In this paper, the governing equations and finite element formulations for a microstructure-dependent unified beam theory with the von Kármán nonlinearity are developed. The unified beam theory includes the three familiar beam theories (namely, Euler-Bernoulli beam theory, Timoshenko beam theory, and third-order Reddy beam theory) as special cases. The unified beam formulation can be used to facilitate the development of general finite element codes for different beam theories. Nonlocal size-dependent properties are introduced through classical strain gradient theories. The von Kármán nonlinearity which accounts for the coupling between extensional and bending responses in beams with moderately large rotations but small strains is included. Equations for each beam theory can be deduced by setting the values of certain parameters. Newton's iterative scheme is used to solve the resulting nonlinear set of finite element equations. The numerical results show that both the strain gradient theory and the von Kármán nonlinearity have a stiffening effect, and therefore, reduce the displacements. The influence is more prominent in thin beams when compared to thick beams.

]]>We provide precise estimates of the Poincaré constants firstly for scalar functions and secondly for solenoidal (i.e. divergence free) vector fields (in both cases with vanishing Dirichlet traces on the boundary) on 2d-annuli by the use of the first eigenvalues of the scalar Laplacian and the Stokes operator, respectively. In our non-dimensional setting each annulus is defined via two concentrical circles with radii and . Additionally, corresponding problems on domains , the 2d-annuli from , are investigated - for comparison but also to provide limits for . In particular, the Green's function of the Laplacian on with vanishing Dirichlet traces on is used to show that for the first eigenvalue here tends to the first eigenvalue of the corresponding problem on the open unit circle. On the other hand, we take advantage of the so-called small-gap limit for .

]]>Starting from constrained three-dimensional Cosserat elasticity and in a total Lagrangian framework, we derive and discuss a hierarchy of static non-linear micropolar shell models in which the thickness can stretch, and the independence of the micro-rotation on the macro-rotation is not limited to the drilling degree of freedom as in the classical six-parameter theory. If the thickness stretch is considered, unconstrained quaternions allow a singularity-free, non redundant and rational parametrization of the strain measures. Emphasis is placed in the discussion of the differences between curvatures, wryness and director deformation strains.

]]>We determine the wave front sets of solutions to two special cases of the Cauchy problem for the space-time fractional Zener wave equation, one being fractional in space, the other being fractional in time. For the case of the space fractional wave equation, we show that no spatial propagation of singularities occurs. For the time fractional Zener wave equation, we show an analogue of non-characteristic regularity.

]]>The paper presents a numerical study of the tangential contact between a rigid indenter and an elastomer with linear rheology. Occurring friction forces are caused exclusively by internal dissipative losses. Especially the combinations of a conical or paraboloid indenter and the Maxwell or standard body with fixed ratio of moduli are studied. One result is, that by the use of proper chosen dimensionless variables, for each of the combinations, the contact forces depend explicitly on the indentation depth only. Curves describing this dependencies are given as Bézier function fits. Application of these methods to an indenter with arbitrary shape and materials with a discrete relaxation spectrum described by Prony series is possible. Although only stationary solutions are analysed here, the method enables us to examine the transient process as well.

]]>Recently, Ferrero and Gazzola, [Disc. Cont. Dyn. Syst. 35: 5879–5908 (2015)], suggested and investigated a rectangular plate model describing the statics and dynamics of a suspension bridge. The plate is assumed to be hinged on its vertical edges and free on its remaining horizontal edges. This reliable model aims to describe more accurately the motion of suspension bridges compared to all previous known models. In the present paper, we consider a plate equation in the presence of memory and subject to the above-mentioned boundary conditions. We give a rigorous well-posedness result and establish the existence of a global attractor.

]]>It is astonishing that after more than half a century intensive research in the area of non-conservative systems of second order differential equations still new interesting results appear, see . In that paper an old stability criterion by Metelitsyn and Frik was reinvented. We shortly repeat this result in order to emphasize that the criterion is sufficient but not necessary for stability. Afterwards we concentrate on circulatory systems with purely imaginary eigenvalues and investigate the influence of indefinite damping. Finally the possibility of stabilizing circulatory systems by gyroscopic forces will be commented. Examples will demonstrate the developed theory.

]]>A multielement clamp-knife-base tribo-fatigue system is considered. This system is the main part of a harvester's cutting instrument. Boundary element approach in indirect statement is used to model boundary conditions for system's elements taking into account contact interaction between them, cutting force applied to the knife and fixings. Stress-strain state of the system is calculated as superposition of solutions for each boundary element to which predetermined surfaces tractions are applied. Damage state of each element of the system is calculated using the model of a solid with dangerous volume. According to this model dangerous volume is a set of elementary volumes in which acting stresses (strains) are greater than the limiting stress (strain). Obtained results show that the choice of material of the knife (steel or high strength cast iron) and the value of cutting force has significant effect on system's stress-strain state and state of damage.

]]>Airborne wind energy systems are capable of extracting energy from higher wind speeds at higher altitudes. The configuration considered in this paper is based on a tethered kite flown in a pumping orbit. This pumping cycle generates energy by winching out at high tether forces and driving a generator while flying figures-of-eight, or lemniscates, as crosswind pattern. Then, the tether is reeled in while keeping the kite at a neutral position, thus leaving a net amount of generated energy. In order to achieve an economic operation, optimization of pumping cycles is of great interest. In this paper, first the principles of airborne wind energy will be briefly revisited. The first contribution is a singularity-free model for the tethered kite dynamics in quaternion representation, where the model is derived from first principles. The second contribution is an optimal control formulation and numerical results for complete pumping cycles. Based on the developed model, the setup of the optimal control problem (OCP) is described in detail along with its numerical solution based on the direct multiple shooting method in the CasADi optimization environment. Optimization results for a pumping cycle consisting of six lemniscates show that the approach is capable to find an optimal orbit in a few minutes of computation time. For this optimal orbit, the power output is increased by a factor of two compared to a sophisticated initial guess for the considered test scenario.

]]>It was proved by the author that in the case of **normal** contact and crack problems for anisotropic bodies the kernels of the integral transforms of the responses to the singular loadings are inverse to one another. It does not look like this property was previously noticed by other authors. The case of the relationship between **tangential** contact and crack problems is obviously more complicated and deserves a separate investigation. Such relationship is established here. Several particular cases of anisotropy are considered as illustrative examples. It is also shown that the kernels of the governing integral equations can be computed exactly, using the theory of generalized functions.

The fully developed regime of mixed convection in a vertical plane channel with symmetric and uniform temperature prescribed on the bounding walls is studied. The effect of viscous dissipation is taken into account and the Oberbeck-Boussinesq approximation is adopted by choosing the average fluid temperature as the reference temperature. The viscous dissipation effect induces the existence of dual branches of stationary solutions. A nonlinear stability analysis versus fully-developed modes of perturbation is carried out showing that the second branch of dual stationary solutions is unstable.

]]>In this paper we explore a numerical solution to elastoplastic constitutive initial value problems. An improved form of the implicit return-mapping scheme for nonsmooth yield surfaces is proposed that systematically builds upon a subdifferential formulation of the flow rule. The main advantage of this approach is that the treatment of singular points – apices or edges at which the flow direction is multivalued – only involves a uniquely defined set of non-linear equations, similarly to smooth yield surfaces. This paper focuses on isotropic models containing: *a*) yield surfaces with one or two apices (singular points) on the hydrostatic axis, *b*) plastic pseudo-potentials that are independent of the Lode angle, and *c*) possibly nonlinear isotropic hardening. We show that for some models the improved integration scheme also enables us to a priori decide about a type of the return and to investigate the existence, uniqueness, and semismoothness of discretized constitutive operators. The semismooth Newton method is also introduced for solving the incremental boundary-value problems. The paper contains numerical examples related to slope stability with publicly available Matlab implementations.

We consider an infinitely long cylinder, whose inner and outer surfaces are subjected to known surrounding temperatures and are traction free. The problem is in the context of the fractional order thermoelasticity theory. The medium is assumed to be initially quiescent.

]]>We present a new discretization method for convection-diffusion-reaction boundary value problems in 3D with PDE-harmonic shape functions on polyhedral elements. The element stiffness matrices are constructed by means of local boundary element techniques. Our method, which we refer to as a BEM-based FEM, can therefore be considered as a local Trefftz method with element-wise (locally) PDE-harmonic shape functions. The Dirichlet boundary data for these shape functions is chosen according to a convection-adapted procedure which solves projections of the PDE onto the edges and faces of the elements. This improves the stability of the discretization method for convection-dominated problems both when compared to a standard FEM and to previous BEM-based FEM approaches, as we demonstrate in several numerical experiments. Our experiments also show an improved resolution of the exponential layer at the outflow boundary for our proposed method when compared to the SUPG method.

]]>The paper shows an analytical rotordynamic model for induction rotors, supported in sleeve bearings, especially focusing on the influence of electromagnetic field damping on the forced rotor vibrations. The analytical model contains electromagnetic forces in respect of electromagnetic field damping, stiffness and internal material damping of the rotor, typical dynamic eccentricities of an induction rotor – mass eccentricity, bent rotor deflection and magnetic eccentricity –, stiffness and damping of the support, and stiffness and damping of the oil film of the sleeve bearings. With this rotordynamic model a useful possibility is shown, how to consider electromagnetic field damping, when analyzing forced vibrations caused by dynamic eccentricities. The aim of the paper is to show a method – based on a simple rotor dynamic model – how to consider electromagnetic field damping, which can also be adopted in Finite-Element-Analysis.

]]>An exact analytical solution has been constructed for the plane problem on action of a non-stationary load on the surface of an elastic layer for conditions of a ‘mixed’ boundary problem when normal stress and tangent displacement (the fourth boundary problem) are specified on one boundary and normal displacement and shear stress (the second boundary problem) for another boundary. Laplace and Fourier integral transforms are used. Their inversions were obtained with the help of tabular relationships and the convolution theorem for a wide range of acting non-stationary loads. Expressions for stresses (displacements) were obtained in explicit form. The obtained expressions allow determining the wave process characteristics in any point of the layer at an arbitrary moment of time. Some variants of non-stationary loads acting on an area with fixed boundaries or an area with boundaries changing by a known function are considered. Distinctive features of wave processes are analyzed.

]]>In isotropic strain gradient elasticity, we decompose the strain gradient tensor into its irreducible pieces under the *n*-dimensional orthogonal group . Using the Young tableau method for traceless tensors, four irreducible pieces (), which are canonical, are obtained. In three dimensions, the strain gradient tensor can be decomposed into four irreducible pieces with independent components whereas in two dimensions, the strain gradient tensor can be decomposed into three irreducible pieces with independent components. The knowledge of these irreducible pieces is extremely useful when setting up constitutive relations and strain energy.

We present numerical methods for model reduction in the numerical simulation of disk brake squeal. Automotive disk brake squeal is a high frequency noise phenomenon based on self excited vibrations. Our method is based on a variation of the proper orthogonal decomposition method and involves the solution of a large scale, parametric eigenvalue problem. Several important challenges arise, some of which can be traced back to the finite element modeling stage. Compared to the current industrial standard our new approach is more accurate in vibration prediction and achieves a better reduction in model size. This comes at the price of an increased computational cost, but it still gives useful results when the classical modal reduction method fails to do so. We illustrate the results with several numerical experiments, some from real industrial models, some from simpler academic models. These results indicate where improvements of the current black box industrial codes are advisable.

]]>In this paper, the well-known Hencky problem, the large deflection problem of axisymmetric deformation of uniformly loaded circular membranes, was resolved, where the small-rotation-angle assumption usually adopted in membrane problems was given up. The presented closed-form solution has a higher accuracy than well-known Hencky solution, a better understanding of the non-linear behavior of the considered problem could thus be reached. The presented numerical example shows that, the important integral constant controlling differential equations should change along with the increase of the applied transverse loads, but in well-known Hencky solution it becomes a constant due to the adopted small-rotation-angle assumption, resulting in the calculation error to increase, especially when the applied transverse loads is relatively large the well-known Hencky solution will no longer apply.

]]>Recently it has been proposed the study of a new class of bidimensional metamaterials, which have been called extensible pantographic sheets. Such bidimensional continua are the generalization of the continua introduced in as they take simultaneously into account the elastic energy in the extensional deformation and geodesic bending of constituting fibers. In the present paper we consider in the deformation energy a term accounting for the effect of shear deformation. The phenomena which we highlight can be of relevance to model the mechanical behavior of composite fiber reinforcements and of some lattices of beams constrained by internal pivots. We compare the effect of different linear or cubic shear stiffnesses i) on the deformation of pairs of pantographic sheets suitably interconnected and ii) on the deformation of pantographic sheets deformed under the action of forces concentrated on points.

]]>We develop a new method to construct periodic inclusions with uniform internal hydrostatic stress in an elastic plane subjected to uniform remote in-plane loading. The method is based on two particular conformal mappings which lead to a system of nonlinear equations from which the inclusion shapes are determined. We illustrate our results with several examples. In particular, we show that the ratio of the inclusion size to the period of the inclusion-matrix system changes with the inclusion aspect ratio and, in the specific case of uniform remote shear loading, the orientation of the inclusions is also altered. Finally, we show that if the period of the inclusion-matrix system exceeds roughly six times the inclusion size, such periodic inclusions can be treated approximately as periodic elliptical inclusions with specific aspect ratio and orientation determined by the corresponding elastic constants and uniform remote loading.

]]>The Hertz-type three-dimensional frictionless contact problem with a single controlling parameter is considered through a prism of the method of dimensionality reduction (MDR). The corresponding MDR formalism has been developed with respect to the main contact parameters (contact force and contact approach). It is shown that the half-length of the 1D contact interval can be uniquely interpreted as the harmonic radius of the original contact area. Also, the case of self-similar contact is studied in detail, and the obtained relations are applied to this case.

]]>We present analytic and computational studies of the dynamical behavior of an undamped electrostatic MEMS actuator with one-degree of freedom subject to a Casimir force. In such a situation, the well-known mathematical difficulty associated with an inverse quadratic term due to a Coulomb force is supplemented with an inverse quartic term due to the joint application of a Casimir force. We show that the small Coulomb and Casimir force situations, described by sufficiently low values of two positive parameters, λ and μ, respectively, are characterized by one-stagnation-point periodic motions and there exists a unique critical pull-in curve in the coordinate quadrant beyond which a finite-time touch down or collapse of the actuator takes place. We demonstrate how to locate and approximate the pull-in curve. When mechanical nonlinearity such as that due to the presence of a cubic elastic force term is considered in the equation of motion, we show that a similar three-phase oscillation-pull-in-finite-time-touchdown phenomenon occurs and that pull-in curves are actually enhanced or elevated by nonlinear elasticity. Furthermore, we compute solutions of the MEMS wave equations and show that the same characteristic phenomena of subcritical periodic motions and loss of periodicity of motion and onset of a critical pull-in curve occur as one increases the levels of the Coulomb and Casimir forces as in the one-degree-of-freedom case.

]]>Influence of thermal source on growth of a curved crack in stretchable plane with regard to cohesive forces at the crack's end zone, is studied. It is assumed that the areas of action of tractions adjoin to the crack's tips. For inhibiting the growth of curvilinear crack, on the path of its extension in the vicinity of the crack's end, we form a zone of compressible stresses by heating the domain *S* by thermal source to temperature *T*_{0}. The goal of local change of temperature is to delay or retardation of the crack growth. A boundary value problem on equilibrium of a bridged curvilinear crack with interfacial bonds under the action of external tensile loads, induced thermoelastic temperature field and tractions in the bonds preventing its opening is reduced to the system of nonlinear singular integral equations with a Cauchy-type integral. Normal and tangential tractions in the bonds are found from the solution of this system. The condition of limit equilibrium of a curvilinear crack with interfacial bonds is formed with regard to criterion of limit stretching of the bonds.

The paper contains an analysis of a two-dimensional equilibrium problem for an elastic body with a thin elastic inclusion. The thin elastic inclusion is modeled within the framework of Timoshenko beam theory. There is a crack on the interface between two media, displacements of the opposite crack faces are constrained with nonpenetration conditions. We derive the Griffith formula, which gives the first derivative of the energy functional with respect to the crack length. It is proved that the formula for the derivative can be represented as a path-independent integral along a smooth curve surrounding the crack tip. The invariant integral consists of a regular part and a singular part and is an analogue of the classical Eshelby–Cherepanov–Rice *J*-integral.

In this paper, we are concerned with the multi-dimensional () compressible viscoelastic flows in the whole space. We prove the optimal convergence rates of strong solutions to the system for the initial data close to a stable equilibrium state in critical Besov spaces. Our main ideas are based on the low-high frequency decomposition and the smooth effect of dissipative operator.

We show the existence of an energetic solution to a quasistatic evolutionary model of shape memory alloys. Elastic behavior of each material phase/variant is described by polyconvex energy density. Additionally, to every phase boundary, there is an interface-polyconvex energy assigned, introduced by M. Šilhavý in . The model considers internal variables describing the evolving spatial arrangement of the material phases and a deformation mapping with its first-order gradients. It allows for injectivity and orientation-preservation of deformations. Moreover, the resulting material microstructures have finite length scales.

]]>In the present paper the behavior of the piezoelectric response of smart lightweight structures consisting in a piezoelectric patch over a host layer under static load and affected by electrical load at environment conditions is studied. The shear lag analysis is applied to investigate the possible interface delamination and to calculate analytically the interface debond length. It has been demonstrated that the roots of respective characteristic equation play a leading role for place of the interface delamination in the overlap zone of the structure under consideration. This leads to the conditions for the actual deboning existence and opens the possibility of an optimal analysis. The proposed approach consists in involving the shear lag model in a global optimization framework where simultaneously the investigation of all model parameters can be carried out. The solution of that problem gives the values of the parameters at which a vanishing/minimal debond length is ensured. The efficiency of the proposed method is proved on three different examples as the optimal geometrical characteristics and effects ensuring no delamination in the structures are obtained.

]]>A problem of scattering by a resonator connecting two 2D waveguides is studied. The incident wave is one of the waveguide modes taken at the spectral parameter close to a threshold of the continuous spectrum. It is shown that in the general case the reflection coefficient for such a mode is close to -1 (this case corresponds to an almost perfect reflection). Also it is shown that in some special cases the reflection coefficient is close to 0, and an almost perfect transmission is observed. The behavior of scattering at near-threshold frequencies is determined by solutions corresponding to the threshold spectral parameter and crucially depends on whether stabilizing solutions do exist or not. Anomalous transmission is observed when there exist only solutions growing at infinity.

]]>Buckling of short multi-walled carbon nanotubes (MWCNTs) under external radial pressure is studied on the base of a multiple-shell model. The modified Mushtari-Donell-Vlasov type equations taking into account the van der Waals (vdW) interaction forces between adjacent tubes are used as the governing ones. In contrast to a majority of available studies on buckling of MWCNTs, which consider only the simply supported boundary conditions, this paper based on the asymptotic approach allows for the study of the buckling behavior of MWCNTs with different variants of the boundary conditions at the tube edges. At first, the pre-buckling membrane hoop stress-resultants induced by radial pressure are determined for each wall. Then, introducing a small parameter defined as a thickness-to-radius ratio, the asymptotic solutions of the boundary value problem are constructed for different cases which depend on the outermost radius of a MWCNT. The relevance of the present approach is confirmed by good agreement between asymptotic estimates and exact values of the buckling radial pressure for simply supported double- and triple-walled nanotubes determined on the base of the accepted shell model. In addition, the validity of the asymptotic estimates is justified by comparing theirs with existing data obtained on the base of the available multiple-shell model taking into account the pressure dependence of the interlayer vdW forces. The influence of the outermost radius, aspect ratio and boundary conditions as well on the buckling radial pressure is analyzed in this study.

]]>The theoretical results relevant to the vibration modes of Timoshenko beams are here used as benchmarks for assessing the correctness of the numerical values provided by several finite element models, based on either the traditional Lagrangian interpolation or on the recently developed isogeometric approach. Comparison of results is performed on both spectrum error (in terms of the detected natural frequencies) and on the *l*^{2} relative error (in terms of the computed eigenmodes): this double check allows detecting for each finite element model, and for a discretization based on the *same* number of degrees-of-freedom, *N*, the frequency threshold above which some prescribed accuracy level is lost, and results become more and more unreliable. Hence a quantitative way of measuring the finite element performance in modeling a Timoshenko beam is proposed. The use of Fast Fourier Transform is finally employed, for a selected set of vibration modes, to explain the reasons of the accuracy decay, mostly linked to a poor separation of the natural frequencies in the spectrum, which is responsible of some aliasing of modes.

This study considers the linear stability of two-layer films of immiscible liquids confined between an upper impermeable solid plate and a lower porous rigid substrate. The fluids are subjected to a periodic electric field. Based on the von Kármán-Pohlhausen method an integral boundary-layer model for the film thickness, surface charge and the flow rate is derived. The dynamics of the liquid-liquid interface is described for arbitrary amplitudes by evolution equations derived from the basic hydrodynamic equations using long wave approximation. The parameters governing the film flow system and the permeable substrate strongly effect the wave forms and their amplitudes and hence the stability of the fluid. Analytical and numerical simulations of this system of linear evolution equations are performed. The case of uniform electric field is considered as special case, it is found that, the permeability of the porous medium promotes the oscillatory behavior. While a stabilizing influence is observed for increasing of both the non-dimensional conductivity and the electric conductivity ratio. When the case of alternating electric field is taken into account, the method of multiple scales is applied to obtain approximate solutions and analyze the stability picture. Stability behavior is noticed for the decreasing of the permeability parameter and the dielectric constant ratio of the fluids.

]]>In this paper we give an explicit formula for the homogenization limit of Poisson's equation for a wide range of non-periodic problems including self-similarly ramified domains. This work was motivated by the modelling of the diffusion of medical sprays in lungs, which can be approximated by a self-similarly ramified domain. Such motivation also led us to consider the influence that a continuous scaling of the size of holes towards a chosen direction (e.g. towards the fractal boundary) has on the homogenization limit. It turned out that our strategy to explicitly calculate a formula for the homogenization limit could be applied beyond self-similar perforations as presented in this article.

]]>In this study, the existence of Hamiltonian structures for a two-dimensional, linear-elastic model is considered. We show that this model admits the so-called noncanonical singular Poisson bracket. Casimir functionals are found by using the singularity properties of the Poisson bracket obtained. We also demonstrate that these functionals are conserved for an arbitrary choice of Hamiltonian. Excepting the energy functional, we prove that there no exists conservation laws of zero order.

]]>The two-dimensional transient problem that is studied concerns a semi-infinite crack in an isotropic solid comprising an infinite strip and a half-plane joined together and having the same elastic constants. The crack propagates along the interface at constant speed subject to time-independent loading. By means of the Laplace and Fourier transforms the problem is formulated as a vector Riemann-Hilbert problem. When the distance from the crack to the boundary grows to infinity the problem admits a closed-form solution. In the general case, a method of partial matrix factorization is proposed. In addition to factorizing some scalar functions it requires solving a certain system of integral equations whose numerical solution is computed by the collocation method. The stress intensity factors and the associated weight functions are derived. Numerical results for the weight functions are reported and the boundary effects are discussed. The weight functions are employed to describe propagation of a semi-infinite crack beneath the half-plane boundary at piecewise constant speed.

]]>This paper investigates the radial breathing mode (RBM) vibration of single-walled carbon nanotubes (SWCNTs) based on doublet mechanics (DM) with a length scale parameter. A second order partial differential equation that governs the RBM vibration of SWCNTs is derived. Using DM, the relation between natural frequency and length scale parameter is derived in the RBM mode of vibration. It is shown that the length scale parameter plays significant role in the RBM vibration response of SWCNTs. The length scale parameter decreases the natural frequency of vibration compared to the predictions of the classical continuum models. However, with increase in tube radius, the effect of the scale parameter on the natural frequency decreases. The results obtained herein are compared with the existing theoretical and experimental results and good agreement with the latter is observed.

This paper investigates the radial breathing mode (RBM) vibration of single-walled carbon nanotubes (SWCNTs) based on doublet mechanics (DM) with a length scale parameter. A second order partial differential equation that governs the RBM vibration of SWCNTs is derived. Using DM, the relation between natural frequency and length scale parameter is derived in the RBM mode of vibration. It is shown that the length scale parameter plays significant role in the RBM vibration response of SWCNTs. The length scale parameter decreases the natural frequency of vibration compared to the predictions of the classical continuum models.

A method of the description of piezoelectric effect in non-polar materials is suggested. This method is based on the model of unit cell possessing a non-zero quadrupole moment and the zero dipole moment. The suggested theory contains several additional material tensors. A comparison of the suggested theory with the classical theory of piezoelectricity is carried out. The structure of the additional material tensors is determined for the crystal lattice with the hexagonal crystal symmetry of quartz. An analysis of dispersion relations is carried out. This analysis reveals that the quadrupole moment tensor has a qualitative influence on the behavior of solution. Namely, the quadrupole moments cause rather the redistribution of energy between the waves of different type than the wave damping. A method of determination of the additional material moduli is suggested. The method consists in using the dispersion relations containing the unknown material moduli and the experimental data on attenuation factor versus frequency.

]]>We study the Stokes problem in a bounded planar domain Ω with a friction type boundary condition that switches between a slip and no-slip stage. Unlike our previous work , in the present paper the threshold value may depend on the velocity field. Besides the usual velocity-pressure formulation, we introduce an alternative formulation with three Lagrange multipliers which allows a more flexible treatment of the impermeability condition as well as optimum design problems with cost functions depending on the shear and/or normal stress. Our main goal is to determine under which conditions concerning smoothness of the boundary of Ω, solutions to the Stokes system depend continuously on variations of Ω. Having this result at our disposal, we easily prove the existence of a solution to optimal shape design problems for a large class of cost functionals.

]]>Coupling multiphysical systems by means of a co-simulation, the data between the subsystems are interchanged at a discrete macro time grid, also denoted as communication time grid. Between the communication points the coupling variables are approximated so that the numerical solvers in the subsystems can calculate the differential equations. Classical approximation techniques based on extrapolation entail a discontinuity in the equations in each macro time step which can slow down the numerical time integration. In the paper at hand a *C*^{0}-continuous technique for approximating the coupling variables is reconsidered and the numerical stability and the local error of the method are compared to the classical Lagrange approximation approach. Further, the method is enhanced to a *C*^{1}-continuous (continuous and differentiable) approximation technique. Both methods are investigated in combination with different numerical coupling approaches (sequential *Gauss-Seidel scheme*, parallel *Jacobi scheme*, *force/displacement coupling*, *displacement/displacement coupling*) which are commonly applied for co-simulation in technical applications. It is shown that the *C*^{1}-continuous approximation technique yields a similar numerical stability and a similar local error as the Lagrange approach which results in a comparable or even better overall performance (taking into account the advantage of continuity at the numerical calculation of the subsystem differential equations). Applying the *C*^{0}-continuous approach, a similar numerical stability is obtained. However, the order of the local error is significantly lower than for the *C*^{1}-continuous method and the Lagrange approach.

Classical approximation techniques based on extrapolation entail a discontinuity in the equations in each macro time step which can slow down the numerical time integration. In the paper at hand a *C*_{0}-continuous technique for approximating the coupling variables is reconsidered and the numerical stability and the local error of the method are compared to the classical Lagrange approximation approach. Further, the method is enhanced to a *C*_{1}-continuous (continuous and differentiable) approximation technique. Both methods are investigated in combination with different numerical coupling approaches (sequential *Gauss-Seidel scheme*, parallel *Jacobi scheme*, *force/displacement coupling*, *displacement/displacement coupling*) which are commonly applied for co-simulation in technical applications.

In the present paper on the basis of the linear theory of thermoelasticity of homogeneous isotropic bodies with microtemperatures the zeroth order approximation of hierarchical models of elastic prismatic shells with microtemperatures in the case of constant thickness (but, in general, with bent face surfaces) is considered. The existence and uniqueness of solutions of basic boundary value problems when the projections of the bodies under consideration are bounded and unbounded domains with closed contours are established. The ways of solving boundary value problems in explicit forms and of their numerical solution are indicated.

]]>An impact of an elastic sphere with an elastic half space with a constant coefficient of friction is studied numerically using the method of dimensionality reduction. It is shown that the rebound velocity, angular velocity and hence the loss of kinetic energy during the impact, if written as proper dimensionless variables, are determined by a function depending only on the ratio of tangential and normal stiffness and a second parameter describing the friction properties of the contact.

An impact of an elastic sphere with an elastic half space with a constant coefficient of friction is studied numerically using the method of dimensionality reduction. It is shown that the rebound velocity, angular velocity and hence the loss of kinetic energy during the impact, if written as proper dimensionless variables, are determined by a function depending only on the ratio of tangential and normal stiffness and a second parameter describing the friction properties of the contact.

Within the framework of linear elasticity of anisotropic solids, the strong ellipticity (SE) condition is discussing. In this paper, the SE-condition is deriving for 19 classes of 32 known ones of anisotropy. For each class SE-condition reduces to a finite set of elementary inequalities. As a special case, incompressible materials are also considered.

]]>We study the effects of small boundary perturbation on the flow of viscous fluid using asymptotic analysis. A small perturbation of magnitude is applied on part of the boundary of the fluid domain. The complete asymptotic expansion of the solution of the Stokes system, in powers of ε is derived. First two terms are explicitly computed. A simple example shows that the perturbation is nonlocal, i.e. not concentrated near the boundary. The convergence of the expansion is proved. The results are applied on a simple optimal shape design problem.

The author studies the effects of small boundary perturbation on the flow of viscous fluid using asymptotic analysis. A small perturbation of magnitude is applied on part of the boundary of the fluid domain. The complete asymptotic expansion of the solution of the Stokes system, in powers of ε is derived. First two terms are explicitly computed. A simple example shows that the perturbation is nonlocal, i.e. not concentrated near the boundary. The convergence of the expansion is proved. The results are applied on a simple optimal shape design problem.

We study the nonexistence of global solutions to the Cauchy problem for systems of parabolic-hyperbolic or hyperbolic thermo-elasticity equations posed in . For power nonlinearities, we present threshold critical exponents depending on the space dimension *N*. Our method of proof rests on the nonlinear capacity method.

The plane strain problem of determining stress intensity factors and stress magnification factors for an interfacial Griffith crack situated at the interface of two bonded dissimilar orthotropic media having sub-interfacial Griffith crack is considered. The problem is reduced to the solution of two pair of simultaneous singular integral equations which are finally been solved by using Jacobi polynomials. The propagation of interfacial crack through amplification and shielding factors are shown graphically for different particular cases.

The plane strain problem of determining stress intensity factors and stress magnification factors for an interfacial Griffith crack situated at the interface of two bonded dissimilar orthotropic media having sub-interfacial Griffith crack is considered. The problem is reduced to the solution of two pair of simultaneous singular integral equations which are finally been solved by using Jacobi polynomials. The propagation of interfacial crack through amplification and shielding factors are shown graphically for different particular cases.