The contribution is devoted to formulate an averaged mathematical model describing the dynamic behaviour of the composite annular plates resting on elastic heterogeneous foundation with two foundation modules. The plates are made of two-phased, functionally graded — type composites. In contrast to most of the papers in which material properties vary through the plate thickness, in the presented study we have dealt with the plate and foundation in which effective properties vary in a radial direction of the plate. The formulation of the macroscopic mathematical model for the analysis of the dynamic behaviour of these plates will be based on the tolerance averaging technique (Woźniak, Michalak, Jędrysiak, [ed]). This averaging method is an alternative to known asymptotic homogenization. The general results of the contribution will be illustrated by the analysis of free vibrations of the composite plates on heterogeneous foundation. The results obtained from the tolerance model were compared with the results obtained from FEM. There were compared the first four natural frequencies. A good consistency of the results from both methods was obtained.

The boundary element method (BEM) is easier than the finite element method (FEM) on the viewpoint of the discretization of one dimension reduction rather than the domain discretization of finite element method. The disadvantage of BEM is the rank deficiency in the influence matrix, e.g., degenerate boundary, degenerate scale, spurious eigenvalues and fictitious frequencies, which do not occur in the FEM. The conventional BEM can not be straightforward applied to solve a problem which contains a degenerate boundary without decomposing the domain to multi-regions. A hypersingular integral equation is used to ensure a unique solution for the problem containing a degenerate boundary. By combining the singular and hypersingular equations, it’s termed the dual BEM due to its dual frame. Following the successful experience on the retrieval of information using the singular value decomposition (SVD) updating term and updating document, this technique is also used to extract out the degenerate-boundary information and the rigid-body information in the dual BEM. It is interesting to find that true information due to a rigid-body mode in physics is found in the right singular vector with respect to the corresponding zero singular value while the degenerate-boundary mode (geometry degeneracy) in mathematics is imbedded in the left singular vector with respect to the corresponding zero singular value. The role of the common right and left singular vectors of SVD for the four influence matrices in the dual BEM is also discussed in this paper. Two examples, a potential flow problem across a cutoff wall and a cracked bar under torsion were demonstrated to see the mathematical SVD structure of four influence matrices in the dual BEM.

The inverse problem of the Mei symmetry for nonholonomic systems with variable mass is studied. Firstly, the authors discuss the Mei symmetry of the holonomic system opposite to a nonholonomic system. Secondly, weak and strong Mei symmetries of a nonholonomic system are concluded through restriction equations and additional restriction equations. Thirdly, the relevant conserved quantity is deduced by means of the structure equation for the gauge function. Fourthly, the inverse problem of the Mei symmetry is obtained by the Noether symmetry. Finally, the paper offers an example to illustrate the application of the research result.

The accurate and effective prediction of the failure for an inelastic structure, such as a solder joint in an electronic chip packaging, remains a current issue. Subjected to sub-critical cyclic loading, the solder can undergo fatigue cracks, leading to the failure of the whole system after a certain number of power cycles. In this paper, a model for describing the viscoplastic behavior of the solder material under power cycling is implemented in the finite element code Abaqus and a continuum damage procedure is used for lifetime prediction. Damage initiation criterion and damage evolution law, based both on the inelastic strain energy per stabilized cycle as proposed by Darveaux, are used in conjunction with the direct cyclic procedure available in Abaqus. This latter technique allows reducing the considerable computation time needed to obtain the stabilized states during the repetitive loading cycles.

This paper presents a regularization scheme for the nearly singular integrals used for 3D elastostatic boundary element analysis. For the regularization process, the local projection coordinates of the source point are first located via an iteration procedure. For planar elements, the boundary integrals are analytically integrated by parts to smooth the drastic fluctuations of their integrands so that the regularized forms can be numerically integrated by any conventional schemes in an usual manner. The validity of the formulations is numerically tested using the Gauss Quadrature scheme. The test shows the accuracy is satisfactory for the distance ratio (distance: Element characteristic length) falling below micro-scale. To further demonstrate our successful implementation, a numerical example is studied with verifications compared with ANSYS analysis.

In this paper, we present an analytical solution to the general three-dimensional elasticity problem in a layer, whose Young’s modulus varies exponentially within the thickness coordinate and the Poisson’s ratio is constant. By making use of the direct integration method, the complete set of the governing equations in terms of stresses has been formed. The latter equations were reduced to separate equations for each stress-tensor component and then solved by means of the Fourier double-integral transformation with respect to the planar coordinates.

We present a general solution approach for analysis of transversely isotropic cylindrical tubes and circular plates. On the basis of Hamiltonian state space formalism in a systematic way, rigorous solutions of the twisting problems are determined by means of separation of variables and symplectic eigenfunction expansion.

This paper concerns with the unsteady MHD flow of a second grade fluid between two parallel walls through porous media induced by rectified sine pulses shear stress. The analytical expressions for the velocity field and the adequate shear stress are determined by means of the Laplace transform technique and Fourier cosine and sine transforms and are written as a sum of steady state and transient solutions. The influence of side walls on the fluid motion, the distance between walls for which the velocity of the fluid in the middle of the channel is negligible, and the required time to reach the steady state are presented by graphical illustrations. As the second grade fluid parameter → 0 the problem reduces to the Newtonian fluids performing the same motion.

In the provided research, the design of CHE (Compact Heat Exchanger) is evaluated and discussed from the heat transfer aspect. Benefiting from present equations and considering the objective concepts, the procedural chart is proposed for achieving optimal design. The main goal of this research study is implementing a new algorithm for optimization to modify a conventional design of CHE. Nonlinear gradient mathematical modeling with different scenarios on free or related variables is developed to cover the purpose of maximizing total heat transfer capacity. By mathematical programming analysis, a model has been provided for optimal design and developed in the GAMS (Generalized Algebraic Modelling System) software. Also for further model test rig development purpose, the proposed model has been incorporated in Matlab software using independent variants and the accuracy of the responses was again evaluated. The comparison indicated 109W/K difference in the exchanged thermal energy rate compared to the optimal exchanger operation conditions. After introducing case study to this model, an acceptable response with 0.997W/K difference on optimal point was achieved. Solving the model indicated 0.833W/K difference with the optimal point, which confirms the resulted technical responses.

The paper studied the effect of an array of hemispherical substance-units (HSUs) immersed in inviscid and incompressible liquid on resonance frequency of a quartz crystal resonator (QCR) under the thickness-shear mode (TSM) vibrations. A dynamic model of the compound QCR system, consisting of a QCR and HSUs, was established in advance. The frequency equation of the compound QCR system including the liquid-solid coupling was derived subsequently. Numerical results showed the frequency-shift characteristics of the compound system. The obtained results are useful in resonator design and applications.

The stability of pressure-driven liquid-liquid viscosity-stratified microchannel flow is investigated by considering the combined effects of electrical double layer (EDL) and boundary slip. The electrical streaming currents determined by the Streaming Electrical Current Balance (ECB) and the boundary slip are considered by Navier slip assumption. The stability of flow is studied by the small disturbance theory. Numerical results indicate that the effect of boundary slip on the flow stability is strongly depended on the EDL. The boundary slip stables the flow when EDL effect is weak and destabilizes the flow when EDL effect is strong. The effects of boundary slip and EDL on flow stability is also determined by viscosity ratio, the height of channel, interface position and conductivity ratio.

A numerical analysis was performed to study the effects of combined magnetohydrodynamic and thermal radiation under convective boundary condition over a semi infinite vertical plate embedded in a non-Darcy porous medium. Coupled heat and mass transfer of free convective boundary layer with viscous nanofluid are considered. The model used for the nanofluid includes the effects of Brownian motion and thermophoresis mechanisms, while the Darcy-Forchheimer model is used for the porous medium. The governing partial differential equations are transformed into the ordinary differential equations using the similarity transformations. The accuracy of the method is observed by a comparison with other works reduced to a common case. Many results are tabulated and representative set is displayed graphically to illustrate the influence of the various parameters of interest on different profiles. Extensive numerical investigations show that the flow field, temperature, concentration and nanoparticle volume fraction shapes are significantly influenced by magnetic parameter, regular Lewis number, Brownian motion parameter, thermophoresis parameter, regular buoyancy ratio parameter and Biot number. Heat and mass transfer rates are significantly affected by the level of the applied magnetic field and the convective heat coefficient.