a correlation-based method, which uses information from the Proper Orthogonal Decomposition (POD). The theoretical background shows that the snapshot POD coefficients can be used to recover flow phase information. Once this information is recovered, the real snapshots are used to reconstruct the flow history and characteristics, avoiding neither the use of POD modes nor any associated artifact. The proposed time reconstruction algorithm is in agreement with the experimental evidence given by the practical implementation proposed in the second part of this work (Legrand et al. in Exp Fluids, 2011), using the coefficients corresponding to the first three POD modes. It also agrees with the results on similar issues by other authors (Ben Chiekh et al. in 9 Congrès Francophone de Vélocimétrie Laser, Bruxelles, Belgium, 2004; Van Oudheusden et al. in Exp Fluids 39-1:86?C98, 2005; Meyer et al. in 7th International Symposium on Particle Image Velocimetry, Rome, Italy, 2007a; in J Fluid Mech 583:199?C227, 2007b; Perrin et al. in Exp Fluids 43-2:341?C355, 2007). Computer time to perform the reconstruction is relatively short, of the order of minutes with current PC technology.     

Asymptotic analysis of a multimaterial with a thin piezoelectric interphase

We study the electromechanical behavior of a multimaterial constituted by a linear piezoelectric transversely isotropic plate-like body with high rigidity embedded between two generic three-dimensional piezoelectric bodies by means of the asymptotic expansion method. After defining a small real dimensionless parameter \(\varepsilon \) , which will tend to zero, we characterize the limit model and the associated limit problem. We give also a mathematical justification of the model by means of a functional convergence argument. Moreover, we identify the non classical electromechanical transmission conditions between the two three-dimensional bodies.